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WL ComponentFourier components of uTMLP(a,b) or uTMLP(a, b) FVE
cos (W14C)44.6cos(w14a) cos(w14b)- 43.6 sin(w14a) sin(w14b)~ 44.1 cos (w14(a + b))93.2%
sin (w14c)44.1 sin(w14a) cos(w14b) + 44.1 cos(w14a) sin(w14b) ~ 44.1 sin (w14(a + b))93.5%
cos (W35C)40.7 cos(w35a) cos(w35b)- 43.6 sin(w35a) sin(w35b) ~ 42.2 cos (w35(a + b))96.8%
sin (w35C)41.8 sin(w35a) cos(w35b) + 41.8 cos(w35a) sin(w35b) ~ 41.8 sin (w35(a + b))96.5%
cos (w41c)44.8 cos(w41a) cos(w41b) - 44.8 sin(w41a) sin(w41b) ~ 44.8 cos (w41(a + b))97.0%
sin (w41c)44.5 sin(w41a) cos(w41b) + 44.5 cos(w41a) sin(w41b) ~ 44.5 sin (w41(a + b))97.0%
cos (W42C)64.6 cos(w42a) cos(w42b) - 68.5 sin(w42a) sin(w42b) ~ 66.6 cos (w42(a + b))96.4%
sin (w42c)67.8 sin(w42a) cos(w42b) + 67.8 cos(w42a) sin(w42b) ~ 67.8 sin (w42(a + b))96.4%
cos (W52C)60.5 cos(w52a) cos(w52b) - 65.5 sin(w52a) sin(w52b) ~ 63.0cos (w52(a + b))97.4%
sin (w52C)64.5 sin(w52a) cos(w52b) + 64.5 cos(w52a) sin(w52b) ~ 64.5 sin (w52(a + b))98.2%
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WL Component Fourier components of u MLP(a,b) or uT MLP(a,b) FVEcos(w2c) 147.4cos (w2a) cos (w2b)-145.8sin(w2a)sin (w2b) ~ 146.6cos (w2(a +b)) 99.2%sin(w2c) 145.5 cos (wza) sin (w2b) + 145.6 sin(w2a) cos (w2b) ≈ 145.5 sin (w2(a + b)) 99.1%cos (wgc) 49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b)) 96.4%sin (wgc) 48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b)) 96.7%
WL ComponentFourier components of u MLP(a,b) or uT MLP(a,b)oruMLP(a,b)
COSW2C147.4cosw2b)≈146.6cosw2(a+b))
sinW2C145.5 cosw2a) sin(w2b)+145.6sin(w2a) cos(w2b)~145.5sinw2(a+b)）
COSwgC49.3cos49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b))48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b))
sinwgC48.6cosgb)≈48.5sin(wg(a+b))
COSW19c)58.0cos (wiga) cos (w1gb)- 58.3 sin (wiga) sin(w1gb) ~ 58.2 cos(w19(a + b))59.3 cos (w1ga) sin (w1gb) + 59.4 sin (w19a) cos (w19b) ~ 59.4sin(w1g(a + b))94.4cos(w31a) cos (w31b) - 96.4sin (w31a) sin(w3ib) ~ 95.4 cos (w31(a + b))97.2 cos (w31a) sin(w31b) + 97.1 sin (w31a) cos (w3ib) ~ 97.2 sin (w31(a +b)58.0 cos(w
sin(w19c）
sin (w31c)cos (w31c)sin (w31c)
WL Component(a) Seed 1
Fourier components of uTMLP(a, b) or uTMLP(a, b)
COSW40C97.0cOsw40aCOs(w40b)-99.4sin(w4oasinW40b) ~ 98.2 cos (w40(a + b))97.3%92.7%
sinW40C81.3cosw40asinw40b+81.3sin(w4oa)COSw40b)≈ 81.3sin(w4o(a+b))
COSW44C309.1cos(w44a)COsW44b-338.7 sin(w44asins (w44a) sin(w44b) + 327.2 sin (w44a) cos (w44b) ~ 327.3 sin (w44(a + b))））
sinW44C327.3cosW44asin (W44b)+ 327.2 sinw44aCos
COSW53C192.1cos (W53a)COSW53b-192.2 sin(w53asin97.3%95.7%FVE
sinW53C166.7cos(w53asin (W53b+ 166.8 sinw53aCos
WL Component(b) Seed 2
WL ComponentFouriercomponents of uMLPa，borUjMLP(a,b)
COsW31C156.1cos(w31a)cosw31b-156.5 sin(w31asin(w31b~156.3ccs(w31(a+b))99.3%98.9%
sinW31C150.7cos(w31asin (w31b)+ 150.7sinw31aCos(w31b~150.7sin(w31(a+b))
COSW45C72.5 cosW45aW45b）-76.8sinW45asinW45b~ 74.6cos（W45(a+b))95.9%
sinW45C74.7cOsw45asinw45b+74.6sin(w45a)COSw45b~ 74.6sinw45(a+b))96.6%
cos(W49C)45.9 cosw49acosw49b）-45.5sinW49a)sinw49b~45.7cos（w49(a+b))97.0%
sinW49C45.8 cosw49asinw49b+45.8sin(w49a)COsW4gb~ 45.8sinw49(a+b))96.9%
cos((W52C)sin (w52C)71.6cosw52acos(w52b）-72.1sin(W52qsin (W52b~ 71.9 cosw52(a+b))w52(a +b)
68.7cosW52asinW52b）+68.7sin(W52a)CosW52b~ 68.7sin
(c) Seed 3FVE
WLCWL ComponentFouriercomponentsof uMLP(a，b)orvMLP(a,b)
cossin(w17C)cos (w32c)sin(w32c)cos (W42C)(W17C)66.0cosw17acosw17b）-63.5sin(sin(w17b~64.8cosw17(a+b))96.4%94.9%96.2%
W17C66.4cosw17asinw17b）+66.4sin(w17a)Cosw17b~ 66.4sinw1z(a+b))
W32C68.7cosW32acos（w32b-68.4sin(Wa2q)sin (.68.5.cosWaa(a +b))
W32C68.0 cosw32asinW32b+68.0sin(W32a)CosW32b~ 68.0sinw32(a+b))96.3%
W42C100.4cos(w420)cos(w42)-96.0sin(w42asinw42b~98.2cos(w42(a+b))97.9%
W42C100.2 cos (W42a)sin (w42b)+ 100.1 sin(w42gcos(w42b~100.1sii(w42(a+b))98.6%
COSW51C118.0 cos (w51a)CosW51b-116.2 sin(w51asinW51b≈117.1cos(w51(a+b))99.0%
sinW51C114.3cos (w51a)sin (w51b)+ 114.2 sinw51acos(w51b)≈114.2sin(w51(a+b))98.5%
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SeedTest LossLoss (Key frequencies removed)|Loss (All other frequencies removed)
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32.05·10-76.7·1005.5.10-8
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ModelTest LossGini(Wε)Gini(WL)Key FrequenciesLogit FVEMLP FVE
40% Training Data1.98:10-70.760.79[17,43,49,55]94.9%83.3% [26.1%]
50% Training Data1.68 : 10-70.750.77[2,17,31,41, 44]91.2%85.2% [28.2%]
60% Training Data1.23.10-70.790.84[2,23,34,51]96.4%95.7% [1.4%]
70% Training Data9.85.10-80.800.91[14,15,26]99.0%98.9% [0.4%]
80% Training Data5.83.10-70.620.80[38,41]63.9%94.1% [2.5%]
90% Training Data1.11:10-70.790.88[3,26,34,43]98.6%98.7% [0.3%]
2Layer Transformer9.54:10-70.590.80[14,18,29]91.8%95.2%[1.9%]
2 Layer Transformer4.41:10-50.550.73[7,12,35,49]86.1%86.2% [6.4%]
2 Layer Transformer6.50.10-20.660.80[4,9,28]88.5%85.4% [5.9%]
2 Layer Transformer4.18.10-20.560.76[4,5,15,54]91.4%81.2% [17.8%]
2 Layer Transformer1.75:10-20.680.71[3,4,13,30,38]84.0%71.9% [19.5%]
P=533.00:10-40.610.68[6,9,16,21]91.2%90.2% [5.8%]
P=531.03:10-40.560.72[4,13,16]94.8%93.1% [6.4%]
P=531.21:10-50.660.79[13,22,23]98.2%97.6% [0.9%]
P=533.95·10-60.660.74[3,14,15]88.5%91.8% [4.6%]
P=535.56:10-60.670.80[10,14,22]98.1%98.3% [0.6%]
P=1092.02:10-70.760.83[6,7,22,25]98.0%97.3% [1.9%]
P=1092.95.10-70.690.82[8,14,29,32,41]95.2%94.7% [2.3%]
P=1091.66·10-70.780.86[13,23,39,45]98.5%97.6% [0.9%]
P=1092.50.10-70.680.82[8,13,32,41]96.8%95.5% [2.3%]
P=1092.77:10-70.760.85[29,37,38,49]97.9%98.1% [0.8%]
Dropout p= 0.22.65:10-10.190.46[1,4,7,17,22,33,40,49,55]71.3%65.0% [17.5%]
Dropout p = 0.24.52.10-10.190.46[3,8,19,28,32,34,40,44]73.3%71.4% [10.7%]
Dropout p = 0.22.03:10-10.200.45[4,5,32,38,41,44,49,50]74.2%71.1% [10.6%]
Dropout p = 0.5<10-80.260.56[1,4,26,46,47,55]89.4%88.9%[3.5%]
Dropout p = 0.52.01.10-20.200.49[16,21,35,47,53]88.4%88.4% [3.0%]
Dropout p = 0.5<10-80.250.54[1,4,7,19,29,31,42]86.1%85.6%[4.0%]
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Model TypeLossAccuracyAblated LossAblated Acuracy
VaryingData Fraction1.83:10-7 (1.65:10 7100%7.74· 10-7 (6.74· 10-7)100%
2 Layer Transformer1.97·10-2 (2.41·10-299.6%4.63· 10-2 (6.72 : 10-298.7%
P=535.96·10-5 (8.91·10-5)100%1.5·10-4 (2.70· 10-4)100%
P=1091.94· 10-7 (3.74· 10-8)100%6.53· 10-7 (1.41 · 10-7)100%
Dropout p = 0.20.215 (0.091)99.7%0.205 (0.075)99.7%
Dropout p = 0.54.68 · 10-³ (8.11· 10-3)100%3.6:10-3 (5.82.10-3)100%
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+Shixiang Shane Gu Google Research + +# ABSTRACT + +How to extract as much learning signal from each trajectory data has been a key problem in reinforcement learning (RL), where sample inefficiency has posed serious challenges for practical applications. Recent works have shown that using expressive policy function approximators and conditioning on future trajectory information – such as future states in hindsight experience replay (HER) or returnsto-go in Decision Transformer (DT) – enables efficient learning of multi-task policies, where at times online RL is fully replaced by offline behavioral cloning (BC), e.g. sequence modeling. We demonstrate that all these approaches are doing hindsight information matching (HIM) – training policies that can output the rest of trajectory that matches some statistics of future state information. We present Generalized Decision Transformer (GDT) for solving any HIM problem, and show how different choices for the feature function and the anti-causal aggregator not only recover DT as a special case, but also lead to novel Categorical DT (CDT) and Bi-directional DT (BDT) for matching different statistics of the future. For evaluating CDT and BDT, we define offline multi-task state-marginal matching (SMM) and imitation learning (IL) as two generic HIM problems, propose a Wasserstein distance loss as a metric for both, and empirically study them on MuJoCo continuous control benchmarks. Categorical DT, which simply replaces anti-causal summation with anti-causal binning in DT, enables arguably the first effective offline multi-task SMM algorithm that generalizes well to unseen (and even synthetic) multi-modal reward or state-feature distributions. Bi-directional DT, which uses an anti-causal second transformer as the aggregator, can learn to model any statistics of the future and outperforms DT variants in offline multi-task IL, i.e. one-shot IL. Our generalized formulations from HIM and GDT greatly expand the role of powerful sequence modeling architectures in modern RL. + +# 1 INTRODUCTION + +Reinforcement learning (RL) suffers from the problem of sample inefficiency, and a central question is how to extract as much learning signals, or constraint equations (Pong et al., 2018; Tu & Recht, 2019; Dean et al., 2020), from each trajectory data as possible. As dynamics transitions and Bellman equation provide a rich source of supervisory objectives and constraints, many algorithms combined model-free with model-based, and policy-based with value-based in order to achieve maximal sample efficiency, while approximately preserving stable, unbiased policy learning (Heess et al., 2015; Gu et al., 2016; 2017; Buckman et al., 2018; Pong et al., 2018; Tu & Recht, 2019). + +Orthogonal to these, in the recent years we have seen a number of algorithms that are derived from different motivations and frameworks, but share the following common trait: they use future trajectory information $\tau _ { \mathbf { t } ; \mathbf { T } }$ to accelerate optimization of a contextual policy $\pi ( \mathbf { a _ { t } } | \mathbf { s _ { t } } , \mathbf { z } )$ with context $\mathbf { z }$ with respect to a parameterized reward function $\mathbf { r } ( \mathbf { s _ { t } } , \mathbf { a _ { t } } , \mathbf { z } )$ (see Section 3 for notations). These hindsight algorithms have enabled Q-learning with sparse rewards (Andrychowicz et al., 2017), temporally-extended model-based RL with Q-function (Pong et al., 2018), mastery of 6-DoF object manipulation in cluttered scenes from human play (Lynch et al., 2019), efficient multi-task RL (Eysenbach et al., 2020; Li et al., 2020), offline self-supervised discovery of manipulation primitives from pixels (Chebotar et al., 2021), and offline RL using return-conditioned supervised learning with transformers (Chen et al., 2021a; Janner et al., 2021). We derive a generic problem formulation covering all these variants, and observe that this hindsight information matching (HIM) framework, with behavioral cloning (BC) as the learning objective, can learn a conditional policy to generate trajectories that each satisfy any properties, including distributional. + +Given this insight and recent casting of RL as sequence modeling (Chen et al., 2021a; Janner et al., 2021), we propose Generalized Decision Transformer (GDT), a family of algorithms for future information matching using hindsight behavioral cloning with transformers, and greatly expand the applicability of transformers and other powerful sequential modeling architectures within RL with only small architectural changes to DT. In summary, our key contributions are: + +• We introduce hindsight information matching (HIM) (Section 4, Table 1) as a unifying view of existing hindsight-inspired algorithms, and Generalized Decision Transformers (GDT) as a generalization of DT for RL as sequence modeling to solve any HIM problem (Figure 1). Inspired by distribution RL (Bellemare et al., 2017; Dabney et al., 2018) and state-marginal matching (SMM) (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), we define offline multi-task SMM problems, propose Categorical DT (CDT) (Section 5), validate its empirical performance to match feature distributions (even generalizing to a synthetic bi-modal target distribution at times), and construct the first benchmark tasks for offline multi-task SMM. Inspired by one-shot imitation learning (Duan et al., 2017; Finn et al., 2017; Dasari & Gupta, 2020), we define offline multi-task imitation learning $( I L )$ , propose a Wasserstein-distance evaluation metric, develop Bi-directional DT (BDT) as a fully expressive variant of GDT (Section 5), and demonstrate BDT’s competitive performance at offline multi-task IL. + +![](images/f65124c0f2d4f6c2f977e80aa9a54cfbd184001708c35ad790687fd7753196be.jpg) +Figure 1: Generalized Decision Transformer (GDT), where the figure is a minor generalization of the DT architecture (Chen et al., 2021a) and the table summarizes how it leads to different classes of algorithms with only small architectural changes. If the feature function $\Phi ( s , a )$ is reward $r ( s , a )$ and the anti-causal aggregator is $\gamma$ -discounted summation, we recover DT for offline RL. If the aggregator is binning, we get Categorical DT (CDT) for offline multi-task state-marginal matching. If the aggregator is a second transformer, we get Bi-directional DT (BDT) for offline multi-task imitation learning (IL), or equivalently one-shot IL. The choices of $\Phi ( s , a )$ and the aggregator together decide $I ^ { \Phi } ( \tau )$ in Hindsight Information Matching (HIM) objective discussed in Section 4 and Table 1, where conversely GDT can essentially solve any HIM problem with proper choices of $\Phi$ and aggregator. + +
Method(s,a)Aggregator
DT (Chen et al., 2021a) DT-X (Section 5.3)r(s,a)Summation
CDT (Section 5.2)LearnedSummation
r(s,a) or anyBinning
BDT (Section 5.4)LearnedTransformer
+ +# 2 RELATED WORK + +Hindsight Reinforcement Learning and Behavior Cloning Hindsight techniques (Kaelbling, 1993; Andrychowicz et al., 2017; Pong et al., 2018) have revolutionized off-policy optimization with respect to parameterized reward functions. Two key insights were (1) for off-policy algorithms such as Q-learning (Mnih et al., 2015; Gu et al., 2016) and actor-critic methods (Lillicrap et al., 2016; Haarnoja et al., 2018; Fujimoto et al., 2018; Furuta et al., 2021a), the same transition samples can be used to learn with respect to any reward parameters, as long as the reward function is re-computable, i.e. “relabel”-able, like goal reaching rewards, and (2) if policy or Q-functions are smooth with respect to the reward parameter, generalization can speed up learning even with respect to “unexplored” rewards. In goal-based RL where future states can inform “optimal” reward parameters with respect to the transitions’ actions, hindsight methods were applied successfully to enable effective training of goal-based Q-function for sparse rewards (Andrychowicz et al., 2017), derive exact connections between Q-learning and classic model-based RL (Pong et al., 2018), dataefficient off-policy hierarchical RL (Nachum et al., 2018), multi-task RL (Eysenbach et al., 2020; Li et al., 2020), offline RL (Chebotar et al., 2021), and more (Eysenbach et al., 2021; Choi et al., 2021; Ren et al., 2019; Zhao & Tresp, 2018; Ghosh et al., 2021; Nasiriany et al., 2021). Additionally, Lynch et al. (2019) and Gupta et al. (2018) have shown that often BC is sufficient for learning generalizable parameterized policies, due to rich positive examples from future states, and most recently Chen et al. (2021a) and Janner et al. (2021), when combined with powerful transformer architectures (Vaswani et al., 2017), it produced state-of-the-art offline RL and goal-based RL results. Lastly, while motivated from alternative mathematical principles and not for parameterized objectives, future state information was also explored as ways of reducing variance or improving estimations for generic policy gradient methods (Pinto et al., 2017; Guo et al., 2021; Venuto et al., 2021). + +Distributional Reinforcement Learning and State-Marginal Matching Modeling the full distribution of returns instead of the averages led to the development of distributional RL algorithms (Bellemare et al., 2017; Dabney et al., 2018; 2020; Castro et al., 2018; Barth-Maron et al., 2018) such as Categorical Q-learning (Bellemare et al., 2017). While our work shares techniques such as discretization and binning, these works focus on optimizing a non-conditional reward-maximizing RL policy and therefore our problem definition is closer to that of state-marginal matching algorithms (Hazan et al., 2019; Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), or equivalently inverse RL algorithms (Ziebart et al., 2008; Ho & Ermon, 2016; Finn et al., 2016; Fu et al., 2018; Ghasemipour et al., 2020) whose connections to feature-expectation matching have been long discussed (Abbeel & $\mathrm { N g }$ , 2004). However, those are often exclusively online algorithms even sample-efficient variants (Kostrikov et al., 2019), since density-ratio estimations with either discriminative (Ghasemipour et al., 2020) or generative (Lee et al., 2020) approach requires on-policy samples, with a rare exception of Kostrikov et al. (2020). Building on the success of DT and brute-force hindsight imitation learning, our Categorical DT is to the best our knowledge the first method that benchmarks offline state-marginal matching problem in the multi-task settings. + +RL and Imitation Learning as Sequence Modeling When scaled to the extreme levels of data and computing, sequence models such as transformers (Vaswani et al., 2017) can train models to master an impressive range of capabilities in natural language processing and computer vision (Devlin et al., 2019; Radford et al., 2019; Brown et al., 2020; Radford et al., 2021; Ramesh et al., 2021; Chen et al., 2021b; Bommasani et al., 2021; Dosovitskiy et al., 2020). Comparing to their popularity in other areas, the adoption of transformers or architectural innovations in RL have been slow, partially due the difficulty of using transformers over temporal scales for online RL (Parisotto et al., 2020). Recent successes have focused on processing variable-length per-timestep information such as morphology (Kurin et al., 2021), sensory information (Tang & Ha, 2021), one-shot or few-shot imitation learning (Dasari & Gupta, 2020), or leveraged offline learning (Chen et al., 2021a; Janner et al., 2021). Our formulation enables sequence modeling to solve novel RL problems such as statemarginal matching with minimal architectural modifications to DT, greatly expanding the impacts of transformers and other powerful sequence models in RL. + +# 3 PRELIMINARIES + +We consider a Markov Decision Process (MDP) defined by the tuple of action space $\mathcal { A }$ , state space $s$ , transition probability function $p ( s ^ { \prime } | s , a )$ , initial state distribution $p ( s _ { 0 } )$ , reward function $r ( s , a )$ , and discount factor $\gamma \in ( 0 , 1 ]$ . In deep RL, a policy that maps the state space to the action space is parameterized by the function approximators, $\dot { \pi } _ { \boldsymbol { \theta } } ( \dot { a } | s ) ^ { 1 }$ . The RL objective is given by: + +$$ +L _ { \mathrm { R L } } ( \pi ) = \frac { 1 } { 1 - \gamma } \mathbb { E } _ { s \sim \rho ^ { \pi } ( s ) , a \sim \pi ( \cdot | s ) } \left[ r ( s , a ) \right] +$$ + +where $\begin{array} { r } { p _ { t } ^ { \pi } ( s ) = \iint _ { s _ { 0 : t } , a _ { 0 : t - 1 } } \prod _ { t } p ( s _ { t } | s _ { t - 1 } , a _ { t - 1 } ) \pi ( a _ { t } | s _ { t } ) } \end{array}$ and $\begin{array} { r } { \rho ^ { \pi } ( s ) = ( 1 - \gamma ) \sum _ { t ^ { \prime } } \gamma ^ { t ^ { \prime } } p _ { t ^ { \prime } } ^ { \pi } ( s _ { t ^ { \prime } } = s ) } \end{array}$ are short-hands for time-aligned and time-aggregated state marginal distributions following policy $\pi$ . + +# 3.1 STATE MARGINAL MATCHING + +State marginal matching (SMM) (Lee et al., 2020; Hazan et al., 2019; Ghasemipour et al., 2020) has been recently studied as an alternative problem specification in RL, where instead of stationary-reward maximization, the objective is to find a policy minimizing the divergence $D$ between its state marginal distribution $\rho ^ { \pi } ( s )$ to a given target distribution $p ^ { * } ( s ) ^ { 2 }$ : + +$$ +L _ { \mathrm { S M M } } ( \pi ) = - D ( \rho ^ { \pi } ( s ) , p ^ { * } ( s ) ) +$$ + +where $D$ is a divergence measure such as Kullback-Leibler (KL) divergence (Lee et al., 2020; Fu et al., 2018) or, more generally, some $f$ -divergences (Ghasemipour et al., 2020). For the target distribution $p ^ { * } ( s )$ , Lee et al. (2020) set a uniform distribution to enhance the exploration over the entire state space; Ghasemipour et al. (2020) and Gu et al. (2021) set through scripted distribution sketches to generate desired behaviors; and adversarial inverse RL methods (Ho & Ermon, 2016; Fu et al., 2018; Ghasemipour et al., 2020; Kostrikov et al., 2020) set as the expert data for imitation learning. Notably, unlike the RL objective in Eq.1, SMM objectives like Eq.2 no longer depend on task rewards and are only functions of state transition dynamics and target state distribution. + +# 3.2 PARAMETERIZED RL OBJECTIVES + +Lastly, we discuss the basis for methods like HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al., 2019), and return-conditioned or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al., 2021): parameterized RL objectives. Given parameterized reward functions with parameter $z \in { \mathcal { Z } }$ , a conditional policy $\pi ( a | s , z )$ is learned with respect to multiple values of $z$ simultaneously weighted by $p ( z )$ . As examples, the RL objective in Eq.1 becomes: + +$$ +L _ { \mathrm { R L } } ( \pi ) = \mathbb { E } _ { z } \left[ L _ { \mathrm { R L } } ( \pi , z ) \right] = \frac { 1 } { 1 - \gamma } \mathbb { E } _ { z \sim p ( z ) , s \sim \rho _ { z } ^ { \pi } ( s ) , a \sim \pi ( \cdot | s , z ) } \left[ r _ { z } ( s , a ) \right] +$$ + +where the state marginal $\rho _ { z } ^ { \pi }$ is from rolling out a conditioned policy $\pi ( \cdot | \cdot , z )$ . These can be considered as a special case of contextual MDPs (Jiang et al., 2017) and are all multi-task RL problems. + +# 4 HINDSIGHT INFORMATION MATCHING + +We show how HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al., 2019), hindsight multi-task RL (Li et al., 2020; Eysenbach et al., 2020), and return-conditioned or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al., 2021) all belong to hindsight algorithms with a shared idea of using future state information to automatically mine for positive, or “optimal”, examples with respect to certain contextual parameter values, where these examples can accelerate RL or be used for behavior cloning (BC), i.e. supervised learning. We start by defining additional notations. + +Given a partial trajectory from state $s _ { t }$ as $\tau _ { t } = \{ s _ { t } , a _ { t } , s _ { t + 1 } , a _ { t + 1 } , \dots \}$ , we define its information statistics as $I ( \tau _ { t } )$ . $I ( \hat { \tau _ { t } } )$ could be any function of a trajectory that captures some statistical properties in state-space or trajectory-space, such as sufficient statistics of a distribution, like mean, variance or higher-order moments (Wainwright & Jordan, 2008). For convenience, we further define the notion of a feature function $\Phi ( \cdot , \cdot ) : S \times A \to F ^ { 5 }$ , where the trajectory is then noted as $\tau _ { t } ^ { \Phi } = \{ \phi _ { t } , \phi _ { t + 1 } , \ldots , \phi _ { T } \} , \phi _ { t } = \Phi ( s _ { t } , a _ { t } ) \in \stackrel { } { F }$ and the information statistics as $\tilde { I } ^ { \Phi } \bar { ( } \tau _ { t } )$ . $\Phi$ in practice can be an identity function, the reward function $r ( s , a )$ , sub-dimensions of $s$ (e.g. xy-velocities), or a generic parameterized function (e.g. auto-encoder). Generalizing reward-centric intuitions in DT (Chen et al., 2021a), we define information matching (IM) problems as learning a conditional policy $\pi ( a | s , z )$ whose trajectory rollouts satisfy some desired information statistics value $z$ : + +Table 1: A coarse summary of hindsight information matching (HIM) algorithms. The notation follows Section 4. With HIM, all prior works can be categorized to four generic problem types based on $I ^ { \Phi } ( \tau )$ : (1) goal-based $\phi _ { T }$ (Andrychowicz et al., 2017), (2) multi-task arg max $\begin{array} { r } { \sum _ { t } \hat { \gamma } ^ { t } r ( s _ { t } , a _ { t } , \cdot ) } \end{array}$ (Li et al., 2020), (3) return-based $\textstyle \sum _ { t } \gamma ^ { t } r _ { t }$ (Chen et al., 2021a), or (4) full trajectory imitation $\tau$ (Duan et al., 2017). $\Phi$ is the reward function $r ( s , a )$ in (2) and (3), an indexing function for state dimensions (e.g. xy-velocities) or a learned function (Nair et al., 2018) in (1), or an identify function in (4). Our CDT introduces a new category, (5) distribution-based $I ^ { \Phi } ( \tau ) = \mathrm { h i s t o g r a m } ( r _ { t } , \gamma )$ , based on a minimal modification to DT, while our BDT can be considered as DT adapted for (4), the trajectory imitation. + +
MethodAlgo.TypeTraining1(T）Architectures
Andrychowicz et al. (2017) Pong et al. (2018)RL RLOnline OnlineT TMLP MLP
Chebotar et al. (2021)RLOfflineTCNN
Li et al. (2020)RLOnlineMLP
tγtr(st,at,.)
Eysenbach et al. (2020)BC/RLOn/OfflineargmaxMLP
Lynch et al. (2019)BCOfflineΦTStochastic RNN
Ghosh et al. (2021)BCOnlineΦTMLP
Srivastava et al. (2019)BCOnlineMt rtFast Weights
Kumar et al. (2019)BCOnlineMt ytrtMLP
Janner et al. (2021)BCOfflinetrtorTTransformer
Duan et al. (2017)3BCOfflineMtMLP+LSTM
Generalized DT(ours)BCOfflineT AnyTransformer
DT(Chen et al., 2021a)BCOfflineMttrtTransformer
Categorical DT(ours)4BCOfflinehistogram(rt,γ)
Transformer
Bi-Directional DT (ours)BCOfflineTTransformer
+ +$$ +\operatorname* { m i n } _ { \pi } \mathbb { E } _ { z \sim p ( z ) , \tau \sim \rho _ { z } ^ { \pi } ( \tau ) } \left[ D ( I ^ { \Phi } ( \tau ) , z ) \right] +$$ + +An important observation for the IM objective (Eq.4) is that for any given trajectory $\tau$ , setting $\mathbf { z } ^ { * } = \dot { \mathbf { I } ^ { \Phi } } ( \tau )$ will minimize the inner term divergence $\mathbf { D = 0 }$ and therefore $\tau$ states and actions are optimal with respect to $\mathbf { z } = \mathbf { z } ^ { * }$ and samples of $( \tau _ { \mathrm { i } } , \mathbf { z _ { \mathrm { i } } ^ { \ast } } )$ can be used to accelerate RL or do BC. We call these algorithms hindsight information matching (HIM) algorithms. + +Table 1, which classifies prior methods into effectively four categories based on $\mathbf { I } ^ { \Phi } ( \tau )$ , leads us to have the following insights around HIM algorithms: + +• New HIM algorithms can be proposed by simply changing $\mathbf { I } ^ { \Phi } ( \tau )$ , as we did to propose Categorical DT for (5) distribution-based. +• Given a choice of $\mathbf { I } ^ { \Phi } ( \tau )$ , new HIM algorithms can be proposed by changing implementation details (Furuta et al., 2021a), such as using “RL" or “BC" as algorithm type, doing “Online” or “Offline” training (Levine et al., 2020), and network architectures. All “Offline” “BC” methods could be adopted easily to “Online” learning through recursive data collections (Ghosh et al., 2021; Kumar et al., 2019; Matsushima et al., 2021). Only (1) goal-based and (2) multi-task can use “RL” as algorithm type, while all four, plus our (5) distribution-based, can use “BC”, because “RL” requires optimizing Eq. 4 with respect to the policy, which gets non-trivial for some choices of $\dot { \mathbf { I } } ^ { \Phi } ( \tau )$ ; e.g. (3-5) return-based, full trajectory imitation, or distribution-based. “BC” bypasses the need to solve Eq. 4 and therefore is universally applicable to any $\mathbf { I } ^ { \Phi } ( \tau )$ or HIM algorithm6. + +# 5 GENERALIZED DECISION TRANSFORMER + +Following the insights in Section 4, we introduce Generalized Decision Transformer (GDT), which generalizes DT (Chen et al., 2021a) based on different choices of $I ^ { \Phi } ( \tau )$ , as described in Figure 1 and the last rows of Table 1. We chose DT as the base model since it is a simple model that uses “BC” as the algorithm type and “Transformer” as the architecture. The choice of “BC” is a must, so we can tractably train GDT with respect to any $I ^ { \Phi } ( \tau )$ or HIM problem. The choice for the architecture is more flexible; however, we decided to use transformers (Vaswani et al., 2017) in this work due to their enormous scaling successes in language and vision domains (Dosovitskiy et al., 2020; Brown et al., 2020; Ramesh et al., 2021). See Algorithm 1 (in Appendix F) for the full pseudocode. While GDT in Figure 1 can lead to different algorithms depending on different choices of the feature function $\Phi ( s , a )$ and the anti-causal aggregator (which together determine $I ^ { \Phi } ( \tau ) \rangle$ ), in this work we focus our empirical studies on the following two variants: Categorical DT (CDT) and Bi-directional DT (BDT). + +# 5.1 TASK DEFINITIONS AND METRICS + +Before proceeding to define CDT and BDT, we first concretely define the tasks they are designed to solve, namely: offline multi-task state-marginal matching (SMM), and offline multi-task imitation learning $\mathbf { ( I L ) }$ . Given the intrinsic connection or equivalence between distribution matching and IL (Ghasemipour et al., 2020), these two separate terminologies may seem redundant. However, inspired by the initial papers studying SMM problems (Lee et al., 2020; Ghasemipour et al., 2020) which qualitatively evaluates distribution matching results in specified state dimensions (e.g. xypositions), we define the imitation task as offline multi-task SMM if specific $\Phi$ is given, and as offline multi-task $\mathrm { I L }$ if $\Phi$ is an identity (i.e. $\phi = s$ ) or learned (e.g. auto-encoder). We essentially view $\mathrm { I L }$ as SMM evaluation on full state. + +Given these definition, we also define a single metric for both offline multi-task SMM/IL: typical IL assumes some availability of task reward or success evaluation, and indirectly measure the quality of imitation through it (Ho & Ermon, 2016; Fu et al., 2018). Instead, again grounding on its connection to distribution matching (Ghasemipour et al., 2020), we propose a Wasserstein loss between statemarginal and target distributions as SMM-inspired metrics for evaluating offline multi-task SMM or IL tasks. However, it is often intractable to measure such loss for full state or even for some state dimensions analytically because both state-marginal and target distributions can be non-parametric and we cannot access their densities. In practice, we empirically estimate it employing the binning of the feature space we specified. More discussions are included in Appendix C. + +# 5.2 CATEGORICAL DECISION TRANSFORMER FOR DISTRIBUTION MATCHING + +Inspired by the recent successes in distributional RL (Bellemare et al., 2017; Dabney et al., 2018; 2020), offline RL (Fujimoto et al., 2019; Jaques et al., 2020; Ghasemipour et al., 2021; Fujimoto & Gu, 2021) and state-marginal matching (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), we introduce Categorical DT (CDT) for offline state-marginal matching (SMM) problem in Section 5.1. Following the prior works (Bellemare et al., 2017; Furuta et al., 2021b), we assume low-dimensional $\Phi$ , e.g. rewards or state dimensions like xyz-velocities, and employ the discretization of feature spaces to form categorical approximations of continuous distributions. To compute $z _ { t } ^ { * } = I ^ { \Phi } ( \tau _ { t : T } )$ for all timesteps $t$ given a trajectory $\tau _ { 1 : T }$ , we use similar recursive Bellman-like computation inspired by Bellemare et al. (2017) (see Appendix F for the details). To the best of our knowledge, this is the first paper to study offline multi-task SMM and propose an effective algorithm for it. + +# 5.3 DECISION TRANSFORMER WITH LEARNED $\Phi$ + +While CDT assumes some low-dimensional $\Phi$ is provided for tractable binning and distribution approximation, we also study cases where $\Phi$ or $r$ is not provided, and instead $\Phi$ is learned through auto-encoding (Hinton & Salakhutdinov, 2006; Bengio et al., 2012) (DT-AE) or contrastive (van den Oord et al., 2018; Srinivas et al., 2020; Yang & Nachum, 2021) (DT-CPC) losses for DT (see Appendix G for the details). In this case, CDT is unnecessary because if $\Phi$ learns sufficient features of $s$ , matching their means, i.e. moments, through DT is enough to match any distribution to an arbitrary precision (Wainwright & Jordan, 2008; Li et al., 2015). Since $\Phi$ is differentiable with respect to DT’s action-prediction losses, we also compare DT-E2E, where we learn $\Phi$ through end-to-end differentiation. As Section 5.1 defines, these methods do offline multi-task SMM with full state, or offline multi-task IL, a similar objective to state-marginal matching or adversarial inverse RL (Ho & Ermon, 2016; Ghasemipour et al., 2020) in online RL. To the best of our knowledge, this is the first offline multi-task IL method that explicitly accounts for SMM through architectural bottlenecks. + +5.4 BI-DIRECTIONAL DECISION TRANSFORMER FOR ONE-SHOT IMITATION LEARNING + +The absence of given $\Phi$ could be tackled with learning not only parameterized $\Phi$ as in Section 5.3, but also a parameterized aggregator. Building on the connection to one-shot imitation learning (Duan et al., 2017), we provide another natural extension of DT under GDT framework called Bi-directional Decision Transformer (BDT), which assumes an identity $\Phi$ , and learns representation within the aggregator, in this case a second (anti-causal) transformer (Radford et al., 2018) that takes a reverseorder state sequence as an input. See Algorithm 1 in Appendix F for the pseudocode, and Appendix D for further comments on the connection to one-shot or meta learning. While we found some positive results for even simple unsupervised regularizer approaches (e.g. DT-AE), we observe BDT could achieve substantially better offline multi-task IL results than DT- $X$ variants in Section 5.3. + +# 6 EXPERIMENTS + +We empirically investigate the following questions: + +• (SMM) Can CDT match unseen reward distributions? • (SMM) Can CDT match and generalize to unseen 1D/2D state-feature distributions? • (SMM) Can CDT match unseen synthesized state-feature distributions? • (IL) Can BDT perform offline one-shot imitation learning in full state? + +We experiment on the OpenAI Gym, MuJoCo tasks (HalfCheetah, Hopper, Walker2d, Ant-v3), a common benchmark for continuous control (Brockman et al., 2016; Todorov et al., 2012). Through the experiments, we use medium-expert datasets in D4RL (Fu et al., 2020) to ensure the decent data coverage. We sort all the trajectories by their cumulative rewards, hold out five best trajectories and five 50 percentile trajectories as a test set (10 trajectories in total), and use the rest as a train set. We report the results averaged over 20 rollouts every 4 random seed. We share our implementation to ensure the reproducibility8 + +As discussed in Section 5.1 and Appendix C, we evaluate CDT/BDT with approximate distribution matching objective: Wasserstein-1 distance between categorical distributions of features in rollouts or target trajectories. We compare CDT to DT, Meta-BC, and FOCAL (Li et al., 2021), a metric-based offline meta RL method, as baselines. While Meta-BC and FOCAL does not solve the offline distribution matching problem directly, they provide decent baseline performance since their offline one-shot adaptation to the given target trajectories could deal with it (see Appendix B for the details). + +# 6.1 REWARD AND STATE-FEATURE MATCHING + +First, we evaluate whether CDT could match its rollout to the target distribution. We choose reward and state-feature, such as x-velocity of the agents, as feature spaces to match. To specify the target distributions during the evaluation, we feed the categorical representation of the target to CDT. As the same as the reward case, DT takes the summation of the state-feature over a trajectory as an input. + +We quantitatively compare CDT against baselines (Table 2) in x-velocity case, where CDT shows better matching results to the target distributions unseen during training. We provide the reward distribution results and the visualization in Appendix E.1, where CDT performs very well in all cases. To test the generalization additionally, we intervene the target distributions by (1) shifting the target distributions in the test set by constant offsets, and (2) synthesizing novel distributions via python scripts. See Appendix E.7 and E.8 for the results. Furthermore, to investigate the scalability of CDT to multi-dimensional state-features, we experiment 2D state-feature distribution matching (xy-velocities on Ant) in Appendix E.3, where CDT outperforms other baselines. + +
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.633 ± 0.3290.996 ± 1.4670.814 ± 1.0790.139± 0.0430.059± 0.0130.099 ± 0.0510.122 ± 0.0710.136 ± 0.0450.129 ±0.0600.347
DT0.746 ± 0.3801.076 ± 1.5490.911 ± 1.1400.177 ± 0.0530.093± 0.0370.135 ± 0.0630.083 ±0.0310.146 ± 0.0840.115± 0.0700.387
BC (no-context)3.017±0.8913.468 ± 1.2713.242 ± 1.1210.652 ± 0.2640.248 ± 0.1990.450 ± 0.3090.748 ± 0.5290.858 ± 0.6170.803 ± 0.5771.498
Meta-BC0.852 ± 0.6880.840 ± 1.1390.846 ± 0.9410.799 ± 0.5050.130± 0.0560.464 ± 0.4910.110 ± 0.0821.462 ± 1.1360.786 ± 1.0520.699
FOCAL (Li ct al.,2021)1.643 ± 0.4611.123 ± 1.5501.383 ± 0.5181.456 ± 0.4730.484 ± 0.3820.970 ± 0.6491.571 ± 0.5630.603± 0.4271.087 ± 0.6951.147
+ +Table 2: Quantitative evaluation of state-feature distribution matching, measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT against DT, BC, Meta-BC, and FOCAL. CDT achieves better matching than baselines. See Table 9 in Appendix E.1 for the reward distribution results. + +# 6.2 GENERALIZATION TO UNSEEN TARGET DISTRIBUTION + +The performance of offline methods in RL is often restricted by the coverage or quality of datasets. While we demonstrate CDT can perform offline state-marginal matching to unseen target distributions in Section 6.1, the standard offline datasets might not be diverse enough to observe the generalization since those are collected by single-task reward-maximization policies. To test the generalization to more diverse behaviors, we investigate the following tasks: (1) z-velocity distribution matching with synthesized bi-modal behavior and (2) cheetah-velocity matching problem from meta RL/IL literature (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020) + +# 6.2.1 SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION + +To generate diverse behaviors for z-axis, we obtain the expert cheetah that backflips towards -x direction by modifying reward function (see Appendix E.4 for the details). Combining expert backflipping trajectories and expert running forward trajectories from D4RL dataset, we construct a novel dataset with diverse behaviors. We experiment the offline SMM with not only each uni-modal behavior (backflipping or running forward), but also synthesized bi-modal behavior; running forward first, then backflipping during a single rollout, using patchworked target trajectories. + +Table 3 and Figure 2 (a) show that CDT successfully matches the distribution to both uni-modal (running forward or backflipping) and synthesized bi-modal distributions better than DT and FOCAL, and is comparable to Meta-BC that originally designed to deal with such multi-task settings. Due to the difficulty for RL algorithms to maximize Eq. 4, FOCAL struggles to solve the offline multi-task SMM, even though FOCAL uses the same context embedding as Meta-BC. The bi-modal behavior learned by CDT can be seen at https://sites.google.com/view/generalizeddt. + +Table 3: State-feature $\mathbf { \dot { z } }$ -velocity) distribution matching with uni-modal and (synthesized) bi-modal target trajectories in HalfCheetah environment. Categorical DT matches both uni- and bi-modal trajectories better than DT and FOCAL, and is comparable to Meta-BC that originally aims to solve multi-task problem. + +
MethodUni-modalBi-modalAverage
Categorical DT1.562 ± 0.6321.625 ± 0.9021.594
DT2.676 ±0.7652.703 ±0.7032.690
Meta-BC1.519 ±0.6961.655 ± 0.9901.587
FOCAL (Li et al., 2021)2.203 士 1.0501.983 士 0.9482.093
+ +![](images/6122f8e9ffbb9010be05cd17b1ddf24787e73452368349e57817055b17b8fe1b.jpg) +Figure 2: (a) Z-Velocity and (b) Unseen Cheetah-Velocity results. Blue histograms represent target distributions. In (a), CDT (red) can match not only uni-modal behaviors for both running forward and backflipping, but also bi-modal behaviors; during a single rollout running forward first, then backflipping. DT (yellow) tends to lean backflipping and fails to fit neither uni-modal nor bi-modal ones. In (b), CDT successfully handles the trajectories unseen during training, while DT seems to output covering behaviors over the dataset support. + +# 6.2.2 DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK + +Generalization to unknown target demonstrations or tasks has been actively investigated in meta or one-shot RL/IL literature (Duan et al., 2016; Wang et al., 2016). To verify the generalization of CDT to diverse behaviors, we adopt the cheetah-velocity task; a popular task in meta RL/IL (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020), where the cheetah tries to run with the specified velocity. We prepare 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals, and hold out $\bar { \{ 0 . 5 , 1 . 5 , 2 . 5 \} }$ as a test set. See Appendix E.5 for the dataset generation. + +Table 4 and Figure 2 (b) reveal that CDT outperforms DT or FOCAL, and is slightly better than MetaBC through the distribution matching evaluation, which implies CDT could solve offline multi-task SMM generalizing to the unknown target trajectories, given sufficiently diverse offline datasets. + +
Methodx-vel: 0.5 x-vel: 1.5x-vel: 2.5Average
Categorical DT0.060± 0.0260.211 ± 0.0220.149± 0.1100.140
DT1.197 ± 0.2270.533 ± 0.1050.861 ± 0.2470.864
Meta-BC0.150 ± 0.0690.152 ± 0.1270.167 ± 0.0550.156
FOCAL (Li et al., 2021)0.472 ± 0.0050.952 ± 0.0730.346 ± 0.1860.590
+ +Table 4: Generalization to unseen target velocity $\mathbf { \hat { x } } \mathbf { \cdot }$ -velocity $= 0 . 5$ , 1.5, 2.5) among training dataset (x-velocity $\in$ $[ 0 . 0 , 3 . 0 ] \backslash \{ 0 . 5 , 1 . 5 , 2 . 5 \}$ ; uniformly spaced at 0.1 intervals), which is a popular setting in meta RL/IL. CDT successfully deals with unseen target velocities well, outperforming DT and FOCAL, and is slightly better than Meta-BC through the offline multi-task SMM evaluation. + +# 6.3 ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE + +Lastly, we investigate BDT for offline multi-task $\mathrm { I L }$ , where we do not observe rewards nor state features explicitly. Instead, target full state trajectories that the agents are expected to mimic is given. We compare BDT against parameterized $\Phi$ variants; DT-AE, -CPC, and -E2E discussed in Section 5.3. We consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT’s supervised loss jointly (called as “joint”), and pre-training with only unsupervised loss and freezing the weights during DT training (called as “frozen”). We aggregate the learned $\Phi$ by summation within the input sequence (length is 20). We evaluate BDT and DT- $X$ with the same distribution matching evaluation as Section 6.1. + +We sweep $m$ -dim learned feature from the encoder $\Phi ( s )$ with $m \in \{ 1 , 4 , 1 6 \}$ . Table 5 presents the offline multi-task IL results with respect to $\mathbf { X }$ -velocity distribution, using $m = 1 6$ embeddings (see Appendix E.6 for the full results including $m = 1 , 4$ , and the reward distribution case). Similar to the discussion in Yang & Nachum (2021), DT-CPC sometimes fails to obtain sufficient representation for imitation. While even simple approaches, DT-AE and DT-AE (frozen), show positive results compared to no-context BC baselines (in Table 2), BDT outperforms all other learned $\Phi$ variants or Meta-BC and is comparable to CDT or DT (also in Table 2) with longer input $\mathrm { \Delta } N = 5 0 \mathrm { \Omega }$ ). This implies that even though we don’t assume the state-feature specification, aggregator choice in GDT with minimal architectural changes may solve the offline distribution matching problem efficiently. We leave more sophisticated objectives or architectural changes as future work. + +
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610 ± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710± 0.5721.591
DT-AE (joint)8.643 ± 0.6792.260± 0.6905.451 ± 3.2641.255±0.3630.649 ± 0.2410.952 ± 0.4322.104±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-CPC (joint)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575 ± 0.0320.096± 0.0280.335 ± 0.2410.949 ± 0.4840.412 ± 0.3780.680± 0.5101.410
DT-E2E8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ± 0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE (frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385 ± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (frozen)3.489 ± 1.1592.543 ± 0.9033.016 ± 1.1410.631 ±0.0910.171 ± 0.1300.401 ± 0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(N=20)1.592 ± 0.2011.208 ± 1.8541.400 ± 1.3330.318±0.0930.081 ± 0.0130.200± 0.1360.196 ±0.0310.392 ± 0.1840.294± 0.1640.631
BDT(N=50)0.840 ± 0.0631.223 ± 1.8281.031 ± 1.3070.142 ± 0.0100.098 ± 0.0250.120 ± 0.0290.192 ± 0.0510.163 ± 0.0270.178 ± 0.0430.443
+ +Table 5: The results of BDT on the state-feature distribution matching problem $m = 1 6$ ). While even simple auto-encoder regularizers sometimes work well, BDT with longer contexts seems to outperform other strategies and is comparable to CDT or DT (in Table 2). See Appendix E.6 for the full results including $m = 1 , 4$ . + +# 7 CONCLUSION + +We provide a unified perspective on a wide range of hindsight algorithms, and generalize the problem formulation as hindsight information matching (HIM). Inspired by recent successes in RL as sequence modeling, we propose Generalized Decision Transformer (GDT) which includes DT, Categorical DT, and Bi-directional DT as special cases, and is applicable to any HIM with proper choices of $\Phi$ and aggregator. We show how Categorical DT, a minor modification of DT, enables the first effective offline state-marginal matching algorithm and propose new benchmark tasks for this problem class. 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Offline meta-reinforcement learning for industrial insertion. arXiv preprint arXiv:2110.04276, 2021. + +Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, Anind K Dey, et al. Maximum entropy inverse reinforcement learning. In Twenty-Third AAAI Conference on Artificial Intelligence, 2008. + +# APPENDIX + +# A HYPER-PARAMETER OF GENERALIZED DECISION TRANSFORMERS + +We implement Categorical Decision Transformer and Bi-directional Decision Transformer, built upon the official codebase released by Chen et al. (2021a) (https://github.com/kzl/ decision-transformer). We follow the most of hyperparameters as they did (Table 6). + +Table 6: List of hyperparameters for DT, CDT and BDT. We refer Chen et al. (2021a). + +
HyperparameterValue
Number of layers Number of attention heads3 1 128
Embedding dimension
Nonlinearity functionReLU
Batch size64
Context length N20
Dropout0.1
Learning rate1e-4
Grad norm clip0.25
Weight decay1e-4
Learning rate decayLinear warmup for first 1OOk training steps
Training(Gradient) steps Number of bins forcategorical distribution1M
31
Encoder size (for DT-X)2 layer MLP,(128,128)
Coefficient of unsupervised loss0.1
+ +# B DETAILS OF BASELINES + +While, to the best of our knowledge, there are no prior work to tackle the offline state-marginal matching problem, we can regard meta or one-shot imitation learning as the methods solving similar problem (see Appendix D for further discussion). In this work, we choose Meta-BC and FOCAL (Li et al., 2021), a metric-based offline meta RL method, as decent baselines. + +FOCAL FOCAL is a offline meta RL method combining BRAC (Wu et al., 2019) with metricbased approach. It utilizes deterministic context encoder trained with inverse-power distance metric losses and detached from Bellman backup gradients. We follow the hyperparameters in the official implementation (https://github.com/LanqingLi1993/FOCAL-ICLR). Deterministic context encoder takes single-step state-action-reward tuple as a context for task inference. We parameterize it with (200, 200, 200)-layers MLP. We train deterministic context encoder with 100000 iteration first, and then train BRAC agent with task-conditioned policy and value functions with 100000 iteration. We use (256, 256, 256)-layers MLPs for policy and value network, and set batch size to 256. + +Meta-BC We also use metric-based Meta-BC (Duan et al., 2017; Dasari & Gupta, 2020) as a strong baseline method for offline multi-task SMM. We adapt deterministic context encoder from FOCAL to infer the task. We train Meta-BC in the same way as FOCAL just replacing BRAC to BC. The objective is mean-squared error minimization. + +Throughout our experiments, Meta-BC shows better results than FOCAL, because RL algorithms often struggle to optimize Eq. 4 for distribution matching and FOCAL tries to maximize the task reward, which is not necessarily required for distribution matching problem. In addition, BC often converges faster than offline RL methods. + +# C DETAILS OF EVALUATION + +Throughout the paper, we evaluate both CDT and BDT from a distribution matching perspective by formulating them as offline multi-task SMM or offline multi-task IL problem. However, the current distribution matching research in RL (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021) has been suffered from the lack of quantitative metrics for evaluation (while standard RL or BC are typically evaluated on the task reward performance). As summarized in Table 7, prior works (Lee et al., 2020; Ghasemipour et al., 2020) evaluate the SMM performance by qualitative density visualization using rollout particles. + +Although the distance between state-marginal and target distribution seems the most intuitive and suitable metric to quantify the performance of distribution matching algorithms, it is often intractable to measure such distance analytically because both state-marginal and target distribution can be non-parametric; we cannot access their densities, and only their samples are available. While a concurrent work (Gu et al., 2021) tackles this problem by leveraging sample-based energy distance estimation, we introduce a single SMM-inspired evaluation for both offline multi-task SMM and offline multi-task IL via estimating Wasserstein-1 distance between empirical categorical distributions. Since the discretization of low-dimensional features, such as reward, have success in many RL methods (Bellemare et al., 2017; Dabney et al., 2018; 2020; Furuta et al., 2021b), quantification of distribution matching performance by Wasserstein-1 distance could be reliable evaluations (in addition Wasserstein-1 distance is symmetric, different from KL distance that is asymmetric). We note that due to the equivalence between inverse RL and SMM methods, the performance of distribution matching methods may be measured via task reward when assuming the accessivity to the expert trajectories (Ho & Ermon, 2016; Fu et al., 2018; Kostrikov et al., 2019). However, in our experiments, it is not suitable to evaluate the performance based on task reward since we uses the sub-optimal trajectories (Section 6.1) or multi-task trajectories from different reward functions (Section 6.2). + +Wasserstein-1 distance between state-marginal distribution $\rho ^ { \pi } ( s )$ and target distribution $p ^ { * } ( s )$ is defined as: + +$$ +W _ { 1 } ( \rho ^ { \pi } ( s ) , p ^ { * } ( s ) ) = \operatorname* { i n f } _ { \nu \in \Gamma ( \rho ^ { \pi } , p ^ { * } ) } \mathbb { E } _ { ( X , Y ) \sim \nu } [ | X - Y | ] , +$$ + +where $\Gamma$ is the set of all possible joint distributions $\nu$ whose marginals are $\rho ^ { \pi }$ and $p ^ { * }$ respectively $X$ and $Y$ are random variables). In this work, we empirically estimate Eq. 5 by focusing on the “manually-specified” feature $\phi \in F$ (e.g. reward, xyz-velocities, etc; defined in Section 4) and employing binning and discretization. We discretize the feature space $F$ and obtain $B$ bins (per dimension), whose representatives are $\{ \bar { \phi } _ { 1 } , \bar { \phi } _ { 2 } , . . . \bar { \phi } _ { B } \}$ . The range of feature space $[ \phi _ { \mathrm { m i n } } , \phi _ { \mathrm { m a x } } ]$ is pre-defined from the given offline datasets. Then, we get categorical distribution from the target trajectory $\tau ; \hat { p } _ { \tau } ^ { * } ( \phi ) = \breve { \{ } ( \bar { \phi } _ { 1 } , c _ { \tau 1 } ^ { * } ) , \ldots ( \bar { \phi } _ { B } , c _ { \tau B } ^ { * } ) \}$ , and from the rollouts of the policy $\pi$ (using 4 seeds $\times 2 0$ rollouts); $\hat { \rho } ^ { \pi } ( \phi ) = \{ ( \bar { \phi } _ { 1 } , c _ { 1 } ^ { \pi } ) , \ldots ( \bar { \phi } _ { B } , c _ { B } ^ { \pi } ) \}$ , where $c _ { \tau } ^ { * }$ and $c ^ { \pi }$ are the weights of each bin (i.e. $\begin{array} { r } { \sum _ { l = 1 } ^ { B } c _ { \tau { l } } ^ { \ast } = 1 } \end{array}$ and $\begin{array} { r } { \sum _ { l = 1 } ^ { B } c _ { l } ^ { \pi } = 1 } \end{array}$ ). We compute the evaluation metric averaging on the test set of the trajectories $D _ { \mathrm { t e s t } }$ : + +$$ +\begin{array} { r l } & { \mathrm { M e t r i c } ( \pi , D _ { \mathrm { t e s t } } ) = \displaystyle \frac { 1 } { | D _ { \mathrm { t e s t } } | } \sum _ { \tau \in D _ { \mathrm { t e s t } } } \operatorname* { m i n } _ { w _ { i j } } \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { B } w _ { i j } | c _ { \tau i } ^ { * } - c _ { j } ^ { \pi } | , } \\ & { \mathrm { s . t . } w _ { i j } \ge 0 , \displaystyle \sum _ { j = 1 } ^ { B } w _ { i j } \le c _ { \tau i } ^ { * } , \displaystyle \sum _ { i = 1 } ^ { B } w _ { i j } \le c _ { j } ^ { \pi } , \displaystyle \sum _ { i = 1 } ^ { B } \sum _ { j = 1 } ^ { B } w _ { i j } = 1 . } \end{array} +$$ + +In practice, we utilize the python package (https://github.com/pkomiske/ Wasserstein) released by Komiske et al. (2020) to compute it. + +Table 7: A Review of evaluation for distribution matching algorithms. + +
EvaluationTypeReference
Density VisualizationQualitativeLee et al. (2020); Ghasemipour et al. (2020)
Energy DistanceQuantitativeGu et al. (2021)
Task RewardQuantitativeHo& Ermon 1 (2016); Fu et al. (2018), etc.
Wasserstein-1DistanceQuantitativeOurs
+ +# D CONNECTION TO META LEARNING + +While we solve offline information matching problems in this paper, our formulations (especially Bi-directional DT) and experimental settings are similar to offline meta RL (Dorfman et al., 2021; Mitchell et al., 2021; Li et al., 2021) and meta/one-shot imitation learning (Yu et al., 2019; Ghasemipour et al., 2019; Finn et al., 2017; Duan et al., 2017) (especially metric-based approach (Rakelly et al., 2019; Duan et al., 2016; Fakoor et al., 2020)). We briefly clarify the relevance and difference between them (Table 8). + +Offline Meta RL Recently, some works deal with offline meta RL problem, assuming task distribution and offline training with pre-stored transitions collected by the (sub-) optimal or scripted task-conditioned agents (Dorfman et al., 2021; Mitchell et al., 2021; Li et al., 2021; Pong et al., 2021; Zhao et al., 2021). In these works, few test-task trajectories, following the same task distribution but unseen during training phase, are given at test-time (i.e. few-shot), and the agents adapt the given task in an online (Pong et al., 2021; Zhao et al., 2021; Dorfman et al., 2021) or fully-offline (Mitchell et al., 2021; Li et al., 2021) manner. + +Meta Imitation Learning Another related domain is meta imitation learning, also assuming task distribution and offline training with pre-stored expert transitions per tasks (Yu et al., 2019; Ghasemipour et al., 2019; Finn et al., 2017; Duan et al., 2017; Xu et al., 2019; Gleave & Habryka, 2018; Dasari & Gupta, 2020). While meta inverse RL methods (Yu et al., 2019; Xu et al., 2019; Gleave & Habryka, 2018) require online samples during test-time adaptation, its off-policy extension and BC-based approach performs offline adaptation with given unseen trajectories (Ghasemipour et al., 2019; Finn et al., 2017; Dasari & Gupta, 2020). + +Our work, in contrast, assume no-adaptation at test time and evaluation criteria is a distance between the feature distributions, not the task rewards. + +Table 8: Review of problem settings among offline meta RL, meta IL, and ours. In offline meta RL, the agents are trained offline and tested with several demonstrations (few-shot), in a fully-offline manner or allowing online data-collection for adaptation. In meta IL, IRL-based methods train the agents online (Yu et al., 2019; Xu et al., 2019; Ghasemipour et al., 2019) while BC-based methods (Finn et al., 2017; Duan et al., 2017) do offline. Similar to our settings, Duan et al. (2017) test the agents with single demonstration and no-adaptation process, they evaluate the agent on the task reward, while we do on the distance between the two distributions. + +
MethodProblemTrainTestDemo
Pong et al. (2021)Offline Meta RLofflineonlinefew-shot
Zhao et al. (2021)Offline Meta RLofflineonlinefew-shot
Dorfman et al. (2021)Offline Meta RLofflineonlinefew-shot
Mitchell et al. (2021)Offline Meta RLofflineofflinefew-shot
Li et al. (2021)Offline Meta RLofflineofflinefew-shot
Yu et al. (2019)Meta ILonlineonlinefew-shot
Xu et al. (2019)Meta ILonlineonlinefew-shot
Ghasemipour et al. (2019)Meta ILonlineofflinefew-shot
Finn et al. (2017)Meta ILofflineofflineone-shot
Duan et al. (2017)Meta ILoffline(no-adaptation)one-shot
OursOffline multi-task SMM/ILoffline(no-adaptation)one-shot
+ +# E DETAILS OF EXPERIMENTS + +In this section, we provide the details and supplemental quantitative/qualitative results of the experiments in Section 6. + +Figure 3 visualizes the dataset distributions we use in the experiments. When we sort all the trajectories based on their cumulative rewards, each quality of trajectory (best, middle, worst) shows a different shape of distribution, and the distribution of the whole dataset seems a weighted mixture of those. + +![](images/0a12826cb7d345cb11f89b3ba83fd91c7d770dc6a674535f9e40adbb5a6e9357.jpg) +Figure 3: Distributions of the features (reward, and $\mathbf { X } ^ { }$ -velocity) in the D4RL medium-expert datasets. + +# E.1 QUANTITATIVE AND QUALITATIVE RESULTS OF REWARD AND STATE-FEATURE MATCHING + +We provide the quantitative comaprison of Section 6.1 between Categorical DT and DT in the reward (Table 9) matching problem, computing Wasserstein-1 distance between the discretized rollout and target feature distributions. Generally, similar to the $\mathbf { X }$ -velocity case, CDT shows the better results in offline multi-task SMM compared to original DT. We also visualize some of CDT results in Figure 4. + +
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.674 ± 0.2131.002 ± 1.4580.838 ± 1.0540.159 ± 0.0850.064 ± 0.0170.111 ± 0.0770.095 ± 0.0170.114 ± 0.0370.105 ± 0.0300.351
DT0.652 ± 0.3191.039 ± 1.5480.846 ± 1.1340.227 ± 0.1190.091 ± 0.0350.159 ± 0.1110.056 ± 0.0150.626 ± 0.4950.341 ± 0.4520.448
BC (no-context)3.240± 0.5592.880 ± 0.6143.060 ± 0.6140.597 ± 0.0560.119 ± 0.0670.358 ± 0.2470.977 ± 0.5010.431 ± 0.3960.704 ± 0.5281.374
Meta-BC0.839 ±0.6820.830 ± 1.1300.835 ± 0.9330.803 ± 0.5050.134 ± 0.0560.468 ± 0.4910.113 ± 0.0851.441 ± 1.1130.777 ± 1.0320.693
FOCAL (Li et al.,2021)1.623 ± 0.5011.115 ± 1.5341.369 ± 0.5161.463 ± 0.4720.492 ± 0.3840.977 ± 0.6491.584 ± 0.5700.604 ± 0.4211.094 ± 0.7011.147
+ +Table 9: Quantitative evaluation of reward distribution matching via measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT and DT. Since it can capture the multi-modal nature of target distributions, CDT matches the distribution better than the original DT. + +![](images/c15ec828e747cb1d7499910bbb5901116cf3aecc7338dafa323749f7c6941389.jpg) +Figure 4: (a) Reward and (b) state-feature (x-velocity) distribution matching in halfcheetah (top), hopper (middle), and walker2d (bottom). The left two examples are the distributions from the best trajectories, and right two are the distributions from the middle trajectories in the held-out test set. The rollout distributions of CDT (red) match the target distributions (blue) very well in all cases. + +# E.2 EVALUATION ON TASK PERFORMANCE + +While in this paper we focus on the evaluation with the SMM-inspired distribution matching objective, such as Wasserstein-1 distance, we here provide the evaluation on the task rewards. Table 10 shows that DT seems consistently better methods than Categorical DT on the task rewards evaluations, and achieves similar performances to the held-out trajectories. + +
Methodhalfcheetahhopperwalker2d
ExpertMediumExpertMediumExpertMedium
Categorical DT10476.746± 218.9575782.557 ± 1666.7162614.559 ± 657.7221518.757 ± 63.0624907.475 ± 11.7113264.585 ± 209.905
DT10500.273 ± 312.7784985.518 ± 67.3222113.207 ± 807.2461528.743± 30.7804965.024 ± 11.8144055.039 ± 849.109
Held-out11146.200 ± 59.0705237.348 ± 37.9703741.854 ± 7.2231600.196± 0.7724995.553 ± 5.4133801.313 ± 0.994
+ +Table 10: Evaluation on the task rewards; conditioning on the held-out trajectories as done in Section 6. We compare the performance between Categorical DT, and DT. DT seems consistently better methods on the task rewards evaluations, and achieves similar performances to the held-out trajectories (averaged over 5 trajectories). + +# E.3 2D STATE-FEATURE DISTRIBUTION MATCHING + +We also consider two-dimensional state-features (xy-velocities) distribution matching in Ant-v3. Same as 1D state-feature distribution matching in HalfCheetah, Hopper, and Walker2d-v3 (Section 6.1), we also use medium-expert(-v2) datasets from D4RL (Fu et al., 2020). We bin the state-features per dimension separately to reduce the dimension of the categorical distribution that CDT takes as input, while in test-time we evaluate the performance with Wasserstein-1 metric on the joint distribution. For DT, we compute the summation of $\mathbf { X } ^ { - }$ and y-velocity each over trajectories and normalize them with the maximum horizon. DT feeds these two scalars as information statistics to match. + +Table 11 reveals that CDT performs better even in the case of two-dimensional state-features distributions, while DT doesn’t generalize to expert-quality trajectories. As shown in Figure 5, while CDT could cope with the distribution shift between expert and medium target distribution, DT always fits the medium one even if the expert trajectory is given as a target. CDT successfully scales to the offline multi-task SMM problem in the multi-dimensional feature spaces. + +
Methodant
ExpertMediumAverage
Categorical DT0.797 ± 0.2160.244 ± 0.0630.521
DT1.714 ± 0.1210.260 ± 0.0670.987
Meta-BC1.295 ± 0.7080.351 ± 0.2050.823
FOCAL (Li et al., 2021)1.473 ± 0.8920.913 ± 0.4551.193
+ +Table 11: Quantitative evaluation of 2D state-feature (xy-velocities) distribution matching, measuring Wasserstein-1 distance. CDT performs better even in the two-dimensional problem. + +![](images/be07d95b475dbda537b278fbe7b1d838a7ec935c089362e0b31dbcc6b73ffec8.jpg) +Figure 5: Visualization of 2D state-feature (xy-velocities) distribution matching, binning each dimension separately. Top row shows the results from an expert target trajectory, and bottom row from a medium one. + +E.4 DETAILS OF SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION + +To construct the dataset, we modified the original reward function in HalfCheetah-v3, adding absolute z-velocity term (such as $+ \mathrm { n p }$ .abs $( z \_ { \mathrm { v e 1 } } )$ ), where the expert cheetah backflips towards - $\mathbf { - X }$ direction. + +We trained SAC agent until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and then collected 500 trajectories $\times ~ 1 0 0 0$ time steps. We combined them with 500 trajectories $\times ~ 1 0 0 0$ time steps from halfcheetah-expert-v2 dataset in D4RL, which consists of both backflipping and running forward behaviors. + +For the evaluation, we prepared additional 5 trajectories of backflipping and 5 of running forward as uni-modal behaviors (10 test trajectories in total). In addition, we synthesized a bi-modal behavior by dividing each 1000-step trajectories into 500-step sub-trajectories, and concatenating them across different behaviors, which results in the patchworked trajectories of first 500-step running forward and next 500-step backflipping (also 10 trajectories in total). + +# E.5 DETAILS OF DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK + +Following prior meta RL/IL works (Rakelly et al., 2019; Ghasemipour et al., 2019; Pong et al., 2021; Li et al., 2021; Fakoor et al., 2020), we modified the reward function for the cheetah to run with specified velocity (such as -np.abs(x_vel - target_vel)), and set the horizon to 200 steps. + +We prepared 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals. We also trained the SAC agents until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and collected 250 trajectories $\times 2 0 0$ time steps each. To simplify the problem than meta learning settings, we held out the 10 trajectories whose $\mathbf { X }$ -velocity is $\{ 0 . 5 , \bar { 1 } . 5 , \bar { 2 } . 5 \}$ as a test set, and used the rest as a train data. + +E.6 QUANTITATIVE AND QUALITATIVE RESULTS OF ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE + +Table 12 and Table 13 show the one-shot reward and state-feature distribution matching results respectively. In addition to the embedding size $m$ , we also sweep the different context window $N = 2 0 , 5 0 , 1 0 0$ for BDT $m = 1 6$ ). While even simple auto-encoder regularizer (DT-AE or DT-AE (frozen)) sometimes works well compared to no-context BC baselines presented in Table 2 and Table 9, BDT, using (second) anti-causal transformer as an encoder $\Phi$ and its aggregator, seems consistently better than other strategies or Meta-BC baseline for offline multi-task SMM and is comparable to CDT or DT results (also presented in Table 2 and Table 9) with longer context length $( N = 5 0 )$ ). Through the experiment we observe that while there are no clear trends in DT-AE, -CPC or -E2E as the size of context embedding $m$ grows, BDT improves its performance with larger size of embedding. Such intuitive properties might a good features to design the architectures. + +We visualize the qualitative results of BDT $( m = 1 6 , N = 2 0 )$ ) in Figure 6. + +Table 12: The results of BDT and DT- $X$ variants on the reward distribution matching problem $( m = 1 , 4 , 1 6$ + +
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE(m=1)2.494 ± 1.0502.877 ± 0.7622.686 ± 0.9370.586± 0.0160.089 ± 0.0290.337 ± 0.2490.733 ± 0.5220.586 ± 0.4450.660 ± 0.4901.228
DT-CPC (m=1)3.595± 0.6762.275 ±0.3412.935±0.8490.600 ± 0.0070.130± 0.0390.365± 0.2371.180 ± 0.0190.139 ± 0.0320.660 ± 0.5211.320
DT-AE(m=4)0.782 ± 0.3291.720 ± 1.6851.251 ± 1.3010.613 ± 0.1080.140 ± 0.0680.376 ± 0.2530.889 ± 0.4020.265 ± 0.1370.577 ± 0.4330.735
DT-CPC (m=4)5.887 ± 0.3571.370 ± 1.3383.628 ± 2.4620.737 ± 0.0180.238 ± 0.0600.487 ± 0.2541.286 ± 0.0180.145 ± 0.0310.715 ± 0.5711.610
DT-AE(m=16)2.041 ± 1.0801.074 ± 0.8141.558 ± 1.0710.613 ± 0.0600.146 ± 0.0510.379 ± 0.2400.837 ± 0.3450.324 ± 0.1260.581 ± 0.3650.839
DT-CPC (m=16)6.022 ± 0.3161.406 ± 1.3713.714 ± 2.5130.614± 0.0190.104 ± 0.0250.359± 0.2561.284± 0.0200.140 ± 0.0390.712 ± 0.5721.595
DT-AE(m=1,joint)3.824 ± 1.1141.988 ± 0.9702.906 ± 1.3910.687 ± 0.0970.137 ± 0.0570.412 ± 0.2860.723± 0.5960.614 ± 0.4620.668 ± 0.5361.329
DT-CPC (m=1, joint)4.460 ± 1.1062.036 ± 1.0323.248 ± 1.6160.587 ± 0.0150.106 ± 0.0420.347 ± 0.2430.815 ± 0.7110.710 ± 0.3980.763 ± 0.5781.452
DT-AE(m=4,joint)5.563 ± 0.4491.028 ± 1.2653.295 ± 2.4581.320 ± 0.2080.485± 0.2350.902 ± 0.4730.917 ± 0.3190.218 ± 0.1120.567 ± 0.4231.588
DT-CPC (m=4, joint)3.486 ± 1.5032.265 ± 1.0772.876 ± 1.4430.690 ± 0.1590.220 ± 0.1760.455± 0.2881.021 ± 0.7080.554 ± 0.3830.788 ± 0.6151.373
DT-AE (m=16, joint)8.450 ± 0.6232.168±0.7015.309 ± 3.2101.257 ± 0.3610.655 ± 0.2420.956 ± 0.4302.112 ±0.6180.994 ± 0.4131.553 ± 0.7672.606
DT-CPC (m=16, joint)4.543 ± 1.1791.869 ± 1.4533.206 ± 1.8810.577 ± 0.0320.098 ± 0.0280.338 ± 0.2410.953 ± 0.4830.414 ± 0.3760.683 ± 0.5101.409
DT-E2E(m=1)3.220 ± 2.3251.026 ± 1.3982.123 ± 2.2101.615 ± 0.5360.540 ± 0.2931.078 ± 0.6901.079 ± 0.2090.455 ± 0.0850.767 ± 0.3511.323
DT-E2E(m=4)8.076 ± 0.5512.552 ± 0.5715.314 ± 2.8181.205 ± 0.3900.493±0.2470.849 ± 0.4832.835±0.9461.239 ± 0.4602.037 ± 1.0912.733
DT-E2E(m=16)8.049 ± 0.9901.859 ± 0.8394.954 ± 3.2281.102 ± 0.2160.549 ± 0.1890.826 ± 0.3432.095±0.4341.241 ± 0.7091.668 ± 0.7262.482
DT-AE(m=1, frozen)2.225 ± 1.0172.804 ± 1.0512.514 ± 1.0740.582 ± 0.0420.131 ± 0.0580.357 ± 0.2311.396 ± 0.1490.259 ± 0.0830.827 ± 0.5811.233
DT-CPC (m=1,frozen)4.110 ± 0.7992.172 ±0.7893.141 ± 1.2530.582 ± 0.0210.102 ± 0.0410.342 ± 0.2421.422 ± 0.0990.275± 0.1090.848 ± 0.5831.444
DT-AE(m=4, frozen)0.815 ± 0.1831.415 ± 1.7551.115 ± 1.2830.681 ± 0.1060.197 ± 0.0610.439 ± 0.2571.066 ± 0.4950.419 ± 0.3210.742 ± 0.5280.765
DT-CPC (m=4, frozen)3.275 ± 1.1862.752 ± 1.0883.014 ± 1.1680.633 ± 0.0550.139 ± 0.0820.386 ± 0.2571.119 ± 0.5980.508 ± 0.3540.814 ± 0.5781.404
DT-AE(m=16, frozen)1.796 ± 0.5781.312 ± 0.852 2.538±0.9051.554 ± 0.7670.637 ± 0.0390.142 ± 0.0630.389 ±0.2531.184 ± 0.3260.405 ± 0.1690.795 ± 0.4690.913
DT-CPC (m=16, frozen)3.486 ± 1.1643.012 ± 1.1450.636± 0.0930.178 ± 0.1320.407 ± 0.2561.318 ± 0.3280.282 ± 0.1260.800 ± 0.5741.407
BDT (m=1, N=20)1.385 ± 0.207 1.660 ± 0.1671.180 ± 1.753 1.058 ± 1.5801.282 ± 1.253 1.359 ± 1.1630.291 ± 0.096 0.494 ± 0.2810.110 ± 0.043 0.096 ± 0.0290.201 ± 0.117 0.295± 0.2821.113 ± 0.044 0.181 ± 0.0230.155 ± 0.046 0.414 ± 0.5370.634 ± 0.481 0.298 ± 0.3970.706
BDT(m=4,N=20)1.565 ± 0.1901.191 ± 1.8301.378 ± 1.3150.321 ± 0.0930.086 ± 0.0170.204 ± 0.1350.204 ± 0.0330.396 ± 0.1860.300 ± 0.1650.650
BDT(m=16, N=20)0.831 ± 0.0641.204 ± 1.8031.018 ± 1.2900.144 ± 0.0110.104 ± 0.0260.124 ± 0.0280.199 ± 0.0520.167 ± 0.0240.183 ± 0.0440.627
BDT(m=16,N=50)1.280 ± 1.8611.108 ± 1.3320.240 ± 0.0470.162 ± 0.0330.201 ± 0.0560.057 ± 0.0070.442
BDT(m=16,N=100)0.936 ± 0.1660.873 ± 0.6070.465 ± 0.5930.591
+ +Table 13: The results of BDT and DT- $X$ variants on the state-feature $\mathbf { \dot { x } }$ -velocity) distribution matching problem $( m = 1 , 4 , 1 6 )$ ). + +
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE (m=1)2.504 ± 1.0502.880 ± 0.7782.692 ± 0.9430.580 ± 0.0150.084 ± 0.0270.332 ± 0.2490.729 ± 0.5220.584 ± 0.4470.656 ± 0.4911.227
DT-CPC (m=1)3.595 ±0.6702.277 ± 0.3492.936 ± 0.8480.601 ± 0.0080.130 ± 0.0370.365 ± 0.2371.177 ± 0.0210.135 ± 0.0330.656 ± 0.5221.319
DT-AE(m=4)0.789 ± 0.3331.729 ± 1.7141.259 ± 1.3210.612 ± 0.1080.138 ± 0.0660.375 ± 0.2540.884 ± 0.4020.262 ± 0.1380.573 ± 0.4330.736
DT-CPC (m=4)5.883 ± 0.3611.371 ± 1.3473.627 ± 2.4620.731 ± 0.0180.229 ± 0.0570.480 ± 0.2551.282 ± 0.0220.141 ± 0.0320.712 ± 0.5711.606
DT-AE(m=16)2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC (m=16)6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710 ± 0.5721.591
DT-AE (m=1,joint)3.825 ± 1.1151.990 ± 0.9882.908 ± 1.3970.683 ± 0.0940.132 ± 0.0550.407 ± 0.2860.719 ± 0.5970.611 ± 0.4630.665 ± 0.5371.327
DT-CPC (m=1,joint)4.460 ± 1.1012.035 ±1.0553.248 ± 1.6220.583 ± 0.0150.099 ± 0.0410.341 ± 0.2440.811 ± 0.7110.709 ± 0.3990.760 ± 0.5791.450
DT-AE(m=4,joint)5.589 ± 0.4581.040 ± 1.2703.315 ± 2.4671.318 ± 0.2080.481 ± 0.2340.899 ± 0.4730.912 ± 0.3190.216 ± 0.1120.564 ± 0.4221.593
DT-CPC (m=4, joint)3.484 ± 1.4982.270 ± 1.0992.877 ± 1.4480.685 ± 0.1570.214 ± 0.1740.449 ± 0.2881.018 ± 0.7130.555 ± 0.3840.786 ± 0.6181.371
DT-AE(m=16,joint) DT-CPC (m=16, joint)8.643 ± 0.6792.260 ±0.6905.451 ± 3.2641.255 ± 0.3630.649 ± 0.2410.952 ± 0.4322.104 ±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-E2E (m=1)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575± 0.0320.096± 0.0280.335± 0.2410.949 ± 0.4840.412 ± 0.3780.680 ± 0.5101.410
DT-E2E(m=4)3.265 ± 2.3461.050 ± 1.4132.158 ± 2.2311.613 ± 0.5400.534 ± 0.2931.073 ± 0.6921.073 ± 0.2110.451 ± 0.0840.762 ± 0.3501.331
8.215 ± 0.6042.684 ±0.5755.449 ± 2.8281.202 ± 0.3920.487 ± 0.2460.845 ± 0.4852.832 ± 0.9511.235 ± 0.4632.034 ± 1.0942.776
DT-E2E (m=16)8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ±0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE(m=1,frozen)2.238 ± 1.0112.807 ± 1.0562.523 ± 1.0720.577 ±0.0430.126± 0.0560.352 ± 0.2311.391 ± 0.1490.256 ±0.0830.824 ± 0.5801.233
DT-CPC (m=1, frozen)4.110 ± 0.7942.173 ± 0.8063.141 ± 1.2570.578± 0.0220.097 ± 0.0390.338 ± 0.2421.417 ± 0.1000.272 ± 0.1090.845 ± 0.5821.441
DT-AE (m=4, frozen)0.825 ± 0.1891.431 ± 1.7821.128 ± 1.3030.679 ± 0.1060.193 ± 0.0600.436 ± 0.2581.063 ± 0.4950.418 ± 0.3220.740 ± 0.5270.768
DT-CPC (m=4,frozen)3.274 ± 1.1872.756 ± 1.0943.015 ± 1.1700.629 ± 0.0540.135 ± 0.0810.382 ± 0.2571.115 ± 0.6000.507 ± 0.3530.811 ± 0.5781.403
DT-AE(m=16, frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (m=16, frozen)3.489 ±1.1592.543±0.9033.016 ± 1.1410.631±0.0910.171 ± 0.1300.401 ±0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(m=1, N=20)1.414 ± 0.210 1.694 ± 0.1711.197 ± 1.770 1.071 ± 1.5941.305 ± 1.265 1.382 ± 1.1750.288 ± 0.0960.108 ± 0.0410.198 ± 0.1161.108 ± 0.0450.152 ± 0.0510.630± 0.4800.711
BDT (m=4,N=20)1.208 ± 1.8541.400 ± 1.3330.490± 0.2800.092 ± 0.0300.291 ± 0.2810.173 ± 0.0240.411 ± 0.5470.292 ± 0.4050.655
BDT(m=16, N=20) BDT(m=16,N=50)1.592 ± 0.2011.223 ± 1.8281.031 ± 1.3070.318 ± 0.0930.081 ± 0.0130.200 ± 0.1360.196 ± 0.0310.392 ± 0.1840.294 ± 0.1640.631
BDT(m=16,N=100)0.840 ± 0.063 0.953 ± 0.1681.308 ± 1.8811.130 ± 1.3470.142 ± 0.010 0.240 ± 0.0440.098 ± 0.025 0.156 ± 0.0320.120 ± 0.029 0.198 ± 0.0570.192 ± 0.051 0.051 ± 0.0060.163 ± 0.027 0.883 ± 0.6140.178 ± 0.043 0.467 ± 0.6010.443 0.598
+ +![](images/07aa2efe41809e75eecc8006e64e0385c9cea39d36c6b3b79db002ed61ddd233.jpg) +Figure 6: (a) Reward and (b) state-feature distribution matching by Bi-directional Decision Transformer $( m = 1 6 )$ ) in halfcheetah (top), hopper (middle), and walker2d (bottom). The left two examples are the distributions from the best trajectories, and right two are the distributions from the middle trajectories. + +# E.7 SHIFTING THE TARGET DISTRIBUTION + +To generate the unseen but manageable generalization-test trajectories within the support of dataset distribution, we make the reward and velocities values of trajectories in the test set shift with a constant offset: bin_size $\times \{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \}$ . Table 14 shows the quantitative comparison between CDT and DT, based on Wasserstein-1 distance between two distributions. CDT successfully handles the distribution shifts (especially in hopper) better than DT. We also provide the state-feature results (Table 15) and qualitative visualizations (Figure 7 and Figure 8), which reveals that CDT can match the rollouts to the shifted target distributions when they are within the support of dataset distributions. + +
Methodhalfcheetahwalker2dAverage
ExpertMediumTotalExperthopper MediumTotalExpertMediumTotal
Categorical DT1.126 ± 0.2452.026 ± 1.1801.576 ± 0.9640.147 ± 0.0340.302 ± 0.0850.224± 0.1010.285± 0.0441.024 ± 0.0760.655 ± 0.3750.818
DT1.133 ± 0.1971.978 ± 1.1041.555 ± 0.8990.521 ±0.0410.531 ± 0.0450.526 ± 0.0430.656±0.3800.915 ± 0.1060.786 ± 0.3080.956
+ +Table 14: Wasserstein-1 distance between shifted reward distribution and the rollout distributions. Categorical DT handles the target distribution shifts and matches the distributions better than DT, since CDT is aware of distributional information of entire trajectories. + +Table 15: Wasserstein-1 distance between shifted state-feature (x-velocity) distribution and the rollout distributions. Similar to the case of reward, Categorical DT handles the target distribution shifts and matches the distributions better than DT. + +
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
Categorical DT1.270 ± 0.2422.371 ± 1.7471.821 ± 1.3630.157 ± 0.0380.337 ± 0.0880.247 ± 0.1120.289 ±0.0520.964±0.1040.626± 0.3470.898
DT1.173 ± 0.3722.056 ± 1.0451.614 ± 0.9010.432 ±0.0870.531 ± 0.0530.482 ±0.0880.408 ±0.2460.885 ± 0.1430.646 ± 0.3120.914
+ +![](images/f4c6314fde7aaf0038339f982c1f76950e21b022a6b636a1222aba573f75ed64.jpg) +Figure 7: Reward distribution matching in halfcheetah (top two rows; best and middle), hopper (middle two rows; best and middle), and walker2d (bottom two rows; best and middle). We shift the target distribution with (from left to right column); bin_size $\times \{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \}$ (Table 14). Categorical DT (red) can match the rollouts to the shifted target distributions (blue) when the shifted targets are within the support of dataset distributions. For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts. + +![](images/2b8e2f52509043a703d85031dfbd9ba8e0f362e95fd35c8dd8bce30ecfe55cce.jpg) +Figure 8: State-feature distribution matching, especially $\mathbf { X }$ -velocity, in halfcheetah (top two rows; best and middle), hopper (middle two rows; best and middle), and walker2d (bottom two rows; best and middle). We shift the target distribution with constant offset (from left to right column); bin_size $\times \{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \}$ (Table 15). For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts. + +# E.8 SYNTHESIZING UNREALISTIC TARGET DISTRIBUTION + +We synthesize six target distributions manually generating x-velocity samples from Gaussian distributions via python scripts as done in prior works (Ghasemipour et al., 2020; Gu et al., 2021). Since we consider the one-dimensional feature space (x-velocity), we simply specify the mean and standard deviation of Gaussian distributions referring each dataset distribution, as shown in Figure 3, and then generate the 1000 samples per trajectory. While these targets are designed at least within the support of the dataset, we do not consider physical realizability. + +• HalfCheetah: $( \mu , \sigma ) = ( 5 . 0 , 1 . 0 )$ , (13.0, 1.0), (9.0, 1.0), (2.5, 1.0), {(5.0, 1.0), (13.0, 1.0)}, {(9.0, 1.0), (2.5, 1.0)}. + +• Hopper: $\left( \mu , \sigma \right) = \left( 2 . 5 , 1 . 0 \right)$ , (1.5, 1.0), (3.5, 1.0), (2.5, 0.5), {(3.5, 1.0), (1.5, 1.0)}, {(3.5, 0.5), (1.5, 0.5)}. + +• Walker2d: $( \mu , \sigma ) = ( 3 . 5 , 1 . 0 )$ , (2.5, 1.0), (4.5, 1.0), (1.5, 1.0), {(2.5, 0.5), (4.5, 0.5)}, {(1.5, 0.5), (4.5, 0.5)}. + +For the last two sets, that have different distributional parameters $\{ ( \mu _ { 1 } , \sigma _ { 1 } ) , ( \mu _ { 2 } , \sigma _ { 2 } ) \}$ , we sampled from two gaussian distributions 500 samples each, and marge them as one trajectory that has multiple modes. + +Although they might be unrealistic, Figure 9 implies that CDT tends to match the target distributions, even in the cases of bi-modal target distributions. Such generalization to synthesized distribution is an important benefit of distribution-conditioned training. We also quantify the performance of CDT against DT from distributional matching perspective in Table 16. + +![](images/3bd096aee9959fb498de0de06509c1c21e2264b56e46d4acf7d1a1105fac6f2b.jpg) +Figure 9: State-feature distribution matching in halfcheetah (top), hopper (middle), and walker2d (bottom). We synthesize the each target distribution from Gaussian distributions. While the results are worse than for realistic test target distributions in Figure 4, considering that many of these arbitrary synthetic targets could be unrealizable, seeing bi-modal matching results show that indeed CDT has learned to generalize something non-trivial. + +Table 16: State-feature $\mathbf { \widetilde { x } }$ -velocity) distribution matching with synthesized, physically unrealistic target distributions generated from scripted Gaussian distributions. We compare Categorical DT and DT. Categorical DT manages to deal with such unrealistic target matching. + +
Methodhalfcheetahhopperwalker2dAverage
Categorical DT2.059 ± 0.7720.536 ± 0.2250.920 ± 0.4281.172
DT4.256 ± 2.2200.584 ± 0.2980.974 ± 0.5401.938
+ +# F DETAILS FOR CATEGORICAL DECISION TRANSFORMER + +Categorical Decision Transformer (CDT) takes histograms of categorical distribution (i.e. discrete approximations of feature distributions; $B$ -dim vector) as the inputs of the transformer. Here we describe how to compute the distributions for all timesteps given a trajectory $t \in [ 0 , 1 , \ldots , T ]$ . For simplicity, we explain the case of one-dimensional feature (e.g. scalar reward), but for $n$ -dimensional features, we can adopt following procedure per each dimension and arrive at $n$ categorical features. This essentially approximates the full joints with a product of independent marginal distribution per dimension, and ensures that number of samples for getting reasonably discretized approximations do not need to grow exponentially as the dimension grows. + +At first, we discretize the feature space $F$ using $B$ bins (per dimension), and convert the feature $\phi _ { t }$ into the one-hot representation $\tilde { \phi } _ { t }$ . The range of feature space $[ \phi _ { \mathrm { m i n } } , \phi _ { \mathrm { m a x } } ]$ is pre-defined from the given offline datasets. $z _ { \Phi _ { \mathrm { h i s t } } } ( t )$ , the categorical feature distribution at time step $t$ , can be computed recursively following Bellman-like equation: + +$$ +z _ { \Phi _ { \mathrm { h i s t } } } ( t ) \propto \tilde { \phi } _ { t } + \gamma ( 1 - \mathbb { 1 } [ t = T ] ) z _ { \Phi _ { \mathrm { h i s t } } } ( t + 1 ) , +$$ + +where $\mathbb { 1 }$ is the indicator function. We compute a series of $z _ { \Phi _ { \mathrm { h i s t } } }$ in a backward manner starting from $T$ . After we obtain the desired information statistics for all trajectories, we feed them to the categorical transformer during training/test time. We describe the python-like pseudocode in Algorithm 1, coloring the changes from the original Decision Transformer (Chen et al., 2021a). + +# G DETAILS OF DECISION TRANSFORMER WITH LEARNED $\Phi$ + +While any unsupervised regularizer for an encoder could be combined into the action MSE loss of Decision Transformer to obtain the learned $\Phi$ efficiently, we observe that even simple objectives, such as auto-encoder and contrastive loss, sometimes perform well. For auto-encoder regularization (DT-AE), we train the MLP encoder (parameterized by $\psi$ ) and decoder with MSE loss of current state reconstruction: + +$$ +\operatorname* { m i n } _ { \psi } \mathbb { E } _ { s \sim D } \left[ \| s - \mathrm { d e c o d e r } _ { \psi } ( \mathrm { e n c o d e r } _ { \psi } ( s ) ) \| ^ { 2 } \right] . +$$ + +Then, we use the output of the encoder as a learned information statistics. In addition, for contrastive loss (DT-CPC), we adopt CURL objective (Srinivas et al., 2020) for state input while adding gaussian perturbation $\epsilon \sim \mathcal { N } ( \mu = 0 . 0 , \sigma = 0 . 1 )$ as data argumentation (following Sinha et al. (2021)). We train the MLP encoder as a query, use its momentum encoder as a key: + +$$ +\operatorname* { m i n } _ { \psi } \mathbb { E } _ { \stackrel { s \sim D , } { \epsilon , \epsilon ^ { \prime } \sim \mathcal { N } ( 0 , \sigma ) } } \left[ \log \frac { \exp { ( \mathrm { e n c o d e r } _ { \psi } ( s ) ^ { T } W \mathrm { e n c o d e r } _ { \tilde { \psi } } ( s + \epsilon ) ) } } { \sum _ { \epsilon ^ { \prime } \neq \epsilon } \exp { ( \mathrm { e n c o d e r } _ { \psi } ( s ) ^ { T } W \mathrm { e n c o d e r } _ { \tilde { \psi } } ( s + \epsilon ^ { \prime } ) ) } } \right] , +$$ + +where $W$ is a learned parameter matrix for the bi-linear inner-product, and $\tilde { \psi }$ is the weights of the momentum encoder, updated as $\tilde { \psi } m \tilde { \psi } + ( 1 - m ) \psi$ , and $m = 0 . 9 5$ . We treat its trained encoder output as a learned information statistics. It remains as future work to combine more advanced, temporary-extended objectives, such as attentive contrastive learning approach proposed in Yang & Nachum (2021). As described in Section 6.3, we consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT’s supervised loss jointly (called as “joint”), and pre-training with only unsupervised loss and freezing the weights during DT training (called as “frozen”). We train DT-E2E with only DT’s supervised loss as in Algorithm 1 for CDT and BDT. + +Algorithm 1 Categorical/Bi-directional Decision Transformer Pseudocode: Orange and green texts describe additional details on top of the base DT pseudocode from Chen et al. (2021a). + +# z: information statistics (histogram, or learned representation) # s, a, t: states, actions, or timesteps # transformer: transformer with causal masking (GPT) # embed_s, embed_a, embed_z: linear embedding layers # anti_causal_tf: second transformer as encoder and aggregator # # embed_t: learned episode positional embeddingpred_a: linear action prediction layer compute_stats: a function to compute information statistics + +# # main model + +def DecisionTransformer(z, s, a, t): # compute embeddings for tokens pos_embedding $=$ embed_t(t) s_embedding $=$ embed_s(s) $^ +$ pos_embedding a_embedding $=$ embed_a(a) $^ +$ pos_embedding if categorical: z_embedding $=$ embed_z(z) $^ +$ pos_embedding elif bi_directional: # input state sequence in a reverse order # NOTE: z is a target state sequence here reversed $=$ flip(z) z_embedding $=$ embed_z(anti_causal_tf(reversed)) $^ +$ pos_embedding input_embeds $=$ stack(z_embedding, s_embedding, a_embedding) + +# use transformer to get hidden states hidden_states $=$ transformer(input_embeds $=$ input_embeds) + +# select hidden states for action prediction tokens a_hidden $=$ unstack(hidden_states).actions + +# predict action return pred_a(a_hidden) + +# training loop +train_z $=$ compute_stats(train_dataset) +# dims: (batch_size, K, dim) +for z, (s, a, t) in zip(train_z, train_dataset): if unsupervised: $\textbf { z } = \textbf { s }$ a_preds $=$ DecisionTransformer(z, s, a, t) loss $=$ mean((a_preds - a)\*\*2) optimizer.zero_grad(); loss.backward(); optimizer.step() +# evaluation loop +test_z $=$ compute_stats(test_dataset) +n_tests $=$ len(test_dataset) +for index in range(n_tests): test_trajectory $=$ test_dataset[index] max_time_steps $=$ len(test_trajectory) s, a, t, done $=$ [env.reset()], [], [1], False # conditioning on the desired information statistics if categorical: $z =$ [test_z[index][0]] elif bi_directional: z = [test_trajectory[’observations’][0]] for t in range(max_time_steps): action $=$ DecisionTransformer(z, s, a, t)[-1] new_s, r, done, $=$ env.step(action) # append new tokens to sequence if categorical: $\mathrm { ~ \bf ~ z ~ } = \mathrm { ~ \bf ~ z ~ } + \mathrm { ~ \bf ~ \cdot ~ }$ [test_z[index] $[ \ t + \pm ] ]$ elif bi_directional: $\textbf { z } = \textbf { z } +$ [test_trajectory[’observations’][t+1]] s, a, t = s + [new_s], a + [action], t + [len(z)] z, s, a, t = z[-N:], # only keep context length of N \ No newline at end of file diff --git a/parse/dev/CAjxVodl_v/CAjxVodl_v_content_list.json b/parse/dev/CAjxVodl_v/CAjxVodl_v_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7e153cd6a349d3e11687920ea91a8724f27026c3 --- /dev/null +++ b/parse/dev/CAjxVodl_v/CAjxVodl_v_content_list.json @@ -0,0 +1,3228 @@ +[ + { + "type": "text", + "text": "GENERALIZED DECISION TRANSFORMER FOROFFLINE HINDSIGHT INFORMATION MATCHING", + "text_level": 1, + "bbox": [ + 176, + 106, + 746, + 154 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hiroki Furuta \nThe University of Tokyo \nfuruta@weblab.t.u-tokyo.ac.jp ", + "bbox": [ + 186, + 176, + 468, + 219 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yutaka Matsuo The University of Tokyo ", + "bbox": [ + 491, + 178, + 651, + 205 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Shixiang Shane Gu Google Research ", + "bbox": [ + 673, + 178, + 808, + 205 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 256, + 544, + 271 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "How to extract as much learning signal from each trajectory data has been a key problem in reinforcement learning (RL), where sample inefficiency has posed serious challenges for practical applications. Recent works have shown that using expressive policy function approximators and conditioning on future trajectory information – such as future states in hindsight experience replay (HER) or returnsto-go in Decision Transformer (DT) – enables efficient learning of multi-task policies, where at times online RL is fully replaced by offline behavioral cloning (BC), e.g. sequence modeling. We demonstrate that all these approaches are doing hindsight information matching (HIM) – training policies that can output the rest of trajectory that matches some statistics of future state information. We present Generalized Decision Transformer (GDT) for solving any HIM problem, and show how different choices for the feature function and the anti-causal aggregator not only recover DT as a special case, but also lead to novel Categorical DT (CDT) and Bi-directional DT (BDT) for matching different statistics of the future. For evaluating CDT and BDT, we define offline multi-task state-marginal matching (SMM) and imitation learning (IL) as two generic HIM problems, propose a Wasserstein distance loss as a metric for both, and empirically study them on MuJoCo continuous control benchmarks. Categorical DT, which simply replaces anti-causal summation with anti-causal binning in DT, enables arguably the first effective offline multi-task SMM algorithm that generalizes well to unseen (and even synthetic) multi-modal reward or state-feature distributions. Bi-directional DT, which uses an anti-causal second transformer as the aggregator, can learn to model any statistics of the future and outperforms DT variants in offline multi-task IL, i.e. one-shot IL. Our generalized formulations from HIM and GDT greatly expand the role of powerful sequence modeling architectures in modern RL. ", + "bbox": [ + 232, + 290, + 766, + 635 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 662, + 336, + 679 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Reinforcement learning (RL) suffers from the problem of sample inefficiency, and a central question is how to extract as much learning signals, or constraint equations (Pong et al., 2018; Tu & Recht, 2019; Dean et al., 2020), from each trajectory data as possible. As dynamics transitions and Bellman equation provide a rich source of supervisory objectives and constraints, many algorithms combined model-free with model-based, and policy-based with value-based in order to achieve maximal sample efficiency, while approximately preserving stable, unbiased policy learning (Heess et al., 2015; Gu et al., 2016; 2017; Buckman et al., 2018; Pong et al., 2018; Tu & Recht, 2019). ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Orthogonal to these, in the recent years we have seen a number of algorithms that are derived from different motivations and frameworks, but share the following common trait: they use future trajectory information $\\tau _ { \\mathbf { t } ; \\mathbf { T } }$ to accelerate optimization of a contextual policy $\\pi ( \\mathbf { a _ { t } } | \\mathbf { s _ { t } } , \\mathbf { z } )$ with context $\\mathbf { z }$ with respect to a parameterized reward function $\\mathbf { r } ( \\mathbf { s _ { t } } , \\mathbf { a _ { t } } , \\mathbf { z } )$ (see Section 3 for notations). These hindsight algorithms have enabled Q-learning with sparse rewards (Andrychowicz et al., 2017), temporally-extended model-based RL with Q-function (Pong et al., 2018), mastery of 6-DoF object manipulation in cluttered scenes from human play (Lynch et al., 2019), efficient multi-task RL (Eysenbach et al., 2020; Li et al., 2020), offline self-supervised discovery of manipulation primitives from pixels (Chebotar et al., 2021), and offline RL using return-conditioned supervised learning with transformers (Chen et al., 2021a; Janner et al., 2021). We derive a generic problem formulation covering all these variants, and observe that this hindsight information matching (HIM) framework, with behavioral cloning (BC) as the learning objective, can learn a conditional policy to generate trajectories that each satisfy any properties, including distributional. ", + "bbox": [ + 173, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Given this insight and recent casting of RL as sequence modeling (Chen et al., 2021a; Janner et al., 2021), we propose Generalized Decision Transformer (GDT), a family of algorithms for future information matching using hindsight behavioral cloning with transformers, and greatly expand the applicability of transformers and other powerful sequential modeling architectures within RL with only small architectural changes to DT. In summary, our key contributions are: ", + "bbox": [ + 174, + 166, + 825, + 237 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We introduce hindsight information matching (HIM) (Section 4, Table 1) as a unifying view of existing hindsight-inspired algorithms, and Generalized Decision Transformers (GDT) as a generalization of DT for RL as sequence modeling to solve any HIM problem (Figure 1). Inspired by distribution RL (Bellemare et al., 2017; Dabney et al., 2018) and state-marginal matching (SMM) (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), we define offline multi-task SMM problems, propose Categorical DT (CDT) (Section 5), validate its empirical performance to match feature distributions (even generalizing to a synthetic bi-modal target distribution at times), and construct the first benchmark tasks for offline multi-task SMM. Inspired by one-shot imitation learning (Duan et al., 2017; Finn et al., 2017; Dasari & Gupta, 2020), we define offline multi-task imitation learning $( I L )$ , propose a Wasserstein-distance evaluation metric, develop Bi-directional DT (BDT) as a fully expressive variant of GDT (Section 5), and demonstrate BDT’s competitive performance at offline multi-task IL. ", + "bbox": [ + 202, + 243, + 826, + 416 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/f65124c0f2d4f6c2f977e80aa9a54cfbd184001708c35ad790687fd7753196be.jpg", + "image_caption": [ + "Figure 1: Generalized Decision Transformer (GDT), where the figure is a minor generalization of the DT architecture (Chen et al., 2021a) and the table summarizes how it leads to different classes of algorithms with only small architectural changes. If the feature function $\\Phi ( s , a )$ is reward $r ( s , a )$ and the anti-causal aggregator is $\\gamma$ -discounted summation, we recover DT for offline RL. If the aggregator is binning, we get Categorical DT (CDT) for offline multi-task state-marginal matching. If the aggregator is a second transformer, we get Bi-directional DT (BDT) for offline multi-task imitation learning (IL), or equivalently one-shot IL. The choices of $\\Phi ( s , a )$ and the aggregator together decide $I ^ { \\Phi } ( \\tau )$ in Hindsight Information Matching (HIM) objective discussed in Section 4 and Table 1, where conversely GDT can essentially solve any HIM problem with proper choices of $\\Phi$ and aggregator. " + ], + "image_footnote": [], + "bbox": [ + 176, + 430, + 483, + 602 + ], + "page_idx": 1 + }, + { + "type": "table", + "img_path": "images/2b81d9d90d5198990122f5bb89a45d1543ea20fd95cdf7e02c3ef28b57a6a6c9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Method(s,a)Aggregator
DT (Chen et al., 2021a) DT-X (Section 5.3)r(s,a)Summation
CDT (Section 5.2)LearnedSummation
r(s,a) or anyBinning
BDT (Section 5.4)LearnedTransformer
", + "bbox": [ + 491, + 522, + 812, + 585 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 752, + 344, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Hindsight Reinforcement Learning and Behavior Cloning Hindsight techniques (Kaelbling, 1993; Andrychowicz et al., 2017; Pong et al., 2018) have revolutionized off-policy optimization with respect to parameterized reward functions. Two key insights were (1) for off-policy algorithms such as Q-learning (Mnih et al., 2015; Gu et al., 2016) and actor-critic methods (Lillicrap et al., 2016; Haarnoja et al., 2018; Fujimoto et al., 2018; Furuta et al., 2021a), the same transition samples can be used to learn with respect to any reward parameters, as long as the reward function is re-computable, i.e. “relabel”-able, like goal reaching rewards, and (2) if policy or Q-functions are smooth with respect to the reward parameter, generalization can speed up learning even with respect to “unexplored” rewards. In goal-based RL where future states can inform “optimal” reward parameters with respect to the transitions’ actions, hindsight methods were applied successfully to enable effective training of goal-based Q-function for sparse rewards (Andrychowicz et al., 2017), derive exact connections between Q-learning and classic model-based RL (Pong et al., 2018), dataefficient off-policy hierarchical RL (Nachum et al., 2018), multi-task RL (Eysenbach et al., 2020; Li et al., 2020), offline RL (Chebotar et al., 2021), and more (Eysenbach et al., 2021; Choi et al., 2021; Ren et al., 2019; Zhao & Tresp, 2018; Ghosh et al., 2021; Nasiriany et al., 2021). Additionally, Lynch et al. (2019) and Gupta et al. (2018) have shown that often BC is sufficient for learning generalizable parameterized policies, due to rich positive examples from future states, and most recently Chen et al. (2021a) and Janner et al. (2021), when combined with powerful transformer architectures (Vaswani et al., 2017), it produced state-of-the-art offline RL and goal-based RL results. Lastly, while motivated from alternative mathematical principles and not for parameterized objectives, future state information was also explored as ways of reducing variance or improving estimations for generic policy gradient methods (Pinto et al., 2017; Guo et al., 2021; Venuto et al., 2021). ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 826, + 270 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Distributional Reinforcement Learning and State-Marginal Matching Modeling the full distribution of returns instead of the averages led to the development of distributional RL algorithms (Bellemare et al., 2017; Dabney et al., 2018; 2020; Castro et al., 2018; Barth-Maron et al., 2018) such as Categorical Q-learning (Bellemare et al., 2017). While our work shares techniques such as discretization and binning, these works focus on optimizing a non-conditional reward-maximizing RL policy and therefore our problem definition is closer to that of state-marginal matching algorithms (Hazan et al., 2019; Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), or equivalently inverse RL algorithms (Ziebart et al., 2008; Ho & Ermon, 2016; Finn et al., 2016; Fu et al., 2018; Ghasemipour et al., 2020) whose connections to feature-expectation matching have been long discussed (Abbeel & $\\mathrm { N g }$ , 2004). However, those are often exclusively online algorithms even sample-efficient variants (Kostrikov et al., 2019), since density-ratio estimations with either discriminative (Ghasemipour et al., 2020) or generative (Lee et al., 2020) approach requires on-policy samples, with a rare exception of Kostrikov et al. (2020). Building on the success of DT and brute-force hindsight imitation learning, our Categorical DT is to the best our knowledge the first method that benchmarks offline state-marginal matching problem in the multi-task settings. ", + "bbox": [ + 173, + 277, + 825, + 486 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "RL and Imitation Learning as Sequence Modeling When scaled to the extreme levels of data and computing, sequence models such as transformers (Vaswani et al., 2017) can train models to master an impressive range of capabilities in natural language processing and computer vision (Devlin et al., 2019; Radford et al., 2019; Brown et al., 2020; Radford et al., 2021; Ramesh et al., 2021; Chen et al., 2021b; Bommasani et al., 2021; Dosovitskiy et al., 2020). Comparing to their popularity in other areas, the adoption of transformers or architectural innovations in RL have been slow, partially due the difficulty of using transformers over temporal scales for online RL (Parisotto et al., 2020). Recent successes have focused on processing variable-length per-timestep information such as morphology (Kurin et al., 2021), sensory information (Tang & Ha, 2021), one-shot or few-shot imitation learning (Dasari & Gupta, 2020), or leveraged offline learning (Chen et al., 2021a; Janner et al., 2021). Our formulation enables sequence modeling to solve novel RL problems such as statemarginal matching with minimal architectural modifications to DT, greatly expanding the impacts of transformers and other powerful sequence models in RL. ", + "bbox": [ + 173, + 492, + 826, + 672 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 696, + 339, + 712 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider a Markov Decision Process (MDP) defined by the tuple of action space $\\mathcal { A }$ , state space $s$ , transition probability function $p ( s ^ { \\prime } | s , a )$ , initial state distribution $p ( s _ { 0 } )$ , reward function $r ( s , a )$ , and discount factor $\\gamma \\in ( 0 , 1 ]$ . In deep RL, a policy that maps the state space to the action space is parameterized by the function approximators, $\\dot { \\pi } _ { \\boldsymbol { \\theta } } ( \\dot { a } | s ) ^ { 1 }$ . The RL objective is given by: ", + "bbox": [ + 174, + 728, + 825, + 786 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f3551f570439a82a43fd9ca1f14caba561d788db3066c210dc00ef37921af258.jpg", + "text": "$$\nL _ { \\mathrm { R L } } ( \\pi ) = \\frac { 1 } { 1 - \\gamma } \\mathbb { E } _ { s \\sim \\rho ^ { \\pi } ( s ) , a \\sim \\pi ( \\cdot | s ) } \\left[ r ( s , a ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 356, + 795, + 642, + 827 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { p _ { t } ^ { \\pi } ( s ) = \\iint _ { s _ { 0 : t } , a _ { 0 : t - 1 } } \\prod _ { t } p ( s _ { t } | s _ { t - 1 } , a _ { t - 1 } ) \\pi ( a _ { t } | s _ { t } ) } \\end{array}$ and $\\begin{array} { r } { \\rho ^ { \\pi } ( s ) = ( 1 - \\gamma ) \\sum _ { t ^ { \\prime } } \\gamma ^ { t ^ { \\prime } } p _ { t ^ { \\prime } } ^ { \\pi } ( s _ { t ^ { \\prime } } = s ) } \\end{array}$ are short-hands for time-aligned and time-aggregated state marginal distributions following policy $\\pi$ . ", + "bbox": [ + 173, + 837, + 828, + 871 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 STATE MARGINAL MATCHING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 424, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "State marginal matching (SMM) (Lee et al., 2020; Hazan et al., 2019; Ghasemipour et al., 2020) has been recently studied as an alternative problem specification in RL, where instead of stationary-reward maximization, the objective is to find a policy minimizing the divergence $D$ between its state marginal distribution $\\rho ^ { \\pi } ( s )$ to a given target distribution $p ^ { * } ( s ) ^ { 2 }$ : ", + "bbox": [ + 173, + 128, + 825, + 185 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2a1b3fd5b00115be16d4a7463fda89bfa059cbc4baece45293731ad689c1b1c3.jpg", + "text": "$$\nL _ { \\mathrm { S M M } } ( \\pi ) = - D ( \\rho ^ { \\pi } ( s ) , p ^ { * } ( s ) )\n$$", + "text_format": "latex", + "bbox": [ + 395, + 193, + 602, + 210 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D$ is a divergence measure such as Kullback-Leibler (KL) divergence (Lee et al., 2020; Fu et al., 2018) or, more generally, some $f$ -divergences (Ghasemipour et al., 2020). For the target distribution $p ^ { * } ( s )$ , Lee et al. (2020) set a uniform distribution to enhance the exploration over the entire state space; Ghasemipour et al. (2020) and Gu et al. (2021) set through scripted distribution sketches to generate desired behaviors; and adversarial inverse RL methods (Ho & Ermon, 2016; Fu et al., 2018; Ghasemipour et al., 2020; Kostrikov et al., 2020) set as the expert data for imitation learning. Notably, unlike the RL objective in Eq.1, SMM objectives like Eq.2 no longer depend on task rewards and are only functions of state transition dynamics and target state distribution. ", + "bbox": [ + 173, + 215, + 825, + 328 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 PARAMETERIZED RL OBJECTIVES ", + "text_level": 1, + "bbox": [ + 176, + 344, + 452, + 359 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Lastly, we discuss the basis for methods like HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al., 2019), and return-conditioned or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al., 2021): parameterized RL objectives. Given parameterized reward functions with parameter $z \\in { \\mathcal { Z } }$ , a conditional policy $\\pi ( a | s , z )$ is learned with respect to multiple values of $z$ simultaneously weighted by $p ( z )$ . As examples, the RL objective in Eq.1 becomes: ", + "bbox": [ + 173, + 369, + 826, + 455 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/746b993baf462f1c4b39701d72c4bbd71882aa02220343884b5ca33d28152772.jpg", + "text": "$$\nL _ { \\mathrm { R L } } ( \\pi ) = \\mathbb { E } _ { z } \\left[ L _ { \\mathrm { R L } } ( \\pi , z ) \\right] = \\frac { 1 } { 1 - \\gamma } \\mathbb { E } _ { z \\sim p ( z ) , s \\sim \\rho _ { z } ^ { \\pi } ( s ) , a \\sim \\pi ( \\cdot | s , z ) } \\left[ r _ { z } ( s , a ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 267, + 460, + 730, + 493 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the state marginal $\\rho _ { z } ^ { \\pi }$ is from rolling out a conditioned policy $\\pi ( \\cdot | \\cdot , z )$ . These can be considered as a special case of contextual MDPs (Jiang et al., 2017) and are all multi-task RL problems. ", + "bbox": [ + 173, + 500, + 826, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 HINDSIGHT INFORMATION MATCHING", + "text_level": 1, + "bbox": [ + 174, + 547, + 526, + 565 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We show how HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al., 2019), hindsight multi-task RL (Li et al., 2020; Eysenbach et al., 2020), and return-conditioned or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al., 2021) all belong to hindsight algorithms with a shared idea of using future state information to automatically mine for positive, or “optimal”, examples with respect to certain contextual parameter values, where these examples can accelerate RL or be used for behavior cloning (BC), i.e. supervised learning. We start by defining additional notations. ", + "bbox": [ + 173, + 579, + 826, + 679 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given a partial trajectory from state $s _ { t }$ as $\\tau _ { t } = \\{ s _ { t } , a _ { t } , s _ { t + 1 } , a _ { t + 1 } , \\dots \\}$ , we define its information statistics as $I ( \\tau _ { t } )$ . $I ( \\hat { \\tau _ { t } } )$ could be any function of a trajectory that captures some statistical properties in state-space or trajectory-space, such as sufficient statistics of a distribution, like mean, variance or higher-order moments (Wainwright & Jordan, 2008). For convenience, we further define the notion of a feature function $\\Phi ( \\cdot , \\cdot ) : S \\times A \\to F ^ { 5 }$ , where the trajectory is then noted as $\\tau _ { t } ^ { \\Phi } = \\{ \\phi _ { t } , \\phi _ { t + 1 } , \\ldots , \\phi _ { T } \\} , \\phi _ { t } = \\Phi ( s _ { t } , a _ { t } ) \\in \\stackrel { } { F }$ and the information statistics as $\\tilde { I } ^ { \\Phi } \\bar { ( } \\tau _ { t } )$ . $\\Phi$ in practice can be an identity function, the reward function $r ( s , a )$ , sub-dimensions of $s$ (e.g. xy-velocities), or a generic parameterized function (e.g. auto-encoder). Generalizing reward-centric intuitions in DT (Chen et al., 2021a), we define information matching (IM) problems as learning a conditional policy $\\pi ( a | s , z )$ whose trajectory rollouts satisfy some desired information statistics value $z$ : ", + "bbox": [ + 173, + 684, + 825, + 770 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/94a63d673cafcb8da046ec87de3dc11eee062efe246552697db71151673aff96.jpg", + "table_caption": [ + "Table 1: A coarse summary of hindsight information matching (HIM) algorithms. The notation follows Section 4. With HIM, all prior works can be categorized to four generic problem types based on $I ^ { \\Phi } ( \\tau )$ : (1) goal-based $\\phi _ { T }$ (Andrychowicz et al., 2017), (2) multi-task arg max $\\begin{array} { r } { \\sum _ { t } \\hat { \\gamma } ^ { t } r ( s _ { t } , a _ { t } , \\cdot ) } \\end{array}$ (Li et al., 2020), (3) return-based $\\textstyle \\sum _ { t } \\gamma ^ { t } r _ { t }$ (Chen et al., 2021a), or (4) full trajectory imitation $\\tau$ (Duan et al., 2017). $\\Phi$ is the reward function $r ( s , a )$ in (2) and (3), an indexing function for state dimensions (e.g. xy-velocities) or a learned function (Nair et al., 2018) in (1), or an identify function in (4). Our CDT introduces a new category, (5) distribution-based $I ^ { \\Phi } ( \\tau ) = \\mathrm { h i s t o g r a m } ( r _ { t } , \\gamma )$ , based on a minimal modification to DT, while our BDT can be considered as DT adapted for (4), the trajectory imitation. " + ], + "table_footnote": [], + "table_body": "
MethodAlgo.TypeTraining1(T）Architectures
Andrychowicz et al. (2017) Pong et al. (2018)RL RLOnline OnlineT TMLP MLP
Chebotar et al. (2021)RLOfflineTCNN
Li et al. (2020)RLOnlineMLP
tγtr(st,at,.)
Eysenbach et al. (2020)BC/RLOn/OfflineargmaxMLP
Lynch et al. (2019)BCOfflineΦTStochastic RNN
Ghosh et al. (2021)BCOnlineΦTMLP
Srivastava et al. (2019)BCOnlineMt rtFast Weights
Kumar et al. (2019)BCOnlineMt ytrtMLP
Janner et al. (2021)BCOfflinetrtorTTransformer
Duan et al. (2017)3BCOfflineMtMLP+LSTM
Generalized DT(ours)BCOfflineT AnyTransformer
DT(Chen et al., 2021a)BCOfflineMttrtTransformer
Categorical DT(ours)4BCOfflinehistogram(rt,γ)
Transformer
Bi-Directional DT (ours)BCOfflineTTransformer
", + "bbox": [ + 254, + 102, + 741, + 258 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 392, + 825, + 448 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b252eccea7f7a47d35b7316557ab57b6e0d80b6b1bba0ae95e14cd41471b2bad.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { z \\sim p ( z ) , \\tau \\sim \\rho _ { z } ^ { \\pi } ( \\tau ) } \\left[ D ( I ^ { \\Phi } ( \\tau ) , z ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 379, + 455, + 617, + 479 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "An important observation for the IM objective (Eq.4) is that for any given trajectory $\\tau$ , setting $\\mathbf { z } ^ { * } = \\dot { \\mathbf { I } ^ { \\Phi } } ( \\tau )$ will minimize the inner term divergence $\\mathbf { D = 0 }$ and therefore $\\tau$ states and actions are optimal with respect to $\\mathbf { z } = \\mathbf { z } ^ { * }$ and samples of $( \\tau _ { \\mathrm { i } } , \\mathbf { z _ { \\mathrm { i } } ^ { \\ast } } )$ can be used to accelerate RL or do BC. We call these algorithms hindsight information matching (HIM) algorithms. ", + "bbox": [ + 173, + 493, + 825, + 550 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1, which classifies prior methods into effectively four categories based on $\\mathbf { I } ^ { \\Phi } ( \\tau )$ , leads us to have the following insights around HIM algorithms: ", + "bbox": [ + 169, + 556, + 823, + 585 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• New HIM algorithms can be proposed by simply changing $\\mathbf { I } ^ { \\Phi } ( \\tau )$ , as we did to propose Categorical DT for (5) distribution-based. \n• Given a choice of $\\mathbf { I } ^ { \\Phi } ( \\tau )$ , new HIM algorithms can be proposed by changing implementation details (Furuta et al., 2021a), such as using “RL\" or “BC\" as algorithm type, doing “Online” or “Offline” training (Levine et al., 2020), and network architectures. All “Offline” “BC” methods could be adopted easily to “Online” learning through recursive data collections (Ghosh et al., 2021; Kumar et al., 2019; Matsushima et al., 2021). Only (1) goal-based and (2) multi-task can use “RL” as algorithm type, while all four, plus our (5) distribution-based, can use “BC”, because “RL” requires optimizing Eq. 4 with respect to the policy, which gets non-trivial for some choices of $\\dot { \\mathbf { I } } ^ { \\Phi } ( \\tau )$ ; e.g. (3-5) return-based, full trajectory imitation, or distribution-based. “BC” bypasses the need to solve Eq. 4 and therefore is universally applicable to any $\\mathbf { I } ^ { \\Phi } ( \\tau )$ or HIM algorithm6. ", + "bbox": [ + 205, + 592, + 825, + 765 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 GENERALIZED DECISION TRANSFORMER ", + "text_level": 1, + "bbox": [ + 174, + 784, + 549, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Following the insights in Section 4, we introduce Generalized Decision Transformer (GDT), which generalizes DT (Chen et al., 2021a) based on different choices of $I ^ { \\Phi } ( \\tau )$ , as described in Figure 1 and the last rows of Table 1. We chose DT as the base model since it is a simple model that uses “BC” as the algorithm type and “Transformer” as the architecture. The choice of “BC” is a must, so we can tractably train GDT with respect to any $I ^ { \\Phi } ( \\tau )$ or HIM problem. The choice for the architecture is more flexible; however, we decided to use transformers (Vaswani et al., 2017) in this work due to their enormous scaling successes in language and vision domains (Dosovitskiy et al., 2020; Brown et al., 2020; Ramesh et al., 2021). See Algorithm 1 (in Appendix F) for the full pseudocode. While GDT in Figure 1 can lead to different algorithms depending on different choices of the feature function $\\Phi ( s , a )$ and the anti-causal aggregator (which together determine $I ^ { \\Phi } ( \\tau ) \\rangle$ ), in this work we focus our empirical studies on the following two variants: Categorical DT (CDT) and Bi-directional DT (BDT). ", + "bbox": [ + 174, + 815, + 825, + 887 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 TASK DEFINITIONS AND METRICS ", + "text_level": 1, + "bbox": [ + 176, + 205, + 450, + 219 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Before proceeding to define CDT and BDT, we first concretely define the tasks they are designed to solve, namely: offline multi-task state-marginal matching (SMM), and offline multi-task imitation learning $\\mathbf { ( I L ) }$ . Given the intrinsic connection or equivalence between distribution matching and IL (Ghasemipour et al., 2020), these two separate terminologies may seem redundant. However, inspired by the initial papers studying SMM problems (Lee et al., 2020; Ghasemipour et al., 2020) which qualitatively evaluates distribution matching results in specified state dimensions (e.g. xypositions), we define the imitation task as offline multi-task SMM if specific $\\Phi$ is given, and as offline multi-task $\\mathrm { I L }$ if $\\Phi$ is an identity (i.e. $\\phi = s$ ) or learned (e.g. auto-encoder). We essentially view $\\mathrm { I L }$ as SMM evaluation on full state. ", + "bbox": [ + 174, + 232, + 825, + 357 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Given these definition, we also define a single metric for both offline multi-task SMM/IL: typical IL assumes some availability of task reward or success evaluation, and indirectly measure the quality of imitation through it (Ho & Ermon, 2016; Fu et al., 2018). Instead, again grounding on its connection to distribution matching (Ghasemipour et al., 2020), we propose a Wasserstein loss between statemarginal and target distributions as SMM-inspired metrics for evaluating offline multi-task SMM or IL tasks. However, it is often intractable to measure such loss for full state or even for some state dimensions analytically because both state-marginal and target distributions can be non-parametric and we cannot access their densities. In practice, we empirically estimate it employing the binning of the feature space we specified. More discussions are included in Appendix C. ", + "bbox": [ + 174, + 364, + 826, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 CATEGORICAL DECISION TRANSFORMER FOR DISTRIBUTION MATCHING ", + "text_level": 1, + "bbox": [ + 178, + 508, + 722, + 522 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Inspired by the recent successes in distributional RL (Bellemare et al., 2017; Dabney et al., 2018; 2020), offline RL (Fujimoto et al., 2019; Jaques et al., 2020; Ghasemipour et al., 2021; Fujimoto & Gu, 2021) and state-marginal matching (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), we introduce Categorical DT (CDT) for offline state-marginal matching (SMM) problem in Section 5.1. Following the prior works (Bellemare et al., 2017; Furuta et al., 2021b), we assume low-dimensional $\\Phi$ , e.g. rewards or state dimensions like xyz-velocities, and employ the discretization of feature spaces to form categorical approximations of continuous distributions. To compute $z _ { t } ^ { * } = I ^ { \\Phi } ( \\tau _ { t : T } )$ for all timesteps $t$ given a trajectory $\\tau _ { 1 : T }$ , we use similar recursive Bellman-like computation inspired by Bellemare et al. (2017) (see Appendix F for the details). To the best of our knowledge, this is the first paper to study offline multi-task SMM and propose an effective algorithm for it. ", + "bbox": [ + 174, + 535, + 826, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.3 DECISION TRANSFORMER WITH LEARNED $\\Phi$ ", + "text_level": 1, + "bbox": [ + 176, + 693, + 524, + 707 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "While CDT assumes some low-dimensional $\\Phi$ is provided for tractable binning and distribution approximation, we also study cases where $\\Phi$ or $r$ is not provided, and instead $\\Phi$ is learned through auto-encoding (Hinton & Salakhutdinov, 2006; Bengio et al., 2012) (DT-AE) or contrastive (van den Oord et al., 2018; Srinivas et al., 2020; Yang & Nachum, 2021) (DT-CPC) losses for DT (see Appendix G for the details). In this case, CDT is unnecessary because if $\\Phi$ learns sufficient features of $s$ , matching their means, i.e. moments, through DT is enough to match any distribution to an arbitrary precision (Wainwright & Jordan, 2008; Li et al., 2015). Since $\\Phi$ is differentiable with respect to DT’s action-prediction losses, we also compare DT-E2E, where we learn $\\Phi$ through end-to-end differentiation. As Section 5.1 defines, these methods do offline multi-task SMM with full state, or offline multi-task IL, a similar objective to state-marginal matching or adversarial inverse RL (Ho & Ermon, 2016; Ghasemipour et al., 2020) in online RL. To the best of our knowledge, this is the first offline multi-task IL method that explicitly accounts for SMM through architectural bottlenecks. ", + "bbox": [ + 173, + 718, + 825, + 885 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.4 BI-DIRECTIONAL DECISION TRANSFORMER FOR ONE-SHOT IMITATION LEARNING", + "bbox": [ + 171, + 904, + 790, + 919 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The absence of given $\\Phi$ could be tackled with learning not only parameterized $\\Phi$ as in Section 5.3, but also a parameterized aggregator. Building on the connection to one-shot imitation learning (Duan et al., 2017), we provide another natural extension of DT under GDT framework called Bi-directional Decision Transformer (BDT), which assumes an identity $\\Phi$ , and learns representation within the aggregator, in this case a second (anti-causal) transformer (Radford et al., 2018) that takes a reverseorder state sequence as an input. See Algorithm 1 in Appendix F for the pseudocode, and Appendix D for further comments on the connection to one-shot or meta learning. While we found some positive results for even simple unsupervised regularizer approaches (e.g. DT-AE), we observe BDT could achieve substantially better offline multi-task IL results than DT- $X$ variants in Section 5.3. ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 250, + 326, + 266 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We empirically investigate the following questions: ", + "bbox": [ + 174, + 281, + 509, + 296 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• (SMM) Can CDT match unseen reward distributions? • (SMM) Can CDT match and generalize to unseen 1D/2D state-feature distributions? • (SMM) Can CDT match unseen synthesized state-feature distributions? • (IL) Can BDT perform offline one-shot imitation learning in full state? ", + "bbox": [ + 205, + 303, + 772, + 367 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We experiment on the OpenAI Gym, MuJoCo tasks (HalfCheetah, Hopper, Walker2d, Ant-v3), a common benchmark for continuous control (Brockman et al., 2016; Todorov et al., 2012). Through the experiments, we use medium-expert datasets in D4RL (Fu et al., 2020) to ensure the decent data coverage. We sort all the trajectories by their cumulative rewards, hold out five best trajectories and five 50 percentile trajectories as a test set (10 trajectories in total), and use the rest as a train set. We report the results averaged over 20 rollouts every 4 random seed. We share our implementation to ensure the reproducibility8 ", + "bbox": [ + 173, + 373, + 825, + 472 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As discussed in Section 5.1 and Appendix C, we evaluate CDT/BDT with approximate distribution matching objective: Wasserstein-1 distance between categorical distributions of features in rollouts or target trajectories. We compare CDT to DT, Meta-BC, and FOCAL (Li et al., 2021), a metric-based offline meta RL method, as baselines. While Meta-BC and FOCAL does not solve the offline distribution matching problem directly, they provide decent baseline performance since their offline one-shot adaptation to the given target trajectories could deal with it (see Appendix B for the details). ", + "bbox": [ + 174, + 478, + 825, + 563 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 REWARD AND STATE-FEATURE MATCHING ", + "text_level": 1, + "bbox": [ + 178, + 579, + 509, + 594 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, we evaluate whether CDT could match its rollout to the target distribution. We choose reward and state-feature, such as x-velocity of the agents, as feature spaces to match. To specify the target distributions during the evaluation, we feed the categorical representation of the target to CDT. As the same as the reward case, DT takes the summation of the state-feature over a trajectory as an input. ", + "bbox": [ + 174, + 606, + 825, + 661 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We quantitatively compare CDT against baselines (Table 2) in x-velocity case, where CDT shows better matching results to the target distributions unseen during training. We provide the reward distribution results and the visualization in Appendix E.1, where CDT performs very well in all cases. To test the generalization additionally, we intervene the target distributions by (1) shifting the target distributions in the test set by constant offsets, and (2) synthesizing novel distributions via python scripts. See Appendix E.7 and E.8 for the results. Furthermore, to investigate the scalability of CDT to multi-dimensional state-features, we experiment 2D state-feature distribution matching (xy-velocities on Ant) in Appendix E.3, where CDT outperforms other baselines. ", + "bbox": [ + 173, + 669, + 825, + 781 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/db6308c6f6a1c0b6014b8b8c69bcb4e8fe1d109603b2af68fcf7cc5b3a6999fd.jpg", + "table_caption": [], + "table_footnote": [ + "Table 2: Quantitative evaluation of state-feature distribution matching, measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT against DT, BC, Meta-BC, and FOCAL. CDT achieves better matching than baselines. See Table 9 in Appendix E.1 for the reward distribution results. " + ], + "table_body": "
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.633 ± 0.3290.996 ± 1.4670.814 ± 1.0790.139± 0.0430.059± 0.0130.099 ± 0.0510.122 ± 0.0710.136 ± 0.0450.129 ±0.0600.347
DT0.746 ± 0.3801.076 ± 1.5490.911 ± 1.1400.177 ± 0.0530.093± 0.0370.135 ± 0.0630.083 ±0.0310.146 ± 0.0840.115± 0.0700.387
BC (no-context)3.017±0.8913.468 ± 1.2713.242 ± 1.1210.652 ± 0.2640.248 ± 0.1990.450 ± 0.3090.748 ± 0.5290.858 ± 0.6170.803 ± 0.5771.498
Meta-BC0.852 ± 0.6880.840 ± 1.1390.846 ± 0.9410.799 ± 0.5050.130± 0.0560.464 ± 0.4910.110 ± 0.0821.462 ± 1.1360.786 ± 1.0520.699
FOCAL (Li ct al.,2021)1.643 ± 0.4611.123 ± 1.5501.383 ± 0.5181.456 ± 0.4730.484 ± 0.3820.970 ± 0.6491.571 ± 0.5630.603± 0.4271.087 ± 0.6951.147
", + "bbox": [ + 176, + 795, + 825, + 849 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.2 GENERALIZATION TO UNSEEN TARGET DISTRIBUTION", + "text_level": 1, + "bbox": [ + 173, + 103, + 594, + 118 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The performance of offline methods in RL is often restricted by the coverage or quality of datasets. While we demonstrate CDT can perform offline state-marginal matching to unseen target distributions in Section 6.1, the standard offline datasets might not be diverse enough to observe the generalization since those are collected by single-task reward-maximization policies. To test the generalization to more diverse behaviors, we investigate the following tasks: (1) z-velocity distribution matching with synthesized bi-modal behavior and (2) cheetah-velocity matching problem from meta RL/IL literature (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020) ", + "bbox": [ + 173, + 128, + 825, + 227 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2.1 SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 174, + 242, + 584, + 257 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To generate diverse behaviors for z-axis, we obtain the expert cheetah that backflips towards -x direction by modifying reward function (see Appendix E.4 for the details). Combining expert backflipping trajectories and expert running forward trajectories from D4RL dataset, we construct a novel dataset with diverse behaviors. We experiment the offline SMM with not only each uni-modal behavior (backflipping or running forward), but also synthesized bi-modal behavior; running forward first, then backflipping during a single rollout, using patchworked target trajectories. ", + "bbox": [ + 174, + 266, + 825, + 351 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 and Figure 2 (a) show that CDT successfully matches the distribution to both uni-modal (running forward or backflipping) and synthesized bi-modal distributions better than DT and FOCAL, and is comparable to Meta-BC that originally designed to deal with such multi-task settings. Due to the difficulty for RL algorithms to maximize Eq. 4, FOCAL struggles to solve the offline multi-task SMM, even though FOCAL uses the same context embedding as Meta-BC. The bi-modal behavior learned by CDT can be seen at https://sites.google.com/view/generalizeddt. ", + "bbox": [ + 174, + 356, + 825, + 440 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/9d01fe212134c915640785a9a7a1c1ba24fa11558b84ebe3f1519c2890fc17da.jpg", + "table_caption": [ + "Table 3: State-feature $\\mathbf { \\dot { z } }$ -velocity) distribution matching with uni-modal and (synthesized) bi-modal target trajectories in HalfCheetah environment. Categorical DT matches both uni- and bi-modal trajectories better than DT and FOCAL, and is comparable to Meta-BC that originally aims to solve multi-task problem. " + ], + "table_footnote": [], + "table_body": "
MethodUni-modalBi-modalAverage
Categorical DT1.562 ± 0.6321.625 ± 0.9021.594
DT2.676 ±0.7652.703 ±0.7032.690
Meta-BC1.519 ±0.6961.655 ± 0.9901.587
FOCAL (Li et al., 2021)2.203 士 1.0501.983 士 0.9482.093
", + "bbox": [ + 274, + 454, + 722, + 525 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6122f8e9ffbb9010be05cd17b1ddf24787e73452368349e57817055b17b8fe1b.jpg", + "image_caption": [ + "Figure 2: (a) Z-Velocity and (b) Unseen Cheetah-Velocity results. Blue histograms represent target distributions. In (a), CDT (red) can match not only uni-modal behaviors for both running forward and backflipping, but also bi-modal behaviors; during a single rollout running forward first, then backflipping. DT (yellow) tends to lean backflipping and fails to fit neither uni-modal nor bi-modal ones. In (b), CDT successfully handles the trajectories unseen during training, while DT seems to output covering behaviors over the dataset support. " + ], + "image_footnote": [], + "bbox": [ + 173, + 587, + 820, + 727 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2.2 DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "text_level": 1, + "bbox": [ + 176, + 815, + 671, + 830 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Generalization to unknown target demonstrations or tasks has been actively investigated in meta or one-shot RL/IL literature (Duan et al., 2016; Wang et al., 2016). To verify the generalization of CDT to diverse behaviors, we adopt the cheetah-velocity task; a popular task in meta RL/IL (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020), where the cheetah tries to run with the specified velocity. We prepare 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals, and hold out $\\bar { \\{ 0 . 5 , 1 . 5 , 2 . 5 \\} }$ as a test set. See Appendix E.5 for the dataset generation. ", + "bbox": [ + 173, + 840, + 826, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/75d2d25f8b7fc19452c9a4da271824f2652e4cb6b7c68ba9271051d104cc5069.jpg", + "table_caption": [ + "Table 4 and Figure 2 (b) reveal that CDT outperforms DT or FOCAL, and is slightly better than MetaBC through the distribution matching evaluation, which implies CDT could solve offline multi-task SMM generalizing to the unknown target trajectories, given sufficiently diverse offline datasets. " + ], + "table_footnote": [ + "Table 4: Generalization to unseen target velocity $\\mathbf { \\hat { x } } \\mathbf { \\cdot }$ -velocity $= 0 . 5$ , 1.5, 2.5) among training dataset (x-velocity $\\in$ $[ 0 . 0 , 3 . 0 ] \\backslash \\{ 0 . 5 , 1 . 5 , 2 . 5 \\}$ ; uniformly spaced at 0.1 intervals), which is a popular setting in meta RL/IL. CDT successfully deals with unseen target velocities well, outperforming DT and FOCAL, and is slightly better than Meta-BC through the offline multi-task SMM evaluation. " + ], + "table_body": "
Methodx-vel: 0.5 x-vel: 1.5x-vel: 2.5Average
Categorical DT0.060± 0.0260.211 ± 0.0220.149± 0.1100.140
DT1.197 ± 0.2270.533 ± 0.1050.861 ± 0.2470.864
Meta-BC0.150 ± 0.0690.152 ± 0.1270.167 ± 0.0550.156
FOCAL (Li et al., 2021)0.472 ± 0.0050.952 ± 0.0730.346 ± 0.1860.590
", + "bbox": [ + 223, + 159, + 774, + 231 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.3 ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE ", + "text_level": 1, + "bbox": [ + 173, + 308, + 588, + 321 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Lastly, we investigate BDT for offline multi-task $\\mathrm { I L }$ , where we do not observe rewards nor state features explicitly. Instead, target full state trajectories that the agents are expected to mimic is given. We compare BDT against parameterized $\\Phi$ variants; DT-AE, -CPC, and -E2E discussed in Section 5.3. We consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT’s supervised loss jointly (called as “joint”), and pre-training with only unsupervised loss and freezing the weights during DT training (called as “frozen”). We aggregate the learned $\\Phi$ by summation within the input sequence (length is 20). We evaluate BDT and DT- $X$ with the same distribution matching evaluation as Section 6.1. ", + "bbox": [ + 173, + 333, + 825, + 445 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We sweep $m$ -dim learned feature from the encoder $\\Phi ( s )$ with $m \\in \\{ 1 , 4 , 1 6 \\}$ . Table 5 presents the offline multi-task IL results with respect to $\\mathbf { X }$ -velocity distribution, using $m = 1 6$ embeddings (see Appendix E.6 for the full results including $m = 1 , 4$ , and the reward distribution case). Similar to the discussion in Yang & Nachum (2021), DT-CPC sometimes fails to obtain sufficient representation for imitation. While even simple approaches, DT-AE and DT-AE (frozen), show positive results compared to no-context BC baselines (in Table 2), BDT outperforms all other learned $\\Phi$ variants or Meta-BC and is comparable to CDT or DT (also in Table 2) with longer input $\\mathrm { \\Delta } N = 5 0 \\mathrm { \\Omega }$ ). This implies that even though we don’t assume the state-feature specification, aggregator choice in GDT with minimal architectural changes may solve the offline distribution matching problem efficiently. We leave more sophisticated objectives or architectural changes as future work. ", + "bbox": [ + 173, + 452, + 825, + 592 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5b3956a29256a25c162a1a77a916567fad897a8de3405e07c9fc1b9c0ebb09b6.jpg", + "table_caption": [], + "table_footnote": [ + "Table 5: The results of BDT on the state-feature distribution matching problem $m = 1 6$ ). While even simple auto-encoder regularizers sometimes work well, BDT with longer contexts seems to outperform other strategies and is comparable to CDT or DT (in Table 2). See Appendix E.6 for the full results including $m = 1 , 4$ . " + ], + "table_body": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610 ± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710± 0.5721.591
DT-AE (joint)8.643 ± 0.6792.260± 0.6905.451 ± 3.2641.255±0.3630.649 ± 0.2410.952 ± 0.4322.104±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-CPC (joint)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575 ± 0.0320.096± 0.0280.335 ± 0.2410.949 ± 0.4840.412 ± 0.3780.680± 0.5101.410
DT-E2E8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ± 0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE (frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385 ± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (frozen)3.489 ± 1.1592.543 ± 0.9033.016 ± 1.1410.631 ±0.0910.171 ± 0.1300.401 ± 0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(N=20)1.592 ± 0.2011.208 ± 1.8541.400 ± 1.3330.318±0.0930.081 ± 0.0130.200± 0.1360.196 ±0.0310.392 ± 0.1840.294± 0.1640.631
BDT(N=50)0.840 ± 0.0631.223 ± 1.8281.031 ± 1.3070.142 ± 0.0100.098 ± 0.0250.120 ± 0.0290.192 ± 0.0510.163 ± 0.0270.178 ± 0.0430.443
", + "bbox": [ + 184, + 603, + 813, + 686 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 753, + 318, + 770 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We provide a unified perspective on a wide range of hindsight algorithms, and generalize the problem formulation as hindsight information matching (HIM). Inspired by recent successes in RL as sequence modeling, we propose Generalized Decision Transformer (GDT) which includes DT, Categorical DT, and Bi-directional DT as special cases, and is applicable to any HIM with proper choices of $\\Phi$ and aggregator. We show how Categorical DT, a minor modification of DT, enables the first effective offline state-marginal matching algorithm and propose new benchmark tasks for this problem class. We also demonstrate the effectiveness of Bi-directional DT as a one-shot imitation learner, significantly outperforming simple variants based on DT. We hope our proposed HIM and GDT frameworks shed new perspectives on hindsight algorithms and the applicability of sequence modeling to much broader classes of RL problems beyond classic reward-based RL. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ETHICS STATEMENT ", + "text_level": 1, + "bbox": [ + 174, + 103, + 316, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Since this paper mainly focuses on the reinterpretation of hindsight RL algorithms and experiments on existing benchmark datasets, we believe there are no ethical concerns. ", + "bbox": [ + 174, + 128, + 823, + 156 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 171, + 393, + 186 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We share our codes to ensure the reproductivity. The details of hyperparameters are described in Appendix A and B. 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", + "bbox": [ + 176, + 166, + 825, + 209 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/b931d7a829874247dba1034e5b6d8746c87d560f15db1d1482dae7e2dd42af93.jpg", + "table_caption": [ + "Table 6: List of hyperparameters for DT, CDT and BDT. We refer Chen et al. (2021a). " + ], + "table_footnote": [], + "table_body": "
HyperparameterValue
Number of layers Number of attention heads3 1 128
Embedding dimension
Nonlinearity functionReLU
Batch size64
Context length N20
Dropout0.1
Learning rate1e-4
Grad norm clip0.25
Weight decay1e-4
Learning rate decayLinear warmup for first 1OOk training steps
Training(Gradient) steps Number of bins forcategorical distribution1M
31
Encoder size (for DT-X)2 layer MLP,(128,128)
Coefficient of unsupervised loss0.1
", + "bbox": [ + 225, + 223, + 771, + 433 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B DETAILS OF BASELINES", + "text_level": 1, + "bbox": [ + 176, + 484, + 408, + 502 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "While, to the best of our knowledge, there are no prior work to tackle the offline state-marginal matching problem, we can regard meta or one-shot imitation learning as the methods solving similar problem (see Appendix D for further discussion). In this work, we choose Meta-BC and FOCAL (Li et al., 2021), a metric-based offline meta RL method, as decent baselines. ", + "bbox": [ + 173, + 517, + 825, + 573 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "FOCAL FOCAL is a offline meta RL method combining BRAC (Wu et al., 2019) with metricbased approach. It utilizes deterministic context encoder trained with inverse-power distance metric losses and detached from Bellman backup gradients. We follow the hyperparameters in the official implementation (https://github.com/LanqingLi1993/FOCAL-ICLR). Deterministic context encoder takes single-step state-action-reward tuple as a context for task inference. We parameterize it with (200, 200, 200)-layers MLP. We train deterministic context encoder with 100000 iteration first, and then train BRAC agent with task-conditioned policy and value functions with 100000 iteration. We use (256, 256, 256)-layers MLPs for policy and value network, and set batch size to 256. ", + "bbox": [ + 173, + 588, + 825, + 713 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Meta-BC We also use metric-based Meta-BC (Duan et al., 2017; Dasari & Gupta, 2020) as a strong baseline method for offline multi-task SMM. We adapt deterministic context encoder from FOCAL to infer the task. We train Meta-BC in the same way as FOCAL just replacing BRAC to BC. The objective is mean-squared error minimization. ", + "bbox": [ + 174, + 728, + 825, + 785 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Throughout our experiments, Meta-BC shows better results than FOCAL, because RL algorithms often struggle to optimize Eq. 4 for distribution matching and FOCAL tries to maximize the task reward, which is not necessarily required for distribution matching problem. In addition, BC often converges faster than offline RL methods. ", + "bbox": [ + 174, + 792, + 825, + 848 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C DETAILS OF EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 423, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Throughout the paper, we evaluate both CDT and BDT from a distribution matching perspective by formulating them as offline multi-task SMM or offline multi-task IL problem. However, the current distribution matching research in RL (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021) has been suffered from the lack of quantitative metrics for evaluation (while standard RL or BC are typically evaluated on the task reward performance). As summarized in Table 7, prior works (Lee et al., 2020; Ghasemipour et al., 2020) evaluate the SMM performance by qualitative density visualization using rollout particles. ", + "bbox": [ + 173, + 133, + 825, + 231 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Although the distance between state-marginal and target distribution seems the most intuitive and suitable metric to quantify the performance of distribution matching algorithms, it is often intractable to measure such distance analytically because both state-marginal and target distribution can be non-parametric; we cannot access their densities, and only their samples are available. While a concurrent work (Gu et al., 2021) tackles this problem by leveraging sample-based energy distance estimation, we introduce a single SMM-inspired evaluation for both offline multi-task SMM and offline multi-task IL via estimating Wasserstein-1 distance between empirical categorical distributions. Since the discretization of low-dimensional features, such as reward, have success in many RL methods (Bellemare et al., 2017; Dabney et al., 2018; 2020; Furuta et al., 2021b), quantification of distribution matching performance by Wasserstein-1 distance could be reliable evaluations (in addition Wasserstein-1 distance is symmetric, different from KL distance that is asymmetric). We note that due to the equivalence between inverse RL and SMM methods, the performance of distribution matching methods may be measured via task reward when assuming the accessivity to the expert trajectories (Ho & Ermon, 2016; Fu et al., 2018; Kostrikov et al., 2019). However, in our experiments, it is not suitable to evaluate the performance based on task reward since we uses the sub-optimal trajectories (Section 6.1) or multi-task trajectories from different reward functions (Section 6.2). ", + "bbox": [ + 174, + 237, + 826, + 460 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Wasserstein-1 distance between state-marginal distribution $\\rho ^ { \\pi } ( s )$ and target distribution $p ^ { * } ( s )$ is defined as: ", + "bbox": [ + 176, + 465, + 823, + 494 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7974f92a4521db8e1bdee5e14957f8227b08c107e631c50580ea1af5616949a1.jpg", + "text": "$$\nW _ { 1 } ( \\rho ^ { \\pi } ( s ) , p ^ { * } ( s ) ) = \\operatorname* { i n f } _ { \\nu \\in \\Gamma ( \\rho ^ { \\pi } , p ^ { * } ) } \\mathbb { E } _ { ( X , Y ) \\sim \\nu } [ | X - Y | ] ,\n$$", + "text_format": "latex", + "bbox": [ + 326, + 493, + 669, + 518 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $\\Gamma$ is the set of all possible joint distributions $\\nu$ whose marginals are $\\rho ^ { \\pi }$ and $p ^ { * }$ respectively $X$ and $Y$ are random variables). In this work, we empirically estimate Eq. 5 by focusing on the “manually-specified” feature $\\phi \\in F$ (e.g. reward, xyz-velocities, etc; defined in Section 4) and employing binning and discretization. We discretize the feature space $F$ and obtain $B$ bins (per dimension), whose representatives are $\\{ \\bar { \\phi } _ { 1 } , \\bar { \\phi } _ { 2 } , . . . \\bar { \\phi } _ { B } \\}$ . The range of feature space $[ \\phi _ { \\mathrm { m i n } } , \\phi _ { \\mathrm { m a x } } ]$ is pre-defined from the given offline datasets. Then, we get categorical distribution from the target trajectory $\\tau ; \\hat { p } _ { \\tau } ^ { * } ( \\phi ) = \\breve { \\{ } ( \\bar { \\phi } _ { 1 } , c _ { \\tau 1 } ^ { * } ) , \\ldots ( \\bar { \\phi } _ { B } , c _ { \\tau B } ^ { * } ) \\}$ , and from the rollouts of the policy $\\pi$ (using 4 seeds $\\times 2 0$ rollouts); $\\hat { \\rho } ^ { \\pi } ( \\phi ) = \\{ ( \\bar { \\phi } _ { 1 } , c _ { 1 } ^ { \\pi } ) , \\ldots ( \\bar { \\phi } _ { B } , c _ { B } ^ { \\pi } ) \\}$ , where $c _ { \\tau } ^ { * }$ and $c ^ { \\pi }$ are the weights of each bin (i.e. $\\begin{array} { r } { \\sum _ { l = 1 } ^ { B } c _ { \\tau { l } } ^ { \\ast } = 1 } \\end{array}$ and $\\begin{array} { r } { \\sum _ { l = 1 } ^ { B } c _ { l } ^ { \\pi } = 1 } \\end{array}$ ). We compute the evaluation metric averaging on the test set of the trajectories $D _ { \\mathrm { t e s t } }$ : ", + "bbox": [ + 173, + 521, + 825, + 666 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a36d4bfa20fe6a3e6f0ec02ada0091d82d01aeb4883eb9a2fe5f7472e49ac141.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { M e t r i c } ( \\pi , D _ { \\mathrm { t e s t } } ) = \\displaystyle \\frac { 1 } { | D _ { \\mathrm { t e s t } } | } \\sum _ { \\tau \\in D _ { \\mathrm { t e s t } } } \\operatorname* { m i n } _ { w _ { i j } } \\sum _ { i = 1 } ^ { B } \\sum _ { j = 1 } ^ { B } w _ { i j } | c _ { \\tau i } ^ { * } - c _ { j } ^ { \\pi } | , } \\\\ & { \\mathrm { s . t . } w _ { i j } \\ge 0 , \\displaystyle \\sum _ { j = 1 } ^ { B } w _ { i j } \\le c _ { \\tau i } ^ { * } , \\displaystyle \\sum _ { i = 1 } ^ { B } w _ { i j } \\le c _ { j } ^ { \\pi } , \\displaystyle \\sum _ { i = 1 } ^ { B } \\sum _ { j = 1 } ^ { B } w _ { i j } = 1 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 671, + 707, + 763 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In practice, we utilize the python package (https://github.com/pkomiske/ Wasserstein) released by Komiske et al. (2020) to compute it. ", + "bbox": [ + 173, + 768, + 825, + 797 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/4470bb6dd2e9594900449bf17bf2cfccdd7376a0cfe0d62b0143a63931af8c63.jpg", + "table_caption": [ + "Table 7: A Review of evaluation for distribution matching algorithms. " + ], + "table_footnote": [], + "table_body": "
EvaluationTypeReference
Density VisualizationQualitativeLee et al. (2020); Ghasemipour et al. (2020)
Energy DistanceQuantitativeGu et al. (2021)
Task RewardQuantitativeHo& Ermon 1 (2016); Fu et al. (2018), etc.
Wasserstein-1DistanceQuantitativeOurs
", + "bbox": [ + 233, + 809, + 763, + 882 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D CONNECTION TO META LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 102, + 501, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "While we solve offline information matching problems in this paper, our formulations (especially Bi-directional DT) and experimental settings are similar to offline meta RL (Dorfman et al., 2021; Mitchell et al., 2021; Li et al., 2021) and meta/one-shot imitation learning (Yu et al., 2019; Ghasemipour et al., 2019; Finn et al., 2017; Duan et al., 2017) (especially metric-based approach (Rakelly et al., 2019; Duan et al., 2016; Fakoor et al., 2020)). We briefly clarify the relevance and difference between them (Table 8). ", + "bbox": [ + 173, + 133, + 826, + 217 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Offline Meta RL Recently, some works deal with offline meta RL problem, assuming task distribution and offline training with pre-stored transitions collected by the (sub-) optimal or scripted task-conditioned agents (Dorfman et al., 2021; Mitchell et al., 2021; Li et al., 2021; Pong et al., 2021; Zhao et al., 2021). In these works, few test-task trajectories, following the same task distribution but unseen during training phase, are given at test-time (i.e. few-shot), and the agents adapt the given task in an online (Pong et al., 2021; Zhao et al., 2021; Dorfman et al., 2021) or fully-offline (Mitchell et al., 2021; Li et al., 2021) manner. ", + "bbox": [ + 173, + 223, + 825, + 321 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Meta Imitation Learning Another related domain is meta imitation learning, also assuming task distribution and offline training with pre-stored expert transitions per tasks (Yu et al., 2019; Ghasemipour et al., 2019; Finn et al., 2017; Duan et al., 2017; Xu et al., 2019; Gleave & Habryka, 2018; Dasari & Gupta, 2020). While meta inverse RL methods (Yu et al., 2019; Xu et al., 2019; Gleave & Habryka, 2018) require online samples during test-time adaptation, its off-policy extension and BC-based approach performs offline adaptation with given unseen trajectories (Ghasemipour et al., 2019; Finn et al., 2017; Dasari & Gupta, 2020). ", + "bbox": [ + 173, + 328, + 826, + 426 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Our work, in contrast, assume no-adaptation at test time and evaluation criteria is a distance between the feature distributions, not the task rewards. ", + "bbox": [ + 174, + 433, + 821, + 462 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/5c4328b174faf078d209c417dec2a70799ec0d8aca39c603ee663155fe7d084f.jpg", + "table_caption": [ + "Table 8: Review of problem settings among offline meta RL, meta IL, and ours. In offline meta RL, the agents are trained offline and tested with several demonstrations (few-shot), in a fully-offline manner or allowing online data-collection for adaptation. In meta IL, IRL-based methods train the agents online (Yu et al., 2019; Xu et al., 2019; Ghasemipour et al., 2019) while BC-based methods (Finn et al., 2017; Duan et al., 2017) do offline. Similar to our settings, Duan et al. (2017) test the agents with single demonstration and no-adaptation process, they evaluate the agent on the task reward, while we do on the distance between the two distributions. " + ], + "table_footnote": [], + "table_body": "
MethodProblemTrainTestDemo
Pong et al. (2021)Offline Meta RLofflineonlinefew-shot
Zhao et al. (2021)Offline Meta RLofflineonlinefew-shot
Dorfman et al. (2021)Offline Meta RLofflineonlinefew-shot
Mitchell et al. (2021)Offline Meta RLofflineofflinefew-shot
Li et al. (2021)Offline Meta RLofflineofflinefew-shot
Yu et al. (2019)Meta ILonlineonlinefew-shot
Xu et al. (2019)Meta ILonlineonlinefew-shot
Ghasemipour et al. (2019)Meta ILonlineofflinefew-shot
Finn et al. (2017)Meta ILofflineofflineone-shot
Duan et al. (2017)Meta ILoffline(no-adaptation)one-shot
OursOffline multi-task SMM/ILoffline(no-adaptation)one-shot
", + "bbox": [ + 200, + 474, + 795, + 632 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E DETAILS OF EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 434, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section, we provide the details and supplemental quantitative/qualitative results of the experiments in Section 6. ", + "bbox": [ + 173, + 133, + 825, + 161 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Figure 3 visualizes the dataset distributions we use in the experiments. When we sort all the trajectories based on their cumulative rewards, each quality of trajectory (best, middle, worst) shows a different shape of distribution, and the distribution of the whole dataset seems a weighted mixture of those. ", + "bbox": [ + 173, + 167, + 825, + 224 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/0a12826cb7d345cb11f89b3ba83fd91c7d770dc6a674535f9e40adbb5a6e9357.jpg", + "image_caption": [ + "Figure 3: Distributions of the features (reward, and $\\mathbf { X } ^ { }$ -velocity) in the D4RL medium-expert datasets. " + ], + "image_footnote": [], + "bbox": [ + 181, + 234, + 812, + 343 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.1 QUANTITATIVE AND QUALITATIVE RESULTS OF REWARD AND STATE-FEATURE MATCHING ", + "text_level": 1, + "bbox": [ + 174, + 395, + 763, + 422 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We provide the quantitative comaprison of Section 6.1 between Categorical DT and DT in the reward (Table 9) matching problem, computing Wasserstein-1 distance between the discretized rollout and target feature distributions. Generally, similar to the $\\mathbf { X }$ -velocity case, CDT shows the better results in offline multi-task SMM compared to original DT. We also visualize some of CDT results in Figure 4. ", + "bbox": [ + 173, + 434, + 823, + 491 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/d024bea0c8817fac97f37a8d384a81db42a17e85280433d1ef413f6a31373a03.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.674 ± 0.2131.002 ± 1.4580.838 ± 1.0540.159 ± 0.0850.064 ± 0.0170.111 ± 0.0770.095 ± 0.0170.114 ± 0.0370.105 ± 0.0300.351
DT0.652 ± 0.3191.039 ± 1.5480.846 ± 1.1340.227 ± 0.1190.091 ± 0.0350.159 ± 0.1110.056 ± 0.0150.626 ± 0.4950.341 ± 0.4520.448
BC (no-context)3.240± 0.5592.880 ± 0.6143.060 ± 0.6140.597 ± 0.0560.119 ± 0.0670.358 ± 0.2470.977 ± 0.5010.431 ± 0.3960.704 ± 0.5281.374
Meta-BC0.839 ±0.6820.830 ± 1.1300.835 ± 0.9330.803 ± 0.5050.134 ± 0.0560.468 ± 0.4910.113 ± 0.0851.441 ± 1.1130.777 ± 1.0320.693
FOCAL (Li et al.,2021)1.623 ± 0.5011.115 ± 1.5341.369 ± 0.5161.463 ± 0.4720.492 ± 0.3840.977 ± 0.6491.584 ± 0.5700.604 ± 0.4211.094 ± 0.7011.147
", + "bbox": [ + 176, + 505, + 823, + 559 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Table 9: Quantitative evaluation of reward distribution matching via measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT and DT. Since it can capture the multi-modal nature of target distributions, CDT matches the distribution better than the original DT. ", + "bbox": [ + 174, + 570, + 823, + 609 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/c15ec828e747cb1d7499910bbb5901116cf3aecc7338dafa323749f7c6941389.jpg", + "image_caption": [ + "Figure 4: (a) Reward and (b) state-feature (x-velocity) distribution matching in halfcheetah (top), hopper (middle), and walker2d (bottom). The left two examples are the distributions from the best trajectories, and right two are the distributions from the middle trajectories in the held-out test set. The rollout distributions of CDT (red) match the target distributions (blue) very well in all cases. " + ], + "image_footnote": [], + "bbox": [ + 179, + 628, + 813, + 796 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.2 EVALUATION ON TASK PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 103, + 485, + 118 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "While in this paper we focus on the evaluation with the SMM-inspired distribution matching objective, such as Wasserstein-1 distance, we here provide the evaluation on the task rewards. Table 10 shows that DT seems consistently better methods than Categorical DT on the task rewards evaluations, and achieves similar performances to the held-out trajectories. ", + "bbox": [ + 174, + 128, + 825, + 185 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/5d74301f949071f3d8b1fea32c3f7aaed2b1d5ee6199e60a001f50f8d8d9eb47.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2d
ExpertMediumExpertMediumExpertMedium
Categorical DT10476.746± 218.9575782.557 ± 1666.7162614.559 ± 657.7221518.757 ± 63.0624907.475 ± 11.7113264.585 ± 209.905
DT10500.273 ± 312.7784985.518 ± 67.3222113.207 ± 807.2461528.743± 30.7804965.024 ± 11.8144055.039 ± 849.109
Held-out11146.200 ± 59.0705237.348 ± 37.9703741.854 ± 7.2231600.196± 0.7724995.553 ± 5.4133801.313 ± 0.994
", + "bbox": [ + 184, + 199, + 813, + 244 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 10: Evaluation on the task rewards; conditioning on the held-out trajectories as done in Section 6. We compare the performance between Categorical DT, and DT. DT seems consistently better methods on the task rewards evaluations, and achieves similar performances to the held-out trajectories (averaged over 5 trajectories). ", + "bbox": [ + 174, + 256, + 825, + 295 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.3 2D STATE-FEATURE DISTRIBUTION MATCHING ", + "text_level": 1, + "bbox": [ + 174, + 332, + 545, + 347 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We also consider two-dimensional state-features (xy-velocities) distribution matching in Ant-v3. Same as 1D state-feature distribution matching in HalfCheetah, Hopper, and Walker2d-v3 (Section 6.1), we also use medium-expert(-v2) datasets from D4RL (Fu et al., 2020). We bin the state-features per dimension separately to reduce the dimension of the categorical distribution that CDT takes as input, while in test-time we evaluate the performance with Wasserstein-1 metric on the joint distribution. For DT, we compute the summation of $\\mathbf { X } ^ { - }$ and y-velocity each over trajectories and normalize them with the maximum horizon. DT feeds these two scalars as information statistics to match. ", + "bbox": [ + 173, + 358, + 826, + 457 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 11 reveals that CDT performs better even in the case of two-dimensional state-features distributions, while DT doesn’t generalize to expert-quality trajectories. As shown in Figure 5, while CDT could cope with the distribution shift between expert and medium target distribution, DT always fits the medium one even if the expert trajectory is given as a target. CDT successfully scales to the offline multi-task SMM problem in the multi-dimensional feature spaces. ", + "bbox": [ + 174, + 462, + 825, + 534 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/fd666a38b459b1261377224f7e99a0c975abee8dfa7e3b62a8d7f62fdc9d885a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Methodant
ExpertMediumAverage
Categorical DT0.797 ± 0.2160.244 ± 0.0630.521
DT1.714 ± 0.1210.260 ± 0.0670.987
Meta-BC1.295 ± 0.7080.351 ± 0.2050.823
FOCAL (Li et al., 2021)1.473 ± 0.8920.913 ± 0.4551.193
", + "bbox": [ + 276, + 546, + 720, + 631 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Table 11: Quantitative evaluation of 2D state-feature (xy-velocities) distribution matching, measuring Wasserstein-1 distance. CDT performs better even in the two-dimensional problem. ", + "bbox": [ + 171, + 642, + 823, + 669 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/be07d95b475dbda537b278fbe7b1d838a7ec935c089362e0b31dbcc6b73ffec8.jpg", + "image_caption": [ + "Figure 5: Visualization of 2D state-feature (xy-velocities) distribution matching, binning each dimension separately. Top row shows the results from an expert target trajectory, and bottom row from a medium one. " + ], + "image_footnote": [], + "bbox": [ + 214, + 690, + 784, + 876 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.4 DETAILS OF SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION ", + "bbox": [ + 174, + 103, + 661, + 118 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To construct the dataset, we modified the original reward function in HalfCheetah-v3, adding absolute z-velocity term (such as $+ \\mathrm { n p }$ .abs $( z \\_ { \\mathrm { v e 1 } } )$ ), where the expert cheetah backflips towards - $\\mathbf { - X }$ direction. ", + "bbox": [ + 176, + 128, + 823, + 171 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We trained SAC agent until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and then collected 500 trajectories $\\times ~ 1 0 0 0$ time steps. We combined them with 500 trajectories $\\times ~ 1 0 0 0$ time steps from halfcheetah-expert-v2 dataset in D4RL, which consists of both backflipping and running forward behaviors. ", + "bbox": [ + 174, + 179, + 825, + 234 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For the evaluation, we prepared additional 5 trajectories of backflipping and 5 of running forward as uni-modal behaviors (10 test trajectories in total). In addition, we synthesized a bi-modal behavior by dividing each 1000-step trajectories into 500-step sub-trajectories, and concatenating them across different behaviors, which results in the patchworked trajectories of first 500-step running forward and next 500-step backflipping (also 10 trajectories in total). ", + "bbox": [ + 174, + 241, + 825, + 311 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E.5 DETAILS OF DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "text_level": 1, + "bbox": [ + 176, + 328, + 746, + 343 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Following prior meta RL/IL works (Rakelly et al., 2019; Ghasemipour et al., 2019; Pong et al., 2021; Li et al., 2021; Fakoor et al., 2020), we modified the reward function for the cheetah to run with specified velocity (such as -np.abs(x_vel - target_vel)), and set the horizon to 200 steps. ", + "bbox": [ + 174, + 354, + 825, + 410 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We prepared 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals. We also trained the SAC agents until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and collected 250 trajectories $\\times 2 0 0$ time steps each. To simplify the problem than meta learning settings, we held out the 10 trajectories whose $\\mathbf { X }$ -velocity is $\\{ 0 . 5 , \\bar { 1 } . 5 , \\bar { 2 } . 5 \\}$ as a test set, and used the rest as a train data. ", + "bbox": [ + 174, + 416, + 825, + 487 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "E.6 QUANTITATIVE AND QUALITATIVE RESULTS OF ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE ", + "bbox": [ + 176, + 503, + 823, + 531 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Table 12 and Table 13 show the one-shot reward and state-feature distribution matching results respectively. In addition to the embedding size $m$ , we also sweep the different context window $N = 2 0 , 5 0 , 1 0 0$ for BDT $m = 1 6$ ). While even simple auto-encoder regularizer (DT-AE or DT-AE (frozen)) sometimes works well compared to no-context BC baselines presented in Table 2 and Table 9, BDT, using (second) anti-causal transformer as an encoder $\\Phi$ and its aggregator, seems consistently better than other strategies or Meta-BC baseline for offline multi-task SMM and is comparable to CDT or DT results (also presented in Table 2 and Table 9) with longer context length $( N = 5 0 )$ ). Through the experiment we observe that while there are no clear trends in DT-AE, -CPC or -E2E as the size of context embedding $m$ grows, BDT improves its performance with larger size of embedding. Such intuitive properties might a good features to design the architectures. ", + "bbox": [ + 173, + 544, + 825, + 683 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We visualize the qualitative results of BDT $( m = 1 6 , N = 2 0 )$ ) in Figure 6. ", + "bbox": [ + 174, + 689, + 661, + 704 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/4c8423fed7b86bbd0827bb02ba6bdbdd63bc74db7fb23e6958356840ff4fdc52.jpg", + "table_caption": [ + "Table 12: The results of BDT and DT- $X$ variants on the reward distribution matching problem $( m = 1 , 4 , 1 6$ " + ], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE(m=1)2.494 ± 1.0502.877 ± 0.7622.686 ± 0.9370.586± 0.0160.089 ± 0.0290.337 ± 0.2490.733 ± 0.5220.586 ± 0.4450.660 ± 0.4901.228
DT-CPC (m=1)3.595± 0.6762.275 ±0.3412.935±0.8490.600 ± 0.0070.130± 0.0390.365± 0.2371.180 ± 0.0190.139 ± 0.0320.660 ± 0.5211.320
DT-AE(m=4)0.782 ± 0.3291.720 ± 1.6851.251 ± 1.3010.613 ± 0.1080.140 ± 0.0680.376 ± 0.2530.889 ± 0.4020.265 ± 0.1370.577 ± 0.4330.735
DT-CPC (m=4)5.887 ± 0.3571.370 ± 1.3383.628 ± 2.4620.737 ± 0.0180.238 ± 0.0600.487 ± 0.2541.286 ± 0.0180.145 ± 0.0310.715 ± 0.5711.610
DT-AE(m=16)2.041 ± 1.0801.074 ± 0.8141.558 ± 1.0710.613 ± 0.0600.146 ± 0.0510.379 ± 0.2400.837 ± 0.3450.324 ± 0.1260.581 ± 0.3650.839
DT-CPC (m=16)6.022 ± 0.3161.406 ± 1.3713.714 ± 2.5130.614± 0.0190.104 ± 0.0250.359± 0.2561.284± 0.0200.140 ± 0.0390.712 ± 0.5721.595
DT-AE(m=1,joint)3.824 ± 1.1141.988 ± 0.9702.906 ± 1.3910.687 ± 0.0970.137 ± 0.0570.412 ± 0.2860.723± 0.5960.614 ± 0.4620.668 ± 0.5361.329
DT-CPC (m=1, joint)4.460 ± 1.1062.036 ± 1.0323.248 ± 1.6160.587 ± 0.0150.106 ± 0.0420.347 ± 0.2430.815 ± 0.7110.710 ± 0.3980.763 ± 0.5781.452
DT-AE(m=4,joint)5.563 ± 0.4491.028 ± 1.2653.295 ± 2.4581.320 ± 0.2080.485± 0.2350.902 ± 0.4730.917 ± 0.3190.218 ± 0.1120.567 ± 0.4231.588
DT-CPC (m=4, joint)3.486 ± 1.5032.265 ± 1.0772.876 ± 1.4430.690 ± 0.1590.220 ± 0.1760.455± 0.2881.021 ± 0.7080.554 ± 0.3830.788 ± 0.6151.373
DT-AE (m=16, joint)8.450 ± 0.6232.168±0.7015.309 ± 3.2101.257 ± 0.3610.655 ± 0.2420.956 ± 0.4302.112 ±0.6180.994 ± 0.4131.553 ± 0.7672.606
DT-CPC (m=16, joint)4.543 ± 1.1791.869 ± 1.4533.206 ± 1.8810.577 ± 0.0320.098 ± 0.0280.338 ± 0.2410.953 ± 0.4830.414 ± 0.3760.683 ± 0.5101.409
DT-E2E(m=1)3.220 ± 2.3251.026 ± 1.3982.123 ± 2.2101.615 ± 0.5360.540 ± 0.2931.078 ± 0.6901.079 ± 0.2090.455 ± 0.0850.767 ± 0.3511.323
DT-E2E(m=4)8.076 ± 0.5512.552 ± 0.5715.314 ± 2.8181.205 ± 0.3900.493±0.2470.849 ± 0.4832.835±0.9461.239 ± 0.4602.037 ± 1.0912.733
DT-E2E(m=16)8.049 ± 0.9901.859 ± 0.8394.954 ± 3.2281.102 ± 0.2160.549 ± 0.1890.826 ± 0.3432.095±0.4341.241 ± 0.7091.668 ± 0.7262.482
DT-AE(m=1, frozen)2.225 ± 1.0172.804 ± 1.0512.514 ± 1.0740.582 ± 0.0420.131 ± 0.0580.357 ± 0.2311.396 ± 0.1490.259 ± 0.0830.827 ± 0.5811.233
DT-CPC (m=1,frozen)4.110 ± 0.7992.172 ±0.7893.141 ± 1.2530.582 ± 0.0210.102 ± 0.0410.342 ± 0.2421.422 ± 0.0990.275± 0.1090.848 ± 0.5831.444
DT-AE(m=4, frozen)0.815 ± 0.1831.415 ± 1.7551.115 ± 1.2830.681 ± 0.1060.197 ± 0.0610.439 ± 0.2571.066 ± 0.4950.419 ± 0.3210.742 ± 0.5280.765
DT-CPC (m=4, frozen)3.275 ± 1.1862.752 ± 1.0883.014 ± 1.1680.633 ± 0.0550.139 ± 0.0820.386 ± 0.2571.119 ± 0.5980.508 ± 0.3540.814 ± 0.5781.404
DT-AE(m=16, frozen)1.796 ± 0.5781.312 ± 0.852 2.538±0.9051.554 ± 0.7670.637 ± 0.0390.142 ± 0.0630.389 ±0.2531.184 ± 0.3260.405 ± 0.1690.795 ± 0.4690.913
DT-CPC (m=16, frozen)3.486 ± 1.1643.012 ± 1.1450.636± 0.0930.178 ± 0.1320.407 ± 0.2561.318 ± 0.3280.282 ± 0.1260.800 ± 0.5741.407
BDT (m=1, N=20)1.385 ± 0.207 1.660 ± 0.1671.180 ± 1.753 1.058 ± 1.5801.282 ± 1.253 1.359 ± 1.1630.291 ± 0.096 0.494 ± 0.2810.110 ± 0.043 0.096 ± 0.0290.201 ± 0.117 0.295± 0.2821.113 ± 0.044 0.181 ± 0.0230.155 ± 0.046 0.414 ± 0.5370.634 ± 0.481 0.298 ± 0.3970.706
BDT(m=4,N=20)1.565 ± 0.1901.191 ± 1.8301.378 ± 1.3150.321 ± 0.0930.086 ± 0.0170.204 ± 0.1350.204 ± 0.0330.396 ± 0.1860.300 ± 0.1650.650
BDT(m=16, N=20)0.831 ± 0.0641.204 ± 1.8031.018 ± 1.2900.144 ± 0.0110.104 ± 0.0260.124 ± 0.0280.199 ± 0.0520.167 ± 0.0240.183 ± 0.0440.627
BDT(m=16,N=50)1.280 ± 1.8611.108 ± 1.3320.240 ± 0.0470.162 ± 0.0330.201 ± 0.0560.057 ± 0.0070.442
BDT(m=16,N=100)0.936 ± 0.1660.873 ± 0.6070.465 ± 0.5930.591
", + "bbox": [ + 176, + 121, + 826, + 324 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/b486f31962d281d9c8354da603dbac1291dfd112a314fab4c69f6a8c4e5a2c47.jpg", + "table_caption": [ + "Table 13: The results of BDT and DT- $X$ variants on the state-feature $\\mathbf { \\dot { x } }$ -velocity) distribution matching problem $( m = 1 , 4 , 1 6 )$ ). " + ], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE (m=1)2.504 ± 1.0502.880 ± 0.7782.692 ± 0.9430.580 ± 0.0150.084 ± 0.0270.332 ± 0.2490.729 ± 0.5220.584 ± 0.4470.656 ± 0.4911.227
DT-CPC (m=1)3.595 ±0.6702.277 ± 0.3492.936 ± 0.8480.601 ± 0.0080.130 ± 0.0370.365 ± 0.2371.177 ± 0.0210.135 ± 0.0330.656 ± 0.5221.319
DT-AE(m=4)0.789 ± 0.3331.729 ± 1.7141.259 ± 1.3210.612 ± 0.1080.138 ± 0.0660.375 ± 0.2540.884 ± 0.4020.262 ± 0.1380.573 ± 0.4330.736
DT-CPC (m=4)5.883 ± 0.3611.371 ± 1.3473.627 ± 2.4620.731 ± 0.0180.229 ± 0.0570.480 ± 0.2551.282 ± 0.0220.141 ± 0.0320.712 ± 0.5711.606
DT-AE(m=16)2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC (m=16)6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710 ± 0.5721.591
DT-AE (m=1,joint)3.825 ± 1.1151.990 ± 0.9882.908 ± 1.3970.683 ± 0.0940.132 ± 0.0550.407 ± 0.2860.719 ± 0.5970.611 ± 0.4630.665 ± 0.5371.327
DT-CPC (m=1,joint)4.460 ± 1.1012.035 ±1.0553.248 ± 1.6220.583 ± 0.0150.099 ± 0.0410.341 ± 0.2440.811 ± 0.7110.709 ± 0.3990.760 ± 0.5791.450
DT-AE(m=4,joint)5.589 ± 0.4581.040 ± 1.2703.315 ± 2.4671.318 ± 0.2080.481 ± 0.2340.899 ± 0.4730.912 ± 0.3190.216 ± 0.1120.564 ± 0.4221.593
DT-CPC (m=4, joint)3.484 ± 1.4982.270 ± 1.0992.877 ± 1.4480.685 ± 0.1570.214 ± 0.1740.449 ± 0.2881.018 ± 0.7130.555 ± 0.3840.786 ± 0.6181.371
DT-AE(m=16,joint) DT-CPC (m=16, joint)8.643 ± 0.6792.260 ±0.6905.451 ± 3.2641.255 ± 0.3630.649 ± 0.2410.952 ± 0.4322.104 ±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-E2E (m=1)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575± 0.0320.096± 0.0280.335± 0.2410.949 ± 0.4840.412 ± 0.3780.680 ± 0.5101.410
DT-E2E(m=4)3.265 ± 2.3461.050 ± 1.4132.158 ± 2.2311.613 ± 0.5400.534 ± 0.2931.073 ± 0.6921.073 ± 0.2110.451 ± 0.0840.762 ± 0.3501.331
8.215 ± 0.6042.684 ±0.5755.449 ± 2.8281.202 ± 0.3920.487 ± 0.2460.845 ± 0.4852.832 ± 0.9511.235 ± 0.4632.034 ± 1.0942.776
DT-E2E (m=16)8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ±0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE(m=1,frozen)2.238 ± 1.0112.807 ± 1.0562.523 ± 1.0720.577 ±0.0430.126± 0.0560.352 ± 0.2311.391 ± 0.1490.256 ±0.0830.824 ± 0.5801.233
DT-CPC (m=1, frozen)4.110 ± 0.7942.173 ± 0.8063.141 ± 1.2570.578± 0.0220.097 ± 0.0390.338 ± 0.2421.417 ± 0.1000.272 ± 0.1090.845 ± 0.5821.441
DT-AE (m=4, frozen)0.825 ± 0.1891.431 ± 1.7821.128 ± 1.3030.679 ± 0.1060.193 ± 0.0600.436 ± 0.2581.063 ± 0.4950.418 ± 0.3220.740 ± 0.5270.768
DT-CPC (m=4,frozen)3.274 ± 1.1872.756 ± 1.0943.015 ± 1.1700.629 ± 0.0540.135 ± 0.0810.382 ± 0.2571.115 ± 0.6000.507 ± 0.3530.811 ± 0.5781.403
DT-AE(m=16, frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (m=16, frozen)3.489 ±1.1592.543±0.9033.016 ± 1.1410.631±0.0910.171 ± 0.1300.401 ±0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(m=1, N=20)1.414 ± 0.210 1.694 ± 0.1711.197 ± 1.770 1.071 ± 1.5941.305 ± 1.265 1.382 ± 1.1750.288 ± 0.0960.108 ± 0.0410.198 ± 0.1161.108 ± 0.0450.152 ± 0.0510.630± 0.4800.711
BDT (m=4,N=20)1.208 ± 1.8541.400 ± 1.3330.490± 0.2800.092 ± 0.0300.291 ± 0.2810.173 ± 0.0240.411 ± 0.5470.292 ± 0.4050.655
BDT(m=16, N=20) BDT(m=16,N=50)1.592 ± 0.2011.223 ± 1.8281.031 ± 1.3070.318 ± 0.0930.081 ± 0.0130.200 ± 0.1360.196 ± 0.0310.392 ± 0.1840.294 ± 0.1640.631
BDT(m=16,N=100)0.840 ± 0.063 0.953 ± 0.1681.308 ± 1.8811.130 ± 1.3470.142 ± 0.010 0.240 ± 0.0440.098 ± 0.025 0.156 ± 0.0320.120 ± 0.029 0.198 ± 0.0570.192 ± 0.051 0.051 ± 0.0060.163 ± 0.027 0.883 ± 0.6140.178 ± 0.043 0.467 ± 0.6010.443 0.598
", + "bbox": [ + 176, + 397, + 826, + 601 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/07aa2efe41809e75eecc8006e64e0385c9cea39d36c6b3b79db002ed61ddd233.jpg", + "image_caption": [ + "Figure 6: (a) Reward and (b) state-feature distribution matching by Bi-directional Decision Transformer $( m = 1 6 )$ ) in halfcheetah (top), hopper (middle), and walker2d (bottom). The left two examples are the distributions from the best trajectories, and right two are the distributions from the middle trajectories. " + ], + "image_footnote": [], + "bbox": [ + 181, + 684, + 813, + 851 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E.7 SHIFTING THE TARGET DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 483, + 118 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To generate the unseen but manageable generalization-test trajectories within the support of dataset distribution, we make the reward and velocities values of trajectories in the test set shift with a constant offset: bin_size $\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}$ . Table 14 shows the quantitative comparison between CDT and DT, based on Wasserstein-1 distance between two distributions. CDT successfully handles the distribution shifts (especially in hopper) better than DT. We also provide the state-feature results (Table 15) and qualitative visualizations (Figure 7 and Figure 8), which reveals that CDT can match the rollouts to the shifted target distributions when they are within the support of dataset distributions. ", + "bbox": [ + 173, + 128, + 825, + 241 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/06930919804dd12de2f5bf0dc078f7e98e5460d1869dfbbde41e2ae519a0bf0b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Methodhalfcheetahwalker2dAverage
ExpertMediumTotalExperthopper MediumTotalExpertMediumTotal
Categorical DT1.126 ± 0.2452.026 ± 1.1801.576 ± 0.9640.147 ± 0.0340.302 ± 0.0850.224± 0.1010.285± 0.0441.024 ± 0.0760.655 ± 0.3750.818
DT1.133 ± 0.1971.978 ± 1.1041.555 ± 0.8990.521 ±0.0410.531 ± 0.0450.526 ± 0.0430.656±0.3800.915 ± 0.1060.786 ± 0.3080.956
", + "bbox": [ + 181, + 252, + 810, + 286 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Table 14: Wasserstein-1 distance between shifted reward distribution and the rollout distributions. Categorical DT handles the target distribution shifts and matches the distributions better than DT, since CDT is aware of distributional information of entire trajectories. ", + "bbox": [ + 176, + 296, + 823, + 335 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/8bb281c38a60fcf9f46d443afe9e63501615364aed67368783027c15a1827da6.jpg", + "table_caption": [ + "Table 15: Wasserstein-1 distance between shifted state-feature (x-velocity) distribution and the rollout distributions. Similar to the case of reward, Categorical DT handles the target distribution shifts and matches the distributions better than DT. " + ], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
Categorical DT1.270 ± 0.2422.371 ± 1.7471.821 ± 1.3630.157 ± 0.0380.337 ± 0.0880.247 ± 0.1120.289 ±0.0520.964±0.1040.626± 0.3470.898
DT1.173 ± 0.3722.056 ± 1.0451.614 ± 0.9010.432 ±0.0870.531 ± 0.0530.482 ±0.0880.408 ±0.2460.885 ± 0.1430.646 ± 0.3120.914
", + "bbox": [ + 184, + 358, + 810, + 392 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/f4c6314fde7aaf0038339f982c1f76950e21b022a6b636a1222aba573f75ed64.jpg", + "image_caption": [ + "Figure 7: Reward distribution matching in halfcheetah (top two rows; best and middle), hopper (middle two rows; best and middle), and walker2d (bottom two rows; best and middle). We shift the target distribution with (from left to right column); bin_size $\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}$ (Table 14). Categorical DT (red) can match the rollouts to the shifted target distributions (blue) when the shifted targets are within the support of dataset distributions. For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts. " + ], + "image_footnote": [], + "bbox": [ + 171, + 462, + 825, + 801 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/2b8e2f52509043a703d85031dfbd9ba8e0f362e95fd35c8dd8bce30ecfe55cce.jpg", + "image_caption": [ + "Figure 8: State-feature distribution matching, especially $\\mathbf { X }$ -velocity, in halfcheetah (top two rows; best and middle), hopper (middle two rows; best and middle), and walker2d (bottom two rows; best and middle). We shift the target distribution with constant offset (from left to right column); bin_size $\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}$ (Table 15). For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts. " + ], + "image_footnote": [], + "bbox": [ + 171, + 303, + 825, + 642 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.8 SYNTHESIZING UNREALISTIC TARGET DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 173, + 103, + 591, + 118 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We synthesize six target distributions manually generating x-velocity samples from Gaussian distributions via python scripts as done in prior works (Ghasemipour et al., 2020; Gu et al., 2021). Since we consider the one-dimensional feature space (x-velocity), we simply specify the mean and standard deviation of Gaussian distributions referring each dataset distribution, as shown in Figure 3, and then generate the 1000 samples per trajectory. While these targets are designed at least within the support of the dataset, we do not consider physical realizability. ", + "bbox": [ + 173, + 128, + 826, + 213 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "• HalfCheetah: $( \\mu , \\sigma ) = ( 5 . 0 , 1 . 0 )$ , (13.0, 1.0), (9.0, 1.0), (2.5, 1.0), {(5.0, 1.0), (13.0, 1.0)}, {(9.0, 1.0), (2.5, 1.0)}. ", + "bbox": [ + 171, + 219, + 826, + 248 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "• Hopper: $\\left( \\mu , \\sigma \\right) = \\left( 2 . 5 , 1 . 0 \\right)$ , (1.5, 1.0), (3.5, 1.0), (2.5, 0.5), {(3.5, 1.0), (1.5, 1.0)}, {(3.5, 0.5), (1.5, 0.5)}. ", + "bbox": [ + 174, + 250, + 826, + 279 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "• Walker2d: $( \\mu , \\sigma ) = ( 3 . 5 , 1 . 0 )$ , (2.5, 1.0), (4.5, 1.0), (1.5, 1.0), {(2.5, 0.5), (4.5, 0.5)}, {(1.5, 0.5), (4.5, 0.5)}. ", + "bbox": [ + 173, + 281, + 826, + 309 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For the last two sets, that have different distributional parameters $\\{ ( \\mu _ { 1 } , \\sigma _ { 1 } ) , ( \\mu _ { 2 } , \\sigma _ { 2 } ) \\}$ , we sampled from two gaussian distributions 500 samples each, and marge them as one trajectory that has multiple modes. ", + "bbox": [ + 174, + 315, + 823, + 358 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Although they might be unrealistic, Figure 9 implies that CDT tends to match the target distributions, even in the cases of bi-modal target distributions. Such generalization to synthesized distribution is an important benefit of distribution-conditioned training. We also quantify the performance of CDT against DT from distributional matching perspective in Table 16. ", + "bbox": [ + 173, + 364, + 825, + 421 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/3bd096aee9959fb498de0de06509c1c21e2264b56e46d4acf7d1a1105fac6f2b.jpg", + "image_caption": [ + "Figure 9: State-feature distribution matching in halfcheetah (top), hopper (middle), and walker2d (bottom). We synthesize the each target distribution from Gaussian distributions. While the results are worse than for realistic test target distributions in Figure 4, considering that many of these arbitrary synthetic targets could be unrealizable, seeing bi-modal matching results show that indeed CDT has learned to generalize something non-trivial. " + ], + "image_footnote": [], + "bbox": [ + 171, + 435, + 823, + 636 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/816a004fc1bf092326a151f0e81f2e9760ec85bcc7bc1858d69bca5004245a96.jpg", + "table_caption": [ + "Table 16: State-feature $\\mathbf { \\widetilde { x } }$ -velocity) distribution matching with synthesized, physically unrealistic target distributions generated from scripted Gaussian distributions. We compare Categorical DT and DT. Categorical DT manages to deal with such unrealistic target matching. " + ], + "table_footnote": [], + "table_body": "
Methodhalfcheetahhopperwalker2dAverage
Categorical DT2.059 ± 0.7720.536 ± 0.2250.920 ± 0.4281.172
DT4.256 ± 2.2200.584 ± 0.2980.974 ± 0.5401.938
", + "bbox": [ + 250, + 733, + 745, + 780 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "F DETAILS FOR CATEGORICAL DECISION TRANSFORMER ", + "text_level": 1, + "bbox": [ + 173, + 102, + 666, + 118 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Categorical Decision Transformer (CDT) takes histograms of categorical distribution (i.e. discrete approximations of feature distributions; $B$ -dim vector) as the inputs of the transformer. Here we describe how to compute the distributions for all timesteps given a trajectory $t \\in [ 0 , 1 , \\ldots , T ]$ . For simplicity, we explain the case of one-dimensional feature (e.g. scalar reward), but for $n$ -dimensional features, we can adopt following procedure per each dimension and arrive at $n$ categorical features. This essentially approximates the full joints with a product of independent marginal distribution per dimension, and ensures that number of samples for getting reasonably discretized approximations do not need to grow exponentially as the dimension grows. ", + "bbox": [ + 173, + 132, + 825, + 246 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "At first, we discretize the feature space $F$ using $B$ bins (per dimension), and convert the feature $\\phi _ { t }$ into the one-hot representation $\\tilde { \\phi } _ { t }$ . The range of feature space $[ \\phi _ { \\mathrm { m i n } } , \\phi _ { \\mathrm { m a x } } ]$ is pre-defined from the given offline datasets. $z _ { \\Phi _ { \\mathrm { h i s t } } } ( t )$ , the categorical feature distribution at time step $t$ , can be computed recursively following Bellman-like equation: ", + "bbox": [ + 173, + 251, + 825, + 310 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/5a62cf0a101050942e30220229498763fb6911dc6f90c4ec4286234d29e7da47.jpg", + "text": "$$\nz _ { \\Phi _ { \\mathrm { h i s t } } } ( t ) \\propto \\tilde { \\phi } _ { t } + \\gamma ( 1 - \\mathbb { 1 } [ t = T ] ) z _ { \\Phi _ { \\mathrm { h i s t } } } ( t + 1 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 315, + 648, + 335 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where $\\mathbb { 1 }$ is the indicator function. We compute a series of $z _ { \\Phi _ { \\mathrm { h i s t } } }$ in a backward manner starting from $T$ . After we obtain the desired information statistics for all trajectories, we feed them to the categorical transformer during training/test time. We describe the python-like pseudocode in Algorithm 1, coloring the changes from the original Decision Transformer (Chen et al., 2021a). ", + "bbox": [ + 173, + 342, + 826, + 398 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "G DETAILS OF DECISION TRANSFORMER WITH LEARNED $\\Phi$ ", + "text_level": 1, + "bbox": [ + 174, + 417, + 687, + 435 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "While any unsupervised regularizer for an encoder could be combined into the action MSE loss of Decision Transformer to obtain the learned $\\Phi$ efficiently, we observe that even simple objectives, such as auto-encoder and contrastive loss, sometimes perform well. For auto-encoder regularization (DT-AE), we train the MLP encoder (parameterized by $\\psi$ ) and decoder with MSE loss of current state reconstruction: ", + "bbox": [ + 173, + 449, + 826, + 518 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/512df1449e1103a78b3aaf3a11c5508106652abc4a42fc6449d756b5bc08e9eb.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\psi } \\mathbb { E } _ { s \\sim D } \\left[ \\| s - \\mathrm { d e c o d e r } _ { \\psi } ( \\mathrm { e n c o d e r } _ { \\psi } ( s ) ) \\| ^ { 2 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 516, + 650, + 542 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Then, we use the output of the encoder as a learned information statistics. In addition, for contrastive loss (DT-CPC), we adopt CURL objective (Srinivas et al., 2020) for state input while adding gaussian perturbation $\\epsilon \\sim \\mathcal { N } ( \\mu = 0 . 0 , \\sigma = 0 . 1 )$ as data argumentation (following Sinha et al. (2021)). We train the MLP encoder as a query, use its momentum encoder as a key: ", + "bbox": [ + 174, + 545, + 825, + 602 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/8ce1ca4cf6103aff1b2f0d1ce616c369d5fcda787b992c9251cb3ec7ee777861.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\psi } \\mathbb { E } _ { \\stackrel { s \\sim D , } { \\epsilon , \\epsilon ^ { \\prime } \\sim \\mathcal { N } ( 0 , \\sigma ) } } \\left[ \\log \\frac { \\exp { ( \\mathrm { e n c o d e r } _ { \\psi } ( s ) ^ { T } W \\mathrm { e n c o d e r } _ { \\tilde { \\psi } } ( s + \\epsilon ) ) } } { \\sum _ { \\epsilon ^ { \\prime } \\neq \\epsilon } \\exp { ( \\mathrm { e n c o d e r } _ { \\psi } ( s ) ^ { T } W \\mathrm { e n c o d e r } _ { \\tilde { \\psi } } ( s + \\epsilon ^ { \\prime } ) ) } } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 254, + 608, + 738, + 650 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where $W$ is a learned parameter matrix for the bi-linear inner-product, and $\\tilde { \\psi }$ is the weights of the momentum encoder, updated as $\\tilde { \\psi } m \\tilde { \\psi } + ( 1 - m ) \\psi$ , and $m = 0 . 9 5$ . We treat its trained encoder output as a learned information statistics. It remains as future work to combine more advanced, temporary-extended objectives, such as attentive contrastive learning approach proposed in Yang & Nachum (2021). As described in Section 6.3, we consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT’s supervised loss jointly (called as “joint”), and pre-training with only unsupervised loss and freezing the weights during DT training (called as “frozen”). We train DT-E2E with only DT’s supervised loss as in Algorithm 1 for CDT and BDT. ", + "bbox": [ + 173, + 657, + 826, + 787 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Algorithm 1 Categorical/Bi-directional Decision Transformer Pseudocode: Orange and green texts describe additional details on top of the base DT pseudocode from Chen et al. (2021a). ", + "bbox": [ + 173, + 112, + 823, + 142 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# z: information statistics (histogram, or learned representation) # s, a, t: states, actions, or timesteps # transformer: transformer with causal masking (GPT) # embed_s, embed_a, embed_z: linear embedding layers # anti_causal_tf: second transformer as encoder and aggregator # # embed_t: learned episode positional embeddingpred_a: linear action prediction layer compute_stats: a function to compute information statistics ", + "bbox": [ + 174, + 151, + 754, + 244 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# main model ", + "text_level": 1, + "bbox": [ + 174, + 255, + 281, + 265 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "def DecisionTransformer(z, s, a, t): # compute embeddings for tokens pos_embedding $=$ embed_t(t) s_embedding $=$ embed_s(s) $^ +$ pos_embedding a_embedding $=$ embed_a(a) $^ +$ pos_embedding if categorical: z_embedding $=$ embed_z(z) $^ +$ pos_embedding elif bi_directional: # input state sequence in a reverse order # NOTE: z is a target state sequence here reversed $=$ flip(z) z_embedding $=$ embed_z(anti_causal_tf(reversed)) $^ +$ pos_embedding input_embeds $=$ stack(z_embedding, s_embedding, a_embedding) ", + "bbox": [ + 179, + 265, + 807, + 404 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 209, + 411, + 730, + 425 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# use transformer to get hidden states hidden_states $=$ transformer(input_embeds $=$ input_embeds) ", + "bbox": [ + 209, + 435, + 686, + 459 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# select hidden states for action prediction tokens a_hidden $=$ unstack(hidden_states).actions ", + "bbox": [ + 207, + 469, + 660, + 492 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# predict action return pred_a(a_hidden) ", + "bbox": [ + 209, + 503, + 411, + 526 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "# training loop \ntrain_z $=$ compute_stats(train_dataset) \n# dims: (batch_size, K, dim) \nfor z, (s, a, t) in zip(train_z, train_dataset): if unsupervised: $\\textbf { z } = \\textbf { s }$ a_preds $=$ DecisionTransformer(z, s, a, t) loss $=$ mean((a_preds - a)\\*\\*2) optimizer.zero_grad(); loss.backward(); optimizer.step() \n# evaluation loop \ntest_z $=$ compute_stats(test_dataset) \nn_tests $=$ len(test_dataset) \nfor index in range(n_tests): test_trajectory $=$ test_dataset[index] max_time_steps $=$ len(test_trajectory) s, a, t, done $=$ [env.reset()], [], [1], False # conditioning on the desired information statistics if categorical: $z =$ [test_z[index][0]] elif bi_directional: z = [test_trajectory[’observations’][0]] for t in range(max_time_steps): action $=$ DecisionTransformer(z, s, a, t)[-1] new_s, r, done, $=$ env.step(action) # append new tokens to sequence if categorical: $\\mathrm { ~ \\bf ~ z ~ } = \\mathrm { ~ \\bf ~ z ~ } + \\mathrm { ~ \\bf ~ \\cdot ~ }$ [test_z[index] $[ \\ t + \\pm ] ]$ elif bi_directional: $\\textbf { z } = \\textbf { z } +$ [test_trajectory[’observations’][t+1]] s, a, t = s + [new_s], a + [action], t + [len(z)] z, s, a, t = z[-N:], # only keep context length of N ", + "bbox": [ + 173, + 537, + 702, + 640 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 650, + 759, + 901 + ], + "page_idx": 27 + } +] \ No newline at end of file diff --git a/parse/dev/CAjxVodl_v/CAjxVodl_v_middle.json b/parse/dev/CAjxVodl_v/CAjxVodl_v_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..96b7759112a93e0b73e6aac2e581dec1be8a9130 --- /dev/null +++ b/parse/dev/CAjxVodl_v/CAjxVodl_v_middle.json @@ -0,0 +1,69287 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 84, + 457, + 122 + ], + "lines": [ + { + "bbox": [ + 106, + 84, + 445, + 102 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 445, + 102 + ], + "score": 1.0, + "content": "GENERALIZED DECISION TRANSFORMER FOR", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 104, + 459, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 459, + 123 + ], + "score": 1.0, + "content": "OFFLINE HINDSIGHT INFORMATION MATCHING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 140, + 287, + 174 + ], + "lines": [ + { + "bbox": [ + 114, + 140, + 178, + 153 + ], + "spans": [ + { + "bbox": [ + 114, + 140, + 178, + 153 + ], + "score": 1.0, + "content": "Hiroki Furuta", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 150, + 214, + 165 + ], + "spans": [ + { + "bbox": [ + 112, + 150, + 214, + 165 + ], + "score": 1.0, + "content": "The University of Tokyo", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 112, + 162, + 288, + 176 + ], + "spans": [ + { + "bbox": [ + 112, + 162, + 288, + 176 + ], + "score": 1.0, + "content": "furuta@weblab.t.u-tokyo.ac.jp", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 301, + 141, + 399, + 163 + ], + "lines": [ + { + "bbox": [ + 300, + 140, + 368, + 153 + ], + "spans": [ + { + "bbox": [ + 300, + 140, + 368, + 153 + ], + "score": 1.0, + "content": "Yutaka Matsuo", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 299, + 151, + 401, + 165 + ], + "spans": [ + { + "bbox": [ + 299, + 151, + 401, + 165 + ], + "score": 1.0, + "content": "The University of Tokyo", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 412, + 141, + 495, + 163 + ], + "lines": [ + { + "bbox": [ + 412, + 141, + 497, + 154 + ], + "spans": [ + { + "bbox": [ + 412, + 141, + 497, + 154 + ], + "score": 1.0, + "content": "Shixiang Shane Gu", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 412, + 151, + 484, + 164 + ], + "spans": [ + { + "bbox": [ + 412, + 151, + 484, + 164 + ], + "score": 1.0, + "content": "Google Research", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 278, + 203, + 333, + 215 + ], + "lines": [ + { + "bbox": [ + 276, + 202, + 336, + 217 + ], + "spans": [ + { + "bbox": [ + 276, + 202, + 336, + 217 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 142, + 230, + 469, + 503 + ], + "lines": [ + { + "bbox": [ + 141, + 228, + 469, + 242 + ], + "spans": [ + { + "bbox": [ + 141, + 228, + 469, + 242 + ], + "score": 1.0, + "content": "How to extract as much learning signal from each trajectory data has been a key", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 240, + 469, + 252 + ], + "spans": [ + { + "bbox": [ + 141, + 240, + 469, + 252 + ], + "score": 1.0, + "content": "problem in reinforcement learning (RL), where sample inefficiency has posed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 250, + 469, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 250, + 469, + 264 + ], + "score": 1.0, + "content": "serious challenges for practical applications. 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For the target", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 156, + 207 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 194, + 179, + 206 + ], + "score": 0.91, + "content": "p ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 192, + 506, + 207 + ], + "score": 1.0, + "content": ", Lee et al. (2020) set a uniform distribution to enhance the exploration over the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "entire state space; Ghasemipour et al. (2020) and Gu et al. 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Given", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 295, + 338 + ], + "score": 1.0, + "content": "parameterized reward functions with parameter", + "type": "text" + }, + { + "bbox": [ + 295, + 329, + 321, + 337 + ], + "score": 0.89, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 327, + 406, + 338 + ], + "score": 1.0, + "content": ", a conditional policy", + "type": "text" + }, + { + "bbox": [ + 406, + 327, + 443, + 339 + ], + "score": 0.92, + "content": "\\pi ( a | s , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "is learned with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 226, + 351 + ], + "score": 1.0, + "content": "respect to multiple values of", + "type": "text" + }, + { + "bbox": [ + 226, + 340, + 232, + 348 + ], + "score": 0.73, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 338, + 352, + 351 + ], + "score": 1.0, + "content": "simultaneously weighted by", + "type": "text" + }, + { + "bbox": [ + 352, + 338, + 370, + 350 + ], + "score": 0.91, + "content": "p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ". As examples, the RL objective", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 178, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 178, + 362 + ], + "score": 1.0, + "content": "in Eq.1 becomes:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 365, + 447, + 391 + ], + "lines": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "spans": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "score": 0.92, + "content": "L _ { \\mathrm { R L } } ( \\pi ) = \\mathbb { E } _ { z } \\left[ L _ { \\mathrm { R L } } ( \\pi , z ) \\right] = \\frac { 1 } { 1 - \\gamma } \\mathbb { E } _ { z \\sim p ( z ) , s \\sim \\rho _ { z } ^ { \\pi } ( s ) , a \\sim \\pi ( \\cdot | s , z ) } \\left[ r _ { z } ( s , a ) \\right]", + "type": "interline_equation", + "image_path": "746b993baf462f1c4b39701d72c4bbd71882aa02220343884b5ca33d28152772.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 506, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 203, + 409 + ], + "score": 1.0, + "content": "where the state marginal", + "type": "text" + }, + { + "bbox": [ + 204, + 398, + 215, + 408 + ], + "score": 0.89, + "content": "\\rho _ { z } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 396, + 370, + 409 + ], + "score": 1.0, + "content": "is from rolling out a conditioned policy", + "type": "text" + }, + { + "bbox": [ + 371, + 397, + 403, + 409 + ], + "score": 0.91, + "content": "\\pi ( \\cdot | \\cdot , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 396, + 505, + 409 + ], + "score": 1.0, + "content": ". These can be considered", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 477, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 477, + 420 + ], + "score": 1.0, + "content": "as a special case of contextual MDPs (Jiang et al., 2017) and are all multi-task RL problems.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 322, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 323, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 323, + 449 + ], + "score": 1.0, + "content": "4 HINDSIGHT INFORMATION MATCHING", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 506, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "We show how HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "2019), hindsight multi-task RL (Li et al., 2020; Eysenbach et al., 2020), and return-conditioned", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "2021) all belong to hindsight algorithms with a shared idea of using future state information", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "to automatically mine for positive, or “optimal”, examples with respect to certain contextual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "parameter values, where these examples can accelerate RL or be used for behavior cloning (BC),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 526, + 372, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 372, + 539 + ], + "score": 1.0, + "content": "i.e. supervised learning. We start by defining additional notations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 255, + 556 + ], + "score": 1.0, + "content": "Given a partial trajectory from state", + "type": "text" + }, + { + "bbox": [ + 255, + 545, + 265, + 554 + ], + "score": 0.84, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 542, + 277, + 556 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 277, + 543, + 396, + 555 + ], + "score": 0.9, + "content": "\\tau _ { t } = \\{ s _ { t } , a _ { t } , s _ { t + 1 } , a _ { t + 1 } , \\dots \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 542, + 506, + 556 + ], + "score": 1.0, + "content": ", we define its information", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 158, + 565 + ], + "score": 1.0, + "content": "statistics as", + "type": "text" + }, + { + "bbox": [ + 158, + 554, + 180, + 566 + ], + "score": 0.88, + "content": "I ( \\tau _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 552, + 189, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 189, + 554, + 211, + 566 + ], + "score": 0.85, + "content": "I ( \\hat { \\tau _ { t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "could be any function of a trajectory that captures some statistical", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "properties in state-space or trajectory-space, such as sufficient statistics of a distribution, like mean,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "variance or higher-order moments (Wainwright & Jordan, 2008). For convenience, we further", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 263, + 599 + ], + "score": 1.0, + "content": "define the notion of a feature function", + "type": "text" + }, + { + "bbox": [ + 263, + 586, + 351, + 599 + ], + "score": 0.91, + "content": "\\Phi ( \\cdot , \\cdot ) : S \\times A \\to F ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ", where the trajectory is then noted as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 594, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 294, + 610 + ], + "score": 0.83, + "content": "\\tau _ { t } ^ { \\Phi } = \\{ \\phi _ { t } , \\phi _ { t + 1 } , \\ldots , \\phi _ { T } \\} , \\phi _ { t } = \\Phi ( s _ { t } , a _ { t } ) \\in \\stackrel { } { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 594, + 420, + 612 + ], + "score": 1.0, + "content": "and the information statistics as", + "type": "text" + }, + { + "bbox": [ + 420, + 597, + 448, + 610 + ], + "score": 0.91, + "content": "\\tilde { I } ^ { \\Phi } \\bar { ( } \\tau _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 594, + 452, + 612 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 452, + 598, + 461, + 608 + ], + "score": 0.73, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 594, + 506, + 612 + ], + "score": 1.0, + "content": "in practice", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 618, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 616, + 507, + 630 + ], + "spans": [ + { + "bbox": [ + 118, + 616, + 507, + 630 + ], + "score": 1.0, + "content": "2As discussed in Fu et al. (2018) and Ghasemipour et al. (2020), it’s also straight-forward to define state-", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 627, + 480, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 256, + 640 + ], + "score": 1.0, + "content": "action-marginal matching with respect to", + "type": "text" + }, + { + "bbox": [ + 256, + 628, + 345, + 639 + ], + "score": 0.92, + "content": "\\rho ^ { \\pi } ( s , a ) = \\rho ^ { \\pi } ( s ) \\pi ( a | s )", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 627, + 480, + 640 + ], + "score": 1.0, + "content": "and the exact same algorithms apply.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 636, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 118, + 636, + 506, + 651 + ], + "score": 1.0, + "content": "3There is a rich literature on one-shot, or few-shot, imitation learning through meta learning. For closer", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "connections to parameterized policies and relabeling, we mainly discuss metric-based (or amortization-based)", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 659, + 428, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 428, + 670 + ], + "score": 1.0, + "content": "methods (Duan et al., 2016), as opposed to gradient-based approaches (Finn et al., 2017).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 667, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 118, + 667, + 506, + 682 + ], + "score": 1.0, + "content": "4DisCo RL (Nasiriany et al., 2021) conditions on a parameterized goal distribution and uses hindsight", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 297, + 691 + ], + "score": 1.0, + "content": "techniques; however, their RL objective in Equation", + "type": "text" + }, + { + "bbox": [ + 297, + 680, + 379, + 691 + ], + "score": 0.3, + "content": "1 \\mathbb { E } _ { s \\sim \\rho _ { z } ^ { \\pi } ( s ) } \\left[ \\log p _ { z } ^ { * } ( s ) \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 678, + 505, + 691 + ], + "score": 1.0, + "content": ", in contrast to a proper divergence", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 689, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 312, + 703 + ], + "score": 1.0, + "content": "objective like in Ghasemipour et al. (2020), is missing the", + "type": "text" + }, + { + "bbox": [ + 313, + 691, + 350, + 702 + ], + "score": 0.88, + "content": "\\mathcal { H } \\left( \\rho _ { z } ^ { \\pi } ( s ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 689, + 506, + 703 + ], + "score": 1.0, + "content": "entropy term and is essentially just solving", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 700, + 408, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 408, + 712 + ], + "score": 1.0, + "content": "for parameterized stationary reward maximization, as also stated in their Remark 1.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "5Recent mutual information maximization or empowerment methods (Eysenbach et al., 2019; Sharma et al.,", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 442, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 442, + 732 + ], + "score": 1.0, + "content": "2020; Choi et al., 2021) also make similar assumptions; see Gu et al. (2021) for more details.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 260, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 260, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 260, + 94 + ], + "score": 1.0, + "content": "3.1 STATE MARGINAL MATCHING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "State marginal matching (SMM) (Lee et al., 2020; Hazan et al., 2019; Ghasemipour et al., 2020) has", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "been recently studied as an alternative problem specification in RL, where instead of stationary-reward", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 391, + 137 + ], + "score": 1.0, + "content": "maximization, the objective is to find a policy minimizing the divergence", + "type": "text" + }, + { + "bbox": [ + 391, + 126, + 401, + 135 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "between its state marginal", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 327, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 155, + 148 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 155, + 136, + 179, + 148 + ], + "score": 0.92, + "content": "\\rho ^ { \\pi } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 135, + 295, + 148 + ], + "score": 1.0, + "content": "to a given target distribution", + "type": "text" + }, + { + "bbox": [ + 295, + 135, + 322, + 148 + ], + "score": 0.91, + "content": "p ^ { * } ( s ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 135, + 327, + 148 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 102, + 506, + 148 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 153, + 369, + 167 + ], + "lines": [ + { + "bbox": [ + 242, + 153, + 369, + 167 + ], + "spans": [ + { + "bbox": [ + 242, + 153, + 369, + 167 + ], + "score": 0.92, + "content": "L _ { \\mathrm { S M M } } ( \\pi ) = - D ( \\rho ^ { \\pi } ( s ) , p ^ { * } ( s ) )", + "type": "interline_equation", + "image_path": "2a1b3fd5b00115be16d4a7463fda89bfa059cbc4baece45293731ad689c1b1c3.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 242, + 153, + 369, + 167 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 505, + 260 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 133, + 184 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 172, + 144, + 182 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 172, + 505, + 184 + ], + "score": 1.0, + "content": "is a divergence measure such as Kullback-Leibler (KL) divergence (Lee et al., 2020; Fu", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 266, + 195 + ], + "score": 1.0, + "content": "et al., 2018) or, more generally, some", + "type": "text" + }, + { + "bbox": [ + 266, + 183, + 273, + 195 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "-divergences (Ghasemipour et al., 2020). For the target", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 156, + 207 + ], + "score": 1.0, + "content": "distribution", + "type": "text" + }, + { + "bbox": [ + 156, + 194, + 179, + 206 + ], + "score": 0.91, + "content": "p ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 192, + 506, + 207 + ], + "score": 1.0, + "content": ", Lee et al. (2020) set a uniform distribution to enhance the exploration over the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "entire state space; Ghasemipour et al. (2020) and Gu et al. (2021) set through scripted distribution", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "sketches to generate desired behaviors; and adversarial inverse RL methods (Ho & Ermon, 2016;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 238 + ], + "score": 1.0, + "content": "Fu et al., 2018; Ghasemipour et al., 2020; Kostrikov et al., 2020) set as the expert data for imitation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 505, + 249 + ], + "score": 1.0, + "content": "learning. Notably, unlike the RL objective in Eq.1, SMM objectives like Eq.2 no longer depend on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 475, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 475, + 260 + ], + "score": 1.0, + "content": "task rewards and are only functions of state transition dynamics and target state distribution.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 172, + 506, + 260 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 273, + 277, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 278, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 278, + 286 + ], + "score": 1.0, + "content": "3.2 PARAMETERIZED RL OBJECTIVES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 506, + 361 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "Lastly, we discuss the basis for methods like HER and TDM (Andrychowicz et al., 2017; Pong et al.,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "2018), LfP (Lynch et al., 2019), and return-conditioned or upside-down RL (Srivastava et al., 2019;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "Kumar et al., 2019; Chen et al., 2021a; Janner et al., 2021): parameterized RL objectives. Given", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 295, + 338 + ], + "score": 1.0, + "content": "parameterized reward functions with parameter", + "type": "text" + }, + { + "bbox": [ + 295, + 329, + 321, + 337 + ], + "score": 0.89, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 327, + 406, + 338 + ], + "score": 1.0, + "content": ", a conditional policy", + "type": "text" + }, + { + "bbox": [ + 406, + 327, + 443, + 339 + ], + "score": 0.92, + "content": "\\pi ( a | s , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 327, + 505, + 338 + ], + "score": 1.0, + "content": "is learned with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 226, + 351 + ], + "score": 1.0, + "content": "respect to multiple values of", + "type": "text" + }, + { + "bbox": [ + 226, + 340, + 232, + 348 + ], + "score": 0.73, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 338, + 352, + 351 + ], + "score": 1.0, + "content": "simultaneously weighted by", + "type": "text" + }, + { + "bbox": [ + 352, + 338, + 370, + 350 + ], + "score": 0.91, + "content": "p ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ". As examples, the RL objective", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 348, + 178, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 178, + 362 + ], + "score": 1.0, + "content": "in Eq.1 becomes:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 293, + 506, + 362 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 164, + 365, + 447, + 391 + ], + "lines": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "spans": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "score": 0.92, + "content": "L _ { \\mathrm { R L } } ( \\pi ) = \\mathbb { E } _ { z } \\left[ L _ { \\mathrm { R L } } ( \\pi , z ) \\right] = \\frac { 1 } { 1 - \\gamma } \\mathbb { E } _ { z \\sim p ( z ) , s \\sim \\rho _ { z } ^ { \\pi } ( s ) , a \\sim \\pi ( \\cdot | s , z ) } \\left[ r _ { z } ( s , a ) \\right]", + "type": "interline_equation", + "image_path": "746b993baf462f1c4b39701d72c4bbd71882aa02220343884b5ca33d28152772.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 164, + 365, + 447, + 391 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 506, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 203, + 409 + ], + "score": 1.0, + "content": "where the state marginal", + "type": "text" + }, + { + "bbox": [ + 204, + 398, + 215, + 408 + ], + "score": 0.89, + "content": "\\rho _ { z } ^ { \\pi }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 396, + 370, + 409 + ], + "score": 1.0, + "content": "is from rolling out a conditioned policy", + "type": "text" + }, + { + "bbox": [ + 371, + 397, + 403, + 409 + ], + "score": 0.91, + "content": "\\pi ( \\cdot | \\cdot , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 396, + 505, + 409 + ], + "score": 1.0, + "content": ". These can be considered", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 407, + 477, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 477, + 420 + ], + "score": 1.0, + "content": "as a special case of contextual MDPs (Jiang et al., 2017) and are all multi-task RL problems.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 396, + 505, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 322, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 323, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 323, + 449 + ], + "score": 1.0, + "content": "4 HINDSIGHT INFORMATION MATCHING", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 506, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "We show how HER and TDM (Andrychowicz et al., 2017; Pong et al., 2018), LfP (Lynch et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "2019), hindsight multi-task RL (Li et al., 2020; Eysenbach et al., 2020), and return-conditioned", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "or upside-down RL (Srivastava et al., 2019; Kumar et al., 2019; Chen et al., 2021a; Janner et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "2021) all belong to hindsight algorithms with a shared idea of using future state information", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "to automatically mine for positive, or “optimal”, examples with respect to certain contextual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "parameter values, where these examples can accelerate RL or be used for behavior cloning (BC),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 526, + 372, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 372, + 539 + ], + "score": 1.0, + "content": "i.e. supervised learning. We start by defining additional notations.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 460, + 506, + 539 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 255, + 556 + ], + "score": 1.0, + "content": "Given a partial trajectory from state", + "type": "text" + }, + { + "bbox": [ + 255, + 545, + 265, + 554 + ], + "score": 0.84, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 542, + 277, + 556 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 277, + 543, + 396, + 555 + ], + "score": 0.9, + "content": "\\tau _ { t } = \\{ s _ { t } , a _ { t } , s _ { t + 1 } , a _ { t + 1 } , \\dots \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 542, + 506, + 556 + ], + "score": 1.0, + "content": ", we define its information", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 158, + 565 + ], + "score": 1.0, + "content": "statistics as", + "type": "text" + }, + { + "bbox": [ + 158, + 554, + 180, + 566 + ], + "score": 0.88, + "content": "I ( \\tau _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 552, + 189, + 565 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 189, + 554, + 211, + 566 + ], + "score": 0.85, + "content": "I ( \\hat { \\tau _ { t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "could be any function of a trajectory that captures some statistical", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "properties in state-space or trajectory-space, such as sufficient statistics of a distribution, like mean,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 506, + 587 + ], + "score": 1.0, + "content": "variance or higher-order moments (Wainwright & Jordan, 2008). For convenience, we further", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 263, + 599 + ], + "score": 1.0, + "content": "define the notion of a feature function", + "type": "text" + }, + { + "bbox": [ + 263, + 586, + 351, + 599 + ], + "score": 0.91, + "content": "\\Phi ( \\cdot , \\cdot ) : S \\times A \\to F ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ", where the trajectory is then noted as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 594, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 294, + 610 + ], + "score": 0.83, + "content": "\\tau _ { t } ^ { \\Phi } = \\{ \\phi _ { t } , \\phi _ { t + 1 } , \\ldots , \\phi _ { T } \\} , \\phi _ { t } = \\Phi ( s _ { t } , a _ { t } ) \\in \\stackrel { } { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 594, + 420, + 612 + ], + "score": 1.0, + "content": "and the information statistics as", + "type": "text" + }, + { + "bbox": [ + 420, + 597, + 448, + 610 + ], + "score": 0.91, + "content": "\\tilde { I } ^ { \\Phi } \\bar { ( } \\tau _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 594, + 452, + 612 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 452, + 598, + 461, + 608 + ], + "score": 0.73, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 594, + 506, + 612 + ], + "score": 1.0, + "content": "in practice", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 311, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 305, + 324 + ], + "score": 1.0, + "content": "can be an identity function, the reward function", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 305, + 311, + 333, + 323 + ], + "score": 0.92, + "content": "r ( s , a )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 333, + 311, + 415, + 324 + ], + "score": 1.0, + "content": ", sub-dimensions of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 416, + 314, + 421, + 321 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 422, + 311, + 507, + 324 + ], + "score": 1.0, + "content": "(e.g. xy-velocities),", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "or a generic parameterized function (e.g. auto-encoder). Generalizing reward-centric intuitions in", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 506, + 346 + ], + "score": 1.0, + "content": "DT (Chen et al., 2021a), we define information matching (IM) problems as learning a conditional", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 344, + 478, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 134, + 356 + ], + "score": 1.0, + "content": "policy", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 134, + 344, + 171, + 356 + ], + "score": 0.93, + "content": "\\pi ( a | s , z )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 171, + 344, + 467, + 356 + ], + "score": 1.0, + "content": "whose trajectory rollouts satisfy some desired information statistics value", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 467, + 346, + 474, + 354 + ], + "score": 0.73, + "content": "z", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 474, + 344, + 478, + 356 + ], + "score": 1.0, + "content": ":", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 542, + 506, + 612 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 156, + 81, + 454, + 205 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 156, + 81, + 454, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 156, + 81, + 454, + 205 + ], + "spans": [ + { + "bbox": [ + 156, + 81, + 454, + 205 + ], + "score": 0.982, + "html": "
MethodAlgo.TypeTraining1(T）Architectures
Andrychowicz et al. (2017) Pong et al. (2018)RL RLOnline OnlineT TMLP MLP
Chebotar et al. (2021)RLOfflineTCNN
Li et al. (2020)RLOnlineMLP
tγtr(st,at,.)
Eysenbach et al. (2020)BC/RLOn/OfflineargmaxMLP
Lynch et al. (2019)BCOfflineΦTStochastic RNN
Ghosh et al. (2021)BCOnlineΦTMLP
Srivastava et al. (2019)BCOnlineMt rtFast Weights
Kumar et al. (2019)BCOnlineMt ytrtMLP
Janner et al. (2021)BCOfflinetrtorTTransformer
Duan et al. (2017)3BCOfflineMtMLP+LSTM
Generalized DT(ours)BCOfflineT AnyTransformer
DT(Chen et al., 2021a)BCOfflineMttrtTransformer
Categorical DT(ours)4BCOfflinehistogram(rt,γ)
Transformer
Bi-Directional DT (ours)BCOfflineTTransformer
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The notation follows Section 4.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 420, + 229 + ], + "score": 1.0, + "content": "With HIM, all prior works can be categorized to four generic problem types based on", + "type": "text" + }, + { + "bbox": [ + 420, + 217, + 444, + 228 + ], + "score": 0.92, + "content": "I ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 217, + 505, + 229 + ], + "score": 1.0, + "content": ": (1) goal-based", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 107, + 228, + 119, + 238 + ], + "score": 0.84, + "content": "\\phi _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 227, + 315, + 240 + ], + "score": 1.0, + "content": "(Andrychowicz et al., 2017), (2) multi-task arg max", + "type": "text" + }, + { + "bbox": [ + 315, + 227, + 377, + 239 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { t } \\hat { \\gamma } ^ { t } r ( s _ { t } , a _ { t } , \\cdot ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "(Li et al., 2020), (3) return-based", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 138, + 250 + ], + "score": 0.87, + "content": "\\textstyle \\sum _ { t } \\gamma ^ { t } r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 238, + 332, + 251 + ], + "score": 1.0, + "content": "(Chen et al., 2021a), or (4) full trajectory imitation", + "type": "text" + }, + { + "bbox": [ + 332, + 241, + 339, + 248 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 238, + 414, + 251 + ], + "score": 1.0, + "content": "(Duan et al., 2017).", + "type": "text" + }, + { + "bbox": [ + 414, + 239, + 422, + 248 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "is the reward function", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 107, + 250, + 132, + 260 + ], + "score": 0.88, + "content": "r ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "in (2) and (3), an indexing function for state dimensions (e.g. xy-velocities) or a learned function (Nair", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 259, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 270 + ], + "score": 1.0, + "content": "et al., 2018) in (1), or an identify function in (4). 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We call these algorithms hindsight information matching (HIM) algorithms.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 104, + 441, + 504, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 430, + 454 + ], + "score": 1.0, + "content": "Table 1, which classifies prior methods into effectively four categories based on", + "type": "text" + }, + { + "bbox": [ + 430, + 441, + 456, + 453 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 441, + 505, + 454 + ], + "score": 1.0, + "content": ", leads us to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 452, + 316, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 316, + 465 + ], + "score": 1.0, + "content": "have the following insights around HIM algorithms:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 126, + 469, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 125, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 125, + 468, + 384, + 482 + ], + "score": 1.0, + "content": "• New HIM algorithms can be proposed by simply changing", + "type": "text" + }, + { + "bbox": [ + 384, + 468, + 410, + 481 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 468, + 505, + 482 + ], + "score": 1.0, + "content": ", as we did to propose", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 134, + 480, + 304, + 492 + ], + "spans": [ + { + "bbox": [ + 134, + 480, + 304, + 492 + ], + "score": 1.0, + "content": "Categorical DT for (5) distribution-based.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 125, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 125, + 491, + 210, + 506 + ], + "score": 1.0, + "content": "• Given a choice of", + "type": "text" + }, + { + "bbox": [ + 211, + 492, + 236, + 505 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ", new HIM algorithms can be proposed by changing implementa-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 133, + 502, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 133, + 502, + 506, + 518 + ], + "score": 1.0, + "content": "tion details (Furuta et al., 2021a), such as using “RL\" or “BC\" as algorithm type, doing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 133, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 133, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "“Online” or “Offline” training (Levine et al., 2020), and network architectures. All “Of-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 133, + 524, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "fline” “BC” methods could be adopted easily to “Online” learning through recursive data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 133, + 536, + 444, + 549 + ], + "spans": [ + { + "bbox": [ + 133, + 536, + 444, + 549 + ], + "score": 1.0, + "content": "collections (Ghosh et al., 2021; Kumar et al., 2019; Matsushima et al., 2021).", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Only (1) goal-based and (2) multi-task can use “RL” as algorithm type, while all four, plus", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 133, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 133, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "our (5) distribution-based, can use “BC”, because “RL” requires optimizing Eq. 4 with respect", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 133, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 133, + 570, + 362, + 585 + ], + "score": 1.0, + "content": "to the policy, which gets non-trivial for some choices of", + "type": "text" + }, + { + "bbox": [ + 362, + 571, + 388, + 584 + ], + "score": 0.92, + "content": "\\dot { \\mathbf { I } } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "; e.g. (3-5) return-based, full", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 133, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 133, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "trajectory imitation, or distribution-based. “BC” bypasses the need to solve Eq. 4 and therefore", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 133, + 593, + 367, + 606 + ], + "spans": [ + { + "bbox": [ + 133, + 594, + 260, + 606 + ], + "score": 1.0, + "content": "is universally applicable to any", + "type": "text" + }, + { + "bbox": [ + 261, + 593, + 286, + 606 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 594, + 367, + 606 + ], + "score": 1.0, + "content": "or HIM algorithm6.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 336, + 635 + ], + "lines": [ + { + "bbox": [ + 104, + 620, + 338, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 338, + 637 + ], + "score": 1.0, + "content": "5 GENERALIZED DECISION TRANSFORMER", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 504, + 657 + ], + "score": 1.0, + "content": "Following the insights in Section 4, we introduce Generalized Decision Transformer (GDT), which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 363, + 670 + ], + "score": 1.0, + "content": "generalizes DT (Chen et al., 2021a) based on different choices of", + "type": "text" + }, + { + "bbox": [ + 363, + 657, + 389, + 670 + ], + "score": 0.93, + "content": "I ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 657, + 506, + 670 + ], + "score": 1.0, + "content": ", as described in Figure 1 and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "the last rows of Table 1. 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MethodAlgo.TypeTraining1(T）Architectures
Andrychowicz et al. (2017) Pong et al. (2018)RL RLOnline OnlineT TMLP MLP
Chebotar et al. (2021)RLOfflineTCNN
Li et al. (2020)RLOnlineMLP
tγtr(st,at,.)
Eysenbach et al. (2020)BC/RLOn/OfflineargmaxMLP
Lynch et al. (2019)BCOfflineΦTStochastic RNN
Ghosh et al. (2021)BCOnlineΦTMLP
Srivastava et al. (2019)BCOnlineMt rtFast Weights
Kumar et al. (2019)BCOnlineMt ytrtMLP
Janner et al. (2021)BCOfflinetrtorTTransformer
Duan et al. (2017)3BCOfflineMtMLP+LSTM
Generalized DT(ours)BCOfflineT AnyTransformer
DT(Chen et al., 2021a)BCOfflineMttrtTransformer
Categorical DT(ours)4BCOfflinehistogram(rt,γ)
Transformer
Bi-Directional DT (ours)BCOfflineTTransformer
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The notation follows Section 4.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 420, + 229 + ], + "score": 1.0, + "content": "With HIM, all prior works can be categorized to four generic problem types based on", + "type": "text" + }, + { + "bbox": [ + 420, + 217, + 444, + 228 + ], + "score": 0.92, + "content": "I ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 217, + 505, + 229 + ], + "score": 1.0, + "content": ": (1) goal-based", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 107, + 228, + 119, + 238 + ], + "score": 0.84, + "content": "\\phi _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 227, + 315, + 240 + ], + "score": 1.0, + "content": "(Andrychowicz et al., 2017), (2) multi-task arg max", + "type": "text" + }, + { + "bbox": [ + 315, + 227, + 377, + 239 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\sum _ { t } \\hat { \\gamma } ^ { t } r ( s _ { t } , a _ { t } , \\cdot ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "(Li et al., 2020), (3) return-based", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 238, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 238, + 138, + 250 + ], + "score": 0.87, + "content": "\\textstyle \\sum _ { t } \\gamma ^ { t } r _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 238, + 332, + 251 + ], + "score": 1.0, + "content": "(Chen et al., 2021a), or (4) full trajectory imitation", + "type": "text" + }, + { + "bbox": [ + 332, + 241, + 339, + 248 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 238, + 414, + 251 + ], + "score": 1.0, + "content": "(Duan et al., 2017).", + "type": "text" + }, + { + "bbox": [ + 414, + 239, + 422, + 248 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 238, + 506, + 251 + ], + "score": 1.0, + "content": "is the reward function", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 107, + 250, + 132, + 260 + ], + "score": 0.88, + "content": "r ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "in (2) and (3), an indexing function for state dimensions (e.g. xy-velocities) or a learned function (Nair", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 259, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 270 + ], + "score": 1.0, + "content": "et al., 2018) in (1), or an identify function in (4). Our CDT introduces a new category, (5) distribution-based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 107, + 269, + 203, + 280 + ], + "score": 0.84, + "content": "I ^ { \\Phi } ( \\tau ) = \\mathrm { h i s t o g r a m } ( r _ { t } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 267, + 506, + 281 + ], + "score": 1.0, + "content": ", based on a minimal modification to DT, while our BDT can be considered as DT", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 279, + 251, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 251, + 290 + ], + "score": 1.0, + "content": "adapted for (4), the trajectory imitation.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 355 + ], + "lines": [], + "index": 12.5, + "bbox_fs": [ + 104, + 311, + 507, + 356 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 361, + 378, + 380 + ], + "lines": [ + { + "bbox": [ + 232, + 361, + 378, + 380 + ], + "spans": [ + { + "bbox": [ + 232, + 361, + 378, + 380 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { z \\sim p ( z ) , \\tau \\sim \\rho _ { z } ^ { \\pi } ( \\tau ) } \\left[ D ( I ^ { \\Phi } ( \\tau ) , z ) \\right]", + "type": "interline_equation", + "image_path": "b252eccea7f7a47d35b7316557ab57b6e0d80b6b1bba0ae95e14cd41471b2bad.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 232, + 361, + 378, + 380 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 505, + 436 + ], + "lines": [ + { + "bbox": [ + 104, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 391, + 463, + 405 + ], + "score": 1.0, + "content": "An important observation for the IM objective (Eq.4) is that for any given trajectory", + "type": "text" + }, + { + "bbox": [ + 463, + 394, + 470, + 402 + ], + "score": 0.71, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 391, + 505, + 405 + ], + "score": 1.0, + "content": ", setting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 401, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 155, + 415 + ], + "score": 0.92, + "content": "\\mathbf { z } ^ { * } = \\dot { \\mathbf { I } ^ { \\Phi } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 401, + 329, + 416 + ], + "score": 1.0, + "content": "will minimize the inner term divergence", + "type": "text" + }, + { + "bbox": [ + 329, + 403, + 358, + 413 + ], + "score": 0.89, + "content": "\\mathbf { D = 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 401, + 419, + 416 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 419, + 405, + 426, + 412 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 401, + 506, + 416 + ], + "score": 1.0, + "content": "states and actions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 224, + 426 + ], + "score": 1.0, + "content": "are optimal with respect to", + "type": "text" + }, + { + "bbox": [ + 224, + 414, + 253, + 424 + ], + "score": 0.9, + "content": "\\mathbf { z } = \\mathbf { z } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 413, + 322, + 426 + ], + "score": 1.0, + "content": "and samples of", + "type": "text" + }, + { + "bbox": [ + 322, + 414, + 352, + 425 + ], + "score": 0.92, + "content": "( \\tau _ { \\mathrm { i } } , \\mathbf { z _ { \\mathrm { i } } ^ { \\ast } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "can be used to accelerate RL or do", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 424, + 437, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 437, + 437 + ], + "score": 1.0, + "content": "BC. We call these algorithms hindsight information matching (HIM) algorithms.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 391, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 441, + 504, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 430, + 454 + ], + "score": 1.0, + "content": "Table 1, which classifies prior methods into effectively four categories based on", + "type": "text" + }, + { + "bbox": [ + 430, + 441, + 456, + 453 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 441, + 505, + 454 + ], + "score": 1.0, + "content": ", leads us to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 452, + 316, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 316, + 465 + ], + "score": 1.0, + "content": "have the following insights around HIM algorithms:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 441, + 505, + 465 + ] + }, + { + "type": "list", + "bbox": [ + 126, + 469, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 125, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 125, + 468, + 384, + 482 + ], + "score": 1.0, + "content": "• New HIM algorithms can be proposed by simply changing", + "type": "text" + }, + { + "bbox": [ + 384, + 468, + 410, + 481 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 468, + 505, + 482 + ], + "score": 1.0, + "content": ", as we did to propose", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 134, + 480, + 304, + 492 + ], + "spans": [ + { + "bbox": [ + 134, + 480, + 304, + 492 + ], + "score": 1.0, + "content": "Categorical DT for (5) distribution-based.", + "type": "text" + } + ], + "index": 23, + "is_list_end_line": true + }, + { + "bbox": [ + 125, + 491, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 125, + 491, + 210, + 506 + ], + "score": 1.0, + "content": "• Given a choice of", + "type": "text" + }, + { + "bbox": [ + 211, + 492, + 236, + 505 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 491, + 506, + 506 + ], + "score": 1.0, + "content": ", new HIM algorithms can be proposed by changing implementa-", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 133, + 502, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 133, + 502, + 506, + 518 + ], + "score": 1.0, + "content": "tion details (Furuta et al., 2021a), such as using “RL\" or “BC\" as algorithm type, doing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 133, + 515, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 133, + 515, + 506, + 527 + ], + "score": 1.0, + "content": "“Online” or “Offline” training (Levine et al., 2020), and network architectures. All “Of-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 133, + 524, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 133, + 524, + 506, + 540 + ], + "score": 1.0, + "content": "fline” “BC” methods could be adopted easily to “Online” learning through recursive data", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 133, + 536, + 444, + 549 + ], + "spans": [ + { + "bbox": [ + 133, + 536, + 444, + 549 + ], + "score": 1.0, + "content": "collections (Ghosh et al., 2021; Kumar et al., 2019; Matsushima et al., 2021).", + "type": "text" + } + ], + "index": 28, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 132, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Only (1) goal-based and (2) multi-task can use “RL” as algorithm type, while all four, plus", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 133, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 133, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "our (5) distribution-based, can use “BC”, because “RL” requires optimizing Eq. 4 with respect", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 133, + 570, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 133, + 570, + 362, + 585 + ], + "score": 1.0, + "content": "to the policy, which gets non-trivial for some choices of", + "type": "text" + }, + { + "bbox": [ + 362, + 571, + 388, + 584 + ], + "score": 0.92, + "content": "\\dot { \\mathbf { I } } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 570, + 506, + 585 + ], + "score": 1.0, + "content": "; e.g. (3-5) return-based, full", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 133, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 133, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "trajectory imitation, or distribution-based. “BC” bypasses the need to solve Eq. 4 and therefore", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 133, + 593, + 367, + 606 + ], + "spans": [ + { + "bbox": [ + 133, + 594, + 260, + 606 + ], + "score": 1.0, + "content": "is universally applicable to any", + "type": "text" + }, + { + "bbox": [ + 261, + 593, + 286, + 606 + ], + "score": 0.92, + "content": "\\mathbf { I } ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 594, + 367, + 606 + ], + "score": 1.0, + "content": "or HIM algorithm6.", + "type": "text" + } + ], + "index": 33, + "is_list_end_line": true + } + ], + "index": 27.5, + "bbox_fs": [ + 125, + 468, + 506, + 606 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 621, + 336, + 635 + ], + "lines": [ + { + "bbox": [ + 104, + 620, + 338, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 338, + 637 + ], + "score": 1.0, + "content": "5 GENERALIZED DECISION TRANSFORMER", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 647, + 504, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 504, + 657 + ], + "score": 1.0, + "content": "Following the insights in Section 4, we introduce Generalized Decision Transformer (GDT), which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 363, + 670 + ], + "score": 1.0, + "content": "generalizes DT (Chen et al., 2021a) based on different choices of", + "type": "text" + }, + { + "bbox": [ + 363, + 657, + 389, + 670 + ], + "score": 0.93, + "content": "I ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 657, + 506, + 670 + ], + "score": 1.0, + "content": ", as described in Figure 1 and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "the last rows of Table 1. We chose DT as the base model since it is a simple model that uses “BC” as", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 693 + ], + "score": 1.0, + "content": "the algorithm type and “Transformer” as the architecture. The choice of “BC” is a must, so we can", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 268, + 703 + ], + "score": 1.0, + "content": "tractably train GDT with respect to any", + "type": "text" + }, + { + "bbox": [ + 269, + 690, + 294, + 703 + ], + "score": 0.92, + "content": "I ^ { \\Phi } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "or HIM problem. The choice for the architecture is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "more flexible; however, we decided to use transformers (Vaswani et al., 2017) in this work due to their", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "enormous scaling successes in language and vision domains (Dosovitskiy et al., 2020; Brown et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "2020; Ramesh et al., 2021). See Algorithm 1 (in Appendix F) for the full pseudocode. While GDT", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "in Figure 1 can lead to different algorithms depending on different choices of the feature function", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 137, + 138 + ], + "score": 0.93, + "content": "\\Phi ( s , a )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 137, + 125, + 368, + 140 + ], + "score": 1.0, + "content": "and the anti-causal aggregator (which together determine", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 368, + 126, + 395, + 138 + ], + "score": 0.9, + "content": "I ^ { \\Phi } ( \\tau ) \\rangle", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 395, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "), in this work we focus our", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "empirical studies on the following two variants: Categorical DT (CDT) and Bi-directional DT (BDT).", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 647, + 506, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "more flexible; however, we decided to use transformers (Vaswani et al., 2017) in this work due to their", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "enormous scaling successes in language and vision domains (Dosovitskiy et al., 2020; Brown et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "2020; Ramesh et al., 2021). See Algorithm 1 (in Appendix F) for the full pseudocode. While GDT", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "in Figure 1 can lead to different algorithms depending on different choices of the feature function", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 137, + 138 + ], + "score": 0.93, + "content": "\\Phi ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 125, + 368, + 140 + ], + "score": 1.0, + "content": "and the anti-causal aggregator (which together determine", + "type": "text" + }, + { + "bbox": [ + 368, + 126, + 395, + 138 + ], + "score": 0.9, + "content": "I ^ { \\Phi } ( \\tau ) \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "), in this work we focus our", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "empirical studies on the following two variants: Categorical DT (CDT) and Bi-directional DT (BDT).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 163, + 276, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 278, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 278, + 176 + ], + "score": 1.0, + "content": "5.1 TASK DEFINITIONS AND METRICS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "Before proceeding to define CDT and BDT, we first concretely define the tasks they are designed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "to solve, namely: offline multi-task state-marginal matching (SMM), and offline multi-task", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 185, + 219 + ], + "score": 1.0, + "content": "imitation learning", + "type": "text" + }, + { + "bbox": [ + 185, + 207, + 203, + 218 + ], + "score": 0.44, + "content": "\\mathbf { ( I L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ". Given the intrinsic connection or equivalence between distribution matching", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "score": 1.0, + "content": "and IL (Ghasemipour et al., 2020), these two separate terminologies may seem redundant. However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "inspired by the initial papers studying SMM problems (Lee et al., 2020; Ghasemipour et al., 2020)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 507, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 507, + 253 + ], + "score": 1.0, + "content": "which qualitatively evaluates distribution matching results in specified state dimensions (e.g. xy-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 406, + 263 + ], + "score": 1.0, + "content": "positions), we define the imitation task as offline multi-task SMM if specific", + "type": "text" + }, + { + "bbox": [ + 406, + 251, + 414, + 260 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "is given, and as offline", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 148, + 274 + ], + "score": 1.0, + "content": "multi-task", + "type": "text" + }, + { + "bbox": [ + 149, + 262, + 159, + 271 + ], + "score": 0.35, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 261, + 169, + 274 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 169, + 262, + 178, + 271 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 261, + 250, + 274 + ], + "score": 1.0, + "content": "is an identity (i.e.", + "type": "text" + }, + { + "bbox": [ + 250, + 262, + 276, + 273 + ], + "score": 0.9, + "content": "\\phi = s", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 261, + 482, + 274 + ], + "score": 1.0, + "content": ") or learned (e.g. auto-encoder). We essentially view", + "type": "text" + }, + { + "bbox": [ + 483, + 261, + 493, + 271 + ], + "score": 0.36, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 228, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 228, + 284 + ], + "score": 1.0, + "content": "SMM evaluation on full state.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 506, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "Given these definition, we also define a single metric for both offline multi-task SMM/IL: typical IL", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "assumes some availability of task reward or success evaluation, and indirectly measure the quality of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "imitation through it (Ho & Ermon, 2016; Fu et al., 2018). Instead, again grounding on its connection", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "to distribution matching (Ghasemipour et al., 2020), we propose a Wasserstein loss between state-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "marginal and target distributions as SMM-inspired metrics for evaluating offline multi-task SMM or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "IL tasks. However, it is often intractable to measure such loss for full state or even for some state", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "dimensions analytically because both state-marginal and target distributions can be non-parametric", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "and we cannot access their densities. In practice, we empirically estimate it employing the binning of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 418, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 418, + 389 + ], + "score": 1.0, + "content": "the feature space we specified. More discussions are included in Appendix C.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 109, + 403, + 442, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 444, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 444, + 415 + ], + "score": 1.0, + "content": "5.2 CATEGORICAL DECISION TRANSFORMER FOR DISTRIBUTION MATCHING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 506, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 507, + 437 + ], + "score": 1.0, + "content": "Inspired by the recent successes in distributional RL (Bellemare et al., 2017; Dabney et al., 2018;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 507, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 507, + 448 + ], + "score": 1.0, + "content": "2020), offline RL (Fujimoto et al., 2019; Jaques et al., 2020; Ghasemipour et al., 2021; Fujimoto & Gu,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "2021) and state-marginal matching (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al., 2021), we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "introduce Categorical DT (CDT) for offline state-marginal matching (SMM) problem in Section 5.1.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "Following the prior works (Bellemare et al., 2017; Furuta et al., 2021b), we assume low-dimensional", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 115, + 489 + ], + "score": 0.74, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 478, + 505, + 491 + ], + "score": 1.0, + "content": ", e.g. rewards or state dimensions like xyz-velocities, and employ the discretization of feature", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 446, + 504 + ], + "score": 1.0, + "content": "spaces to form categorical approximations of continuous distributions. To compute", + "type": "text" + }, + { + "bbox": [ + 446, + 489, + 505, + 502 + ], + "score": 0.93, + "content": "z _ { t } ^ { * } = I ^ { \\Phi } ( \\tau _ { t : T } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 171, + 514 + ], + "score": 1.0, + "content": "for all timesteps", + "type": "text" + }, + { + "bbox": [ + 172, + 502, + 177, + 511 + ], + "score": 0.54, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 500, + 248, + 514 + ], + "score": 1.0, + "content": "given a trajectory", + "type": "text" + }, + { + "bbox": [ + 248, + 502, + 266, + 512 + ], + "score": 0.87, + "content": "\\tau _ { 1 : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ", we use similar recursive Bellman-like computation inspired", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "by Bellemare et al. (2017) (see Appendix F for the details). To the best of our knowledge, this is the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 446, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 446, + 535 + ], + "score": 1.0, + "content": "first paper to study offline multi-task SMM and propose an effective algorithm for it.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 549, + 321, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 322, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 313, + 561 + ], + "score": 1.0, + "content": "5.3 DECISION TRANSFORMER WITH LEARNED", + "type": "text" + }, + { + "bbox": [ + 313, + 549, + 322, + 559 + ], + "score": 0.75, + "content": "\\Phi", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 569, + 505, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 290, + 581 + ], + "score": 1.0, + "content": "While CDT assumes some low-dimensional", + "type": "text" + }, + { + "bbox": [ + 291, + 570, + 299, + 579 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 570, + 505, + 581 + ], + "score": 1.0, + "content": "is provided for tractable binning and distribution", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 581, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 279, + 592 + ], + "score": 1.0, + "content": "approximation, we also study cases where", + "type": "text" + }, + { + "bbox": [ + 279, + 581, + 287, + 591 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 581, + 299, + 592 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 299, + 583, + 306, + 591 + ], + "score": 0.73, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 581, + 420, + 592 + ], + "score": 1.0, + "content": "is not provided, and instead", + "type": "text" + }, + { + "bbox": [ + 420, + 581, + 429, + 591 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 581, + 505, + 592 + ], + "score": 1.0, + "content": "is learned through", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "auto-encoding (Hinton & Salakhutdinov, 2006; Bengio et al., 2012) (DT-AE) or contrastive (van den", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "Oord et al., 2018; Srinivas et al., 2020; Yang & Nachum, 2021) (DT-CPC) losses for DT (see", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 396, + 626 + ], + "score": 1.0, + "content": "Appendix G for the details). In this case, CDT is unnecessary because if", + "type": "text" + }, + { + "bbox": [ + 396, + 614, + 405, + 624 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "learns sufficient features", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 118, + 637 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 627, + 124, + 635 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 624, + 505, + 637 + ], + "score": 1.0, + "content": ", matching their means, i.e. moments, through DT is enough to match any distribution to an", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 383, + 648 + ], + "score": 1.0, + "content": "arbitrary precision (Wainwright & Jordan, 2008; Li et al., 2015). 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As Section 5.1 defines, these methods do offline multi-task SMM with full state, or", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "offline multi-task IL, a similar objective to state-marginal matching or adversarial inverse RL (Ho &", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "Ermon, 2016; Ghasemipour et al., 2020) in online RL. To the best of our knowledge, this is the first", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 492, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 492, + 702 + ], + "score": 1.0, + "content": "offline multi-task IL method that explicitly accounts for SMM through architectural bottlenecks.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 105, + 716, + 484, + 728 + ], + "lines": [ + { + "bbox": [ + 106, + 716, + 484, + 729 + ], + "spans": [ + { + "bbox": [ + 106, + 716, + 484, + 729 + ], + "score": 1.0, + "content": "5.4 BI-DIRECTIONAL DECISION TRANSFORMER FOR ONE-SHOT IMITATION LEARNING", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 163, + 276, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 278, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 278, + 176 + ], + "score": 1.0, + "content": "5.1 TASK DEFINITIONS AND METRICS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 184, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "Before proceeding to define CDT and BDT, we first concretely define the tasks they are designed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "to solve, namely: offline multi-task state-marginal matching (SMM), and offline multi-task", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 185, + 219 + ], + "score": 1.0, + "content": "imitation learning", + "type": "text" + }, + { + "bbox": [ + 185, + 207, + 203, + 218 + ], + "score": 0.44, + "content": "\\mathbf { ( I L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 205, + 506, + 219 + ], + "score": 1.0, + "content": ". Given the intrinsic connection or equivalence between distribution matching", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "score": 1.0, + "content": "and IL (Ghasemipour et al., 2020), these two separate terminologies may seem redundant. 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We essentially view", + "type": "text" + }, + { + "bbox": [ + 483, + 261, + 493, + 271 + ], + "score": 0.36, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 272, + 228, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 228, + 284 + ], + "score": 1.0, + "content": "SMM evaluation on full state.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 184, + 507, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 289, + 506, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "Given these definition, we also define a single metric for both offline multi-task SMM/IL: typical IL", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "assumes some availability of task reward or success evaluation, and indirectly measure the quality of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "imitation through it (Ho & Ermon, 2016; Fu et al., 2018). Instead, again grounding on its connection", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "to distribution matching (Ghasemipour et al., 2020), we propose a Wasserstein loss between state-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "marginal and target distributions as SMM-inspired metrics for evaluating offline multi-task SMM or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "IL tasks. However, it is often intractable to measure such loss for full state or even for some state", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "dimensions analytically because both state-marginal and target distributions can be non-parametric", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "and we cannot access their densities. In practice, we empirically estimate it employing the binning of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 418, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 418, + 389 + ], + "score": 1.0, + "content": "the feature space we specified. 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To compute", + "type": "text" + }, + { + "bbox": [ + 446, + 489, + 505, + 502 + ], + "score": 0.93, + "content": "z _ { t } ^ { * } = I ^ { \\Phi } ( \\tau _ { t : T } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 171, + 514 + ], + "score": 1.0, + "content": "for all timesteps", + "type": "text" + }, + { + "bbox": [ + 172, + 502, + 177, + 511 + ], + "score": 0.54, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 500, + 248, + 514 + ], + "score": 1.0, + "content": "given a trajectory", + "type": "text" + }, + { + "bbox": [ + 248, + 502, + 266, + 512 + ], + "score": 0.87, + "content": "\\tau _ { 1 : T }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ", we use similar recursive Bellman-like computation inspired", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "by Bellemare et al. (2017) (see Appendix F for the details). 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In this case, CDT is unnecessary because if", + "type": "text" + }, + { + "bbox": [ + 396, + 614, + 405, + 624 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "learns sufficient features", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 118, + 637 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 627, + 124, + 635 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 624, + 505, + 637 + ], + "score": 1.0, + "content": ", matching their means, i.e. moments, through DT is enough to match any distribution to an", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 383, + 648 + ], + "score": 1.0, + "content": "arbitrary precision (Wainwright & Jordan, 2008; Li et al., 2015). Since", + "type": "text" + }, + { + "bbox": [ + 383, + 636, + 391, + 645 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "is differentiable with respect", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 646, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 414, + 658 + ], + "score": 1.0, + "content": "to DT’s action-prediction losses, we also compare DT-E2E, where we learn", + "type": "text" + }, + { + "bbox": [ + 415, + 647, + 423, + 657 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 646, + 506, + 658 + ], + "score": 1.0, + "content": "through end-to-end", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "differentiation. As Section 5.1 defines, these methods do offline multi-task SMM with full state, or", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "offline multi-task IL, a similar objective to state-marginal matching or adversarial inverse RL (Ho &", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "Ermon, 2016; Ghasemipour et al., 2020) in online RL. To the best of our knowledge, this is the first", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 689, + 492, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 492, + 702 + ], + "score": 1.0, + "content": "offline multi-task IL method that explicitly accounts for SMM through architectural bottlenecks.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 570, + 506, + 702 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 716, + 484, + 728 + ], + "lines": [ + { + "bbox": [ + 106, + 716, + 484, + 729 + ], + "spans": [ + { + "bbox": [ + 106, + 716, + 484, + 729 + ], + "score": 1.0, + "content": "5.4 BI-DIRECTIONAL DECISION TRANSFORMER FOR ONE-SHOT IMITATION LEARNING", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49, + "bbox_fs": [ + 106, + 716, + 484, + 729 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 194, + 94 + ], + "score": 1.0, + "content": "The absence of given", + "type": "text" + }, + { + "bbox": [ + 195, + 83, + 203, + 92 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 82, + 424, + 94 + ], + "score": 1.0, + "content": "could be tackled with learning not only parameterized", + "type": "text" + }, + { + "bbox": [ + 425, + 83, + 433, + 93 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "as in Section 5.3,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "but also a parameterized aggregator. Building on the connection to one-shot imitation learning (Duan", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "et al., 2017), we provide another natural extension of DT under GDT framework called Bi-directional", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 344, + 127 + ], + "score": 1.0, + "content": "Decision Transformer (BDT), which assumes an identity", + "type": "text" + }, + { + "bbox": [ + 345, + 116, + 353, + 126 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 115, + 504, + 127 + ], + "score": 1.0, + "content": ", and learns representation within the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "score": 1.0, + "content": "aggregator, in this case a second (anti-causal) transformer (Radford et al., 2018) that takes a reverse-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "order state sequence as an input. See Algorithm 1 in Appendix F for the pseudocode, and Appendix D", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "for further comments on the connection to one-shot or meta learning. While we found some positive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "results for even simple unsupervised regularizer approaches (e.g. DT-AE), we observe BDT could", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 469, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 364, + 182 + ], + "score": 1.0, + "content": "achieve substantially better offline multi-task IL results than DT-", + "type": "text" + }, + { + "bbox": [ + 365, + 171, + 374, + 180 + ], + "score": 0.64, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 171, + 469, + 182 + ], + "score": 1.0, + "content": "variants in Section 5.3.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 198, + 200, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 201, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 201, + 212 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 312, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 313, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 313, + 238 + ], + "score": 1.0, + "content": "We empirically investigate the following questions:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 126, + 240, + 473, + 291 + ], + "lines": [ + { + "bbox": [ + 126, + 240, + 352, + 253 + ], + "spans": [ + { + "bbox": [ + 126, + 240, + 352, + 253 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match unseen reward distributions?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 126, + 254, + 474, + 265 + ], + "spans": [ + { + "bbox": [ + 126, + 254, + 474, + 265 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match and generalize to unseen 1D/2D state-feature distributions?", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 129, + 267, + 423, + 278 + ], + "spans": [ + { + "bbox": [ + 129, + 267, + 423, + 278 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match unseen synthesized state-feature distributions?", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 126, + 280, + 419, + 292 + ], + "spans": [ + { + "bbox": [ + 126, + 280, + 419, + 292 + ], + "score": 1.0, + "content": "• (IL) Can BDT perform offline one-shot imitation learning in full state?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "We experiment on the OpenAI Gym, MuJoCo tasks (HalfCheetah, Hopper, Walker2d, Ant-v3), a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "common benchmark for continuous control (Brockman et al., 2016; Todorov et al., 2012). Through", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "the experiments, we use medium-expert datasets in D4RL (Fu et al., 2020) to ensure the decent data", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "score": 1.0, + "content": "coverage. We sort all the trajectories by their cumulative rewards, hold out five best trajectories and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "five 50 percentile trajectories as a test set (10 trajectories in total), and use the rest as a train set. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "score": 1.0, + "content": "report the results averaged over 20 rollouts every 4 random seed. We share our implementation to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 214, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 214, + 375 + ], + "score": 1.0, + "content": "ensure the reproducibility8", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "As discussed in Section 5.1 and Appendix C, we evaluate CDT/BDT with approximate distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "matching objective: Wasserstein-1 distance between categorical distributions of features in rollouts or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "target trajectories. We compare CDT to DT, Meta-BC, and FOCAL (Li et al., 2021), a metric-based", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "offline meta RL method, as baselines. While Meta-BC and FOCAL does not solve the offline", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "distribution matching problem directly, they provide decent baseline performance since their offline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 446 + ], + "score": 1.0, + "content": "one-shot adaptation to the given target trajectories could deal with it (see Appendix B for the details).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 109, + 459, + 312, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 314, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 314, + 472 + ], + "score": 1.0, + "content": "6.1 REWARD AND STATE-FEATURE MATCHING", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "First, we evaluate whether CDT could match its rollout to the target distribution. We choose reward", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "and state-feature, such as x-velocity of the agents, as feature spaces to match. To specify the target", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "distributions during the evaluation, we feed the categorical representation of the target to CDT. As the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "score": 1.0, + "content": "same as the reward case, DT takes the summation of the state-feature over a trajectory as an input.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "We quantitatively compare CDT against baselines (Table 2) in x-velocity case, where CDT shows", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "better matching results to the target distributions unseen during training. We provide the reward", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "distribution results and the visualization in Appendix E.1, where CDT performs very well in all", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 562, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 577 + ], + "score": 1.0, + "content": "cases. To test the generalization additionally, we intervene the target distributions by (1) shifting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the target distributions in the test set by constant offsets, and (2) synthesizing novel distributions via", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "python scripts. See Appendix E.7 and E.8 for the results. Furthermore, to investigate the scalability", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "of CDT to multi-dimensional state-features, we experiment 2D state-feature distribution matching", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 607, + 432, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 432, + 619 + ], + "score": 1.0, + "content": "(xy-velocities on Ant) in Appendix E.3, where CDT outperforms other baselines.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5 + }, + { + "type": "table", + "bbox": [ + 108, + 630, + 505, + 673 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 630, + 505, + 673 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 630, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 108, + 630, + 505, + 673 + ], + "score": 0.969, + "html": "
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.633 ± 0.3290.996 ± 1.4670.814 ± 1.0790.139± 0.0430.059± 0.0130.099 ± 0.0510.122 ± 0.0710.136 ± 0.0450.129 ±0.0600.347
DT0.746 ± 0.3801.076 ± 1.5490.911 ± 1.1400.177 ± 0.0530.093± 0.0370.135 ± 0.0630.083 ±0.0310.146 ± 0.0840.115± 0.0700.387
BC (no-context)3.017±0.8913.468 ± 1.2713.242 ± 1.1210.652 ± 0.2640.248 ± 0.1990.450 ± 0.3090.748 ± 0.5290.858 ± 0.6170.803 ± 0.5771.498
Meta-BC0.852 ± 0.6880.840 ± 1.1390.846 ± 0.9410.799 ± 0.5050.130± 0.0560.464 ± 0.4910.110 ± 0.0821.462 ± 1.1360.786 ± 1.0520.699
FOCAL (Li ct al.,2021)1.643 ± 0.4611.123 ± 1.5501.383 ± 0.5181.456 ± 0.4730.484 ± 0.3820.970 ± 0.6491.571 ± 0.5630.603± 0.4271.087 ± 0.6951.147
", + "type": "table", + "image_path": "db6308c6f6a1c0b6014b8b8c69bcb4e8fe1d109603b2af68fcf7cc5b3a6999fd.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 108, + 630, + 505, + 644.3333333333334 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 108, + 644.3333333333334, + 505, + 658.6666666666667 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 108, + 658.6666666666667, + 505, + 673.0000000000001 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 675, + 505, + 705 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 685 + ], + "score": 1.0, + "content": "Table 2: Quantitative evaluation of state-feature distribution matching, measuring Wasserstein-1 distance between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "score": 1.0, + "content": "the rollout and target distributions. We compare Categorical DT against DT, BC, Meta-BC, and FOCAL. CDT", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 694, + 482, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 482, + 705 + ], + "score": 1.0, + "content": "achieves better matching than baselines. See Table 9 in Appendix E.1 for the reward distribution results.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 722, + 333, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 721, + 335, + 732 + ], + "spans": [ + { + "bbox": [ + 119, + 721, + 335, + 732 + ], + "score": 1.0, + "content": "8https://github.com/frt03/generalized_dt", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 194, + 94 + ], + "score": 1.0, + "content": "The absence of given", + "type": "text" + }, + { + "bbox": [ + 195, + 83, + 203, + 92 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 82, + 424, + 94 + ], + "score": 1.0, + "content": "could be tackled with learning not only parameterized", + "type": "text" + }, + { + "bbox": [ + 425, + 83, + 433, + 93 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "as in Section 5.3,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "but also a parameterized aggregator. Building on the connection to one-shot imitation learning (Duan", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "et al., 2017), we provide another natural extension of DT under GDT framework called Bi-directional", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 344, + 127 + ], + "score": 1.0, + "content": "Decision Transformer (BDT), which assumes an identity", + "type": "text" + }, + { + "bbox": [ + 345, + 116, + 353, + 126 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 115, + 504, + 127 + ], + "score": 1.0, + "content": ", and learns representation within the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "score": 1.0, + "content": "aggregator, in this case a second (anti-causal) transformer (Radford et al., 2018) that takes a reverse-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "order state sequence as an input. See Algorithm 1 in Appendix F for the pseudocode, and Appendix D", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "for further comments on the connection to one-shot or meta learning. While we found some positive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 171 + ], + "score": 1.0, + "content": "results for even simple unsupervised regularizer approaches (e.g. DT-AE), we observe BDT could", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 469, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 364, + 182 + ], + "score": 1.0, + "content": "achieve substantially better offline multi-task IL results than DT-", + "type": "text" + }, + { + "bbox": [ + 365, + 171, + 374, + 180 + ], + "score": 0.64, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 171, + 469, + 182 + ], + "score": 1.0, + "content": "variants in Section 5.3.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 507, + 182 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 198, + 200, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 201, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 201, + 212 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 223, + 312, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 222, + 313, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 313, + 238 + ], + "score": 1.0, + "content": "We empirically investigate the following questions:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 222, + 313, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 126, + 240, + 473, + 291 + ], + "lines": [ + { + "bbox": [ + 126, + 240, + 352, + 253 + ], + "spans": [ + { + "bbox": [ + 126, + 240, + 352, + 253 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match unseen reward distributions?", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 126, + 254, + 474, + 265 + ], + "spans": [ + { + "bbox": [ + 126, + 254, + 474, + 265 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match and generalize to unseen 1D/2D state-feature distributions?", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 129, + 267, + 423, + 278 + ], + "spans": [ + { + "bbox": [ + 129, + 267, + 423, + 278 + ], + "score": 1.0, + "content": "• (SMM) Can CDT match unseen synthesized state-feature distributions?", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 126, + 280, + 419, + 292 + ], + "spans": [ + { + "bbox": [ + 126, + 280, + 419, + 292 + ], + "score": 1.0, + "content": "• (IL) Can BDT perform offline one-shot imitation learning in full state?", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 126, + 240, + 474, + 292 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "We experiment on the OpenAI Gym, MuJoCo tasks (HalfCheetah, Hopper, Walker2d, Ant-v3), a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "common benchmark for continuous control (Brockman et al., 2016; Todorov et al., 2012). Through", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "the experiments, we use medium-expert datasets in D4RL (Fu et al., 2020) to ensure the decent data", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 341 + ], + "score": 1.0, + "content": "coverage. We sort all the trajectories by their cumulative rewards, hold out five best trajectories and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "five 50 percentile trajectories as a test set (10 trajectories in total), and use the rest as a train set. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 506, + 363 + ], + "score": 1.0, + "content": "report the results averaged over 20 rollouts every 4 random seed. We share our implementation to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 214, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 214, + 375 + ], + "score": 1.0, + "content": "ensure the reproducibility8", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 297, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "As discussed in Section 5.1 and Appendix C, we evaluate CDT/BDT with approximate distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "matching objective: Wasserstein-1 distance between categorical distributions of features in rollouts or", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "target trajectories. We compare CDT to DT, Meta-BC, and FOCAL (Li et al., 2021), a metric-based", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "offline meta RL method, as baselines. While Meta-BC and FOCAL does not solve the offline", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "distribution matching problem directly, they provide decent baseline performance since their offline", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 435, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 446 + ], + "score": 1.0, + "content": "one-shot adaptation to the given target trajectories could deal with it (see Appendix B for the details).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 378, + 506, + 446 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 459, + 312, + 471 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 314, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 314, + 472 + ], + "score": 1.0, + "content": "6.1 REWARD AND STATE-FEATURE MATCHING", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 480, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "First, we evaluate whether CDT could match its rollout to the target distribution. We choose reward", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "and state-feature, such as x-velocity of the agents, as feature spaces to match. To specify the target", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "distributions during the evaluation, we feed the categorical representation of the target to CDT. As the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 500, + 526 + ], + "score": 1.0, + "content": "same as the reward case, DT takes the summation of the state-feature over a trajectory as an input.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 480, + 505, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 530, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "We quantitatively compare CDT against baselines (Table 2) in x-velocity case, where CDT shows", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "better matching results to the target distributions unseen during training. We provide the reward", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "distribution results and the visualization in Appendix E.1, where CDT performs very well in all", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 562, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 577 + ], + "score": 1.0, + "content": "cases. To test the generalization additionally, we intervene the target distributions by (1) shifting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "the target distributions in the test set by constant offsets, and (2) synthesizing novel distributions via", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "python scripts. See Appendix E.7 and E.8 for the results. Furthermore, to investigate the scalability", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "of CDT to multi-dimensional state-features, we experiment 2D state-feature distribution matching", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 607, + 432, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 432, + 619 + ], + "score": 1.0, + "content": "(xy-velocities on Ant) in Appendix E.3, where CDT outperforms other baselines.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 530, + 506, + 619 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 630, + 505, + 673 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 630, + 505, + 673 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 630, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 108, + 630, + 505, + 673 + ], + "score": 0.969, + "html": "
Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.633 ± 0.3290.996 ± 1.4670.814 ± 1.0790.139± 0.0430.059± 0.0130.099 ± 0.0510.122 ± 0.0710.136 ± 0.0450.129 ±0.0600.347
DT0.746 ± 0.3801.076 ± 1.5490.911 ± 1.1400.177 ± 0.0530.093± 0.0370.135 ± 0.0630.083 ±0.0310.146 ± 0.0840.115± 0.0700.387
BC (no-context)3.017±0.8913.468 ± 1.2713.242 ± 1.1210.652 ± 0.2640.248 ± 0.1990.450 ± 0.3090.748 ± 0.5290.858 ± 0.6170.803 ± 0.5771.498
Meta-BC0.852 ± 0.6880.840 ± 1.1390.846 ± 0.9410.799 ± 0.5050.130± 0.0560.464 ± 0.4910.110 ± 0.0821.462 ± 1.1360.786 ± 1.0520.699
FOCAL (Li ct al.,2021)1.643 ± 0.4611.123 ± 1.5501.383 ± 0.5181.456 ± 0.4730.484 ± 0.3820.970 ± 0.6491.571 ± 0.5630.603± 0.4271.087 ± 0.6951.147
", + "type": "table", + "image_path": "db6308c6f6a1c0b6014b8b8c69bcb4e8fe1d109603b2af68fcf7cc5b3a6999fd.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 108, + 630, + 505, + 644.3333333333334 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 108, + 644.3333333333334, + 505, + 658.6666666666667 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 108, + 658.6666666666667, + 505, + 673.0000000000001 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 107, + 675, + 505, + 705 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 675, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 685 + ], + "score": 1.0, + "content": "Table 2: Quantitative evaluation of state-feature distribution matching, measuring Wasserstein-1 distance between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 696 + ], + "score": 1.0, + "content": "the rollout and target distributions. We compare Categorical DT against DT, BC, Meta-BC, and FOCAL. CDT", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 694, + 482, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 482, + 705 + ], + "score": 1.0, + "content": "achieves better matching than baselines. See Table 9 in Appendix E.1 for the reward distribution results.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 364, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "score": 1.0, + "content": "6.2 GENERALIZATION TO UNSEEN TARGET DISTRIBUTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "The performance of offline methods in RL is often restricted by the coverage or quality of datasets.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "score": 1.0, + "content": "While we demonstrate CDT can perform offline state-marginal matching to unseen target distributions", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "score": 1.0, + "content": "in Section 6.1, the standard offline datasets might not be diverse enough to observe the generalization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "since those are collected by single-task reward-maximization policies. To test the generalization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "score": 1.0, + "content": "to more diverse behaviors, we investigate the following tasks: (1) z-velocity distribution matching", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 506, + 169 + ], + "score": 1.0, + "content": "with synthesized bi-modal behavior and (2) cheetah-velocity matching problem from meta RL/IL", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 372, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 372, + 181 + ], + "score": 1.0, + "content": "literature (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 192, + 358, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 360, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 360, + 205 + ], + "score": 1.0, + "content": "6.2.1 SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "To generate diverse behaviors for z-axis, we obtain the expert cheetah that backflips towards -x", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "direction by modifying reward function (see Appendix E.4 for the details). Combining expert", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 234, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 506, + 245 + ], + "score": 1.0, + "content": "backflipping trajectories and expert running forward trajectories from D4RL dataset, we construct a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "novel dataset with diverse behaviors. We experiment the offline SMM with not only each uni-modal", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "behavior (backflipping or running forward), but also synthesized bi-modal behavior; running forward", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 266, + 444, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 444, + 279 + ], + "score": 1.0, + "content": "first, then backflipping during a single rollout, using patchworked target trajectories.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "Table 3 and Figure 2 (a) show that CDT successfully matches the distribution to both uni-modal", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 306 + ], + "score": 1.0, + "content": "(running forward or backflipping) and synthesized bi-modal distributions better than DT and FOCAL,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 305, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 505, + 316 + ], + "score": 1.0, + "content": "and is comparable to Meta-BC that originally designed to deal with such multi-task settings. Due to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 507, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 507, + 329 + ], + "score": 1.0, + "content": "the difficulty for RL algorithms to maximize Eq. 4, FOCAL struggles to solve the offline multi-task", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 327, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 506, + 339 + ], + "score": 1.0, + "content": "SMM, even though FOCAL uses the same context embedding as Meta-BC. The bi-modal behavior", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 493, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 493, + 349 + ], + "score": 1.0, + "content": "learned by CDT can be seen at https://sites.google.com/view/generalizeddt.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "table", + "bbox": [ + 168, + 360, + 442, + 416 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 168, + 360, + 442, + 416 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 360, + 442, + 416 + ], + "spans": [ + { + "bbox": [ + 168, + 360, + 442, + 416 + ], + "score": 0.959, + "html": "
MethodUni-modalBi-modalAverage
Categorical DT1.562 ± 0.6321.625 ± 0.9021.594
DT2.676 ±0.7652.703 ±0.7032.690
Meta-BC1.519 ±0.6961.655 ± 0.9901.587
FOCAL (Li et al., 2021)2.203 士 1.0501.983 士 0.9482.093
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Categorical DT matches both uni- and bi-modal trajectories better than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 438, + 457, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 457, + 450 + ], + "score": 1.0, + "content": "DT and FOCAL, and is comparable to Meta-BC that originally aims to solve multi-task problem.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "image", + "bbox": [ + 106, + 465, + 502, + 576 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 465, + 502, + 576 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 465, + 502, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 502, + 576 + ], + "score": 0.967, + "type": "image", + "image_path": "6122f8e9ffbb9010be05cd17b1ddf24787e73452368349e57817055b17b8fe1b.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 106, + 465, + 502, + 502.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 106, + 502.0, + 502, + 539.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 106, + 539.0, + 502, + 576.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 578, + 506, + 628 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 578, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 589 + ], + "score": 1.0, + "content": "Figure 2: (a) Z-Velocity and (b) Unseen Cheetah-Velocity results. Blue histograms represent target distributions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "In (a), CDT (red) can match not only uni-modal behaviors for both running forward and backflipping, but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 609 + ], + "score": 1.0, + "content": "also bi-modal behaviors; during a single rollout running forward first, then backflipping. DT (yellow) tends to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "lean backflipping and fails to fit neither uni-modal nor bi-modal ones. In (b), CDT successfully handles the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 617, + 489, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 489, + 631 + ], + "score": 1.0, + "content": "trajectories unseen during training, while DT seems to output covering behaviors over the dataset support.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + } + ], + "index": 30.0 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 411, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 413, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 413, + 659 + ], + "score": 1.0, + "content": "6.2.2 DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Generalization to unknown target demonstrations or tasks has been actively investigated in meta or", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "one-shot RL/IL literature (Duan et al., 2016; Wang et al., 2016). To verify the generalization of CDT", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "to diverse behaviors, we adopt the cheetah-velocity task; a popular task in meta RL/IL (Rakelly et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 507, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 711 + ], + "score": 1.0, + "content": "2019; Li et al., 2021; Fakoor et al., 2020), where the cheetah tries to run with the specified velocity.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We prepare 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals, and hold", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 412, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 122, + 733 + ], + "score": 1.0, + "content": "out", + "type": "text" + }, + { + "bbox": [ + 122, + 720, + 180, + 732 + ], + "score": 0.87, + "content": "\\bar { \\{ 0 . 5 , 1 . 5 , 2 . 5 \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 720, + 412, + 733 + ], + "score": 1.0, + "content": "as a test set. See Appendix E.5 for the dataset generation.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 364, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "score": 1.0, + "content": "6.2 GENERALIZATION TO UNSEEN TARGET DISTRIBUTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "The performance of offline methods in RL is often restricted by the coverage or quality of datasets.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "score": 1.0, + "content": "While we demonstrate CDT can perform offline state-marginal matching to unseen target distributions", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 137 + ], + "score": 1.0, + "content": "in Section 6.1, the standard offline datasets might not be diverse enough to observe the generalization", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "since those are collected by single-task reward-maximization policies. To test the generalization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "score": 1.0, + "content": "to more diverse behaviors, we investigate the following tasks: (1) z-velocity distribution matching", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 506, + 169 + ], + "score": 1.0, + "content": "with synthesized bi-modal behavior and (2) cheetah-velocity matching problem from meta RL/IL", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 372, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 372, + 181 + ], + "score": 1.0, + "content": "literature (Rakelly et al., 2019; Li et al., 2021; Fakoor et al., 2020)", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 102, + 506, + 181 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 192, + 358, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 360, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 360, + 205 + ], + "score": 1.0, + "content": "6.2.1 SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "To generate diverse behaviors for z-axis, we obtain the expert cheetah that backflips towards -x", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "direction by modifying reward function (see Appendix E.4 for the details). 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Due to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 507, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 507, + 329 + ], + "score": 1.0, + "content": "the difficulty for RL algorithms to maximize Eq. 4, FOCAL struggles to solve the offline multi-task", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 327, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 506, + 339 + ], + "score": 1.0, + "content": "SMM, even though FOCAL uses the same context embedding as Meta-BC. The bi-modal behavior", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 493, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 493, + 349 + ], + "score": 1.0, + "content": "learned by CDT can be seen at https://sites.google.com/view/generalizeddt.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 281, + 507, + 349 + ] + }, + { + "type": "table", + "bbox": [ + 168, + 360, + 442, + 416 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 168, + 360, + 442, + 416 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 360, + 442, + 416 + ], + "spans": [ + { + "bbox": [ + 168, + 360, + 442, + 416 + ], + "score": 0.959, + "html": "
MethodUni-modalBi-modalAverage
Categorical DT1.562 ± 0.6321.625 ± 0.9021.594
DT2.676 ±0.7652.703 ±0.7032.690
Meta-BC1.519 ±0.6961.655 ± 0.9901.587
FOCAL (Li et al., 2021)2.203 士 1.0501.983 士 0.9482.093
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Categorical DT matches both uni- and bi-modal trajectories better than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 438, + 457, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 457, + 450 + ], + "score": 1.0, + "content": "DT and FOCAL, and is comparable to Meta-BC that originally aims to solve multi-task problem.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "image", + "bbox": [ + 106, + 465, + 502, + 576 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 465, + 502, + 576 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 465, + 502, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 502, + 576 + ], + "score": 0.967, + "type": "image", + "image_path": "6122f8e9ffbb9010be05cd17b1ddf24787e73452368349e57817055b17b8fe1b.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 106, + 465, + 502, + 502.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 106, + 502.0, + 502, + 539.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 106, + 539.0, + 502, + 576.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 578, + 506, + 628 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 578, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 506, + 589 + ], + "score": 1.0, + "content": "Figure 2: (a) Z-Velocity and (b) Unseen Cheetah-Velocity results. Blue histograms represent target distributions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "In (a), CDT (red) can match not only uni-modal behaviors for both running forward and backflipping, but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 598, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 506, + 609 + ], + "score": 1.0, + "content": "also bi-modal behaviors; during a single rollout running forward first, then backflipping. DT (yellow) tends to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "lean backflipping and fails to fit neither uni-modal nor bi-modal ones. In (b), CDT successfully handles the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 617, + 489, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 489, + 631 + ], + "score": 1.0, + "content": "trajectories unseen during training, while DT seems to output covering behaviors over the dataset support.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + } + ], + "index": 30.0 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 411, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 413, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 413, + 659 + ], + "score": 1.0, + "content": "6.2.2 DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 664, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 506, + 678 + ], + "score": 1.0, + "content": "Generalization to unknown target demonstrations or tasks has been actively investigated in meta or", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "one-shot RL/IL literature (Duan et al., 2016; Wang et al., 2016). To verify the generalization of CDT", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "to diverse behaviors, we adopt the cheetah-velocity task; a popular task in meta RL/IL (Rakelly et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 507, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 507, + 711 + ], + "score": 1.0, + "content": "2019; Li et al., 2021; Fakoor et al., 2020), where the cheetah tries to run with the specified velocity.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We prepare 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals, and hold", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 412, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 122, + 733 + ], + "score": 1.0, + "content": "out", + "type": "text" + }, + { + "bbox": [ + 122, + 720, + 180, + 732 + ], + "score": 0.87, + "content": "\\bar { \\{ 0 . 5 , 1 . 5 , 2 . 5 \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 720, + 412, + 733 + ], + "score": 1.0, + "content": "as a test set. See Appendix E.5 for the dataset generation.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 664, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 137, + 126, + 474, + 183 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 82, + 506, + 117 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Table 4 and Figure 2 (b) reveal that CDT outperforms DT or FOCAL, and is slightly better than Meta-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "BC through the distribution matching evaluation, which implies CDT could solve offline multi-task", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 489, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 489, + 117 + ], + "score": 1.0, + "content": "SMM generalizing to the unknown target trajectories, given sufficiently diverse offline datasets.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 137, + 126, + 474, + 183 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 126, + 474, + 183 + ], + "spans": [ + { + "bbox": [ + 137, + 126, + 474, + 183 + ], + "score": 0.977, + "html": "
Methodx-vel: 0.5 x-vel: 1.5x-vel: 2.5Average
Categorical DT0.060± 0.0260.211 ± 0.0220.149± 0.1100.140
DT1.197 ± 0.2270.533 ± 0.1050.861 ± 0.2470.864
Meta-BC0.150 ± 0.0690.152 ± 0.1270.167 ± 0.0550.156
FOCAL (Li et al., 2021)0.472 ± 0.0050.952 ± 0.0730.346 ± 0.1860.590
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CDT", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "score": 1.0, + "content": "successfully deals with unseen target velocities well, outperforming DT and FOCAL, and is slightly better than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 214, + 314, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 314, + 227 + ], + "score": 1.0, + "content": "Meta-BC through the offline multi-task SMM evaluation.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 106, + 244, + 360, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 361, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 361, + 257 + ], + "score": 1.0, + "content": "6.3 ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 310, + 277 + ], + "score": 1.0, + "content": "Lastly, we investigate BDT for offline multi-task", + "type": "text" + }, + { + "bbox": [ + 310, + 265, + 321, + 275 + ], + "score": 0.42, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 264, + 505, + 277 + ], + "score": 1.0, + "content": ", where we do not observe rewards nor state", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "features explicitly. Instead, target full state trajectories that the agents are expected to mimic is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 302, + 299 + ], + "score": 1.0, + "content": "given. We compare BDT against parameterized", + "type": "text" + }, + { + "bbox": [ + 303, + 287, + 311, + 297 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "variants; DT-AE, -CPC, and -E2E discussed in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 311 + ], + "score": 1.0, + "content": "Section 5.3. 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Similar to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "discussion in Yang & Nachum (2021), DT-CPC sometimes fails to obtain sufficient representation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "for imitation. While even simple approaches, DT-AE and DT-AE (frozen), show positive results", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 460, + 425 + ], + "score": 1.0, + "content": "compared to no-context BC baselines (in Table 2), BDT outperforms all other learned", + "type": "text" + }, + { + "bbox": [ + 461, + 414, + 469, + 423 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "variants", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 441, + 436 + ], + "score": 1.0, + "content": "or Meta-BC and is comparable to CDT or DT (also in Table 2) with longer input", + "type": "text" + }, + { + "bbox": [ + 441, + 424, + 477, + 434 + ], + "score": 0.86, + "content": "\\mathrm { \\Delta } N = 5 0 \\mathrm { \\Omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "). This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "implies that even though we don’t assume the state-feature specification, aggregator choice in GDT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "with minimal architectural changes may solve the offline distribution matching problem efficiently.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 425, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 425, + 469 + ], + "score": 1.0, + "content": "We leave more sophisticated objectives or architectural changes as future work.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "table", + "bbox": [ + 113, + 478, + 498, + 544 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 478, + 498, + 544 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 113, + 478, + 498, + 544 + ], + "spans": [ + { + "bbox": [ + 113, + 478, + 498, + 544 + ], + "score": 0.978, + "html": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610 ± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710± 0.5721.591
DT-AE (joint)8.643 ± 0.6792.260± 0.6905.451 ± 3.2641.255±0.3630.649 ± 0.2410.952 ± 0.4322.104±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-CPC (joint)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575 ± 0.0320.096± 0.0280.335 ± 0.2410.949 ± 0.4840.412 ± 0.3780.680± 0.5101.410
DT-E2E8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ± 0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE (frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385 ± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (frozen)3.489 ± 1.1592.543 ± 0.9033.016 ± 1.1410.631 ±0.0910.171 ± 0.1300.401 ± 0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(N=20)1.592 ± 0.2011.208 ± 1.8541.400 ± 1.3330.318±0.0930.081 ± 0.0130.200± 0.1360.196 ±0.0310.392 ± 0.1840.294± 0.1640.631
BDT(N=50)0.840 ± 0.0631.223 ± 1.8281.031 ± 1.3070.142 ± 0.0100.098 ± 0.0250.120 ± 0.0290.192 ± 0.0510.163 ± 0.0270.178 ± 0.0430.443
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While even simple", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "auto-encoder regularizers sometimes work well, BDT with longer contexts seems to outperform other strategies", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 482, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 443, + 577 + ], + "score": 1.0, + "content": "and is comparable to CDT or DT (in Table 2). See Appendix E.6 for the full results including", + "type": "text" + }, + { + "bbox": [ + 443, + 566, + 478, + 576 + ], + "score": 0.91, + "content": "m = 1 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 565, + 482, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 597, + 195, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 198, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 198, + 614 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "We provide a unified perspective on a wide range of hindsight algorithms, and generalize the problem", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "formulation as hindsight information matching (HIM). Inspired by recent successes in RL as sequence", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "modeling, we propose Generalized Decision Transformer (GDT) which includes DT, Categorical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "DT, and Bi-directional DT as special cases, and is applicable to any HIM with proper choices", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 118, + 678 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 666, + 126, + 676 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "and aggregator. We show how Categorical DT, a minor modification of DT, enables the first", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "effective offline state-marginal matching algorithm and propose new benchmark tasks for this problem", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 686, + 507, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 507, + 701 + ], + "score": 1.0, + "content": "class. We also demonstrate the effectiveness of Bi-directional DT as a one-shot imitation learner,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "significantly outperforming simple variants based on DT. We hope our proposed HIM and GDT", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "frameworks shed new perspectives on hindsight algorithms and the applicability of sequence modeling", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 404, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 404, + 732 + ], + "score": 1.0, + "content": "to much broader classes of RL problems beyond classic reward-based RL.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 137, + 126, + 474, + 183 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 82, + 506, + 117 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Table 4 and Figure 2 (b) reveal that CDT outperforms DT or FOCAL, and is slightly better than Meta-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "BC through the distribution matching evaluation, which implies CDT could solve offline multi-task", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 489, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 489, + 117 + ], + "score": 1.0, + "content": "SMM generalizing to the unknown target trajectories, given sufficiently diverse offline datasets.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 137, + 126, + 474, + 183 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 126, + 474, + 183 + ], + "spans": [ + { + "bbox": [ + 137, + 126, + 474, + 183 + ], + "score": 0.977, + "html": "
Methodx-vel: 0.5 x-vel: 1.5x-vel: 2.5Average
Categorical DT0.060± 0.0260.211 ± 0.0220.149± 0.1100.140
DT1.197 ± 0.2270.533 ± 0.1050.861 ± 0.2470.864
Meta-BC0.150 ± 0.0690.152 ± 0.1270.167 ± 0.0550.156
FOCAL (Li et al., 2021)0.472 ± 0.0050.952 ± 0.0730.346 ± 0.1860.590
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CDT", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "score": 1.0, + "content": "successfully deals with unseen target velocities well, outperforming DT and FOCAL, and is slightly better than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 214, + 314, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 314, + 227 + ], + "score": 1.0, + "content": "Meta-BC through the offline multi-task SMM evaluation.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 106, + 244, + 360, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 361, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 361, + 257 + ], + "score": 1.0, + "content": "6.3 ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 310, + 277 + ], + "score": 1.0, + "content": "Lastly, we investigate BDT for offline multi-task", + "type": "text" + }, + { + "bbox": [ + 310, + 265, + 321, + 275 + ], + "score": 0.42, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 264, + 505, + 277 + ], + "score": 1.0, + "content": ", where we do not observe rewards nor state", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "features explicitly. Instead, target full state trajectories that the agents are expected to mimic is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 302, + 299 + ], + "score": 1.0, + "content": "given. We compare BDT against parameterized", + "type": "text" + }, + { + "bbox": [ + 303, + 287, + 311, + 297 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "variants; DT-AE, -CPC, and -E2E discussed in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 296, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 311 + ], + "score": 1.0, + "content": "Section 5.3. We consider three strategies to train the encoder for DT-AE and -CPC: training with only", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 307, + 507, + 322 + ], + "score": 1.0, + "content": "unsupervised loss, training with unsupervised and DT’s supervised loss jointly (called as “joint”),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "and pre-training with only unsupervised loss and freezing the weights during DT training (called as", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 253, + 343 + ], + "score": 1.0, + "content": "“frozen”). 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Similar to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "discussion in Yang & Nachum (2021), DT-CPC sometimes fails to obtain sufficient representation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "for imitation. 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This", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "implies that even though we don’t assume the state-feature specification, aggregator choice in GDT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "with minimal architectural changes may solve the offline distribution matching problem efficiently.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 425, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 425, + 469 + ], + "score": 1.0, + "content": "We leave more sophisticated objectives or architectural changes as future work.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 357, + 506, + 469 + ] + }, + { + "type": "table", + "bbox": [ + 113, + 478, + 498, + 544 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 478, + 498, + 544 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 113, + 478, + 498, + 544 + ], + "spans": [ + { + "bbox": [ + 113, + 478, + 498, + 544 + ], + "score": 0.978, + "html": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610 ± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710± 0.5721.591
DT-AE (joint)8.643 ± 0.6792.260± 0.6905.451 ± 3.2641.255±0.3630.649 ± 0.2410.952 ± 0.4322.104±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-CPC (joint)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575 ± 0.0320.096± 0.0280.335 ± 0.2410.949 ± 0.4840.412 ± 0.3780.680± 0.5101.410
DT-E2E8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ± 0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE (frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385 ± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (frozen)3.489 ± 1.1592.543 ± 0.9033.016 ± 1.1410.631 ±0.0910.171 ± 0.1300.401 ± 0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(N=20)1.592 ± 0.2011.208 ± 1.8541.400 ± 1.3330.318±0.0930.081 ± 0.0130.200± 0.1360.196 ±0.0310.392 ± 0.1840.294± 0.1640.631
BDT(N=50)0.840 ± 0.0631.223 ± 1.8281.031 ± 1.3070.142 ± 0.0100.098 ± 0.0250.120 ± 0.0290.192 ± 0.0510.163 ± 0.0270.178 ± 0.0430.443
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While even simple", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "auto-encoder regularizers sometimes work well, BDT with longer contexts seems to outperform other strategies", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 565, + 482, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 443, + 577 + ], + "score": 1.0, + "content": "and is comparable to CDT or DT (in Table 2). See Appendix E.6 for the full results including", + "type": "text" + }, + { + "bbox": [ + 443, + 566, + 478, + 576 + ], + "score": 0.91, + "content": "m = 1 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 565, + 482, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 597, + 195, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 198, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 198, + 614 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "We provide a unified perspective on a wide range of hindsight algorithms, and generalize the problem", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "formulation as hindsight information matching (HIM). 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We hope our proposed HIM and GDT", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 724 + ], + "score": 1.0, + "content": "frameworks shed new perspectives on hindsight algorithms and the applicability of sequence modeling", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 404, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 404, + 732 + ], + "score": 1.0, + "content": "to much broader classes of RL problems beyond classic reward-based RL.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 622, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 194, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 194, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 194, + 94 + ], + "score": 1.0, + "content": "ETHICS STATEMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 504, + 124 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 504, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 504, + 115 + ], + "score": 1.0, + "content": "Since this paper mainly focuses on the reinterpretation of hindsight RL algorithms and experiments", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 400, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 400, + 125 + ], + "score": 1.0, + "content": "on existing benchmark datasets, we believe there are no ethical concerns.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 136, + 241, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 137, + 242, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 242, + 148 + ], + "score": 1.0, + "content": "REPRODUCIBILITY STATEMENT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 156, + 506, + 189 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 506, + 168 + ], + "score": 1.0, + "content": "We share our codes to ensure the reproductivity. The details of hyperparameters are described in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 506, + 179 + ], + "score": 1.0, + "content": "Appendix A and B. The detailed settings of experiments are described in the Section 6, Appendix E,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 177, + 411, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 411, + 190 + ], + "score": 1.0, + "content": "and G. 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(2021a) (https://github.com/kzl/", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 482, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 482, + 168 + ], + "score": 1.0, + "content": "decision-transformer). We follow the most of hyperparameters as they did (Table 6).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "table", + "bbox": [ + 138, + 177, + 472, + 343 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 138, + 177, + 472, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 177, + 472, + 343 + ], + "spans": [ + { + "bbox": [ + 138, + 177, + 472, + 343 + ], + "score": 0.982, + "html": "
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Nonlinearity functionReLU
Batch size64
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Dropout0.1
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Grad norm clip0.25
Weight decay1e-4
Learning rate decayLinear warmup for first 1OOk training steps
Training(Gradient) steps Number of bins forcategorical distribution1M
31
Encoder size (for DT-X)2 layer MLP,(128,128)
Coefficient of unsupervised loss0.1
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(2021a).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + } + ], + "index": 7.0 + }, + { + "type": "title", + "bbox": [ + 108, + 384, + 250, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 252, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 252, + 399 + ], + "score": 1.0, + "content": "B DETAILS OF BASELINES", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "While, to the best of our knowledge, there are no prior work to tackle the offline state-marginal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "matching problem, we can regard meta or one-shot imitation learning as the methods solving similar", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "problem (see Appendix D for further discussion). In this work, we choose Meta-BC and FOCAL (Li", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 443, + 401, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 401, + 455 + ], + "score": 1.0, + "content": "et al., 2021), a metric-based offline meta RL method, as decent baselines.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 507, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 507, + 479 + ], + "score": 1.0, + "content": "FOCAL FOCAL is a offline meta RL method combining BRAC (Wu et al., 2019) with metric-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "based approach. It utilizes deterministic context encoder trained with inverse-power distance metric", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "losses and detached from Bellman backup gradients. We follow the hyperparameters in the official", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "implementation (https://github.com/LanqingLi1993/FOCAL-ICLR). Deterministic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "context encoder takes single-step state-action-reward tuple as a context for task inference. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "parameterize it with (200, 200, 200)-layers MLP. We train deterministic context encoder with 100000", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "iteration first, and then train BRAC agent with task-conditioned policy and value functions with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "100000 iteration. We use (256, 256, 256)-layers MLPs for policy and value network, and set batch", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 554, + 155, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 155, + 566 + ], + "score": 1.0, + "content": "size to 256.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 575, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 507, + 592 + ], + "score": 1.0, + "content": "Meta-BC We also use metric-based Meta-BC (Duan et al., 2017; Dasari & Gupta, 2020) as a strong", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "baseline method for offline multi-task SMM. We adapt deterministic context encoder from FOCAL", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "to infer the task. We train Meta-BC in the same way as FOCAL just replacing BRAC to BC. The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 612, + 292, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 292, + 623 + ], + "score": 1.0, + "content": "objective is mean-squared error minimization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "Throughout our experiments, Meta-BC shows better results than FOCAL, because RL algorithms", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "often struggle to optimize Eq. 4 for distribution matching and FOCAL tries to maximize the task", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "reward, which is not necessarily required for distribution matching problem. In addition, BC often", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 661, + 275, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 275, + 672 + ], + "score": 1.0, + "content": "converges faster than offline RL methods.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 161, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 107, + 468, + 121 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 469, + 122 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 469, + 122 + ], + "score": 1.0, + "content": "A HYPER-PARAMETER OF GENERALIZED DECISION TRANSFORMERS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 132, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "We implement Categorical Decision Transformer and Bi-directional Decision Transformer, built", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "upon the official codebase released by Chen et al. (2021a) (https://github.com/kzl/", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 482, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 482, + 168 + ], + "score": 1.0, + "content": "decision-transformer). We follow the most of hyperparameters as they did (Table 6).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 132, + 505, + 168 + ] + }, + { + "type": "table", + "bbox": [ + 138, + 177, + 472, + 343 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 138, + 177, + 472, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 177, + 472, + 343 + ], + "spans": [ + { + "bbox": [ + 138, + 177, + 472, + 343 + ], + "score": 0.982, + "html": "
HyperparameterValue
Number of layers Number of attention heads3 1 128
Embedding dimension
Nonlinearity functionReLU
Batch size64
Context length N20
Dropout0.1
Learning rate1e-4
Grad norm clip0.25
Weight decay1e-4
Learning rate decayLinear warmup for first 1OOk training steps
Training(Gradient) steps Number of bins forcategorical distribution1M
31
Encoder size (for DT-X)2 layer MLP,(128,128)
Coefficient of unsupervised loss0.1
", + "type": "table", + "image_path": "b931d7a829874247dba1034e5b6d8746c87d560f15db1d1482dae7e2dd42af93.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 138, + 177, + 472, + 232.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 138, + 232.33333333333334, + 472, + 287.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 138, + 287.6666666666667, + 472, + 343.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 150, + 351, + 461, + 362 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 350, + 462, + 364 + ], + "spans": [ + { + "bbox": [ + 149, + 350, + 462, + 364 + ], + "score": 1.0, + "content": "Table 6: List of hyperparameters for DT, CDT and BDT. We refer Chen et al. (2021a).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + } + ], + "index": 7.0 + }, + { + "type": "title", + "bbox": [ + 108, + 384, + 250, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 252, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 252, + 399 + ], + "score": 1.0, + "content": "B DETAILS OF BASELINES", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "While, to the best of our knowledge, there are no prior work to tackle the offline state-marginal", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "matching problem, we can regard meta or one-shot imitation learning as the methods solving similar", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "problem (see Appendix D for further discussion). In this work, we choose Meta-BC and FOCAL (Li", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 443, + 401, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 401, + 455 + ], + "score": 1.0, + "content": "et al., 2021), a metric-based offline meta RL method, as decent baselines.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 410, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 507, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 507, + 479 + ], + "score": 1.0, + "content": "FOCAL FOCAL is a offline meta RL method combining BRAC (Wu et al., 2019) with metric-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "based approach. It utilizes deterministic context encoder trained with inverse-power distance metric", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "losses and detached from Bellman backup gradients. We follow the hyperparameters in the official", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "implementation (https://github.com/LanqingLi1993/FOCAL-ICLR). Deterministic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "context encoder takes single-step state-action-reward tuple as a context for task inference. We", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "parameterize it with (200, 200, 200)-layers MLP. We train deterministic context encoder with 100000", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "iteration first, and then train BRAC agent with task-conditioned policy and value functions with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "100000 iteration. We use (256, 256, 256)-layers MLPs for policy and value network, and set batch", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 554, + 155, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 155, + 566 + ], + "score": 1.0, + "content": "size to 256.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 465, + 507, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 575, + 507, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 507, + 592 + ], + "score": 1.0, + "content": "Meta-BC We also use metric-based Meta-BC (Duan et al., 2017; Dasari & Gupta, 2020) as a strong", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "baseline method for offline multi-task SMM. We adapt deterministic context encoder from FOCAL", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "to infer the task. We train Meta-BC in the same way as FOCAL just replacing BRAC to BC. The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 612, + 292, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 292, + 623 + ], + "score": 1.0, + "content": "objective is mean-squared error minimization.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 575, + 507, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "Throughout our experiments, Meta-BC shows better results than FOCAL, because RL algorithms", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "often struggle to optimize Eq. 4 for distribution matching and FOCAL tries to maximize the task", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "reward, which is not necessarily required for distribution matching problem. In addition, BC often", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 661, + 275, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 275, + 672 + ], + "score": 1.0, + "content": "converges faster than offline RL methods.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 627, + 506, + 672 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 259, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 261, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 261, + 96 + ], + "score": 1.0, + "content": "C DETAILS OF EVALUATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "Throughout the paper, we evaluate both CDT and BDT from a distribution matching perspective", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "by formulating them as offline multi-task SMM or offline multi-task IL problem. However, the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "current distribution matching research in RL (Lee et al., 2020; Ghasemipour et al., 2020; Gu et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 151 + ], + "score": 1.0, + "content": "2021) has been suffered from the lack of quantitative metrics for evaluation (while standard RL", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 506, + 162 + ], + "score": 1.0, + "content": "or BC are typically evaluated on the task reward performance). As summarized in Table 7, prior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "works (Lee et al., 2020; Ghasemipour et al., 2020) evaluate the SMM performance by qualitative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 283, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 283, + 184 + ], + "score": 1.0, + "content": "density visualization using rollout particles.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 506, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "Although the distance between state-marginal and target distribution seems the most intuitive and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "suitable metric to quantify the performance of distribution matching algorithms, it is often intractable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 222 + ], + "score": 1.0, + "content": "to measure such distance analytically because both state-marginal and target distribution can be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "non-parametric; we cannot access their densities, and only their samples are available. While a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "concurrent work (Gu et al., 2021) tackles this problem by leveraging sample-based energy distance", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "score": 1.0, + "content": "estimation, we introduce a single SMM-inspired evaluation for both offline multi-task SMM and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "offline multi-task IL via estimating Wasserstein-1 distance between empirical categorical distributions.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Since the discretization of low-dimensional features, such as reward, have success in many RL", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "methods (Bellemare et al., 2017; Dabney et al., 2018; 2020; Furuta et al., 2021b), quantification", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "of distribution matching performance by Wasserstein-1 distance could be reliable evaluations (in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "addition Wasserstein-1 distance is symmetric, different from KL distance that is asymmetric). We note", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 506, + 321 + ], + "score": 1.0, + "content": "that due to the equivalence between inverse RL and SMM methods, the performance of distribution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "matching methods may be measured via task reward when assuming the accessivity to the expert", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "trajectories (Ho & Ermon, 2016; Fu et al., 2018; Kostrikov et al., 2019). 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EvaluationTypeReference
Density VisualizationQualitativeLee et al. (2020); Ghasemipour et al. (2020)
Energy DistanceQuantitativeGu et al. (2021)
Task RewardQuantitativeHo& Ermon 1 (2016); Fu et al. (2018), etc.
Wasserstein-1DistanceQuantitativeOurs
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As summarized in Table 7, prior", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "works (Lee et al., 2020; Ghasemipour et al., 2020) evaluate the SMM performance by qualitative", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 283, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 283, + 184 + ], + "score": 1.0, + "content": "density visualization using rollout particles.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 105, + 506, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 188, + 506, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "Although the distance between state-marginal and target distribution seems the most intuitive and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "suitable metric to quantify the performance of distribution matching algorithms, it is often intractable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 222 + ], + "score": 1.0, + "content": "to measure such distance analytically because both state-marginal and target distribution can be", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "non-parametric; we cannot access their densities, and only their samples are available. While a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 505, + 245 + ], + "score": 1.0, + "content": "concurrent work (Gu et al., 2021) tackles this problem by leveraging sample-based energy distance", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 255 + ], + "score": 1.0, + "content": "estimation, we introduce a single SMM-inspired evaluation for both offline multi-task SMM and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "offline multi-task IL via estimating Wasserstein-1 distance between empirical categorical distributions.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "Since the discretization of low-dimensional features, such as reward, have success in many RL", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 289 + ], + "score": 1.0, + "content": "methods (Bellemare et al., 2017; Dabney et al., 2018; 2020; Furuta et al., 2021b), quantification", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "of distribution matching performance by Wasserstein-1 distance could be reliable evaluations (in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "addition Wasserstein-1 distance is symmetric, different from KL distance that is asymmetric). 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EvaluationTypeReference
Density VisualizationQualitativeLee et al. (2020); Ghasemipour et al. (2020)
Energy DistanceQuantitativeGu et al. (2021)
Task RewardQuantitativeHo& Ermon 1 (2016); Fu et al. (2018), etc.
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MethodProblemTrainTestDemo
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Zhao et al. (2021)Offline Meta RLofflineonlinefew-shot
Dorfman et al. (2021)Offline Meta RLofflineonlinefew-shot
Mitchell et al. (2021)Offline Meta RLofflineofflinefew-shot
Li et al. (2021)Offline Meta RLofflineofflinefew-shot
Yu et al. (2019)Meta ILonlineonlinefew-shot
Xu et al. (2019)Meta ILonlineonlinefew-shot
Ghasemipour et al. (2019)Meta ILonlineofflinefew-shot
Finn et al. (2017)Meta ILofflineofflineone-shot
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MethodProblemTrainTestDemo
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Zhao et al. (2021)Offline Meta RLofflineonlinefew-shot
Dorfman et al. (2021)Offline Meta RLofflineonlinefew-shot
Mitchell et al. (2021)Offline Meta RLofflineofflinefew-shot
Li et al. (2021)Offline Meta RLofflineofflinefew-shot
Yu et al. (2019)Meta ILonlineonlinefew-shot
Xu et al. (2019)Meta ILonlineonlinefew-shot
Ghasemipour et al. (2019)Meta ILonlineofflinefew-shot
Finn et al. (2017)Meta ILofflineofflineone-shot
Duan et al. (2017)Meta ILoffline(no-adaptation)one-shot
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Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.674 ± 0.2131.002 ± 1.4580.838 ± 1.0540.159 ± 0.0850.064 ± 0.0170.111 ± 0.0770.095 ± 0.0170.114 ± 0.0370.105 ± 0.0300.351
DT0.652 ± 0.3191.039 ± 1.5480.846 ± 1.1340.227 ± 0.1190.091 ± 0.0350.159 ± 0.1110.056 ± 0.0150.626 ± 0.4950.341 ± 0.4520.448
BC (no-context)3.240± 0.5592.880 ± 0.6143.060 ± 0.6140.597 ± 0.0560.119 ± 0.0670.358 ± 0.2470.977 ± 0.5010.431 ± 0.3960.704 ± 0.5281.374
Meta-BC0.839 ±0.6820.830 ± 1.1300.835 ± 0.9330.803 ± 0.5050.134 ± 0.0560.468 ± 0.4910.113 ± 0.0851.441 ± 1.1130.777 ± 1.0320.693
FOCAL (Li et al.,2021)1.623 ± 0.5011.115 ± 1.5341.369 ± 0.5161.463 ± 0.4720.492 ± 0.3840.977 ± 0.6491.584 ± 0.5700.604 ± 0.4211.094 ± 0.7011.147
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Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.674 ± 0.2131.002 ± 1.4580.838 ± 1.0540.159 ± 0.0850.064 ± 0.0170.111 ± 0.0770.095 ± 0.0170.114 ± 0.0370.105 ± 0.0300.351
DT0.652 ± 0.3191.039 ± 1.5480.846 ± 1.1340.227 ± 0.1190.091 ± 0.0350.159 ± 0.1110.056 ± 0.0150.626 ± 0.4950.341 ± 0.4520.448
BC (no-context)3.240± 0.5592.880 ± 0.6143.060 ± 0.6140.597 ± 0.0560.119 ± 0.0670.358 ± 0.2470.977 ± 0.5010.431 ± 0.3960.704 ± 0.5281.374
Meta-BC0.839 ±0.6820.830 ± 1.1300.835 ± 0.9330.803 ± 0.5050.134 ± 0.0560.468 ± 0.4910.113 ± 0.0851.441 ± 1.1130.777 ± 1.0320.693
FOCAL (Li et al.,2021)1.623 ± 0.5011.115 ± 1.5341.369 ± 0.5161.463 ± 0.4720.492 ± 0.3840.977 ± 0.6491.584 ± 0.5700.604 ± 0.4211.094 ± 0.7011.147
", + "type": "table", + "image_path": "d024bea0c8817fac97f37a8d384a81db42a17e85280433d1ef413f6a31373a03.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 108, + 400, + 504, + 414.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 108, + 414.3333333333333, + 504, + 428.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 108, + 428.66666666666663, + 504, + 442.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "Table 9: Quantitative evaluation of reward distribution matching via measuring Wasserstein-1 distance between", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "the rollout and target distributions. We compare Categorical DT and DT. Since it can capture the multi-modal", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 472, + 421, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 421, + 483 + ], + "score": 1.0, + "content": "nature of target distributions, CDT matches the distribution better than the original DT.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 452, + 505, + 483 + ] + }, + { + "type": "image", + "bbox": [ + 110, + 498, + 498, + 631 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 498, + 498, + 631 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 498, + 498, + 631 + ], + "spans": [ + { + "bbox": [ + 110, + 498, + 498, + 631 + ], + "score": 0.969, + "type": "image", + "image_path": "c15ec828e747cb1d7499910bbb5901116cf3aecc7338dafa323749f7c6941389.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 110, + 498, + 498, + 542.3333333333334 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 110, + 542.3333333333334, + 498, + 586.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 110, + 586.6666666666667, + 498, + 631.0000000000001 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 640, + 505, + 680 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "Figure 4: (a) Reward and (b) state-feature (x-velocity) distribution matching in halfcheetah (top), hopper", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 504, + 660 + ], + "score": 1.0, + "content": "(middle), and walker2d (bottom). The left two examples are the distributions from the best trajectories, and right", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "two are the distributions from the middle trajectories in the held-out test set. The rollout distributions of CDT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 669, + 336, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 336, + 682 + ], + "score": 1.0, + "content": "(red) match the target distributions (blue) very well in all cases.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + } + ], + "index": 25.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 297, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "score": 1.0, + "content": "E.2 EVALUATION ON TASK PERFORMANCE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "score": 1.0, + "content": "While in this paper we focus on the evaluation with the SMM-inspired distribution matching objective,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "such as Wasserstein-1 distance, we here provide the evaluation on the task rewards. Table 10 shows", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "that DT seems consistently better methods than Categorical DT on the task rewards evaluations, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 340, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 340, + 148 + ], + "score": 1.0, + "content": "achieves similar performances to the held-out trajectories.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 113, + 158, + 498, + 194 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 158, + 498, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 158, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 498, + 194 + ], + "score": 0.963, + "html": "
Methodhalfcheetahhopperwalker2d
ExpertMediumExpertMediumExpertMedium
Categorical DT10476.746± 218.9575782.557 ± 1666.7162614.559 ± 657.7221518.757 ± 63.0624907.475 ± 11.7113264.585 ± 209.905
DT10500.273 ± 312.7784985.518 ± 67.3222113.207 ± 807.2461528.743± 30.7804965.024 ± 11.8144055.039 ± 849.109
Held-out11146.200 ± 59.0705237.348 ± 37.9703741.854 ± 7.2231600.196± 0.7724995.553 ± 5.4133801.313 ± 0.994
", + "type": "table", + "image_path": "5d74301f949071f3d8b1fea32c3f7aaed2b1d5ee6199e60a001f50f8d8d9eb47.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 113, + 158, + 498, + 170.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 170.0, + 498, + 182.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 113, + 182.0, + 498, + 194.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Table 10: Evaluation on the task rewards; conditioning on the held-out trajectories as done in Section 6. We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 224 + ], + "score": 1.0, + "content": "compare the performance between Categorical DT, and DT. DT seems consistently better methods on the task", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 235 + ], + "score": 1.0, + "content": "rewards evaluations, and achieves similar performances to the held-out trajectories (averaged over 5 trajectories).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 107, + 263, + 334, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 334, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 334, + 276 + ], + "score": 1.0, + "content": "E.3 2D STATE-FEATURE DISTRIBUTION MATCHING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 506, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "We also consider two-dimensional state-features (xy-velocities) distribution matching in Ant-v3. Same", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "as 1D state-feature distribution matching in HalfCheetah, Hopper, and Walker2d-v3 (Section 6.1),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "we also use medium-expert(-v2) datasets from D4RL (Fu et al., 2020). We bin the state-features per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "dimension separately to reduce the dimension of the categorical distribution that CDT takes as input,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 507, + 340 + ], + "score": 1.0, + "content": "while in test-time we evaluate the performance with Wasserstein-1 metric on the joint distribution.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 264, + 351 + ], + "score": 1.0, + "content": "For DT, we compute the summation of", + "type": "text" + }, + { + "bbox": [ + 264, + 341, + 275, + 349 + ], + "score": 0.26, + "content": "\\mathbf { X } ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "and y-velocity each over trajectories and normalize them", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 465, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 465, + 361 + ], + "score": 1.0, + "content": "with the maximum horizon. DT feeds these two scalars as information statistics to match.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 507, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 507, + 379 + ], + "score": 1.0, + "content": "Table 11 reveals that CDT performs better even in the case of two-dimensional state-features dis-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "tributions, while DT doesn’t generalize to expert-quality trajectories. As shown in Figure 5, while", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "CDT could cope with the distribution shift between expert and medium target distribution, DT always", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "fits the medium one even if the expert trajectory is given as a target. CDT successfully scales to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 410, + 401, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 401, + 423 + ], + "score": 1.0, + "content": "offline multi-task SMM problem in the multi-dimensional feature spaces.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 169, + 433, + 441, + 500 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 169, + 433, + 441, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 169, + 433, + 441, + 500 + ], + "spans": [ + { + "bbox": [ + 169, + 433, + 441, + 500 + ], + "score": 0.964, + "html": "
Methodant
ExpertMediumAverage
Categorical DT0.797 ± 0.2160.244 ± 0.0630.521
DT1.714 ± 0.1210.260 ± 0.0670.987
Meta-BC1.295 ± 0.7080.351 ± 0.2050.823
FOCAL (Li et al., 2021)1.473 ± 0.8920.913 ± 0.4551.193
", + "type": "table", + "image_path": "fd666a38b459b1261377224f7e99a0c975abee8dfa7e3b62a8d7f62fdc9d885a.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 169, + 433, + 441, + 455.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 169, + 455.3333333333333, + 441, + 477.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 169, + 477.66666666666663, + 441, + 499.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 105, + 509, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "Table 11: Quantitative evaluation of 2D state-feature (xy-velocities) distribution matching, measuring", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 408, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 408, + 530 + ], + "score": 1.0, + "content": "Wasserstein-1 distance. CDT performs better even in the two-dimensional problem.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "image", + "bbox": [ + 131, + 547, + 480, + 694 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 131, + 547, + 480, + 694 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 131, + 547, + 480, + 694 + ], + "spans": [ + { + "bbox": [ + 131, + 547, + 480, + 694 + ], + "score": 0.972, + "type": "image", + "image_path": "be07d95b475dbda537b278fbe7b1d838a7ec935c089362e0b31dbcc6b73ffec8.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 131, + 547, + 480, + 596.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 131, + 596.0, + 480, + 645.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 131, + 645.0, + 480, + 694.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 703, + 505, + 724 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 703, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 505, + 714 + ], + "score": 1.0, + "content": "Figure 5: Visualization of 2D state-feature (xy-velocities) distribution matching, binning each dimension", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 712, + 492, + 725 + ], + "spans": [ + { + "bbox": [ + 105, + 712, + 492, + 725 + ], + "score": 1.0, + "content": "separately. Top row shows the results from an expert target trajectory, and bottom row from a medium one.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "index": 31.25 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 297, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "score": 1.0, + "content": "E.2 EVALUATION ON TASK PERFORMANCE", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 507, + 116 + ], + "score": 1.0, + "content": "While in this paper we focus on the evaluation with the SMM-inspired distribution matching objective,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 126 + ], + "score": 1.0, + "content": "such as Wasserstein-1 distance, we here provide the evaluation on the task rewards. Table 10 shows", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "that DT seems consistently better methods than Categorical DT on the task rewards evaluations, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 340, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 340, + 148 + ], + "score": 1.0, + "content": "achieves similar performances to the held-out trajectories.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 102, + 507, + 148 + ] + }, + { + "type": "table", + "bbox": [ + 113, + 158, + 498, + 194 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 158, + 498, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 158, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 113, + 158, + 498, + 194 + ], + "score": 0.963, + "html": "
Methodhalfcheetahhopperwalker2d
ExpertMediumExpertMediumExpertMedium
Categorical DT10476.746± 218.9575782.557 ± 1666.7162614.559 ± 657.7221518.757 ± 63.0624907.475 ± 11.7113264.585 ± 209.905
DT10500.273 ± 312.7784985.518 ± 67.3222113.207 ± 807.2461528.743± 30.7804965.024 ± 11.8144055.039 ± 849.109
Held-out11146.200 ± 59.0705237.348 ± 37.9703741.854 ± 7.2231600.196± 0.7724995.553 ± 5.4133801.313 ± 0.994
", + "type": "table", + "image_path": "5d74301f949071f3d8b1fea32c3f7aaed2b1d5ee6199e60a001f50f8d8d9eb47.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 113, + 158, + 498, + 170.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 170.0, + 498, + 182.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 113, + 182.0, + 498, + 194.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Table 10: Evaluation on the task rewards; conditioning on the held-out trajectories as done in Section 6. We", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 213, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 224 + ], + "score": 1.0, + "content": "compare the performance between Categorical DT, and DT. DT seems consistently better methods on the task", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 235 + ], + "score": 1.0, + "content": "rewards evaluations, and achieves similar performances to the held-out trajectories (averaged over 5 trajectories).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 203, + 506, + 235 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 263, + 334, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 334, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 334, + 276 + ], + "score": 1.0, + "content": "E.3 2D STATE-FEATURE DISTRIBUTION MATCHING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 284, + 506, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "We also consider two-dimensional state-features (xy-velocities) distribution matching in Ant-v3. Same", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "as 1D state-feature distribution matching in HalfCheetah, Hopper, and Walker2d-v3 (Section 6.1),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "we also use medium-expert(-v2) datasets from D4RL (Fu et al., 2020). We bin the state-features per", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "dimension separately to reduce the dimension of the categorical distribution that CDT takes as input,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 507, + 340 + ], + "score": 1.0, + "content": "while in test-time we evaluate the performance with Wasserstein-1 metric on the joint distribution.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 264, + 351 + ], + "score": 1.0, + "content": "For DT, we compute the summation of", + "type": "text" + }, + { + "bbox": [ + 264, + 341, + 275, + 349 + ], + "score": 0.26, + "content": "\\mathbf { X } ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "and y-velocity each over trajectories and normalize them", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 351, + 465, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 465, + 361 + ], + "score": 1.0, + "content": "with the maximum horizon. DT feeds these two scalars as information statistics to match.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 284, + 507, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 507, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 507, + 379 + ], + "score": 1.0, + "content": "Table 11 reveals that CDT performs better even in the case of two-dimensional state-features dis-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "tributions, while DT doesn’t generalize to expert-quality trajectories. As shown in Figure 5, while", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "CDT could cope with the distribution shift between expert and medium target distribution, DT always", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 413 + ], + "score": 1.0, + "content": "fits the medium one even if the expert trajectory is given as a target. CDT successfully scales to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 410, + 401, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 401, + 423 + ], + "score": 1.0, + "content": "offline multi-task SMM problem in the multi-dimensional feature spaces.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 365, + 507, + 423 + ] + }, + { + "type": "table", + "bbox": [ + 169, + 433, + 441, + 500 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 169, + 433, + 441, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 169, + 433, + 441, + 500 + ], + "spans": [ + { + "bbox": [ + 169, + 433, + 441, + 500 + ], + "score": 0.964, + "html": "
Methodant
ExpertMediumAverage
Categorical DT0.797 ± 0.2160.244 ± 0.0630.521
DT1.714 ± 0.1210.260 ± 0.0670.987
Meta-BC1.295 ± 0.7080.351 ± 0.2050.823
FOCAL (Li et al., 2021)1.473 ± 0.8920.913 ± 0.4551.193
", + "type": "table", + "image_path": "fd666a38b459b1261377224f7e99a0c975abee8dfa7e3b62a8d7f62fdc9d885a.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 169, + 433, + 441, + 455.3333333333333 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 169, + 455.3333333333333, + 441, + 477.66666666666663 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 169, + 477.66666666666663, + 441, + 499.99999999999994 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 105, + 509, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 523 + ], + "score": 1.0, + "content": "Table 11: Quantitative evaluation of 2D state-feature (xy-velocities) distribution matching, measuring", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 519, + 408, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 408, + 530 + ], + "score": 1.0, + "content": "Wasserstein-1 distance. 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(2021a), and then collected 500 trajectories", + "type": "text" + }, + { + "bbox": [ + 372, + 153, + 404, + 163 + ], + "score": 0.84, + "content": "\\times ~ 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 153, + 505, + 164 + ], + "score": 1.0, + "content": "time steps. We combined", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 215, + 177 + ], + "score": 1.0, + "content": "them with 500 trajectories", + "type": "text" + }, + { + "bbox": [ + 216, + 164, + 248, + 174 + ], + "score": 0.87, + "content": "\\times ~ 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "time steps from halfcheetah-expert-v2 dataset in D4RL, which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 352, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 352, + 187 + ], + "score": 1.0, + "content": "consists of both backflipping and running forward behaviors.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 191, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "For the evaluation, we prepared additional 5 trajectories of backflipping and 5 of running forward as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "uni-modal behaviors (10 test trajectories in total). In addition, we synthesized a bi-modal behavior by", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "dividing each 1000-step trajectories into 500-step sub-trajectories, and concatenating them across", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "different behaviors, which results in the patchworked trajectories of first 500-step running forward", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 349, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 349, + 249 + ], + "score": 1.0, + "content": "and next 500-step backflipping (also 10 trajectories in total).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 260, + 457, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 460, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 460, + 273 + ], + "score": 1.0, + "content": "E.5 DETAILS OF DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "score": 1.0, + "content": "Following prior meta RL/IL works (Rakelly et al., 2019; Ghasemipour et al., 2019; Pong et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "2021; Li et al., 2021; Fakoor et al., 2020), we modified the reward function for the cheetah to run", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "with specified velocity (such as -np.abs(x_vel - target_vel)), and set the horizon to 200", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 313, + 134, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 134, + 328 + ], + "score": 1.0, + "content": "steps.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "We prepared 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals. We also", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "trained the SAC agents until convergence (3 million gradient steps), using pytorch implementation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 352, + 365 + ], + "score": 1.0, + "content": "released by Furuta et al. (2021a), and collected 250 trajectories", + "type": "text" + }, + { + "bbox": [ + 352, + 353, + 379, + 363 + ], + "score": 0.82, + "content": "\\times 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "time steps each. To simplify the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 504, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 395, + 377 + ], + "score": 1.0, + "content": "problem than meta learning settings, we held out the 10 trajectories whose", + "type": "text" + }, + { + "bbox": [ + 395, + 366, + 401, + 374 + ], + "score": 0.37, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 363, + 446, + 377 + ], + "score": 1.0, + "content": "-velocity is", + "type": "text" + }, + { + "bbox": [ + 446, + 363, + 504, + 376 + ], + "score": 0.87, + "content": "\\{ 0 . 5 , \\bar { 1 } . 5 , \\bar { 2 } . 5 \\}", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 285, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 285, + 387 + ], + "score": 1.0, + "content": "as a test set, and used the rest as a train data.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 399, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "E.6 QUANTITATIVE AND QUALITATIVE RESULTS OF ONE-SHOT DISTRIBUTION MATCHING IN", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 129, + 409, + 187, + 423 + ], + "spans": [ + { + "bbox": [ + 129, + 409, + 187, + 423 + ], + "score": 1.0, + "content": "FULL STATE", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 431, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 504, + 443 + ], + "score": 1.0, + "content": "Table 12 and Table 13 show the one-shot reward and state-feature distribution matching results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 305, + 455 + ], + "score": 1.0, + "content": "respectively. In addition to the embedding size", + "type": "text" + }, + { + "bbox": [ + 305, + 444, + 315, + 452 + ], + "score": 0.65, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 442, + 506, + 455 + ], + "score": 1.0, + "content": ", we also sweep the different context window", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 452, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 178, + 465 + ], + "score": 0.73, + "content": "N = 2 0 , 5 0 , 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 452, + 222, + 465 + ], + "score": 1.0, + "content": "for BDT", + "type": "text" + }, + { + "bbox": [ + 223, + 453, + 262, + 464 + ], + "score": 0.85, + "content": "m = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 452, + 506, + 465 + ], + "score": 1.0, + "content": "). 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Through the experiment we observe that while there are no clear trends in DT-AE, -CPC", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 273, + 531 + ], + "score": 1.0, + "content": "or -E2E as the size of context embedding", + "type": "text" + }, + { + "bbox": [ + 273, + 521, + 284, + 529 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "grows, BDT improves its performance with larger size", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 531, + 465, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 465, + 542 + ], + "score": 1.0, + "content": "of embedding. Such intuitive properties might a good features to design the architectures.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 405, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 407, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 282, + 561 + ], + "score": 1.0, + "content": "We visualize the qualitative results of BDT", + "type": "text" + }, + { + "bbox": [ + 282, + 547, + 354, + 558 + ], + "score": 0.87, + "content": "( m = 1 6 , N = 2 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 545, + 407, + 561 + ], + "score": 1.0, + "content": ") in Figure 6.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + } + ], + "page_idx": 21, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 405, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 406, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 406, + 95 + ], + "score": 1.0, + "content": "E.4 DETAILS OF SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 82, + 406, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 102, + 504, + 136 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 506, + 116 + ], + "score": 1.0, + "content": "To construct the dataset, we modified the original reward function in HalfCheetah-v3, adding abso-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 224, + 127 + ], + "score": 1.0, + "content": "lute z-velocity term (such as", + "type": "text" + }, + { + "bbox": [ + 225, + 115, + 244, + 126 + ], + "score": 0.8, + "content": "+ \\mathrm { n p }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 114, + 270, + 127 + ], + "score": 1.0, + "content": ".abs", + "type": "text" + }, + { + "bbox": [ + 270, + 115, + 308, + 126 + ], + "score": 0.45, + "content": "( z \\_ { \\mathrm { v e 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 114, + 496, + 127 + ], + "score": 1.0, + "content": "), where the expert cheetah backflips towards -", + "type": "text" + }, + { + "bbox": [ + 497, + 116, + 504, + 124 + ], + "score": 0.25, + "content": "\\mathbf { - X }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 124, + 147, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 147, + 137 + ], + "score": 1.0, + "content": "direction.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 101, + 506, + 137 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 142, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 155 + ], + "score": 1.0, + "content": "We trained SAC agent until convergence (3 million gradient steps), using pytorch implementation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 372, + 164 + ], + "score": 1.0, + "content": "released by Furuta et al. 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We combined", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 215, + 177 + ], + "score": 1.0, + "content": "them with 500 trajectories", + "type": "text" + }, + { + "bbox": [ + 216, + 164, + 248, + 174 + ], + "score": 0.87, + "content": "\\times ~ 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "time steps from halfcheetah-expert-v2 dataset in D4RL, which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 352, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 352, + 187 + ], + "score": 1.0, + "content": "consists of both backflipping and running forward behaviors.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 141, + 506, + 187 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 191, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "For the evaluation, we prepared additional 5 trajectories of backflipping and 5 of running forward as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "uni-modal behaviors (10 test trajectories in total). In addition, we synthesized a bi-modal behavior by", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "dividing each 1000-step trajectories into 500-step sub-trajectories, and concatenating them across", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "different behaviors, which results in the patchworked trajectories of first 500-step running forward", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 349, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 349, + 249 + ], + "score": 1.0, + "content": "and next 500-step backflipping (also 10 trajectories in total).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 191, + 506, + 249 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 260, + 457, + 272 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 460, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 460, + 273 + ], + "score": 1.0, + "content": "E.5 DETAILS OF DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "score": 1.0, + "content": "Following prior meta RL/IL works (Rakelly et al., 2019; Ghasemipour et al., 2019; Pong et al.,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "2021; Li et al., 2021; Fakoor et al., 2020), we modified the reward function for the cheetah to run", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "with specified velocity (such as -np.abs(x_vel - target_vel)), and set the horizon to 200", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 313, + 134, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 134, + 328 + ], + "score": 1.0, + "content": "steps.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 280, + 507, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "We prepared 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals. We also", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "trained the SAC agents until convergence (3 million gradient steps), using pytorch implementation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 352, + 365 + ], + "score": 1.0, + "content": "released by Furuta et al. (2021a), and collected 250 trajectories", + "type": "text" + }, + { + "bbox": [ + 352, + 353, + 379, + 363 + ], + "score": 0.82, + "content": "\\times 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "time steps each. 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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE(m=1)2.494 ± 1.0502.877 ± 0.7622.686 ± 0.9370.586± 0.0160.089 ± 0.0290.337 ± 0.2490.733 ± 0.5220.586 ± 0.4450.660 ± 0.4901.228
DT-CPC (m=1)3.595± 0.6762.275 ±0.3412.935±0.8490.600 ± 0.0070.130± 0.0390.365± 0.2371.180 ± 0.0190.139 ± 0.0320.660 ± 0.5211.320
DT-AE(m=4)0.782 ± 0.3291.720 ± 1.6851.251 ± 1.3010.613 ± 0.1080.140 ± 0.0680.376 ± 0.2530.889 ± 0.4020.265 ± 0.1370.577 ± 0.4330.735
DT-CPC (m=4)5.887 ± 0.3571.370 ± 1.3383.628 ± 2.4620.737 ± 0.0180.238 ± 0.0600.487 ± 0.2541.286 ± 0.0180.145 ± 0.0310.715 ± 0.5711.610
DT-AE(m=16)2.041 ± 1.0801.074 ± 0.8141.558 ± 1.0710.613 ± 0.0600.146 ± 0.0510.379 ± 0.2400.837 ± 0.3450.324 ± 0.1260.581 ± 0.3650.839
DT-CPC (m=16)6.022 ± 0.3161.406 ± 1.3713.714 ± 2.5130.614± 0.0190.104 ± 0.0250.359± 0.2561.284± 0.0200.140 ± 0.0390.712 ± 0.5721.595
DT-AE(m=1,joint)3.824 ± 1.1141.988 ± 0.9702.906 ± 1.3910.687 ± 0.0970.137 ± 0.0570.412 ± 0.2860.723± 0.5960.614 ± 0.4620.668 ± 0.5361.329
DT-CPC (m=1, joint)4.460 ± 1.1062.036 ± 1.0323.248 ± 1.6160.587 ± 0.0150.106 ± 0.0420.347 ± 0.2430.815 ± 0.7110.710 ± 0.3980.763 ± 0.5781.452
DT-AE(m=4,joint)5.563 ± 0.4491.028 ± 1.2653.295 ± 2.4581.320 ± 0.2080.485± 0.2350.902 ± 0.4730.917 ± 0.3190.218 ± 0.1120.567 ± 0.4231.588
DT-CPC (m=4, joint)3.486 ± 1.5032.265 ± 1.0772.876 ± 1.4430.690 ± 0.1590.220 ± 0.1760.455± 0.2881.021 ± 0.7080.554 ± 0.3830.788 ± 0.6151.373
DT-AE (m=16, joint)8.450 ± 0.6232.168±0.7015.309 ± 3.2101.257 ± 0.3610.655 ± 0.2420.956 ± 0.4302.112 ±0.6180.994 ± 0.4131.553 ± 0.7672.606
DT-CPC (m=16, joint)4.543 ± 1.1791.869 ± 1.4533.206 ± 1.8810.577 ± 0.0320.098 ± 0.0280.338 ± 0.2410.953 ± 0.4830.414 ± 0.3760.683 ± 0.5101.409
DT-E2E(m=1)3.220 ± 2.3251.026 ± 1.3982.123 ± 2.2101.615 ± 0.5360.540 ± 0.2931.078 ± 0.6901.079 ± 0.2090.455 ± 0.0850.767 ± 0.3511.323
DT-E2E(m=4)8.076 ± 0.5512.552 ± 0.5715.314 ± 2.8181.205 ± 0.3900.493±0.2470.849 ± 0.4832.835±0.9461.239 ± 0.4602.037 ± 1.0912.733
DT-E2E(m=16)8.049 ± 0.9901.859 ± 0.8394.954 ± 3.2281.102 ± 0.2160.549 ± 0.1890.826 ± 0.3432.095±0.4341.241 ± 0.7091.668 ± 0.7262.482
DT-AE(m=1, frozen)2.225 ± 1.0172.804 ± 1.0512.514 ± 1.0740.582 ± 0.0420.131 ± 0.0580.357 ± 0.2311.396 ± 0.1490.259 ± 0.0830.827 ± 0.5811.233
DT-CPC (m=1,frozen)4.110 ± 0.7992.172 ±0.7893.141 ± 1.2530.582 ± 0.0210.102 ± 0.0410.342 ± 0.2421.422 ± 0.0990.275± 0.1090.848 ± 0.5831.444
DT-AE(m=4, frozen)0.815 ± 0.1831.415 ± 1.7551.115 ± 1.2830.681 ± 0.1060.197 ± 0.0610.439 ± 0.2571.066 ± 0.4950.419 ± 0.3210.742 ± 0.5280.765
DT-CPC (m=4, frozen)3.275 ± 1.1862.752 ± 1.0883.014 ± 1.1680.633 ± 0.0550.139 ± 0.0820.386 ± 0.2571.119 ± 0.5980.508 ± 0.3540.814 ± 0.5781.404
DT-AE(m=16, frozen)1.796 ± 0.5781.312 ± 0.852 2.538±0.9051.554 ± 0.7670.637 ± 0.0390.142 ± 0.0630.389 ±0.2531.184 ± 0.3260.405 ± 0.1690.795 ± 0.4690.913
DT-CPC (m=16, frozen)3.486 ± 1.1643.012 ± 1.1450.636± 0.0930.178 ± 0.1320.407 ± 0.2561.318 ± 0.3280.282 ± 0.1260.800 ± 0.5741.407
BDT (m=1, N=20)1.385 ± 0.207 1.660 ± 0.1671.180 ± 1.753 1.058 ± 1.5801.282 ± 1.253 1.359 ± 1.1630.291 ± 0.096 0.494 ± 0.2810.110 ± 0.043 0.096 ± 0.0290.201 ± 0.117 0.295± 0.2821.113 ± 0.044 0.181 ± 0.0230.155 ± 0.046 0.414 ± 0.5370.634 ± 0.481 0.298 ± 0.3970.706
BDT(m=4,N=20)1.565 ± 0.1901.191 ± 1.8301.378 ± 1.3150.321 ± 0.0930.086 ± 0.0170.204 ± 0.1350.204 ± 0.0330.396 ± 0.1860.300 ± 0.1650.650
BDT(m=16, N=20)0.831 ± 0.0641.204 ± 1.8031.018 ± 1.2900.144 ± 0.0110.104 ± 0.0260.124 ± 0.0280.199 ± 0.0520.167 ± 0.0240.183 ± 0.0440.627
BDT(m=16,N=50)1.280 ± 1.8611.108 ± 1.3320.240 ± 0.0470.162 ± 0.0330.201 ± 0.0560.057 ± 0.0070.442
BDT(m=16,N=100)0.936 ± 0.1660.873 ± 0.6070.465 ± 0.5930.591
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE (m=1)2.504 ± 1.0502.880 ± 0.7782.692 ± 0.9430.580 ± 0.0150.084 ± 0.0270.332 ± 0.2490.729 ± 0.5220.584 ± 0.4470.656 ± 0.4911.227
DT-CPC (m=1)3.595 ±0.6702.277 ± 0.3492.936 ± 0.8480.601 ± 0.0080.130 ± 0.0370.365 ± 0.2371.177 ± 0.0210.135 ± 0.0330.656 ± 0.5221.319
DT-AE(m=4)0.789 ± 0.3331.729 ± 1.7141.259 ± 1.3210.612 ± 0.1080.138 ± 0.0660.375 ± 0.2540.884 ± 0.4020.262 ± 0.1380.573 ± 0.4330.736
DT-CPC (m=4)5.883 ± 0.3611.371 ± 1.3473.627 ± 2.4620.731 ± 0.0180.229 ± 0.0570.480 ± 0.2551.282 ± 0.0220.141 ± 0.0320.712 ± 0.5711.606
DT-AE(m=16)2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC (m=16)6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710 ± 0.5721.591
DT-AE (m=1,joint)3.825 ± 1.1151.990 ± 0.9882.908 ± 1.3970.683 ± 0.0940.132 ± 0.0550.407 ± 0.2860.719 ± 0.5970.611 ± 0.4630.665 ± 0.5371.327
DT-CPC (m=1,joint)4.460 ± 1.1012.035 ±1.0553.248 ± 1.6220.583 ± 0.0150.099 ± 0.0410.341 ± 0.2440.811 ± 0.7110.709 ± 0.3990.760 ± 0.5791.450
DT-AE(m=4,joint)5.589 ± 0.4581.040 ± 1.2703.315 ± 2.4671.318 ± 0.2080.481 ± 0.2340.899 ± 0.4730.912 ± 0.3190.216 ± 0.1120.564 ± 0.4221.593
DT-CPC (m=4, joint)3.484 ± 1.4982.270 ± 1.0992.877 ± 1.4480.685 ± 0.1570.214 ± 0.1740.449 ± 0.2881.018 ± 0.7130.555 ± 0.3840.786 ± 0.6181.371
DT-AE(m=16,joint) DT-CPC (m=16, joint)8.643 ± 0.6792.260 ±0.6905.451 ± 3.2641.255 ± 0.3630.649 ± 0.2410.952 ± 0.4322.104 ±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-E2E (m=1)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575± 0.0320.096± 0.0280.335± 0.2410.949 ± 0.4840.412 ± 0.3780.680 ± 0.5101.410
DT-E2E(m=4)3.265 ± 2.3461.050 ± 1.4132.158 ± 2.2311.613 ± 0.5400.534 ± 0.2931.073 ± 0.6921.073 ± 0.2110.451 ± 0.0840.762 ± 0.3501.331
8.215 ± 0.6042.684 ±0.5755.449 ± 2.8281.202 ± 0.3920.487 ± 0.2460.845 ± 0.4852.832 ± 0.9511.235 ± 0.4632.034 ± 1.0942.776
DT-E2E (m=16)8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ±0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE(m=1,frozen)2.238 ± 1.0112.807 ± 1.0562.523 ± 1.0720.577 ±0.0430.126± 0.0560.352 ± 0.2311.391 ± 0.1490.256 ±0.0830.824 ± 0.5801.233
DT-CPC (m=1, frozen)4.110 ± 0.7942.173 ± 0.8063.141 ± 1.2570.578± 0.0220.097 ± 0.0390.338 ± 0.2421.417 ± 0.1000.272 ± 0.1090.845 ± 0.5821.441
DT-AE (m=4, frozen)0.825 ± 0.1891.431 ± 1.7821.128 ± 1.3030.679 ± 0.1060.193 ± 0.0600.436 ± 0.2581.063 ± 0.4950.418 ± 0.3220.740 ± 0.5270.768
DT-CPC (m=4,frozen)3.274 ± 1.1872.756 ± 1.0943.015 ± 1.1700.629 ± 0.0540.135 ± 0.0810.382 ± 0.2571.115 ± 0.6000.507 ± 0.3530.811 ± 0.5781.403
DT-AE(m=16, frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (m=16, frozen)3.489 ±1.1592.543±0.9033.016 ± 1.1410.631±0.0910.171 ± 0.1300.401 ±0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(m=1, N=20)1.414 ± 0.210 1.694 ± 0.1711.197 ± 1.770 1.071 ± 1.5941.305 ± 1.265 1.382 ± 1.1750.288 ± 0.0960.108 ± 0.0410.198 ± 0.1161.108 ± 0.0450.152 ± 0.0510.630± 0.4800.711
BDT (m=4,N=20)1.208 ± 1.8541.400 ± 1.3330.490± 0.2800.092 ± 0.0300.291 ± 0.2810.173 ± 0.0240.411 ± 0.5470.292 ± 0.4050.655
BDT(m=16, N=20) BDT(m=16,N=50)1.592 ± 0.2011.223 ± 1.8281.031 ± 1.3070.318 ± 0.0930.081 ± 0.0130.200 ± 0.1360.196 ± 0.0310.392 ± 0.1840.294 ± 0.1640.631
BDT(m=16,N=100)0.840 ± 0.063 0.953 ± 0.1681.308 ± 1.8811.130 ± 1.3470.142 ± 0.010 0.240 ± 0.0440.098 ± 0.025 0.156 ± 0.0320.120 ± 0.029 0.198 ± 0.0570.192 ± 0.051 0.051 ± 0.0060.163 ± 0.027 0.883 ± 0.6140.178 ± 0.043 0.467 ± 0.6010.443 0.598
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE(m=1)2.494 ± 1.0502.877 ± 0.7622.686 ± 0.9370.586± 0.0160.089 ± 0.0290.337 ± 0.2490.733 ± 0.5220.586 ± 0.4450.660 ± 0.4901.228
DT-CPC (m=1)3.595± 0.6762.275 ±0.3412.935±0.8490.600 ± 0.0070.130± 0.0390.365± 0.2371.180 ± 0.0190.139 ± 0.0320.660 ± 0.5211.320
DT-AE(m=4)0.782 ± 0.3291.720 ± 1.6851.251 ± 1.3010.613 ± 0.1080.140 ± 0.0680.376 ± 0.2530.889 ± 0.4020.265 ± 0.1370.577 ± 0.4330.735
DT-CPC (m=4)5.887 ± 0.3571.370 ± 1.3383.628 ± 2.4620.737 ± 0.0180.238 ± 0.0600.487 ± 0.2541.286 ± 0.0180.145 ± 0.0310.715 ± 0.5711.610
DT-AE(m=16)2.041 ± 1.0801.074 ± 0.8141.558 ± 1.0710.613 ± 0.0600.146 ± 0.0510.379 ± 0.2400.837 ± 0.3450.324 ± 0.1260.581 ± 0.3650.839
DT-CPC (m=16)6.022 ± 0.3161.406 ± 1.3713.714 ± 2.5130.614± 0.0190.104 ± 0.0250.359± 0.2561.284± 0.0200.140 ± 0.0390.712 ± 0.5721.595
DT-AE(m=1,joint)3.824 ± 1.1141.988 ± 0.9702.906 ± 1.3910.687 ± 0.0970.137 ± 0.0570.412 ± 0.2860.723± 0.5960.614 ± 0.4620.668 ± 0.5361.329
DT-CPC (m=1, joint)4.460 ± 1.1062.036 ± 1.0323.248 ± 1.6160.587 ± 0.0150.106 ± 0.0420.347 ± 0.2430.815 ± 0.7110.710 ± 0.3980.763 ± 0.5781.452
DT-AE(m=4,joint)5.563 ± 0.4491.028 ± 1.2653.295 ± 2.4581.320 ± 0.2080.485± 0.2350.902 ± 0.4730.917 ± 0.3190.218 ± 0.1120.567 ± 0.4231.588
DT-CPC (m=4, joint)3.486 ± 1.5032.265 ± 1.0772.876 ± 1.4430.690 ± 0.1590.220 ± 0.1760.455± 0.2881.021 ± 0.7080.554 ± 0.3830.788 ± 0.6151.373
DT-AE (m=16, joint)8.450 ± 0.6232.168±0.7015.309 ± 3.2101.257 ± 0.3610.655 ± 0.2420.956 ± 0.4302.112 ±0.6180.994 ± 0.4131.553 ± 0.7672.606
DT-CPC (m=16, joint)4.543 ± 1.1791.869 ± 1.4533.206 ± 1.8810.577 ± 0.0320.098 ± 0.0280.338 ± 0.2410.953 ± 0.4830.414 ± 0.3760.683 ± 0.5101.409
DT-E2E(m=1)3.220 ± 2.3251.026 ± 1.3982.123 ± 2.2101.615 ± 0.5360.540 ± 0.2931.078 ± 0.6901.079 ± 0.2090.455 ± 0.0850.767 ± 0.3511.323
DT-E2E(m=4)8.076 ± 0.5512.552 ± 0.5715.314 ± 2.8181.205 ± 0.3900.493±0.2470.849 ± 0.4832.835±0.9461.239 ± 0.4602.037 ± 1.0912.733
DT-E2E(m=16)8.049 ± 0.9901.859 ± 0.8394.954 ± 3.2281.102 ± 0.2160.549 ± 0.1890.826 ± 0.3432.095±0.4341.241 ± 0.7091.668 ± 0.7262.482
DT-AE(m=1, frozen)2.225 ± 1.0172.804 ± 1.0512.514 ± 1.0740.582 ± 0.0420.131 ± 0.0580.357 ± 0.2311.396 ± 0.1490.259 ± 0.0830.827 ± 0.5811.233
DT-CPC (m=1,frozen)4.110 ± 0.7992.172 ±0.7893.141 ± 1.2530.582 ± 0.0210.102 ± 0.0410.342 ± 0.2421.422 ± 0.0990.275± 0.1090.848 ± 0.5831.444
DT-AE(m=4, frozen)0.815 ± 0.1831.415 ± 1.7551.115 ± 1.2830.681 ± 0.1060.197 ± 0.0610.439 ± 0.2571.066 ± 0.4950.419 ± 0.3210.742 ± 0.5280.765
DT-CPC (m=4, frozen)3.275 ± 1.1862.752 ± 1.0883.014 ± 1.1680.633 ± 0.0550.139 ± 0.0820.386 ± 0.2571.119 ± 0.5980.508 ± 0.3540.814 ± 0.5781.404
DT-AE(m=16, frozen)1.796 ± 0.5781.312 ± 0.852 2.538±0.9051.554 ± 0.7670.637 ± 0.0390.142 ± 0.0630.389 ±0.2531.184 ± 0.3260.405 ± 0.1690.795 ± 0.4690.913
DT-CPC (m=16, frozen)3.486 ± 1.1643.012 ± 1.1450.636± 0.0930.178 ± 0.1320.407 ± 0.2561.318 ± 0.3280.282 ± 0.1260.800 ± 0.5741.407
BDT (m=1, N=20)1.385 ± 0.207 1.660 ± 0.1671.180 ± 1.753 1.058 ± 1.5801.282 ± 1.253 1.359 ± 1.1630.291 ± 0.096 0.494 ± 0.2810.110 ± 0.043 0.096 ± 0.0290.201 ± 0.117 0.295± 0.2821.113 ± 0.044 0.181 ± 0.0230.155 ± 0.046 0.414 ± 0.5370.634 ± 0.481 0.298 ± 0.3970.706
BDT(m=4,N=20)1.565 ± 0.1901.191 ± 1.8301.378 ± 1.3150.321 ± 0.0930.086 ± 0.0170.204 ± 0.1350.204 ± 0.0330.396 ± 0.1860.300 ± 0.1650.650
BDT(m=16, N=20)0.831 ± 0.0641.204 ± 1.8031.018 ± 1.2900.144 ± 0.0110.104 ± 0.0260.124 ± 0.0280.199 ± 0.0520.167 ± 0.0240.183 ± 0.0440.627
BDT(m=16,N=50)1.280 ± 1.8611.108 ± 1.3320.240 ± 0.0470.162 ± 0.0330.201 ± 0.0560.057 ± 0.0070.442
BDT(m=16,N=100)0.936 ± 0.1660.873 ± 0.6070.465 ± 0.5930.591
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE (m=1)2.504 ± 1.0502.880 ± 0.7782.692 ± 0.9430.580 ± 0.0150.084 ± 0.0270.332 ± 0.2490.729 ± 0.5220.584 ± 0.4470.656 ± 0.4911.227
DT-CPC (m=1)3.595 ±0.6702.277 ± 0.3492.936 ± 0.8480.601 ± 0.0080.130 ± 0.0370.365 ± 0.2371.177 ± 0.0210.135 ± 0.0330.656 ± 0.5221.319
DT-AE(m=4)0.789 ± 0.3331.729 ± 1.7141.259 ± 1.3210.612 ± 0.1080.138 ± 0.0660.375 ± 0.2540.884 ± 0.4020.262 ± 0.1380.573 ± 0.4330.736
DT-CPC (m=4)5.883 ± 0.3611.371 ± 1.3473.627 ± 2.4620.731 ± 0.0180.229 ± 0.0570.480 ± 0.2551.282 ± 0.0220.141 ± 0.0320.712 ± 0.5711.606
DT-AE(m=16)2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC (m=16)6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710 ± 0.5721.591
DT-AE (m=1,joint)3.825 ± 1.1151.990 ± 0.9882.908 ± 1.3970.683 ± 0.0940.132 ± 0.0550.407 ± 0.2860.719 ± 0.5970.611 ± 0.4630.665 ± 0.5371.327
DT-CPC (m=1,joint)4.460 ± 1.1012.035 ±1.0553.248 ± 1.6220.583 ± 0.0150.099 ± 0.0410.341 ± 0.2440.811 ± 0.7110.709 ± 0.3990.760 ± 0.5791.450
DT-AE(m=4,joint)5.589 ± 0.4581.040 ± 1.2703.315 ± 2.4671.318 ± 0.2080.481 ± 0.2340.899 ± 0.4730.912 ± 0.3190.216 ± 0.1120.564 ± 0.4221.593
DT-CPC (m=4, joint)3.484 ± 1.4982.270 ± 1.0992.877 ± 1.4480.685 ± 0.1570.214 ± 0.1740.449 ± 0.2881.018 ± 0.7130.555 ± 0.3840.786 ± 0.6181.371
DT-AE(m=16,joint) DT-CPC (m=16, joint)8.643 ± 0.6792.260 ±0.6905.451 ± 3.2641.255 ± 0.3630.649 ± 0.2410.952 ± 0.4322.104 ±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-E2E (m=1)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575± 0.0320.096± 0.0280.335± 0.2410.949 ± 0.4840.412 ± 0.3780.680 ± 0.5101.410
DT-E2E(m=4)3.265 ± 2.3461.050 ± 1.4132.158 ± 2.2311.613 ± 0.5400.534 ± 0.2931.073 ± 0.6921.073 ± 0.2110.451 ± 0.0840.762 ± 0.3501.331
8.215 ± 0.6042.684 ±0.5755.449 ± 2.8281.202 ± 0.3920.487 ± 0.2460.845 ± 0.4852.832 ± 0.9511.235 ± 0.4632.034 ± 1.0942.776
DT-E2E (m=16)8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ±0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE(m=1,frozen)2.238 ± 1.0112.807 ± 1.0562.523 ± 1.0720.577 ±0.0430.126± 0.0560.352 ± 0.2311.391 ± 0.1490.256 ±0.0830.824 ± 0.5801.233
DT-CPC (m=1, frozen)4.110 ± 0.7942.173 ± 0.8063.141 ± 1.2570.578± 0.0220.097 ± 0.0390.338 ± 0.2421.417 ± 0.1000.272 ± 0.1090.845 ± 0.5821.441
DT-AE (m=4, frozen)0.825 ± 0.1891.431 ± 1.7821.128 ± 1.3030.679 ± 0.1060.193 ± 0.0600.436 ± 0.2581.063 ± 0.4950.418 ± 0.3220.740 ± 0.5270.768
DT-CPC (m=4,frozen)3.274 ± 1.1872.756 ± 1.0943.015 ± 1.1700.629 ± 0.0540.135 ± 0.0810.382 ± 0.2571.115 ± 0.6000.507 ± 0.3530.811 ± 0.5781.403
DT-AE(m=16, frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (m=16, frozen)3.489 ±1.1592.543±0.9033.016 ± 1.1410.631±0.0910.171 ± 0.1300.401 ±0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(m=1, N=20)1.414 ± 0.210 1.694 ± 0.1711.197 ± 1.770 1.071 ± 1.5941.305 ± 1.265 1.382 ± 1.1750.288 ± 0.0960.108 ± 0.0410.198 ± 0.1161.108 ± 0.0450.152 ± 0.0510.630± 0.4800.711
BDT (m=4,N=20)1.208 ± 1.8541.400 ± 1.3330.490± 0.2800.092 ± 0.0300.291 ± 0.2810.173 ± 0.0240.411 ± 0.5470.292 ± 0.4050.655
BDT(m=16, N=20) BDT(m=16,N=50)1.592 ± 0.2011.223 ± 1.8281.031 ± 1.3070.318 ± 0.0930.081 ± 0.0130.200 ± 0.1360.196 ± 0.0310.392 ± 0.1840.294 ± 0.1640.631
BDT(m=16,N=100)0.840 ± 0.063 0.953 ± 0.1681.308 ± 1.8811.130 ± 1.3470.142 ± 0.010 0.240 ± 0.0440.098 ± 0.025 0.156 ± 0.0320.120 ± 0.029 0.198 ± 0.0570.192 ± 0.051 0.051 ± 0.0060.163 ± 0.027 0.883 ± 0.6140.178 ± 0.043 0.467 ± 0.6010.443 0.598
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The left two examples are the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 703, + 475, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 475, + 715 + ], + "score": 1.0, + "content": "distributions from the best trajectories, and right two are the distributions from the middle trajectories.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 11.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 296, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "score": 1.0, + "content": "E.7 SHIFTING THE TARGET DISTRIBUTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "To generate the unseen but manageable generalization-test trajectories within the support of dataset", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "distribution, we make the reward and velocities values of trajectories in the test set shift with a constant", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 186, + 137 + ], + "score": 1.0, + "content": "offset: bin_size", + "type": "text" + }, + { + "bbox": [ + 186, + 125, + 368, + 136 + ], + "score": 0.84, + "content": "\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 125, + 505, + 137 + ], + "score": 1.0, + "content": ". Table 14 shows the quantitative", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "comparison between CDT and DT, based on Wasserstein-1 distance between two distributions. CDT", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "successfully handles the distribution shifts (especially in hopper) better than DT. We also provide the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "score": 1.0, + "content": "state-feature results (Table 15) and qualitative visualizations (Figure 7 and Figure 8), which reveals", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "that CDT can match the rollouts to the shifted target distributions when they are within the support of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 191, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 191, + 191 + ], + "score": 1.0, + "content": "dataset distributions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "table", + "bbox": [ + 111, + 200, + 496, + 227 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 111, + 200, + 496, + 227 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 200, + 496, + 227 + ], + "spans": [ + { + "bbox": [ + 111, + 200, + 496, + 227 + ], + "score": 0.957, + "html": "
Methodhalfcheetahwalker2dAverage
ExpertMediumTotalExperthopper MediumTotalExpertMediumTotal
Categorical DT1.126 ± 0.2452.026 ± 1.1801.576 ± 0.9640.147 ± 0.0340.302 ± 0.0850.224± 0.1010.285± 0.0441.024 ± 0.0760.655 ± 0.3750.818
DT1.133 ± 0.1971.978 ± 1.1041.555 ± 0.8990.521 ±0.0410.531 ± 0.0450.526 ± 0.0430.656±0.3800.915 ± 0.1060.786 ± 0.3080.956
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
Categorical DT1.270 ± 0.2422.371 ± 1.7471.821 ± 1.3630.157 ± 0.0380.337 ± 0.0880.247 ± 0.1120.289 ±0.0520.964±0.1040.626± 0.3470.898
DT1.173 ± 0.3722.056 ± 1.0451.614 ± 0.9010.432 ±0.0870.531 ± 0.0530.482 ±0.0880.408 ±0.2460.885 ± 0.1430.646 ± 0.3120.914
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Similar to the case of reward, Categorical DT handles the target distribution shifts and matches the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 339, + 210, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 210, + 349 + ], + "score": 1.0, + "content": "distributions better than DT.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "image", + "bbox": [ + 105, + 366, + 505, + 635 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 105, + 366, + 505, + 635 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 635 + ], + "score": 0.946, + "type": "image", + "image_path": "f4c6314fde7aaf0038339f982c1f76950e21b022a6b636a1222aba573f75ed64.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 105, + 366, + 505, + 455.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 105, + 455.6666666666667, + 505, + 545.3333333333334 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 105, + 545.3333333333334, + 505, + 635.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 642, + 505, + 703 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 643, + 504, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 504, + 654 + ], + "score": 1.0, + "content": "Figure 7: Reward distribution matching in halfcheetah (top two rows; best and middle), hopper (middle two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 653, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 663 + ], + "score": 1.0, + "content": "rows; best and middle), and walker2d (bottom two rows; best and middle). We shift the target distribution with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 662, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 251, + 673 + ], + "score": 1.0, + "content": "(from left to right column); bin_size", + "type": "text" + }, + { + "bbox": [ + 252, + 662, + 419, + 673 + ], + "score": 0.8, + "content": "\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 663, + 505, + 673 + ], + "score": 1.0, + "content": "(Table 14). Categorical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "DT (red) can match the rollouts to the shifted target distributions (blue) when the shifted targets are within", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "score": 1.0, + "content": "the support of dataset distributions. For DT (yellow, captioned as Deterministic), we only visualize the delta", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 692, + 227, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 227, + 703 + ], + "score": 1.0, + "content": "function at the means of rollouts.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + } + ], + "index": 24.25 + } + ], + "page_idx": 23, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 296, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 298, + 95 + ], + "score": 1.0, + "content": "E.7 SHIFTING THE TARGET DISTRIBUTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 191 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 115 + ], + "score": 1.0, + "content": "To generate the unseen but manageable generalization-test trajectories within the support of dataset", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "distribution, we make the reward and velocities values of trajectories in the test set shift with a constant", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 186, + 137 + ], + "score": 1.0, + "content": "offset: bin_size", + "type": "text" + }, + { + "bbox": [ + 186, + 125, + 368, + 136 + ], + "score": 0.84, + "content": "\\times \\{ - 3 . 0 , - 2 . 0 , - 1 . 0 , 0 . 0 , + 1 . 0 , + 2 . 0 , + 3 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 125, + 505, + 137 + ], + "score": 1.0, + "content": ". Table 14 shows the quantitative", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "comparison between CDT and DT, based on Wasserstein-1 distance between two distributions. CDT", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "successfully handles the distribution shifts (especially in hopper) better than DT. We also provide the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 504, + 169 + ], + "score": 1.0, + "content": "state-feature results (Table 15) and qualitative visualizations (Figure 7 and Figure 8), which reveals", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "that CDT can match the rollouts to the shifted target distributions when they are within the support of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 180, + 191, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 191, + 191 + ], + "score": 1.0, + "content": "dataset distributions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 102, + 506, + 191 + ] + }, + { + "type": "table", + "bbox": [ + 111, + 200, + 496, + 227 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 111, + 200, + 496, + 227 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 111, + 200, + 496, + 227 + ], + "spans": [ + { + "bbox": [ + 111, + 200, + 496, + 227 + ], + "score": 0.957, + "html": "
Methodhalfcheetahwalker2dAverage
ExpertMediumTotalExperthopper MediumTotalExpertMediumTotal
Categorical DT1.126 ± 0.2452.026 ± 1.1801.576 ± 0.9640.147 ± 0.0340.302 ± 0.0850.224± 0.1010.285± 0.0441.024 ± 0.0760.655 ± 0.3750.818
DT1.133 ± 0.1971.978 ± 1.1041.555 ± 0.8990.521 ±0.0410.531 ± 0.0450.526 ± 0.0430.656±0.3800.915 ± 0.1060.786 ± 0.3080.956
", + "type": "table", + "image_path": "06930919804dd12de2f5bf0dc078f7e98e5460d1869dfbbde41e2ae519a0bf0b.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 111, + 200, + 496, + 209.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 111, + 209.0, + 496, + 218.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 111, + 218.0, + 496, + 227.0 + ], + "spans": [], + "index": 11 + } + ] + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 235, + 504, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 246 + ], + "score": 1.0, + "content": "Table 14: Wasserstein-1 distance between shifted reward distribution and the rollout distributions. Categorical", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 256 + ], + "score": 1.0, + "content": "DT handles the target distribution shifts and matches the distributions better than DT, since CDT is aware of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 255, + 277, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 277, + 266 + ], + "score": 1.0, + "content": "distributional information of entire trajectories.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 236, + 505, + 266 + ] + }, + { + "type": "table", + "bbox": [ + 113, + 284, + 496, + 311 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 113, + 284, + 496, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 284, + 496, + 311 + ], + "spans": [ + { + "bbox": [ + 113, + 284, + 496, + 311 + ], + "score": 0.96, + "html": "
Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
Categorical DT1.270 ± 0.2422.371 ± 1.7471.821 ± 1.3630.157 ± 0.0380.337 ± 0.0880.247 ± 0.1120.289 ±0.0520.964±0.1040.626± 0.3470.898
DT1.173 ± 0.3722.056 ± 1.0451.614 ± 0.9010.432 ±0.0870.531 ± 0.0530.482 ±0.0880.408 ±0.2460.885 ± 0.1430.646 ± 0.3120.914
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Categorical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "DT (red) can match the rollouts to the shifted target distributions (blue) when the shifted targets are within", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 505, + 694 + ], + "score": 1.0, + "content": "the support of dataset distributions. 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Methodhalfcheetahhopperwalker2dAverage
Categorical DT2.059 ± 0.7720.536 ± 0.2250.920 ± 0.4281.172
DT4.256 ± 2.2200.584 ± 0.2980.974 ± 0.5401.938
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Methodhalfcheetahhopperwalker2dAverage
Categorical DT2.059 ± 0.7720.536 ± 0.2250.920 ± 0.4281.172
DT4.256 ± 2.2200.584 ± 0.2980.974 ± 0.5401.938
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MethodAlgo.TypeTraining1(T）Architectures
Andrychowicz et al. (2017) Pong et al. (2018)RL RLOnline OnlineT TMLP MLP
Chebotar et al. (2021)RLOfflineTCNN
Li et al. (2020)RLOnlineMLP
tγtr(st,at,.)
Eysenbach et al. (2020)BC/RLOn/OfflineargmaxMLP
Lynch et al. (2019)BCOfflineΦTStochastic RNN
Ghosh et al. (2021)BCOnlineΦTMLP
Srivastava et al. (2019)BCOnlineMt rtFast Weights
Kumar et al. (2019)BCOnlineMt ytrtMLP
Janner et al. (2021)BCOfflinetrtorTTransformer
Duan et al. (2017)3BCOfflineMtMLP+LSTM
Generalized DT(ours)BCOfflineT AnyTransformer
DT(Chen et al., 2021a)BCOfflineMttrtTransformer
Categorical DT(ours)4BCOfflinehistogram(rt,γ)
Transformer
Bi-Directional DT (ours)BCOfflineTTransformer
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Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.633 ± 0.3290.996 ± 1.4670.814 ± 1.0790.139± 0.0430.059± 0.0130.099 ± 0.0510.122 ± 0.0710.136 ± 0.0450.129 ±0.0600.347
DT0.746 ± 0.3801.076 ± 1.5490.911 ± 1.1400.177 ± 0.0530.093± 0.0370.135 ± 0.0630.083 ±0.0310.146 ± 0.0840.115± 0.0700.387
BC (no-context)3.017±0.8913.468 ± 1.2713.242 ± 1.1210.652 ± 0.2640.248 ± 0.1990.450 ± 0.3090.748 ± 0.5290.858 ± 0.6170.803 ± 0.5771.498
Meta-BC0.852 ± 0.6880.840 ± 1.1390.846 ± 0.9410.799 ± 0.5050.130± 0.0560.464 ± 0.4910.110 ± 0.0821.462 ± 1.1360.786 ± 1.0520.699
FOCAL (Li ct al.,2021)1.643 ± 0.4611.123 ± 1.5501.383 ± 0.5181.456 ± 0.4730.484 ± 0.3820.970 ± 0.6491.571 ± 0.5630.603± 0.4271.087 ± 0.6951.147
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MethodUni-modalBi-modalAverage
Categorical DT1.562 ± 0.6321.625 ± 0.9021.594
DT2.676 ±0.7652.703 ±0.7032.690
Meta-BC1.519 ±0.6961.655 ± 0.9901.587
FOCAL (Li et al., 2021)2.203 士 1.0501.983 士 0.9482.093
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610 ± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710± 0.5721.591
DT-AE (joint)8.643 ± 0.6792.260± 0.6905.451 ± 3.2641.255±0.3630.649 ± 0.2410.952 ± 0.4322.104±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-CPC (joint)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575 ± 0.0320.096± 0.0280.335 ± 0.2410.949 ± 0.4840.412 ± 0.3780.680± 0.5101.410
DT-E2E8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ± 0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE (frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385 ± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (frozen)3.489 ± 1.1592.543 ± 0.9033.016 ± 1.1410.631 ±0.0910.171 ± 0.1300.401 ± 0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(N=20)1.592 ± 0.2011.208 ± 1.8541.400 ± 1.3330.318±0.0930.081 ± 0.0130.200± 0.1360.196 ±0.0310.392 ± 0.1840.294± 0.1640.631
BDT(N=50)0.840 ± 0.0631.223 ± 1.8281.031 ± 1.3070.142 ± 0.0100.098 ± 0.0250.120 ± 0.0290.192 ± 0.0510.163 ± 0.0270.178 ± 0.0430.443
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Methodx-vel: 0.5 x-vel: 1.5x-vel: 2.5Average
Categorical DT0.060± 0.0260.211 ± 0.0220.149± 0.1100.140
DT1.197 ± 0.2270.533 ± 0.1050.861 ± 0.2470.864
Meta-BC0.150 ± 0.0690.152 ± 0.1270.167 ± 0.0550.156
FOCAL (Li et al., 2021)0.472 ± 0.0050.952 ± 0.0730.346 ± 0.1860.590
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HyperparameterValue
Number of layers Number of attention heads3 1 128
Embedding dimension
Nonlinearity functionReLU
Batch size64
Context length N20
Dropout0.1
Learning rate1e-4
Grad norm clip0.25
Weight decay1e-4
Learning rate decayLinear warmup for first 1OOk training steps
Training(Gradient) steps Number of bins forcategorical distribution1M
31
Encoder size (for DT-X)2 layer MLP,(128,128)
Coefficient of unsupervised loss0.1
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EvaluationTypeReference
Density VisualizationQualitativeLee et al. (2020); Ghasemipour et al. (2020)
Energy DistanceQuantitativeGu et al. (2021)
Task RewardQuantitativeHo& Ermon 1 (2016); Fu et al. (2018), etc.
Wasserstein-1DistanceQuantitativeOurs
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OursOffline multi-task SMM/ILoffline(no-adaptation)one-shot
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Methodhalfcheetahhopperwalker2d MediumAverage
ExpertMediumTotalExpertMediumTotalExpertTotal
Categorical DT0.674 ± 0.2131.002 ± 1.4580.838 ± 1.0540.159 ± 0.0850.064 ± 0.0170.111 ± 0.0770.095 ± 0.0170.114 ± 0.0370.105 ± 0.0300.351
DT0.652 ± 0.3191.039 ± 1.5480.846 ± 1.1340.227 ± 0.1190.091 ± 0.0350.159 ± 0.1110.056 ± 0.0150.626 ± 0.4950.341 ± 0.4520.448
BC (no-context)3.240± 0.5592.880 ± 0.6143.060 ± 0.6140.597 ± 0.0560.119 ± 0.0670.358 ± 0.2470.977 ± 0.5010.431 ± 0.3960.704 ± 0.5281.374
Meta-BC0.839 ±0.6820.830 ± 1.1300.835 ± 0.9330.803 ± 0.5050.134 ± 0.0560.468 ± 0.4910.113 ± 0.0851.441 ± 1.1130.777 ± 1.0320.693
FOCAL (Li et al.,2021)1.623 ± 0.5011.115 ± 1.5341.369 ± 0.5161.463 ± 0.4720.492 ± 0.3840.977 ± 0.6491.584 ± 0.5700.604 ± 0.4211.094 ± 0.7011.147
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Methodant
ExpertMediumAverage
Categorical DT0.797 ± 0.2160.244 ± 0.0630.521
DT1.714 ± 0.1210.260 ± 0.0670.987
Meta-BC1.295 ± 0.7080.351 ± 0.2050.823
FOCAL (Li et al., 2021)1.473 ± 0.8920.913 ± 0.4551.193
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Methodhalfcheetahhopperwalker2d
ExpertMediumExpertMediumExpertMedium
Categorical DT10476.746± 218.9575782.557 ± 1666.7162614.559 ± 657.7221518.757 ± 63.0624907.475 ± 11.7113264.585 ± 209.905
DT10500.273 ± 312.7784985.518 ± 67.3222113.207 ± 807.2461528.743± 30.7804965.024 ± 11.8144055.039 ± 849.109
Held-out11146.200 ± 59.0705237.348 ± 37.9703741.854 ± 7.2231600.196± 0.7724995.553 ± 5.4133801.313 ± 0.994
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE (m=1)2.504 ± 1.0502.880 ± 0.7782.692 ± 0.9430.580 ± 0.0150.084 ± 0.0270.332 ± 0.2490.729 ± 0.5220.584 ± 0.4470.656 ± 0.4911.227
DT-CPC (m=1)3.595 ±0.6702.277 ± 0.3492.936 ± 0.8480.601 ± 0.0080.130 ± 0.0370.365 ± 0.2371.177 ± 0.0210.135 ± 0.0330.656 ± 0.5221.319
DT-AE(m=4)0.789 ± 0.3331.729 ± 1.7141.259 ± 1.3210.612 ± 0.1080.138 ± 0.0660.375 ± 0.2540.884 ± 0.4020.262 ± 0.1380.573 ± 0.4330.736
DT-CPC (m=4)5.883 ± 0.3611.371 ± 1.3473.627 ± 2.4620.731 ± 0.0180.229 ± 0.0570.480 ± 0.2551.282 ± 0.0220.141 ± 0.0320.712 ± 0.5711.606
DT-AE(m=16)2.060 ± 1.0761.089 ± 0.8271.574 ± 1.0750.611 ± 0.0600.142 ± 0.0490.377 ± 0.2410.833 ± 0.3460.321 ± 0.1260.577 ± 0.3650.843
DT-CPC (m=16)6.011 ± 0.3241.403 ± 1.3843.707 ± 2.5140.610± 0.0190.101 ± 0.0240.355 ± 0.2561.281 ± 0.0220.138 ± 0.0400.710 ± 0.5721.591
DT-AE (m=1,joint)3.825 ± 1.1151.990 ± 0.9882.908 ± 1.3970.683 ± 0.0940.132 ± 0.0550.407 ± 0.2860.719 ± 0.5970.611 ± 0.4630.665 ± 0.5371.327
DT-CPC (m=1,joint)4.460 ± 1.1012.035 ±1.0553.248 ± 1.6220.583 ± 0.0150.099 ± 0.0410.341 ± 0.2440.811 ± 0.7110.709 ± 0.3990.760 ± 0.5791.450
DT-AE(m=4,joint)5.589 ± 0.4581.040 ± 1.2703.315 ± 2.4671.318 ± 0.2080.481 ± 0.2340.899 ± 0.4730.912 ± 0.3190.216 ± 0.1120.564 ± 0.4221.593
DT-CPC (m=4, joint)3.484 ± 1.4982.270 ± 1.0992.877 ± 1.4480.685 ± 0.1570.214 ± 0.1740.449 ± 0.2881.018 ± 0.7130.555 ± 0.3840.786 ± 0.6181.371
DT-AE(m=16,joint) DT-CPC (m=16, joint)8.643 ± 0.6792.260 ±0.6905.451 ± 3.2641.255 ± 0.3630.649 ± 0.2410.952 ± 0.4322.104 ±0.6200.988 ± 0.4131.546 ± 0.7672.650
DT-E2E (m=1)4.544 ± 1.1841.884 ± 1.4653.214 ± 1.8820.575± 0.0320.096± 0.0280.335± 0.2410.949 ± 0.4840.412 ± 0.3780.680 ± 0.5101.410
DT-E2E(m=4)3.265 ± 2.3461.050 ± 1.4132.158 ± 2.2311.613 ± 0.5400.534 ± 0.2931.073 ± 0.6921.073 ± 0.2110.451 ± 0.0840.762 ± 0.3501.331
8.215 ± 0.6042.684 ±0.5755.449 ± 2.8281.202 ± 0.3920.487 ± 0.2460.845 ± 0.4852.832 ± 0.9511.235 ± 0.4632.034 ± 1.0942.776
DT-E2E (m=16)8.208 ± 1.0871.928 ± 0.8385.068 ± 3.2871.097 ± 0.2170.542 ± 0.1870.820 ± 0.3442.086 ±0.4371.239 ± 0.7121.662 ± 0.7272.517
DT-AE(m=1,frozen)2.238 ± 1.0112.807 ± 1.0562.523 ± 1.0720.577 ±0.0430.126± 0.0560.352 ± 0.2311.391 ± 0.1490.256 ±0.0830.824 ± 0.5801.233
DT-CPC (m=1, frozen)4.110 ± 0.7942.173 ± 0.8063.141 ± 1.2570.578± 0.0220.097 ± 0.0390.338 ± 0.2421.417 ± 0.1000.272 ± 0.1090.845 ± 0.5821.441
DT-AE (m=4, frozen)0.825 ± 0.1891.431 ± 1.7821.128 ± 1.3030.679 ± 0.1060.193 ± 0.0600.436 ± 0.2581.063 ± 0.4950.418 ± 0.3220.740 ± 0.5270.768
DT-CPC (m=4,frozen)3.274 ± 1.1872.756 ± 1.0943.015 ± 1.1700.629 ± 0.0540.135 ± 0.0810.382 ± 0.2571.115 ± 0.6000.507 ± 0.3530.811 ± 0.5781.403
DT-AE(m=16, frozen)1.821 ± 0.5821.319 ± 0.8631.570 ± 0.7780.634 ± 0.0380.137 ± 0.0620.385± 0.2531.180 ± 0.3280.404 ± 0.1700.792 ± 0.4680.916
DT-CPC (m=16, frozen)3.489 ±1.1592.543±0.9033.016 ± 1.1410.631±0.0910.171 ± 0.1300.401 ±0.2561.315 ± 0.3290.279 ± 0.1270.797 ± 0.5751.405
BDT(m=1, N=20)1.414 ± 0.210 1.694 ± 0.1711.197 ± 1.770 1.071 ± 1.5941.305 ± 1.265 1.382 ± 1.1750.288 ± 0.0960.108 ± 0.0410.198 ± 0.1161.108 ± 0.0450.152 ± 0.0510.630± 0.4800.711
BDT (m=4,N=20)1.208 ± 1.8541.400 ± 1.3330.490± 0.2800.092 ± 0.0300.291 ± 0.2810.173 ± 0.0240.411 ± 0.5470.292 ± 0.4050.655
BDT(m=16, N=20) BDT(m=16,N=50)1.592 ± 0.2011.223 ± 1.8281.031 ± 1.3070.318 ± 0.0930.081 ± 0.0130.200 ± 0.1360.196 ± 0.0310.392 ± 0.1840.294 ± 0.1640.631
BDT(m=16,N=100)0.840 ± 0.063 0.953 ± 0.1681.308 ± 1.8811.130 ± 1.3470.142 ± 0.010 0.240 ± 0.0440.098 ± 0.025 0.156 ± 0.0320.120 ± 0.029 0.198 ± 0.0570.192 ± 0.051 0.051 ± 0.0060.163 ± 0.027 0.883 ± 0.6140.178 ± 0.043 0.467 ± 0.6010.443 0.598
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
DT-AE(m=1)2.494 ± 1.0502.877 ± 0.7622.686 ± 0.9370.586± 0.0160.089 ± 0.0290.337 ± 0.2490.733 ± 0.5220.586 ± 0.4450.660 ± 0.4901.228
DT-CPC (m=1)3.595± 0.6762.275 ±0.3412.935±0.8490.600 ± 0.0070.130± 0.0390.365± 0.2371.180 ± 0.0190.139 ± 0.0320.660 ± 0.5211.320
DT-AE(m=4)0.782 ± 0.3291.720 ± 1.6851.251 ± 1.3010.613 ± 0.1080.140 ± 0.0680.376 ± 0.2530.889 ± 0.4020.265 ± 0.1370.577 ± 0.4330.735
DT-CPC (m=4)5.887 ± 0.3571.370 ± 1.3383.628 ± 2.4620.737 ± 0.0180.238 ± 0.0600.487 ± 0.2541.286 ± 0.0180.145 ± 0.0310.715 ± 0.5711.610
DT-AE(m=16)2.041 ± 1.0801.074 ± 0.8141.558 ± 1.0710.613 ± 0.0600.146 ± 0.0510.379 ± 0.2400.837 ± 0.3450.324 ± 0.1260.581 ± 0.3650.839
DT-CPC (m=16)6.022 ± 0.3161.406 ± 1.3713.714 ± 2.5130.614± 0.0190.104 ± 0.0250.359± 0.2561.284± 0.0200.140 ± 0.0390.712 ± 0.5721.595
DT-AE(m=1,joint)3.824 ± 1.1141.988 ± 0.9702.906 ± 1.3910.687 ± 0.0970.137 ± 0.0570.412 ± 0.2860.723± 0.5960.614 ± 0.4620.668 ± 0.5361.329
DT-CPC (m=1, joint)4.460 ± 1.1062.036 ± 1.0323.248 ± 1.6160.587 ± 0.0150.106 ± 0.0420.347 ± 0.2430.815 ± 0.7110.710 ± 0.3980.763 ± 0.5781.452
DT-AE(m=4,joint)5.563 ± 0.4491.028 ± 1.2653.295 ± 2.4581.320 ± 0.2080.485± 0.2350.902 ± 0.4730.917 ± 0.3190.218 ± 0.1120.567 ± 0.4231.588
DT-CPC (m=4, joint)3.486 ± 1.5032.265 ± 1.0772.876 ± 1.4430.690 ± 0.1590.220 ± 0.1760.455± 0.2881.021 ± 0.7080.554 ± 0.3830.788 ± 0.6151.373
DT-AE (m=16, joint)8.450 ± 0.6232.168±0.7015.309 ± 3.2101.257 ± 0.3610.655 ± 0.2420.956 ± 0.4302.112 ±0.6180.994 ± 0.4131.553 ± 0.7672.606
DT-CPC (m=16, joint)4.543 ± 1.1791.869 ± 1.4533.206 ± 1.8810.577 ± 0.0320.098 ± 0.0280.338 ± 0.2410.953 ± 0.4830.414 ± 0.3760.683 ± 0.5101.409
DT-E2E(m=1)3.220 ± 2.3251.026 ± 1.3982.123 ± 2.2101.615 ± 0.5360.540 ± 0.2931.078 ± 0.6901.079 ± 0.2090.455 ± 0.0850.767 ± 0.3511.323
DT-E2E(m=4)8.076 ± 0.5512.552 ± 0.5715.314 ± 2.8181.205 ± 0.3900.493±0.2470.849 ± 0.4832.835±0.9461.239 ± 0.4602.037 ± 1.0912.733
DT-E2E(m=16)8.049 ± 0.9901.859 ± 0.8394.954 ± 3.2281.102 ± 0.2160.549 ± 0.1890.826 ± 0.3432.095±0.4341.241 ± 0.7091.668 ± 0.7262.482
DT-AE(m=1, frozen)2.225 ± 1.0172.804 ± 1.0512.514 ± 1.0740.582 ± 0.0420.131 ± 0.0580.357 ± 0.2311.396 ± 0.1490.259 ± 0.0830.827 ± 0.5811.233
DT-CPC (m=1,frozen)4.110 ± 0.7992.172 ±0.7893.141 ± 1.2530.582 ± 0.0210.102 ± 0.0410.342 ± 0.2421.422 ± 0.0990.275± 0.1090.848 ± 0.5831.444
DT-AE(m=4, frozen)0.815 ± 0.1831.415 ± 1.7551.115 ± 1.2830.681 ± 0.1060.197 ± 0.0610.439 ± 0.2571.066 ± 0.4950.419 ± 0.3210.742 ± 0.5280.765
DT-CPC (m=4, frozen)3.275 ± 1.1862.752 ± 1.0883.014 ± 1.1680.633 ± 0.0550.139 ± 0.0820.386 ± 0.2571.119 ± 0.5980.508 ± 0.3540.814 ± 0.5781.404
DT-AE(m=16, frozen)1.796 ± 0.5781.312 ± 0.852 2.538±0.9051.554 ± 0.7670.637 ± 0.0390.142 ± 0.0630.389 ±0.2531.184 ± 0.3260.405 ± 0.1690.795 ± 0.4690.913
DT-CPC (m=16, frozen)3.486 ± 1.1643.012 ± 1.1450.636± 0.0930.178 ± 0.1320.407 ± 0.2561.318 ± 0.3280.282 ± 0.1260.800 ± 0.5741.407
BDT (m=1, N=20)1.385 ± 0.207 1.660 ± 0.1671.180 ± 1.753 1.058 ± 1.5801.282 ± 1.253 1.359 ± 1.1630.291 ± 0.096 0.494 ± 0.2810.110 ± 0.043 0.096 ± 0.0290.201 ± 0.117 0.295± 0.2821.113 ± 0.044 0.181 ± 0.0230.155 ± 0.046 0.414 ± 0.5370.634 ± 0.481 0.298 ± 0.3970.706
BDT(m=4,N=20)1.565 ± 0.1901.191 ± 1.8301.378 ± 1.3150.321 ± 0.0930.086 ± 0.0170.204 ± 0.1350.204 ± 0.0330.396 ± 0.1860.300 ± 0.1650.650
BDT(m=16, N=20)0.831 ± 0.0641.204 ± 1.8031.018 ± 1.2900.144 ± 0.0110.104 ± 0.0260.124 ± 0.0280.199 ± 0.0520.167 ± 0.0240.183 ± 0.0440.627
BDT(m=16,N=50)1.280 ± 1.8611.108 ± 1.3320.240 ± 0.0470.162 ± 0.0330.201 ± 0.0560.057 ± 0.0070.442
BDT(m=16,N=100)0.936 ± 0.1660.873 ± 0.6070.465 ± 0.5930.591
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Methodhalfcheetahhopperwalker2dAverage
ExpertMediumTotalExpertMediumTotalExpertMediumTotal
Categorical DT1.270 ± 0.2422.371 ± 1.7471.821 ± 1.3630.157 ± 0.0380.337 ± 0.0880.247 ± 0.1120.289 ±0.0520.964±0.1040.626± 0.3470.898
DT1.173 ± 0.3722.056 ± 1.0451.614 ± 0.9010.432 ±0.0870.531 ± 0.0530.482 ±0.0880.408 ±0.2460.885 ± 0.1430.646 ± 0.3120.914
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Methodhalfcheetahwalker2dAverage
ExpertMediumTotalExperthopper MediumTotalExpertMediumTotal
Categorical DT1.126 ± 0.2452.026 ± 1.1801.576 ± 0.9640.147 ± 0.0340.302 ± 0.0850.224± 0.1010.285± 0.0441.024 ± 0.0760.655 ± 0.3750.818
DT1.133 ± 0.1971.978 ± 1.1041.555 ± 0.8990.521 ±0.0410.531 ± 0.0450.526 ± 0.0430.656±0.3800.915 ± 0.1060.786 ± 0.3080.956
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However, there are still settings where their zero-shot performance is far from optimal. We study model patching, where the goal is to improve accuracy on specific tasks without degrading accuracy on tasks where performance is already adequate. Towards this goal, we introduce PAINT, a patching method that uses interpolations between the weights of a model before fine-tuning and the weights after fine-tuning on a task to be patched. On nine tasks where zeroshot CLIP performs poorly, PAINT increases accuracy by 15 to 60 percentage points while preserving accuracy on ImageNet within one percentage point of the zero-shot model. PAINT also allows a single model to be patched on multiple tasks and improves with model scale. Furthermore, we identify cases of broad transfer, where patching on one task increases accuracy on other tasks even when the tasks have disjoint classes. Finally, we investigate applications beyond common benchmarks such as counting or reducing the impact of typographic attacks on CLIP. Our findings demonstrate that it is possible to expand the set of tasks on which open-vocabulary models achieve high accuracy without re-training them from scratch. + +# 1 Introduction + +Open-vocabulary models are characterized by their ability to perform any image classification task based on text descriptions of the classes [56]. Thanks to advances in large-scale pre-training, recent examples of open-vocabulary models such as CLIP and BASIC have reached parity with or surpassed important task-specific baselines, even when the open-vocabulary models are not fine-tuned on task-specific data (i.e., in a zero-shot setting) [57, 31, 56, 88, 1, 86]. For instance, the largest CLIP model from Radford et al. [57] used in a zero-shot setting matches the ImageNet accuracy of a ResNet-50 trained on 1.2 million ImageNet images [14, 24]. + +Nevertheless, current open-vocabulary models still face challenges. The same CLIP model that matches a ResNet-50 on ImageNet has lower MNIST accuracy than simple logistic regression in pixel space [57]. Moreover, even when zero-shot models achieve good performance, they are usually still worse than models trained or fine-tuned on specific downstream tasks. + +To address these issues, several authors have proposed methods for adapting zero-shot models to a task of interest using labeled data [82, 91, 21, 89, 37, 73]. A common practice is to fine-tune the zero-shot model on the task of interest [82, 56]. However, fine-tuned models can suffer from catastrophic forgetting [48, 76, 20, 33], performing poorly on tasks where the zero-shot model initially performed well [2, 82, 56]. Additionally, fine-tuning typically produces a task-specific classification head, sacrificing the flexible text-based API that makes open-vocabulary models so appealing. Whereas an open-vocabulary model can perform any classification task in a zero-shot fashion, a fine-tuned model with a task-specific head can only process the specific task that it was fine-tuned on. This specialization can prevent knowledge obtained by fine-tuning on one task from transferring to other related tasks with different classes. + +![](images/27454f67282ad59a0cde948bc9974465c3d8f122323b75d055c62f58685f01e7.jpg) +Figure 1: Patching open-vocabulary models by linearly interpolating weights. We wish to improve accuracy on tasks where a model performs poorly (patching tasks), without degrading performance on tasks where accuracy is already adequate (supported tasks). When interpolating weights of fine-tuned models and zeroshot (unpatched) models, there are intermediate solutions where accuracy improves on the patching task without reducing accuracy on supported tasks. Results are shown for CLIP models [57], averaged over nine patching tasks (Stanford Cars, DTD, EuroSAT, GTSRB, KITTI distance, MNIST, RESISC45, SUN397 and SVHN [35, 11, 25, 71, 22, 39, 7, 12, 84, 53]) and five supported tasks (ImageNet, CIFAR-10, CIFAR-100, STL-10 and Food101 [14, 36, 12, 5]). We apply PAINT separately on each patching task and average results across experiments. The dashed lines illustrate vertical movement from the unpatched models and horizontal movement from the fine-tuned models. + +Another approach to adapting zero-shot models would be to add data from the downstream task to the pre-training dataset and train a new open-vocabulary model from scratch. The resulting model could still perform any classification task, and zero-shot performance may improve on related tasks. However, training large image-text models from scratch can require hundreds of thousands of GPU hours [57, 56, 86], which makes this approach practically infeasible in most settings. + +In this paper, we study patching open-vocabulary models, where the goal is to increase accuracy on new target tasks while maintaining the flexibility of the model and its accuracy on other tasks.1 Patching aims to combine the benefits of fine-tuning and re-training from scratch: improved performance on the task of interest, maintaining the flexibility of an open vocabulary, transfer between tasks, and fast adaptation time. Motivated by these goals, we extend existing fine-tuning techniques [82] to open-vocabulary settings, where the class space is not fixed. We introduce Patching with Interpolation (PAINT), a simple, two-step procedure for patching models: first, fine-tune the model on the patching task without introducing any task-specific parameters; then, linearly interpolate between the weights of the model before and after fine-tuning. Linearly interpolating neural network weights [52, 19, 54] has been previously used to improve accuracy on a single task [28, 81] or robustness to distribution shift [82]. Indeed, averaging network weights has been explored in continual learning contexts, although for closed-vocabulary models [40]. + +With PAINT, accuracy can improve on new tasks without degrading accuracy on unrelated tasks, as illustrated in Figure 1. For instance, applying PAINT to a CLIP ViT-L/14 [57] independently on nine image classification tasks [35, 11, 25, 71, 22, 39, 7, 84, 53] improves accuracy by 15 to 60 percentage points compared to the unpatched model, while accuracy on ImageNet [14] decreases by less than one percentage point. We also observe a promising trend: patching becomes more effective with model scale (Section 4.1). + +Beyond single tasks, we show that models can be patched on multiple tasks (Section 5). When patching on nine image classification tasks simultaneously, a single CLIP ViT-L/14 model is competitive with using one specialized model for each task—the average accuracy difference is less than 0.5 percentage points. + +Moreover, PAINT enables broad transfer (Section 6): accuracy on related tasks can increase, even when the class space changes. For instance, we partition EuroSAT [25], a satellite image dataset, into two halves with disjoint labels. Patching a ViT-L/14 model on the first half improves accuracy on the second half by 7.3 percentage points, even though the classes are unseen during patching. + +Finally, we investigate PAINT on case studies including typographic attacks [23], counting [32], and visual question answering [4] (Section 7). For instance, applying PAINT using synthetic typographic attacks leads to a model that is less susceptible to typographic attacks in the real world, improving its accuracy by 41 percentage points. + +In summary: + +• Even the best pre-trained models are not perfect. We introduce PAINT, a method designed to improve accuracy on new tasks without harming accuracy elsewhere. +• PAINT incurs no extra computational cost compared to standard fine-tuning, neither during fine-tuning itself nor at inference time. +• PAINT can also be applied with multiple tasks, providing a single model that is competitive with many specialized models. +• Applying PAINT with one task can improve accuracy on a related task, even when they do not share the same classes. +• PAINT improves with model scale, indicating a promising trend for future models. + +# 2 Patching with interpolation (PAINT) + +This section details our method for patching models on a single and multiple tasks. + +Patching on a single task. Given an open-vocabulary model with weights $\theta _ { \mathrm { z s } }$ and a patching task $\mathcal { D } _ { \mathrm { p a t c h } }$ , our goal is to produce a new model $\theta _ { \mathrm { p a t c h } }$ which achieves high accuracy on $\mathcal { D } _ { \mathrm { { p a t c h } } }$ without decreasing model performance on tasks where accuracy is already acceptable. We let $\mathcal { D } _ { \mathrm { s u p p } }$ denote a representative supported task where model performance is adequate, and later show that the method is stable under different choices of $\mathcal { D } _ { \mathrm { s u p p } }$ (Section 4.2). The two-step procedure we explore for producing $\theta _ { \mathrm { p a t c h } }$ is given below. + +Step 1. Fine-tune $\theta _ { \mathrm { z s } }$ on training data from $\mathcal { D } _ { \mathrm { p a t c h } }$ to produce a model with weights $\theta _ { \mathrm { f t } }$ . Step 2. For mixing coefficient $\alpha \in [ 0 , 1 ]$ , linearly interpolate between $\theta _ { \mathrm { z s } }$ and $\theta _ { \mathrm { f t } }$ to produce $\theta _ { \mathrm { p a t c h } } = ( 1 - \alpha ) \cdot \theta _ { \mathrm { z s } } + \alpha \cdot \theta _ { \mathrm { f t } }$ . The mixing coefficient is determined via held-out validation sets for $\mathcal { D } _ { \mathrm { s u p p } }$ and $\mathcal { D } _ { \mathrm { p a t c h } }$ . We refer to the resulting model as $\theta _ { \mathrm { p a t c h } }$ . + +In our experiments, we do not introduce any additional task-specific parameters when fine-tuning, as discussed in Section 3 and Appendices B and $\textrm { C }$ . + +Patching on a multiple tasks. In practice, we often want to improve model accuracy on multiple patching tasks D(1)patch $\mathcal { D } _ { \mathrm { p a t c h } } ^ { ( 1 ) } , . . . , \mathcal { D } _ { \mathrm { p a t c h } } ^ { ( k ) }$ D(k)patch, which can be accomplished with straightforward modifications to the procedure above. We explore three alternatives and examine their relative trade-offs in Section 5: + +• Joint patching, where we merge all the patching tasks $\mathcal { D } _ { \mathtt { p a t c h } } ^ { ( i ) }$ into a single task $\mathcal { D } _ { \mathrm { p a t c h } }$ before running the patching procedure; +• Sequential patching, where we iteratively repeat the patching procedure above on each new task +D(i) and let $\theta _ { \mathrm { z s } } \theta _ { \mathrm { p a t c h } }$ after each completed iteration; +• Parallel patching, where we apply the first step on each task in parallel to produce fine-tuned $\theta _ { \mathrm { f t } } ^ { ( 1 ) } , . . . , \bar { \theta _ { \mathrm { f t } } ^ { ( k ) } }$ e search for mixing coefficients . $\alpha _ { i }$ to produce $\begin{array} { r } { \theta _ { \mathrm { p a t c h } } = \big ( 1 - \sum _ { i = 1 } ^ { k } \alpha _ { i } \big ) \cdot \theta _ { \mathrm { z s } } + \sum _ { i = 1 } ^ { k } \alpha _ { i } \cdot \theta _ { \mathrm { f t } } ^ { ( i ) } } \end{array}$ + +For joint and parallel patching we assume access to held-out validation sets for all tasks, while in sequential patching we only assume access to held-out validation sets from the tasks seen so far. Unless mentioned otherwise, we pick the mixing coefficient $\alpha$ that optimizes average accuracy on the held-out validation sets from the supported and patching tasks. + +# 3 Experimental setup + +Tasks. We consider a diverse set of image classification tasks from Radford et al. [57]. In most experiments, we use ImageNet [14] as a representative supported task, although we explore other supported tasks in Section 4.2. We categorize tasks into patching tasks or supported tasks based on the accuracy difference between the zero-shot model and a model specialized to the task. A large accuracy difference indicates that the task is a relevant target for patching because the zero-shot model is still far from optimal. Specifically, we consider a subset tasks from Radford et al. [57], categorizing tasks where the linear probes outperform the zero-shot model by over 10 percentage points as patching tasks: Cars [35], DTD [11], EuroSAT [25], GTSRB [71], KITTI [22], MNIST [39], RESISC45 [7], SUN397 [84], and SVHN [53]. We use the remaining tasks as supported tasks: CIFAR10 [36], CIFAR100 [36], Food101 [5], ImageNet [14], and STL10 [12]. We investigate additional patching tasks as case studies in Section 7 and provide further details in Appendix A. + +Models. We primarily use CLIP [57] pre-trained vision transformer (ViT) models [15]. Unless otherwise mentioned our experiments are with the ViT-L/14 model, while Section 4.2 studies ResNets [24]. + +Fine-tuning on patching tasks. Unless otherwise mentioned, we fine-tune with a batch size of 128 for 2000 iterations using learning rate 1e-5 with 200 warm-up steps with a cosine annealing learning rate schedule and the AdamW optimizer [43, 55] (weight decay 0.1). When fine-tuning, we use the frozen final classification layer output by CLIP’s text tower so that we do not introduce additional learnable parameters. This design decision keeps the model open-vocabulary and does not harm accuracy, as discussed in in Appendices B and C. + +Evaluation. We use accuracy as the evaluation metric unless otherwise stated. We refer to the average of the mean accuracy on the patching tasks and the mean accuracy on the supported tasks as combined accuracy.2 + +# 4 Patching models on a single new task + +As shown in Figure 1, when patching a model on a single task, we interpolate the weights of the zero-shot and fine-tuned model, producing a model that achieves high accuracy on both the patching task and the supported task. On the nine tasks, PAINT improves the accuracy of ViT-L/14 by 15 to 60 percentage points, while accuracy on ImageNet decreases by less than one percentage point. PAINT also allows practitioners to control the accuracy trade-off on the patching and supported tasks without re-training a new model, by varying the mixing coefficient $\alpha$ . + +# 4.1 The effect of scale + +We consistently observe that PAINT is more effective for larger models. Our findings are aligned with those of Ramasesh et al. [59], who observed that larger models are less susceptible to catastrophic forgetting. This section formalizes and provides insights for these observations. + +Measuring the effectiveness of patching. We measure the effectiveness of patching via the accuracy difference between the single patched model and two specialized models with the same architecture and initialization. For both the supported task and patching task, we take specialized models that maximize performance on the task, considering the set of all interpolations between the zero-shot and fine-tuned models. We refer to this measure as accuracy distance to optimal. Formally, accuracy distance to optimal is given by + +$$ +\frac { 1 } { 2 } \left[ \operatorname* { m a x } _ { \alpha } \mathsf { A c c } ( \theta _ { \alpha } , { \mathcal D } _ { \mathrm { s u p p } } ) + \operatorname* { m a x } _ { \alpha } \mathsf { A c c } ( \theta _ { \alpha } , { \mathcal D } _ { \mathrm { p a t c h } } ) \right] - \frac { 1 } { 2 } \operatorname* { m a x } _ { \alpha } \left[ \mathsf { A c c } ( \theta _ { \alpha } , { \mathcal D } _ { \mathrm { s u p p } } ) + \mathsf { A c c } ( \theta _ { \alpha } , { \mathcal D } _ { \mathrm { p a t c h } } ) \right] , +$$ + +where $\operatorname { A c c } ( \theta , { \mathcal { D } } )$ represents the accuracy of model $\theta$ on task $\mathcal { D }$ . In Figure 2 (left), we show that accuracy distance to optimal decreases with scale, indicating that patching becomes more effective for larger models. + +Model similarity. Fine-tuning modifies overparameterized models less [9], which provides insights on why larger models are easier to patch: less movement is required to fit new data. We demonstrate this by evaluating representational similarity using Centered Kernel Alignment (CKA) [34] (see Appendix $\mathrm { D }$ for details). As shown in Figure 2 (center), the representations of the unpatched and fine-tuned models become more similar as models grow larger, indicated by larger CKA values. Moreover, Figure 2 (right) shows that the cosine similarity between the weights of the unpatched and fine-tuned models, $\mathrm { c o s } \bar { ( \theta _ { \mathrm { z s } } , \theta _ { \mathrm { f t } } ) } = { \langle \theta _ { \mathrm { z s } } , \theta _ { \mathrm { f t } } \rangle } / ( { | | \theta _ { \mathrm { z s } } | } { | | \theta _ { \mathrm { f t } } | } { | | } )$ , increases with scale. + +![](images/b72c4fb36424a430ec0f1a0abae638b6b4155247529a5a5b91ced0b946da30bc.jpg) +Figure 2: Larger models are easier to patch (left). For larger models, the unpatched and fine-tuned model are more similar with respect to their representations (center) and weights (right). Model scale is measured in Giga Multiply-Accumulate operations (GMACs). + +![](images/2443ac9955361eaece32bc9fb65f1d64c8d05723641e59f827424243e6ead873.jpg) +Figure 3: The frontier of accuracy trade-offs can be recovered by linearly interpolating weights. Interpolating the unpatched and fine-tuned models recovers the accuracy trade-off of early stopping, regularization towards the initialization, and changes in hyperparameters. Additional details and comparisons can be found in Appendix E. + +# 4.2 Baselines and ablations + +Baselines. There are many alternatives which enable a trade-off between accuracy on the supported and patching tasks. These methods include early stopping during fine-tuning, applying a regularization term which penalizes movement from initialization, or training with different hyperparameters including a smaller learning rate. Unlike interpolation, these methods do not enable navigating the accuracy trade-off without fine-tuning the model again many times. Moreover, Figure 3 demonstrates that the accuracy trade-off frontier for early stopping, regularization, or varying hyperparameters can be recovered by interpolating weights with different mixing coefficients. Appendix $\mathrm { E }$ provides additional baselines and discussion, including EMA [74], EWC [33], LwF [41], re-training a model with data from the patching task, and mixing the pre-training and fine-tuning objectives. + +Additional supported tasks. In Figure 1, we use ImageNet as a representative supported task. This section demonstrates that PAINT is stable under different choices of the supported task. Instead of ImageNet, we use CIFAR10, CIFAR100, Food101 and STL10. Figure 4 displays representative results, where performance is averaged over the nine patching tasks (see Appendix $\mathrm { F }$ for additional results). We observe consistent results across supported tasks, and that the optimal mixing coefficients are stable across different choices of supported tasks (Figure 4, right). + +Additional models. In addition to the CLIP ViTs used in the majority of our experiments, we study four ResNet models [24] from Radford et al. [57] in Appendix G. We find that patching is less effective for ResNets compared to ViTs of similar size, which corroborates the findings of Ramasesh et al. [59] that ResNets are generally more susceptible to catastrophic forgetting. However, similarly to ViTs, we still observe improvements with scale. Finally, we show that patching is also effective for closed-vocabulary models in Appendix H. + +![](images/4fadd75b187a9012519acf4b3f049b196caf35139ece0e9667ceb2b469c2d014.jpg) +Figure 4: Results are consistent across supported tasks. For multiple supported tasks, we observe similar accuracy improvements on patching tasks, without substantially decreasing supported task accuracy. Additional results for the supported tasks Food101, STL10 and ImageNet are in Appendix F. Moreover, choosing the mixing coefficients using a different supported task does not substantially decrease combined accuracy on patching and supported tasks (right). + +# 5 Patching models on multiple tasks + +This section details experimental results for patching on multiple datasets. Recall from Section 2 that there are various strategies for extending PAINT to multiple datasets, which we briefly revisit. For joint patching we merge all the datasets into a single fine-tuning task and apply our patching procedure as before. For sequential patching we iteratively perform our procedure once per task, using the patched model at each step as the initialization for the next step.3 We also explore parallel patching, for which we have an unpatched model $\theta _ { \mathrm { z s } }$ and independently fine-tune on each of the tasks in parallel. We then search for mixing coefficients to combine the resulting models. For tasks $1 , . . . , k$ , let θ(1)ft , . $\theta _ { \mathrm { f t } } ^ { ( 1 ) } , . . . , \theta _ { \mathrm { f t } } ^ { ( k ) }$ denote the fine-tuned models for each task. Since it is impractical to exhaustively search over each $\alpha _ { i }$ , we instead search over a one-dimensional scalar $\alpha \in [ 0 , 1 ]$ , which interpolates between $\theta _ { \mathrm { z s } }$ and the average of all fine-tuned solutions $\begin{array} { r } { \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \theta _ { \mathrm { f t } } ^ { ( i ) } } \end{array}$ .4 Appendix J provides further experimental details. + +These methods have various trade-offs and may be applicable for different scenarios. Joint patching is only possible when data from all tasks you wish to patch is available. On the other hand, sequential patching is appropriate when the tasks are observed one after another. Finally, parallel patching can leverage distributed hardware. + +Figure 5 displays experimental results when patching on all nine tasks from Section 4. We observe that joint patching is the best-performing method on average. This is perhaps unsurprising since joint patching has simultaneous access to all patching datasets, unlike other patching strategies. Nevertheless, it is still interesting that for ViT-L/14, joint patching yields a single model with only 0.5 percentage points worse combined accuracy than using multiple specialized models.5 Joint patching also achieves a 15.8 percentage points improvement over the unpatched model. Moreover, patching a ViT-B/32 model with the joint strategy achieves a combined accuracy 6.1 percentage points higher than a ViT-L/14 unpatched model, which requires $1 2 \mathbf { x }$ more GMACs. + +The accuracy of sequential patching approaches that of joint patching, especially for larger models. Note that, unlike in joint patching, forgetting can compound since the patching procedure is applied multiple times in sequence. In sequential patching, weight interpolations do not completely eradicate forgetting, but greatly mitigate it. This is most noticeable for smaller models: sequentially fine-tuning a ViT-B/32 without interpolation reduces the combined accuracy by 4.6 percentage points compared to the unpatched model, as shown in Appendix J. This is compared to a combined accuracy increase of 11 percentage points when using sequential patching. Additional results, including experiments on SplitCIFAR [61], can be found in Appendix J. + +Finally, parallel patching underperforms other patching strategies. Like sequential patching, parallel patching is in the challenging setting where data from all patching tasks is not available simultaneously. + +![](images/ab4c4ccd3a90497f1b687f69c948c94f7552f13a00a84ff2feb8d39fe792a1b1.jpg) +Figure 5: Contrasting various strategies for patching on multiple tasks. On all experiments, ImageNet is used as the supported task while the other nine datasets are used for patching. When data from all patching tasks is available, joint patching yields a single model that is competitive with using ten different specialized models. Weight interpolations greatly mitigate catastrophic forgetting on the sequential case, but do not completely eradicate it. Finally, parallel patching underperforms other patching strategies, but still provides improvements over the unpatched model. + +
CarsDTDEuroSATGTSRBKITTIMNISTRESISC45SUN397SVHN
Unpatched accuracy86.264.979.951.743.482.673.476.972.8
Patched accuracy87.0 (+0.8)66.1 (+1.2)87.2 (+7.3)71.1 (+19.4)60.4 (+17.0)91.3 (+8.7)74.2 (+0.8)79.3 (+2.4)88.9 (+16.1)
+ +Table 1: PAINT can generalize to unseen classes. We randomly partition each dataset into tasks $A$ and $B$ with disjoint class spaces of roughly equal size. This table reports how patching on task $A$ affects accuracy on task $B$ for the ViT-L/14 model. In all cases, accuracy on task $B$ improves when patching on task $A$ even though the classes are unseen during patching. + +Moreover, unlike in joint or sequential patching, no model is optimized on data from all patching tasks. Using a black box optimization algorithm for finding the mixing coefficients did not yield large improvements over using the same mixing coefficient for all models. However, it is possible that more sophisticated search methods could yield better results. In Appendix J, we present additional experiments for a subset of the tasks where exhaustively searching the space of mixing coefficients is tractable, finding headroom for improvement in most cases. + +# 6 Broad transfer + +An alternative to our patching approach is to introduce parameters which are specific to each new task. By contrast, PAINT always maintains a single model. This section describes an additional advantage of the single model approach: patching the model on task $A$ can improve accuracy on task $B$ , even when task $A$ and $B$ do not share the same classes. We refer to this phenomenon as broad transfer. Note that we are able to study this phenomenon because the single patched model remains open-vocabulary throughout the patching procedure. This is a key advantage of PAINT compared to maintaining a collection of task-specific models. + +We now describe two experiments to measure the effects on a task $B$ when patching the model on a task $A$ . First, we explore broad transfer by randomly partitioning datasets into disjoint sets with no class overlap. For a dataset $\mathcal { D }$ we partition the class space $\mathcal { V }$ into two disjoint sets of roughly equal size $\mathcal { V } _ { A }$ and $\mathcal { { V } } _ { B }$ . We build task $A$ with the examples $( x , y ) \in \mathcal { D }$ where $y$ belongs to $\mathcal { V } _ { A }$ , and task $B$ with examples $( x , y )$ where $y$ belongs to $\mathcal { { V } } _ { B }$ . Table 1 shows how patching a model on task $A$ affects the accuracy on task $B$ for nine datasets $\mathcal { D }$ . The accuracy improvements on task $B$ range from 0.8 to 19.4 percentage points, even though the classes from task $B$ are not seen during patching. + +To further understand transfer, we consider additional task pairs $A$ and $B$ , which are now different datasets. While some pairs $A$ , $B$ share classes, there are still instances of broad transfer. Concretely, + +Table 2: Patching on task $A$ can improve accuracy on a related task $B$ . For a pair of tasks $A$ and $B$ , we report accuracy of the ViT-L/14 on task $B$ , after patching on task $A$ , finding improvements on seven out of eight cases. + +
Task A Task BMNIST SVHN SVHNMNISTRESISC45EuroSAT RESISC45MNIST EuroSAT FashionMNISTFashionMNISTGTSRB MNISTMTSD MTSD GTSRB
Unpatched accuracy58.676.4 71.060.267.776.419.3 50.6
Patched accuracy68.9 93.2 (+10.3) ） (+16.8)69.7 (-1.3)70.4 (+10.2)70.8 (+3.1)77.5 (+1.1)30.8 69.8 (+11.5) (+19.2)
+ +![](images/6e73f1e55022da3d02d7e4fcdb5448dcc6e048f006e9441361fb77781b3537fd.jpg) +Figure 6: Guarding against real-world typographic attacks by patching on synthetic data. (a) A sample from our real-world typographic attacks test set. A CLIP ViT-L/14 is “tricked” into classifying this image as a dog instead of a cat. (b) Sample of synthetic typographic attack data. (c) Performance on real-world data with unseen classes after patching on only synthetic typographic attacks (curves produced by interpolating between the unpatched and fine-tuned model). (d) Analogous curves for the test set of the synthetic data used for patching. + +Table 2 examines i) MNIST and SVHN, two digit recognition tasks with shared classes; ii) EuroSAT and RESISC45, two satellite imagery recognition tasks where there are unshared classes but some overlap; iii) GTSRB and MTSD [17], two traffic sign recognition datasets where there are unshared classes but some overlap; and iv) MNIST and FashionMNIST [83], which do not share any classes but appear visually similar. In seven out of eight experiments, patching on task $A$ improves accuracy by 1.1 to 19.2 percentage points on task $B$ . The exception is when $A$ is EuroSAT and $B$ is RESISC45, where accuracy decreases by 1.3 percentage points. + +In all experiments, when patching on task $A$ we choose the mixing coefficient $\alpha$ by optimizing the held-out validation accuracy on task $A$ and a supported task (in this experiment we use ImageNet). While it is possible for a method that introduces new parameters for each task to exhibit broad transfer to new data, this also requires knowing which parameters to apply for the new data. This is not necessary in the single model approach. + +# 7 Case studies + +We further examine the performance of PAINT in three additional settings, which highlight weaknesses of the zero-shot CLIP model and showcase broad transfer (Section 6). + +Typographic attacks. Goh et al. [23] find that CLIP models are susceptible to typographic attacks, where text superimposed on an image leads to misclassification. For example, in Figure $6 ( a )$ , the text on the pink note saying “dog” leads a CLIP to misclassify the image of a cat as a dog. To fix this vulnerability, we procedurally generate typographic attack data by adding text with incorrect class names to SUN397 [84], as seen in Figure 6 $( b )$ . We then collect a test set of 110 real world images by placing notes on objects and taking photos.6 After applying PAINT using the synthetic data, we evaluate on the real-world images (Figure 6 (c)) and synthetic test set (Figure 6 (d)). We observe that while larger models are more susceptible to typographic attacks, they are also more amenable to patching. Furthermore, we see an example of broad transfer between the synthetic and real-world data: when patching ViT-L/14 on synthetic data, its accuracy on real-world typographic attacks improves 41 percentage points even though the real-world classes are unseen. The cost is a reduction of less than 1 percentage point on ImageNet. We present details on the task and data collection in Appendix K. + +Counting. Radford et al. [57] find that CLIP models struggle to count the number of objects in CLEVR [32]. Here, the task is to choose an integer between 3 and 10 for each image, corresponding to the number of visible objects. While a straightforward way to patch such a task is to fine-tune on it directly, we investigate if applying PAINT using a subset of the classes allows the patched model to generalize to other numbers. Specifically, we patch on images with 4, 5, 6, 8, or 9 objects. To evaluate broad transfer, we test on images with 3, 7, and 10 objects (7 for understanding interpolation and 3 and 10 for extrapolation). We find that PAINT improves accuracy from $59 \%$ to over $9 9 \%$ o n unseen classes with less than half a percentage point decrease in ImageNet accuracy. For more details see Appendix L. + +Visual question answering. As shown by Shen et al. [68], zero-shot CLIP models perform poorly on visual question answering [4]. Using CLIP for VQA typically involves additional parameters—for instance, Shen et al. [68] trains a transformer [77] on CLIP features. In contrast, our procedure for patching CLIP on VQA does not introduce new parameters. Following Shen et al. [68], we contrast images with a series of text prompts, where each prompt corresponds to an option in multiple-choice VQA, formed by both the question and a candidate answer using the following template: “Question: [question text] Answer: [answer text]”. We evaluate on multiple-choice VQA v1 [4], where each question is associated with 18 candidate answers. Our results, further detailed in Appendix M, show that patching is effective for visual question answering: PAINT improves the accuracy of a ViT-L/14 model by 18 percentage points, while accuracy drops by less than one percentage point on ImageNet. + +# 8 Related work + +Continual learning and catastrophic forgetting. Learning tasks sequentially remains a challenge for neural networks. When a neural network learns a new task, the accuracy on other tasks often decreases, a phenomenon known as catastrophic forgetting [48, 76, 20, 33]. While forgetting in neural networks may actually aid learning [90], researchers have proposed various approaches for alleviating catastrophic forgetting, including: i) Regularization-based approaches such as elastic weight consolidation (EWC) [33] and synaptic intelligence (SI) [87] which penalize the movement of parameters and are related to weight-interpolation by Lubana et al. [44]; ii) Replay methods [61, 69, 42, 6, 64, 50], which incorporate data or gradient information from previous tasks when learning a new task; and iii) Introducing task-specific parameters [65, 85, 46, 8, 78, 80]. + +In contrast to these approaches, PAINT requires no modification to the standard fine-tuning process besides the later weight interpolation step. Moreover, unlike regularization or replay based methods, PAINT requires no extra computational cost during training. In contrast to methods with task specific parameters, we maintain a single model. Having a single model is beneficial when there is new data which is similar to one of the tasks which have already been patched. Even without explicitly knowing which task the new data is similar to, we can observe accuracy improvements (see Section 6). + +Similar to our work is that of Mirzadeh et al. [50], who observe high accuracy on task A on the linear path between a model which achieves high accuracy on task A and a model which is fine-tuned jointly on task A and B. Moreover, they observe high accuracy on task B on the linear path between a model fine-tuned on task B, and the jointly fine-tuned model. Therefore, there exists a path between a model which achieves good performance on task A and a model fine-tuned on task B along which accuracy is high on both tasks. However, in Mirzadeh et al. [50] this combined path can be non-linear, leading them to propose a regularization and replay based method. In our work, we find that examining models on a linear path between the unpatched model (which has high accuracy on task A) and the model fine-tuned on task B is often sufficient for obtaining a model which achieves high accuracy on both tasks (Figure 1). We speculate that this is due to scale and model architecture: in contrast to Mirzadeh et al. [50], we initialize with a model pre-trained on a large dataset consisting of 400 million images [57], and primarily use vision transformers [15]. As shown in Section 4.2, our method performs substantially worse with ResNets [24], which are used by Mirzadeh et al. [50]. + +Finally, Ramasesh et al. [59] and Mehta et al. [49] also observed that catastrophic forgetting is less problematic for large and pre-trained models. In addition, Ramasesh et al. [59] found—similar to our results—that vision transformers are less susceptible to forgetting than ResNets of the same size. + +Linear mode connectivity and robust fine-tuning. Linearly interpolating neural network weights is a key step in PAINT. Because of the many nonlinear activations in a neural network, it is not clear a priori that linearly interpolating between two sets of weights can result in a high accuracy solution. However, researchers have observed that interpolating neural network weights can achieve high accuracy when training on MNIST from a common initialization [52] or when part of the optimization trajectory is shared [19, 28, 54, 18, 82, 47, 16, 81, 10]. The term linear mode connectivity was coined by Frankle et al. [19]: two networks exhibit linearly mode connectivity if the accuracy does not decrease when using weights on the linear path between them [52, 19]. Weight averaging for continual learning has also been studied by Lee et al. [40] for closed-vocabulary models. + +While Nagarajan and Kolter [52] and Frankle et al. [19] focused on accuracy on a single task, Wortsman et al. [82] use linear mode connectivity to fine-tune models while preserving their robustness to natural distribution shifts. By interpolating the weights of a zero-shot and fine-tuned model, they find a solution which performs well both on the fine-tuning task and under distribution shift. In contrast to Wortsman et al. [82], we do not modify any task-specific parameters when fine-tuning, preserving the open-vocabulary nature of the models we patch. Unlike Wortsman et al. [82], we examine accuracy trade-offs across different tasks with little or no class overlap and adapt a model to multiple tasks. + +In addition, closely related to our work is that of Matena and Raffel [47], who use Fisher-weighted averaging of language models before and after fine-tuning on downstream tasks. Unlike Fisherweighted averaging of Matena and Raffel [47], we do not use different mixing coefficients for each parameter, and thus require no extra compute when patching. Moreover, we explore new strategies for patching on multiple tasks (see Section 5), and focus on open-vocabulary image classifiers. + +Interventions to change the behavior of a trained model. Several authors have studied the problem of updating a model to locally alter its behavior on certain inputs without external disruptions on other inputs [70, 13, 51, 66, 63, 62]. Previous literature uses various terms to refer to this process, including model editing, patching or debugging. A popular use case is to update trained language models to reflect changes in the world (for instance, facts like who is the current president of Brazil) [29, 45, 38, 30]. Moreover, inspired by software engineering practice, previous work explored “debugging” language models through user interaction [63, 62], including providing corrective feedback to the models via natural language [3]. Mitchell et al. [51], De Cao et al. [13] propose training auxiliary networks to perform local edits on pre-trained models. Santurkar et al. [66] introduce a method for rewriting the prediction rules of a classifier, focusing on specific failure modes such as reliance on spurious correlations. In contrast with previous literature, our work explores patching models at the task level, aiming to systemically improve accuracy on a dataset—for instance, enabling a model to recognize dozens of satellite imagery classes with a single patch. + +# 9 Limitations and conclusion + +Limitations. When applying PAINT, accuracy on supported tasks can still decrease, especially for smaller models. This limitation is perhaps best reflected in the case of sequential patching: patched models underperform using multiple specialized models when many tasks are added sequentially. Using larger models and weight interpolations can alleviate this issue, but do not completely resolve it. Finally, better understanding on which datasets patching is more effective is an exciting direction for future research. + +Conclusion. In this work, we explore several techniques for patching open-vocabulary models with the goal of improving accuracy on new tasks without decreasing accuracy elsewhere. PAINT is effective in several scenarios, ranging from classifying digits to defending against typographic attacks. PAINT becomes more effective with scale, and can be applied on multiple tasks sequentially or simultaneously. Our findings demonstrate that in many circumstances it is possible to expand the set of tasks on which models achieve high accuracy, without introducing new parameters, without re-training them from scratch, and without catastrophic forgetting. + +# Acknowledgments + +We thank Akari Asai, Alex Fang, David Fleet, Huy Ha, Ari Holtzman, Pieter-Jan Kindermans, Marco Tulio Ribeiro, Ofir Press, Sarah Pratt, Sewon Min, Thao Nguyen and Tim Dettmers for helpful discussions and feedback, and Hyak at UW for computing support. This work is in part supported by the NSF AI Institute for Foundations of Machine Learning (IFML), Open Philanthropy, NSF IIS 1652052, NSF IIS 17303166, NSF IIS 2044660, NSF IIS 2132519, ONR N00014-18-1-2826, DARPA N66001-19-2-4031, DARPA W911NF-15-1-0543, the Sloan Fellowship and gifts from Allen Institute for AI. + +References +[1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katie Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning, 2022. https://arxiv.org/abs/2204.14198. +[2] Anders Andreassen, Yasaman Bahri, Behnam Neyshabur, and Rebecca Roelofs. The evolution of out-of-distribution robustness throughout fine-tuning, 2021. https://arxiv.org/abs/ 2106.15831. +[3] Anonymous. Fixing model bugs with natural language patches, 2022. https://openreview. net/forum?id=blJrg3WvvDV. +[4] Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C. 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In Conference on Computer Vision and Pattern Recognition (CVPR), 2022. https://arxiv.org/abs/2203.05557. \ No newline at end of file diff --git a/parse/dev/CZZFRxbOLC/CZZFRxbOLC_content_list.json b/parse/dev/CZZFRxbOLC/CZZFRxbOLC_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a25248dc1e4eff64bbe497e12b6e69dcc1043529 --- /dev/null +++ b/parse/dev/CZZFRxbOLC/CZZFRxbOLC_content_list.json @@ -0,0 +1,1689 @@ +[ + { + "type": "text", + "text": "Patching open-vocabulary models by interpolating weights ", + "text_level": 1, + "bbox": [ + 294, + 123, + 702, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gabriel Ilharco∗1 Mitchell Wortsman∗1 Samir Yitzhak Gadre∗2 Shuran Song2 Hannaneh Hajishirzi1,3 Simon Kornblith4 Ali Farhadi1 Ludwig Schmidt1,3 1University of Washington 2Columbia University 3AI2 4Google Research, Brain Team ", + "bbox": [ + 202, + 224, + 795, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 305, + 535, + 321 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Open-vocabulary models like CLIP achieve high accuracy across many image classification tasks. However, there are still settings where their zero-shot performance is far from optimal. We study model patching, where the goal is to improve accuracy on specific tasks without degrading accuracy on tasks where performance is already adequate. Towards this goal, we introduce PAINT, a patching method that uses interpolations between the weights of a model before fine-tuning and the weights after fine-tuning on a task to be patched. On nine tasks where zeroshot CLIP performs poorly, PAINT increases accuracy by 15 to 60 percentage points while preserving accuracy on ImageNet within one percentage point of the zero-shot model. PAINT also allows a single model to be patched on multiple tasks and improves with model scale. Furthermore, we identify cases of broad transfer, where patching on one task increases accuracy on other tasks even when the tasks have disjoint classes. Finally, we investigate applications beyond common benchmarks such as counting or reducing the impact of typographic attacks on CLIP. Our findings demonstrate that it is possible to expand the set of tasks on which open-vocabulary models achieve high accuracy without re-training them from scratch. ", + "bbox": [ + 233, + 330, + 766, + 564 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 579, + 310, + 595 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Open-vocabulary models are characterized by their ability to perform any image classification task based on text descriptions of the classes [56]. Thanks to advances in large-scale pre-training, recent examples of open-vocabulary models such as CLIP and BASIC have reached parity with or surpassed important task-specific baselines, even when the open-vocabulary models are not fine-tuned on task-specific data (i.e., in a zero-shot setting) [57, 31, 56, 88, 1, 86]. For instance, the largest CLIP model from Radford et al. [57] used in a zero-shot setting matches the ImageNet accuracy of a ResNet-50 trained on 1.2 million ImageNet images [14, 24]. ", + "bbox": [ + 174, + 603, + 825, + 700 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nevertheless, current open-vocabulary models still face challenges. The same CLIP model that matches a ResNet-50 on ImageNet has lower MNIST accuracy than simple logistic regression in pixel space [57]. Moreover, even when zero-shot models achieve good performance, they are usually still worse than models trained or fine-tuned on specific downstream tasks. ", + "bbox": [ + 176, + 707, + 825, + 762 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To address these issues, several authors have proposed methods for adapting zero-shot models to a task of interest using labeled data [82, 91, 21, 89, 37, 73]. A common practice is to fine-tune the zero-shot model on the task of interest [82, 56]. However, fine-tuned models can suffer from catastrophic forgetting [48, 76, 20, 33], performing poorly on tasks where the zero-shot model initially performed well [2, 82, 56]. Additionally, fine-tuning typically produces a task-specific classification head, sacrificing the flexible text-based API that makes open-vocabulary models so appealing. Whereas an open-vocabulary model can perform any classification task in a zero-shot fashion, a fine-tuned model with a task-specific head can only process the specific task that it was fine-tuned on. This specialization can prevent knowledge obtained by fine-tuning on one task from transferring to other related tasks with different classes. ", + "bbox": [ + 174, + 768, + 825, + 866 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/27454f67282ad59a0cde948bc9974465c3d8f122323b75d055c62f58685f01e7.jpg", + "image_caption": [ + "Figure 1: Patching open-vocabulary models by linearly interpolating weights. We wish to improve accuracy on tasks where a model performs poorly (patching tasks), without degrading performance on tasks where accuracy is already adequate (supported tasks). When interpolating weights of fine-tuned models and zeroshot (unpatched) models, there are intermediate solutions where accuracy improves on the patching task without reducing accuracy on supported tasks. Results are shown for CLIP models [57], averaged over nine patching tasks (Stanford Cars, DTD, EuroSAT, GTSRB, KITTI distance, MNIST, RESISC45, SUN397 and SVHN [35, 11, 25, 71, 22, 39, 7, 12, 84, 53]) and five supported tasks (ImageNet, CIFAR-10, CIFAR-100, STL-10 and Food101 [14, 36, 12, 5]). We apply PAINT separately on each patching task and average results across experiments. The dashed lines illustrate vertical movement from the unpatched models and horizontal movement from the fine-tuned models. " + ], + "image_footnote": [], + "bbox": [ + 176, + 112, + 477, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 357, + 825, + 398 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Another approach to adapting zero-shot models would be to add data from the downstream task to the pre-training dataset and train a new open-vocabulary model from scratch. The resulting model could still perform any classification task, and zero-shot performance may improve on related tasks. However, training large image-text models from scratch can require hundreds of thousands of GPU hours [57, 56, 86], which makes this approach practically infeasible in most settings. ", + "bbox": [ + 174, + 407, + 825, + 477 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we study patching open-vocabulary models, where the goal is to increase accuracy on new target tasks while maintaining the flexibility of the model and its accuracy on other tasks.1 Patching aims to combine the benefits of fine-tuning and re-training from scratch: improved performance on the task of interest, maintaining the flexibility of an open vocabulary, transfer between tasks, and fast adaptation time. Motivated by these goals, we extend existing fine-tuning techniques [82] to open-vocabulary settings, where the class space is not fixed. We introduce Patching with Interpolation (PAINT), a simple, two-step procedure for patching models: first, fine-tune the model on the patching task without introducing any task-specific parameters; then, linearly interpolate between the weights of the model before and after fine-tuning. Linearly interpolating neural network weights [52, 19, 54] has been previously used to improve accuracy on a single task [28, 81] or robustness to distribution shift [82]. Indeed, averaging network weights has been explored in continual learning contexts, although for closed-vocabulary models [40]. ", + "bbox": [ + 174, + 486, + 825, + 651 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "With PAINT, accuracy can improve on new tasks without degrading accuracy on unrelated tasks, as illustrated in Figure 1. For instance, applying PAINT to a CLIP ViT-L/14 [57] independently on nine image classification tasks [35, 11, 25, 71, 22, 39, 7, 84, 53] improves accuracy by 15 to 60 percentage points compared to the unpatched model, while accuracy on ImageNet [14] decreases by less than one percentage point. We also observe a promising trend: patching becomes more effective with model scale (Section 4.1). ", + "bbox": [ + 174, + 660, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Beyond single tasks, we show that models can be patched on multiple tasks (Section 5). When patching on nine image classification tasks simultaneously, a single CLIP ViT-L/14 model is competitive with using one specialized model for each task—the average accuracy difference is less than 0.5 percentage points. ", + "bbox": [ + 174, + 751, + 825, + 806 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Moreover, PAINT enables broad transfer (Section 6): accuracy on related tasks can increase, even when the class space changes. For instance, we partition EuroSAT [25], a satellite image dataset, into two halves with disjoint labels. Patching a ViT-L/14 model on the first half improves accuracy on the second half by 7.3 percentage points, even though the classes are unseen during patching. ", + "bbox": [ + 174, + 815, + 825, + 871 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Finally, we investigate PAINT on case studies including typographic attacks [23], counting [32], and visual question answering [4] (Section 7). For instance, applying PAINT using synthetic typographic attacks leads to a model that is less susceptible to typographic attacks in the real world, improving its accuracy by 41 percentage points. ", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In summary: ", + "bbox": [ + 174, + 154, + 258, + 167 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Even the best pre-trained models are not perfect. We introduce PAINT, a method designed to improve accuracy on new tasks without harming accuracy elsewhere. \n• PAINT incurs no extra computational cost compared to standard fine-tuning, neither during fine-tuning itself nor at inference time. \n• PAINT can also be applied with multiple tasks, providing a single model that is competitive with many specialized models. \n• Applying PAINT with one task can improve accuracy on a related task, even when they do not share the same classes. \n• PAINT improves with model scale, indicating a promising trend for future models. ", + "bbox": [ + 192, + 174, + 826, + 314 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Patching with interpolation (PAINT) ", + "text_level": 1, + "bbox": [ + 173, + 325, + 513, + 343 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This section details our method for patching models on a single and multiple tasks. ", + "bbox": [ + 176, + 349, + 715, + 364 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Patching on a single task. Given an open-vocabulary model with weights $\\theta _ { \\mathrm { z s } }$ and a patching task $\\mathcal { D } _ { \\mathrm { p a t c h } }$ , our goal is to produce a new model $\\theta _ { \\mathrm { p a t c h } }$ which achieves high accuracy on $\\mathcal { D } _ { \\mathrm { { p a t c h } } }$ without decreasing model performance on tasks where accuracy is already acceptable. We let $\\mathcal { D } _ { \\mathrm { s u p p } }$ denote a representative supported task where model performance is adequate, and later show that the method is stable under different choices of $\\mathcal { D } _ { \\mathrm { s u p p } }$ (Section 4.2). The two-step procedure we explore for producing $\\theta _ { \\mathrm { p a t c h } }$ is given below. ", + "bbox": [ + 173, + 371, + 826, + 455 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Step 1. Fine-tune $\\theta _ { \\mathrm { z s } }$ on training data from $\\mathcal { D } _ { \\mathrm { p a t c h } }$ to produce a model with weights $\\theta _ { \\mathrm { f t } }$ . Step 2. For mixing coefficient $\\alpha \\in [ 0 , 1 ]$ , linearly interpolate between $\\theta _ { \\mathrm { z s } }$ and $\\theta _ { \\mathrm { f t } }$ to produce $\\theta _ { \\mathrm { p a t c h } } = ( 1 - \\alpha ) \\cdot \\theta _ { \\mathrm { z s } } + \\alpha \\cdot \\theta _ { \\mathrm { f t } }$ . The mixing coefficient is determined via held-out validation sets for $\\mathcal { D } _ { \\mathrm { s u p p } }$ and $\\mathcal { D } _ { \\mathrm { p a t c h } }$ . We refer to the resulting model as $\\theta _ { \\mathrm { p a t c h } }$ . ", + "bbox": [ + 196, + 472, + 800, + 531 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In our experiments, we do not introduce any additional task-specific parameters when fine-tuning, as discussed in Section 3 and Appendices B and $\\textrm { C }$ . ", + "bbox": [ + 176, + 547, + 823, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Patching on a multiple tasks. In practice, we often want to improve model accuracy on multiple patching tasks D(1)patch $\\mathcal { D } _ { \\mathrm { p a t c h } } ^ { ( 1 ) } , . . . , \\mathcal { D } _ { \\mathrm { p a t c h } } ^ { ( k ) }$ D(k)patch, which can be accomplished with straightforward modifications to the procedure above. We explore three alternatives and examine their relative trade-offs in Section 5: ", + "bbox": [ + 176, + 582, + 825, + 631 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Joint patching, where we merge all the patching tasks $\\mathcal { D } _ { \\mathtt { p a t c h } } ^ { ( i ) }$ into a single task $\\mathcal { D } _ { \\mathrm { p a t c h } }$ before running the patching procedure; \n• Sequential patching, where we iteratively repeat the patching procedure above on each new task \nD(i) and let $\\theta _ { \\mathrm { z s } } \\theta _ { \\mathrm { p a t c h } }$ after each completed iteration; \n• Parallel patching, where we apply the first step on each task in parallel to produce fine-tuned $\\theta _ { \\mathrm { f t } } ^ { ( 1 ) } , . . . , \\bar { \\theta _ { \\mathrm { f t } } ^ { ( k ) } }$ e search for mixing coefficients . $\\alpha _ { i }$ to produce $\\begin{array} { r } { \\theta _ { \\mathrm { p a t c h } } = \\big ( 1 - \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\big ) \\cdot \\theta _ { \\mathrm { z s } } + \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\cdot \\theta _ { \\mathrm { f t } } ^ { ( i ) } } \\end{array}$ ", + "bbox": [ + 192, + 637, + 825, + 758 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For joint and parallel patching we assume access to held-out validation sets for all tasks, while in sequential patching we only assume access to held-out validation sets from the tasks seen so far. Unless mentioned otherwise, we pick the mixing coefficient $\\alpha$ that optimizes average accuracy on the held-out validation sets from the supported and patching tasks. ", + "bbox": [ + 176, + 762, + 826, + 818 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Experimental setup ", + "text_level": 1, + "bbox": [ + 176, + 830, + 369, + 848 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Tasks. We consider a diverse set of image classification tasks from Radford et al. [57]. In most experiments, we use ImageNet [14] as a representative supported task, although we explore other supported tasks in Section 4.2. We categorize tasks into patching tasks or supported tasks based on the accuracy difference between the zero-shot model and a model specialized to the task. A large accuracy difference indicates that the task is a relevant target for patching because the zero-shot model is still far from optimal. Specifically, we consider a subset tasks from Radford et al. [57], categorizing tasks where the linear probes outperform the zero-shot model by over 10 percentage points as patching tasks: Cars [35], DTD [11], EuroSAT [25], GTSRB [71], KITTI [22], MNIST [39], RESISC45 [7], SUN397 [84], and SVHN [53]. We use the remaining tasks as supported tasks: CIFAR10 [36], CIFAR100 [36], Food101 [5], ImageNet [14], and STL10 [12]. We investigate additional patching tasks as case studies in Section 7 and provide further details in Appendix A. ", + "bbox": [ + 174, + 856, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 189 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Models. We primarily use CLIP [57] pre-trained vision transformer (ViT) models [15]. Unless otherwise mentioned our experiments are with the ViT-L/14 model, while Section 4.2 studies ResNets [24]. ", + "bbox": [ + 174, + 199, + 823, + 228 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Fine-tuning on patching tasks. Unless otherwise mentioned, we fine-tune with a batch size of 128 for 2000 iterations using learning rate 1e-5 with 200 warm-up steps with a cosine annealing learning rate schedule and the AdamW optimizer [43, 55] (weight decay 0.1). When fine-tuning, we use the frozen final classification layer output by CLIP’s text tower so that we do not introduce additional learnable parameters. This design decision keeps the model open-vocabulary and does not harm accuracy, as discussed in in Appendices B and C. ", + "bbox": [ + 174, + 238, + 825, + 321 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Evaluation. We use accuracy as the evaluation metric unless otherwise stated. We refer to the average of the mean accuracy on the patching tasks and the mean accuracy on the supported tasks as combined accuracy.2 ", + "bbox": [ + 174, + 333, + 825, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Patching models on a single new task ", + "text_level": 1, + "bbox": [ + 174, + 391, + 514, + 407 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As shown in Figure 1, when patching a model on a single task, we interpolate the weights of the zero-shot and fine-tuned model, producing a model that achieves high accuracy on both the patching task and the supported task. On the nine tasks, PAINT improves the accuracy of ViT-L/14 by 15 to 60 percentage points, while accuracy on ImageNet decreases by less than one percentage point. PAINT also allows practitioners to control the accuracy trade-off on the patching and supported tasks without re-training a new model, by varying the mixing coefficient $\\alpha$ . ", + "bbox": [ + 174, + 416, + 825, + 501 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 The effect of scale ", + "text_level": 1, + "bbox": [ + 174, + 512, + 338, + 527 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We consistently observe that PAINT is more effective for larger models. Our findings are aligned with those of Ramasesh et al. [59], who observed that larger models are less susceptible to catastrophic forgetting. This section formalizes and provides insights for these observations. ", + "bbox": [ + 174, + 532, + 825, + 575 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Measuring the effectiveness of patching. We measure the effectiveness of patching via the accuracy difference between the single patched model and two specialized models with the same architecture and initialization. For both the supported task and patching task, we take specialized models that maximize performance on the task, considering the set of all interpolations between the zero-shot and fine-tuned models. We refer to this measure as accuracy distance to optimal. Formally, accuracy distance to optimal is given by ", + "bbox": [ + 174, + 585, + 825, + 670 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8b7c17fc63adfde3b41dd8fdcc1945f8f446229365359c91fe06bd64742dcd34.jpg", + "text": "$$\n\\frac { 1 } { 2 } \\left[ \\operatorname* { m a x } _ { \\alpha } \\mathsf { A c c } ( \\theta _ { \\alpha } , { \\mathcal D } _ { \\mathrm { s u p p } } ) + \\operatorname* { m a x } _ { \\alpha } \\mathsf { A c c } ( \\theta _ { \\alpha } , { \\mathcal D } _ { \\mathrm { p a t c h } } ) \\right] - \\frac { 1 } { 2 } \\operatorname* { m a x } _ { \\alpha } \\left[ \\mathsf { A c c } ( \\theta _ { \\alpha } , { \\mathcal D } _ { \\mathrm { s u p p } } ) + \\mathsf { A c c } ( \\theta _ { \\alpha } , { \\mathcal D } _ { \\mathrm { p a t c h } } ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 179, + 678, + 797, + 708 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\operatorname { A c c } ( \\theta , { \\mathcal { D } } )$ represents the accuracy of model $\\theta$ on task $\\mathcal { D }$ . In Figure 2 (left), we show that accuracy distance to optimal decreases with scale, indicating that patching becomes more effective for larger models. ", + "bbox": [ + 176, + 718, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Model similarity. Fine-tuning modifies overparameterized models less [9], which provides insights on why larger models are easier to patch: less movement is required to fit new data. We demonstrate this by evaluating representational similarity using Centered Kernel Alignment (CKA) [34] (see Appendix $\\mathrm { D }$ for details). As shown in Figure 2 (center), the representations of the unpatched and fine-tuned models become more similar as models grow larger, indicated by larger CKA values. Moreover, Figure 2 (right) shows that the cosine similarity between the weights of the unpatched and fine-tuned models, $\\mathrm { c o s } \\bar { ( \\theta _ { \\mathrm { z s } } , \\theta _ { \\mathrm { f t } } ) } = { \\langle \\theta _ { \\mathrm { z s } } , \\theta _ { \\mathrm { f t } } \\rangle } / ( { | | \\theta _ { \\mathrm { z s } } | } { | | \\theta _ { \\mathrm { f t } } | } { | | } )$ , increases with scale. ", + "bbox": [ + 173, + 770, + 825, + 869 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/b72c4fb36424a430ec0f1a0abae638b6b4155247529a5a5b91ced0b946da30bc.jpg", + "image_caption": [ + "Figure 2: Larger models are easier to patch (left). For larger models, the unpatched and fine-tuned model are more similar with respect to their representations (center) and weights (right). Model scale is measured in Giga Multiply-Accumulate operations (GMACs). " + ], + "image_footnote": [], + "bbox": [ + 178, + 93, + 821, + 251 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/2443ac9955361eaece32bc9fb65f1d64c8d05723641e59f827424243e6ead873.jpg", + "image_caption": [ + "Figure 3: The frontier of accuracy trade-offs can be recovered by linearly interpolating weights. Interpolating the unpatched and fine-tuned models recovers the accuracy trade-off of early stopping, regularization towards the initialization, and changes in hyperparameters. Additional details and comparisons can be found in Appendix E. " + ], + "image_footnote": [], + "bbox": [ + 178, + 321, + 820, + 489 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 Baselines and ablations ", + "text_level": 1, + "bbox": [ + 174, + 579, + 375, + 593 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Baselines. There are many alternatives which enable a trade-off between accuracy on the supported and patching tasks. These methods include early stopping during fine-tuning, applying a regularization term which penalizes movement from initialization, or training with different hyperparameters including a smaller learning rate. Unlike interpolation, these methods do not enable navigating the accuracy trade-off without fine-tuning the model again many times. Moreover, Figure 3 demonstrates that the accuracy trade-off frontier for early stopping, regularization, or varying hyperparameters can be recovered by interpolating weights with different mixing coefficients. Appendix $\\mathrm { E }$ provides additional baselines and discussion, including EMA [74], EWC [33], LwF [41], re-training a model with data from the patching task, and mixing the pre-training and fine-tuning objectives. ", + "bbox": [ + 173, + 598, + 825, + 724 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Additional supported tasks. In Figure 1, we use ImageNet as a representative supported task. This section demonstrates that PAINT is stable under different choices of the supported task. Instead of ImageNet, we use CIFAR10, CIFAR100, Food101 and STL10. Figure 4 displays representative results, where performance is averaged over the nine patching tasks (see Appendix $\\mathrm { F }$ for additional results). We observe consistent results across supported tasks, and that the optimal mixing coefficients are stable across different choices of supported tasks (Figure 4, right). ", + "bbox": [ + 174, + 734, + 825, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Additional models. In addition to the CLIP ViTs used in the majority of our experiments, we study four ResNet models [24] from Radford et al. [57] in Appendix G. We find that patching is less effective for ResNets compared to ViTs of similar size, which corroborates the findings of Ramasesh et al. [59] that ResNets are generally more susceptible to catastrophic forgetting. However, similarly to ViTs, we still observe improvements with scale. Finally, we show that patching is also effective for closed-vocabulary models in Appendix H. ", + "bbox": [ + 174, + 828, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/4fadd75b187a9012519acf4b3f049b196caf35139ece0e9667ceb2b469c2d014.jpg", + "image_caption": [ + "Figure 4: Results are consistent across supported tasks. For multiple supported tasks, we observe similar accuracy improvements on patching tasks, without substantially decreasing supported task accuracy. Additional results for the supported tasks Food101, STL10 and ImageNet are in Appendix F. Moreover, choosing the mixing coefficients using a different supported task does not substantially decrease combined accuracy on patching and supported tasks (right). " + ], + "image_footnote": [], + "bbox": [ + 178, + 93, + 818, + 232 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Patching models on multiple tasks ", + "text_level": 1, + "bbox": [ + 174, + 327, + 490, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This section details experimental results for patching on multiple datasets. Recall from Section 2 that there are various strategies for extending PAINT to multiple datasets, which we briefly revisit. For joint patching we merge all the datasets into a single fine-tuning task and apply our patching procedure as before. For sequential patching we iteratively perform our procedure once per task, using the patched model at each step as the initialization for the next step.3 We also explore parallel patching, for which we have an unpatched model $\\theta _ { \\mathrm { z s } }$ and independently fine-tune on each of the tasks in parallel. We then search for mixing coefficients to combine the resulting models. For tasks $1 , . . . , k$ , let θ(1)ft , . $\\theta _ { \\mathrm { f t } } ^ { ( 1 ) } , . . . , \\theta _ { \\mathrm { f t } } ^ { ( k ) }$ denote the fine-tuned models for each task. Since it is impractical to exhaustively search over each $\\alpha _ { i }$ , we instead search over a one-dimensional scalar $\\alpha \\in [ 0 , 1 ]$ , which interpolates between $\\theta _ { \\mathrm { z s } }$ and the average of all fine-tuned solutions $\\begin{array} { r } { \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\theta _ { \\mathrm { f t } } ^ { ( i ) } } \\end{array}$ .4 Appendix J provides further experimental details. ", + "bbox": [ + 173, + 352, + 826, + 512 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These methods have various trade-offs and may be applicable for different scenarios. Joint patching is only possible when data from all tasks you wish to patch is available. On the other hand, sequential patching is appropriate when the tasks are observed one after another. Finally, parallel patching can leverage distributed hardware. ", + "bbox": [ + 174, + 518, + 825, + 574 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 5 displays experimental results when patching on all nine tasks from Section 4. We observe that joint patching is the best-performing method on average. This is perhaps unsurprising since joint patching has simultaneous access to all patching datasets, unlike other patching strategies. Nevertheless, it is still interesting that for ViT-L/14, joint patching yields a single model with only 0.5 percentage points worse combined accuracy than using multiple specialized models.5 Joint patching also achieves a 15.8 percentage points improvement over the unpatched model. Moreover, patching a ViT-B/32 model with the joint strategy achieves a combined accuracy 6.1 percentage points higher than a ViT-L/14 unpatched model, which requires $1 2 \\mathbf { x }$ more GMACs. ", + "bbox": [ + 173, + 582, + 825, + 693 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The accuracy of sequential patching approaches that of joint patching, especially for larger models. Note that, unlike in joint patching, forgetting can compound since the patching procedure is applied multiple times in sequence. In sequential patching, weight interpolations do not completely eradicate forgetting, but greatly mitigate it. This is most noticeable for smaller models: sequentially fine-tuning a ViT-B/32 without interpolation reduces the combined accuracy by 4.6 percentage points compared to the unpatched model, as shown in Appendix J. This is compared to a combined accuracy increase of 11 percentage points when using sequential patching. Additional results, including experiments on SplitCIFAR [61], can be found in Appendix J. ", + "bbox": [ + 173, + 699, + 825, + 810 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Finally, parallel patching underperforms other patching strategies. Like sequential patching, parallel patching is in the challenging setting where data from all patching tasks is not available simultaneously. ", + "bbox": [ + 174, + 818, + 825, + 847 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/ab4c4ccd3a90497f1b687f69c948c94f7552f13a00a84ff2feb8d39fe792a1b1.jpg", + "image_caption": [ + "Figure 5: Contrasting various strategies for patching on multiple tasks. On all experiments, ImageNet is used as the supported task while the other nine datasets are used for patching. When data from all patching tasks is available, joint patching yields a single model that is competitive with using ten different specialized models. Weight interpolations greatly mitigate catastrophic forgetting on the sequential case, but do not completely eradicate it. Finally, parallel patching underperforms other patching strategies, but still provides improvements over the unpatched model. " + ], + "image_footnote": [], + "bbox": [ + 178, + 92, + 816, + 313 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b1b7945c0a9dae4162342796b36ab8d9a3bbc2e1486bcf37a2891a6e0877e091.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CarsDTDEuroSATGTSRBKITTIMNISTRESISC45SUN397SVHN
Unpatched accuracy86.264.979.951.743.482.673.476.972.8
Patched accuracy87.0 (+0.8)66.1 (+1.2)87.2 (+7.3)71.1 (+19.4)60.4 (+17.0)91.3 (+8.7)74.2 (+0.8)79.3 (+2.4)88.9 (+16.1)
", + "bbox": [ + 174, + 411, + 823, + 481 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 1: PAINT can generalize to unseen classes. We randomly partition each dataset into tasks $A$ and $B$ with disjoint class spaces of roughly equal size. This table reports how patching on task $A$ affects accuracy on task $B$ for the ViT-L/14 model. In all cases, accuracy on task $B$ improves when patching on task $A$ even though the classes are unseen during patching. ", + "bbox": [ + 173, + 487, + 825, + 542 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Moreover, unlike in joint or sequential patching, no model is optimized on data from all patching tasks. Using a black box optimization algorithm for finding the mixing coefficients did not yield large improvements over using the same mixing coefficient for all models. However, it is possible that more sophisticated search methods could yield better results. In Appendix J, we present additional experiments for a subset of the tasks where exhaustively searching the space of mixing coefficients is tractable, finding headroom for improvement in most cases. ", + "bbox": [ + 174, + 556, + 825, + 640 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 Broad transfer ", + "text_level": 1, + "bbox": [ + 174, + 652, + 330, + 670 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "An alternative to our patching approach is to introduce parameters which are specific to each new task. By contrast, PAINT always maintains a single model. This section describes an additional advantage of the single model approach: patching the model on task $A$ can improve accuracy on task $B$ , even when task $A$ and $B$ do not share the same classes. We refer to this phenomenon as broad transfer. Note that we are able to study this phenomenon because the single patched model remains open-vocabulary throughout the patching procedure. This is a key advantage of PAINT compared to maintaining a collection of task-specific models. ", + "bbox": [ + 173, + 676, + 825, + 773 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now describe two experiments to measure the effects on a task $B$ when patching the model on a task $A$ . First, we explore broad transfer by randomly partitioning datasets into disjoint sets with no class overlap. For a dataset $\\mathcal { D }$ we partition the class space $\\mathcal { V }$ into two disjoint sets of roughly equal size $\\mathcal { V } _ { A }$ and $\\mathcal { { V } } _ { B }$ . We build task $A$ with the examples $( x , y ) \\in \\mathcal { D }$ where $y$ belongs to $\\mathcal { V } _ { A }$ , and task $B$ with examples $( x , y )$ where $y$ belongs to $\\mathcal { { V } } _ { B }$ . Table 1 shows how patching a model on task $A$ affects the accuracy on task $B$ for nine datasets $\\mathcal { D }$ . The accuracy improvements on task $B$ range from 0.8 to 19.4 percentage points, even though the classes from task $B$ are not seen during patching. ", + "bbox": [ + 173, + 780, + 825, + 877 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To further understand transfer, we consider additional task pairs $A$ and $B$ , which are now different datasets. While some pairs $A$ , $B$ share classes, there are still instances of broad transfer. Concretely, ", + "bbox": [ + 174, + 882, + 821, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/74e1ef3194d2e86f89d81b7c5d43b6d3ff0d39963db8ec0e8d0836f2286d38c9.jpg", + "table_caption": [ + "Table 2: Patching on task $A$ can improve accuracy on a related task $B$ . For a pair of tasks $A$ and $B$ , we report accuracy of the ViT-L/14 on task $B$ , after patching on task $A$ , finding improvements on seven out of eight cases. " + ], + "table_footnote": [], + "table_body": "
Task A Task BMNIST SVHN SVHNMNISTRESISC45EuroSAT RESISC45MNIST EuroSAT FashionMNISTFashionMNISTGTSRB MNISTMTSD MTSD GTSRB
Unpatched accuracy58.676.4 71.060.267.776.419.3 50.6
Patched accuracy68.9 93.2 (+10.3) ） (+16.8)69.7 (-1.3)70.4 (+10.2)70.8 (+3.1)77.5 (+1.1)30.8 69.8 (+11.5) (+19.2)
", + "bbox": [ + 174, + 87, + 825, + 170 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6e73f1e55022da3d02d7e4fcdb5448dcc6e048f006e9441361fb77781b3537fd.jpg", + "image_caption": [ + "Figure 6: Guarding against real-world typographic attacks by patching on synthetic data. (a) A sample from our real-world typographic attacks test set. A CLIP ViT-L/14 is “tricked” into classifying this image as a dog instead of a cat. (b) Sample of synthetic typographic attack data. (c) Performance on real-world data with unseen classes after patching on only synthetic typographic attacks (curves produced by interpolating between the unpatched and fine-tuned model). (d) Analogous curves for the test set of the synthetic data used for patching. " + ], + "image_footnote": [], + "bbox": [ + 181, + 231, + 815, + 359 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2 examines i) MNIST and SVHN, two digit recognition tasks with shared classes; ii) EuroSAT and RESISC45, two satellite imagery recognition tasks where there are unshared classes but some overlap; iii) GTSRB and MTSD [17], two traffic sign recognition datasets where there are unshared classes but some overlap; and iv) MNIST and FashionMNIST [83], which do not share any classes but appear visually similar. In seven out of eight experiments, patching on task $A$ improves accuracy by 1.1 to 19.2 percentage points on task $B$ . The exception is when $A$ is EuroSAT and $B$ is RESISC45, where accuracy decreases by 1.3 percentage points. ", + "bbox": [ + 174, + 477, + 825, + 575 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In all experiments, when patching on task $A$ we choose the mixing coefficient $\\alpha$ by optimizing the held-out validation accuracy on task $A$ and a supported task (in this experiment we use ImageNet). While it is possible for a method that introduces new parameters for each task to exhibit broad transfer to new data, this also requires knowing which parameters to apply for the new data. This is not necessary in the single model approach. ", + "bbox": [ + 174, + 583, + 825, + 652 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 Case studies ", + "text_level": 1, + "bbox": [ + 174, + 667, + 308, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We further examine the performance of PAINT in three additional settings, which highlight weaknesses of the zero-shot CLIP model and showcase broad transfer (Section 6). ", + "bbox": [ + 176, + 694, + 823, + 722 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Typographic attacks. Goh et al. [23] find that CLIP models are susceptible to typographic attacks, where text superimposed on an image leads to misclassification. For example, in Figure $6 ( a )$ , the text on the pink note saying “dog” leads a CLIP to misclassify the image of a cat as a dog. To fix this vulnerability, we procedurally generate typographic attack data by adding text with incorrect class names to SUN397 [84], as seen in Figure 6 $( b )$ . We then collect a test set of 110 real world images by placing notes on objects and taking photos.6 After applying PAINT using the synthetic data, we evaluate on the real-world images (Figure 6 (c)) and synthetic test set (Figure 6 (d)). We observe that while larger models are more susceptible to typographic attacks, they are also more amenable to patching. Furthermore, we see an example of broad transfer between the synthetic and real-world data: when patching ViT-L/14 on synthetic data, its accuracy on real-world typographic attacks improves 41 percentage points even though the real-world classes are unseen. The cost is a reduction of less than 1 percentage point on ImageNet. We present details on the task and data collection in Appendix K. ", + "bbox": [ + 174, + 731, + 825, + 883 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 823, + 119 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Counting. Radford et al. [57] find that CLIP models struggle to count the number of objects in CLEVR [32]. Here, the task is to choose an integer between 3 and 10 for each image, corresponding to the number of visible objects. While a straightforward way to patch such a task is to fine-tune on it directly, we investigate if applying PAINT using a subset of the classes allows the patched model to generalize to other numbers. Specifically, we patch on images with 4, 5, 6, 8, or 9 objects. To evaluate broad transfer, we test on images with 3, 7, and 10 objects (7 for understanding interpolation and 3 and 10 for extrapolation). We find that PAINT improves accuracy from $59 \\%$ to over $9 9 \\%$ o n unseen classes with less than half a percentage point decrease in ImageNet accuracy. For more details see Appendix L. ", + "bbox": [ + 173, + 123, + 825, + 248 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Visual question answering. As shown by Shen et al. [68], zero-shot CLIP models perform poorly on visual question answering [4]. Using CLIP for VQA typically involves additional parameters—for instance, Shen et al. [68] trains a transformer [77] on CLIP features. In contrast, our procedure for patching CLIP on VQA does not introduce new parameters. Following Shen et al. [68], we contrast images with a series of text prompts, where each prompt corresponds to an option in multiple-choice VQA, formed by both the question and a candidate answer using the following template: “Question: [question text] Answer: [answer text]”. We evaluate on multiple-choice VQA v1 [4], where each question is associated with 18 candidate answers. Our results, further detailed in Appendix M, show that patching is effective for visual question answering: PAINT improves the accuracy of a ViT-L/14 model by 18 percentage points, while accuracy drops by less than one percentage point on ImageNet. ", + "bbox": [ + 174, + 253, + 825, + 392 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 Related work ", + "text_level": 1, + "bbox": [ + 174, + 405, + 316, + 421 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Continual learning and catastrophic forgetting. Learning tasks sequentially remains a challenge for neural networks. When a neural network learns a new task, the accuracy on other tasks often decreases, a phenomenon known as catastrophic forgetting [48, 76, 20, 33]. While forgetting in neural networks may actually aid learning [90], researchers have proposed various approaches for alleviating catastrophic forgetting, including: i) Regularization-based approaches such as elastic weight consolidation (EWC) [33] and synaptic intelligence (SI) [87] which penalize the movement of parameters and are related to weight-interpolation by Lubana et al. [44]; ii) Replay methods [61, 69, 42, 6, 64, 50], which incorporate data or gradient information from previous tasks when learning a new task; and iii) Introducing task-specific parameters [65, 85, 46, 8, 78, 80]. ", + "bbox": [ + 174, + 429, + 825, + 554 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In contrast to these approaches, PAINT requires no modification to the standard fine-tuning process besides the later weight interpolation step. Moreover, unlike regularization or replay based methods, PAINT requires no extra computational cost during training. In contrast to methods with task specific parameters, we maintain a single model. Having a single model is beneficial when there is new data which is similar to one of the tasks which have already been patched. Even without explicitly knowing which task the new data is similar to, we can observe accuracy improvements (see Section 6). ", + "bbox": [ + 174, + 560, + 825, + 643 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Similar to our work is that of Mirzadeh et al. [50], who observe high accuracy on task A on the linear path between a model which achieves high accuracy on task A and a model which is fine-tuned jointly on task A and B. Moreover, they observe high accuracy on task B on the linear path between a model fine-tuned on task B, and the jointly fine-tuned model. Therefore, there exists a path between a model which achieves good performance on task A and a model fine-tuned on task B along which accuracy is high on both tasks. However, in Mirzadeh et al. [50] this combined path can be non-linear, leading them to propose a regularization and replay based method. In our work, we find that examining models on a linear path between the unpatched model (which has high accuracy on task A) and the model fine-tuned on task B is often sufficient for obtaining a model which achieves high accuracy on both tasks (Figure 1). We speculate that this is due to scale and model architecture: in contrast to Mirzadeh et al. [50], we initialize with a model pre-trained on a large dataset consisting of 400 million images [57], and primarily use vision transformers [15]. As shown in Section 4.2, our method performs substantially worse with ResNets [24], which are used by Mirzadeh et al. [50]. ", + "bbox": [ + 174, + 650, + 825, + 829 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, Ramasesh et al. [59] and Mehta et al. [49] also observed that catastrophic forgetting is less problematic for large and pre-trained models. In addition, Ramasesh et al. [59] found—similar to our results—that vision transformers are less susceptible to forgetting than ResNets of the same size. ", + "bbox": [ + 176, + 837, + 825, + 877 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Linear mode connectivity and robust fine-tuning. Linearly interpolating neural network weights is a key step in PAINT. Because of the many nonlinear activations in a neural network, it is not clear a priori that linearly interpolating between two sets of weights can result in a high accuracy solution. However, researchers have observed that interpolating neural network weights can achieve high accuracy when training on MNIST from a common initialization [52] or when part of the optimization trajectory is shared [19, 28, 54, 18, 82, 47, 16, 81, 10]. The term linear mode connectivity was coined by Frankle et al. [19]: two networks exhibit linearly mode connectivity if the accuracy does not decrease when using weights on the linear path between them [52, 19]. Weight averaging for continual learning has also been studied by Lee et al. [40] for closed-vocabulary models. ", + "bbox": [ + 173, + 883, + 820, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 188 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "While Nagarajan and Kolter [52] and Frankle et al. [19] focused on accuracy on a single task, Wortsman et al. [82] use linear mode connectivity to fine-tune models while preserving their robustness to natural distribution shifts. By interpolating the weights of a zero-shot and fine-tuned model, they find a solution which performs well both on the fine-tuning task and under distribution shift. In contrast to Wortsman et al. [82], we do not modify any task-specific parameters when fine-tuning, preserving the open-vocabulary nature of the models we patch. Unlike Wortsman et al. [82], we examine accuracy trade-offs across different tasks with little or no class overlap and adapt a model to multiple tasks. ", + "bbox": [ + 174, + 195, + 825, + 291 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In addition, closely related to our work is that of Matena and Raffel [47], who use Fisher-weighted averaging of language models before and after fine-tuning on downstream tasks. Unlike Fisherweighted averaging of Matena and Raffel [47], we do not use different mixing coefficients for each parameter, and thus require no extra compute when patching. Moreover, we explore new strategies for patching on multiple tasks (see Section 5), and focus on open-vocabulary image classifiers. ", + "bbox": [ + 174, + 299, + 825, + 368 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Interventions to change the behavior of a trained model. Several authors have studied the problem of updating a model to locally alter its behavior on certain inputs without external disruptions on other inputs [70, 13, 51, 66, 63, 62]. Previous literature uses various terms to refer to this process, including model editing, patching or debugging. A popular use case is to update trained language models to reflect changes in the world (for instance, facts like who is the current president of Brazil) [29, 45, 38, 30]. Moreover, inspired by software engineering practice, previous work explored “debugging” language models through user interaction [63, 62], including providing corrective feedback to the models via natural language [3]. Mitchell et al. [51], De Cao et al. [13] propose training auxiliary networks to perform local edits on pre-trained models. Santurkar et al. [66] introduce a method for rewriting the prediction rules of a classifier, focusing on specific failure modes such as reliance on spurious correlations. In contrast with previous literature, our work explores patching models at the task level, aiming to systemically improve accuracy on a dataset—for instance, enabling a model to recognize dozens of satellite imagery classes with a single patch. ", + "bbox": [ + 173, + 372, + 825, + 553 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "9 Limitations and conclusion ", + "text_level": 1, + "bbox": [ + 176, + 565, + 431, + 582 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations. When applying PAINT, accuracy on supported tasks can still decrease, especially for smaller models. This limitation is perhaps best reflected in the case of sequential patching: patched models underperform using multiple specialized models when many tasks are added sequentially. Using larger models and weight interpolations can alleviate this issue, but do not completely resolve it. Finally, better understanding on which datasets patching is more effective is an exciting direction for future research. ", + "bbox": [ + 173, + 590, + 825, + 672 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Conclusion. In this work, we explore several techniques for patching open-vocabulary models with the goal of improving accuracy on new tasks without decreasing accuracy elsewhere. PAINT is effective in several scenarios, ranging from classifying digits to defending against typographic attacks. PAINT becomes more effective with scale, and can be applied on multiple tasks sequentially or simultaneously. Our findings demonstrate that in many circumstances it is possible to expand the set of tasks on which models achieve high accuracy, without introducing new parameters, without re-training them from scratch, and without catastrophic forgetting. ", + "bbox": [ + 174, + 679, + 825, + 776 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments ", + "text_level": 1, + "bbox": [ + 176, + 790, + 328, + 806 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Akari Asai, Alex Fang, David Fleet, Huy Ha, Ari Holtzman, Pieter-Jan Kindermans, Marco Tulio Ribeiro, Ofir Press, Sarah Pratt, Sewon Min, Thao Nguyen and Tim Dettmers for helpful discussions and feedback, and Hyak at UW for computing support. This work is in part supported by the NSF AI Institute for Foundations of Machine Learning (IFML), Open Philanthropy, NSF IIS 1652052, NSF IIS 17303166, NSF IIS 2044660, NSF IIS 2132519, ONR N00014-18-1-2826, DARPA N66001-19-2-4031, DARPA W911NF-15-1-0543, the Sloan Fellowship and gifts from Allen Institute for AI. 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", + "bbox": [ + 174, + 869, + 826, + 911 + ], + "page_idx": 15 + } +] \ No newline at end of file diff --git a/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF.md b/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF.md new file mode 100644 index 0000000000000000000000000000000000000000..5f60029c1a4b7c484ee4a9646831779cad465495 --- /dev/null +++ b/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF.md @@ -0,0 +1,365 @@ +# SOM-CPC: UNSUPERVISED CONTRASTIVE LEARNING WITH SELF-ORGANIZING MAPS FOR STRUCTURED REPRESENTATIONS OF HIGH-RATE TIME SERIES + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Continuous monitoring with an ever-increasing number of sensors has become ubiquitous across many application domains. Acquired data are typically highdimensional and difficult to interpret, but they are also hypothesized to lie on a lowdimensional manifold. Dimensionality reduction techniques have, therefore, been sought for. Recently, expressive non-linear deep learning (DL) models have gained popularity over more conventional methods like Principle Component Analysis (PCA) and Self-Organizing Maps (SOMs). However, the resulting latent space of a DL model often remains difficult to interpret. In this work we propose SOM-CPC, a model that jointly optimizes Contrastive Predictive Coding and a SOM to find an organized 2D manifold, while preserving higher-dimensional information. We address a largely unexplored and challenging set of scenarios comprising highrate time series, and show on both synthetic and real-life data (medical sleep data and audio recordings) that SOM-CPC outperforms both DL-based feature extraction, followed by PCA, K-means or a SOM, and strong deep-SOM baselines that jointly optimize a DL model and a SOM. SOM-CPC has great potential to expose latent patterns in high-rate data streams and may therefore contribute to a better understanding of many different processes and systems. + +# 1 INTRODUCTION + +The improvement and abundance of sensor technology has led to large amounts of high-dimensional, information-rich continuous data streams. However, gaining actionable insights from these data is challenging due to their low interpretability. The main objective of this study is, therefore, to develop an algorithm for acquiring a structured and interpretable representation of (high-rate) time series. We define such an interpretable representation as one that has the ability to be informative and to facilitate exploration of the underlying structure (Lipton, 2018). + +According to the manifold hypothesis, high-dimensional real-world data lies on a low-dimensional manifold, comprising disentangled latent factors of variation. The area of unsupervised representation learning is concerned with models that learn this manifold from a set of training data, without the bias of human annotations. Dimensionality reduction techniques like Principle Component Analysis (PCA), possibly in combination with clustering methods like K-means clustering, have conventionally been used for this purpose. Acquiring an interpretable representation with PCA requires omitting many principle components in order to achieve an interpretable number of components. This, however, may discard important information that can not linearly be projected on these few dimensions. A Self-Organizing Map (Kohonen, 1990), on the other hand, is an extension of K-means clustering that creates a low-dimensional interpretable visualization, while still representing the data in multiple dimensions. However, SOMs typically act on features, which need to be selected heuristically and may, therefore, strongly depend on the use case and/or data modality. + +Deep learning (DL) models have become popular alternatives for non-linear dimensionality reduction that can be applied directly on raw data. Such models have been combined with joint clustering objectives in the latent space (Xie et al., 2016; Yang et al., 2017; Madiraju, 2018; Lee & Schaar, 2020). These methods, however, do typically not create a (visually) interpretable representation, and sometimes make use of label information during training (Lee & Schaar, 2020). To enhance interpretability, latent space representations of DL models are often visualized using a t-distributed stochastic neighbor embedding (t-SNE) (Hinton & Roweis, 2002). Albeit its frequent use, t-SNE does not allow a direct deployment on unseen data as it does not learn a reusable mapping between the multi-dimensional and the low-dimensional space. + +To acquire visually interpretable data representations from raw data, without assuming that data must live in two or three dimensions only, non-linear DL encoders have been combined with SOMs (Ferles et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest et al., 2021). In the resulting joint training strategy of these deep-SOM models, the SOM objective can be seen as a regularizer on the encoding procedure, as it promotes a cluster-friendly feature space. Most of these models have focused on autoencoders as feature extractors. However, similar to Mrabah et al. (2020), we hypothesize that their reconstruction objective may hamper the clustering or structured representation learning objective: while within-cluster similarities should remain preserved for latent clustering, reconstruction demands a preservation of all factors of similarity. Moreover, in the context of time series representation learning, other self-supervised models - that take the temporal nature of the data into account during training - might be more suitable. + +Contrastive self-supervised learning approaches have quickly become popular thanks to their superior representation learning performance in many domains (see Le-Khac et al. (2020) for a review). While many of these models rely on data augmentations during training in order to construct pairs of similar data points, Contrastive Predictive Coding (CPC) (Oord et al., 2019) leverages the temporal dimension for this purposes, making it a natural choice for self-supervised representation learning of time series. In CPC, the temporal dimension not only serves as a pretext task, but simultaneously enforces latent smoothness over time. The contributions of this work are as follows: + +• We propose a new model in the deep-SOM family: SOM-CPC, which is suitable for learning structured and interpretable 2D representations of (high-rate) time series by encoding subsequent data windows to a topologically ordered set of quantization vectors. • Using regression and classification probing tasks, we show that SOM-CPC preserves more information in its 2D representation than CPC that is followed by PCA, and a linear classifier or K-means, or directly encoding CPC’s latent space to two dimensions. SOM-CPC’s joint optimization, moreover, facilitates a smooth temporal trajectory through 2D space. • We show that SOM-CPC quantitatively and qualitatively outperforms deep-SOM models with a reconstruction objective in terms of both clustering and topological ordering. It, moreover, requires less auxiliary loss functions (and associated hyperparameter tuning) thanks to its natural tendency to incorporate temporal smoothness. Lastly, SOM-CPC’s training behavior shows that the SOM clustering objective better aligns with the CPC objective than with a reconstruction loss. + +# 2 PRELIMINARIES + +# 2.1 KOHONEN SELF-ORGANIZING MAPS + +Kohonen’s Self-Organizing Map (SOM) (Kohonen, 1990) is an algorithm to find a visually interpretable topological data representation. It has been found useful to reveal intricate patterns and structure in a plethora of applications. The algorithm’s output, the low-dimensional visualization, is often referred to as a SOM as well. We choose to use a use a 2D visualization to enhance interpretability. + +We define a set of data points $\mathcal { Z }$ , and quantized counterparts $q _ { \Phi } ( z ) \in \Phi$ for $z \in { \mathcal { Z } }$ . The set $\Phi : \{ \phi _ { 1 } , \ldots , \phi _ { k } \}$ is a trainable quantization codebook containing $k$ vectors or prototypes $\phi _ { i } \in$ $\mathbb { R } ^ { F } , 1 \le i \le k$ . The jth prototype $\boldsymbol \phi ^ { ( n ) } = q _ { \Phi } ( z )$ is the ‘winning vector’ for data point $_ { z }$ , at iteration $\cdot$ of the training procedure. The learned codebook vectors are placed on a pre-defined 2D grid by assigning an xy-coordinate to each vector at initialization. Note that this creates a 2D representation, while each data point $_ z$ still lives in $\mathbb { R } ^ { F }$ , with $F \gg 2$ . This is conceptually different than the way in which PCA achieves dimensionality reduction to 2D, where all information in the $3 ^ { \mathrm { r d } }$ and higher principle components is strictly omitted. During training of a SOM, each $\phi _ { i }$ is updated as follows (Kohonen, 1990), with $z \in { \mathcal { Z } }$ : + +$$ +\phi _ { i } ^ { ( n + 1 ) } = \phi _ { i } ^ { ( n ) } + \eta ^ { ( n ) } S _ { i } \big ( \phi ^ { ( n ) } \big ) \big ( z - \phi _ { i } ^ { ( n ) } \big ) , +$$ + +where $\eta ^ { ( n ) }$ is a time-decreasing learning rate. Topological neighborhood structure is promoted via a neighbourhood kernel $s$ that weighs nodes inversely proportional to their distance with the winning node. A Gaussian kernel is often used which weighs node $i$ according to: + +$$ +\begin{array} { r l } & { S _ { i } \big ( \phi ^ { ( n ) } \big ) = \exp \Big ( - \frac { d _ { j , i } ^ { ( n ) } } { 2 ( \sigma ^ { ( n ) } ) ^ { 2 } } \Big ) , \quad \mathrm { ~ w i t h ~ } } \\ & { d _ { j , i } ^ { ( n ) } = | | \mathcal { P } \{ \phi ^ { ( n ) } \} , \mathcal { P } \{ \phi _ { i } ^ { ( n ) } \} | | _ { 2 } ^ { 2 } \quad \quad \mathrm { ~ a n d ~ } \quad \quad \sigma ^ { ( n ) } = \sigma ^ { ( 0 ) } \exp ( - n / \lambda ) , } \end{array} +$$ + +where $\mathcal { P }$ projects a codebook vector to its corresponding coordinate on the grid, $\boldsymbol { \sigma } ^ { ( 0 ) }$ denotes the initial standard deviation, and $\lambda$ the decay factor. Setting $\lambda = - n _ { \mathrm { m a x } } / \log \left( \sigma ^ { ( n _ { \mathrm { m a x } } ) } / \sigma ^ { ( 0 ) } \right)$ sweeps $\sigma$ between $\boldsymbol { \sigma } ^ { ( 0 ) }$ and $\sigma ^ { n _ { \mathrm { m a x } } }$ in $n _ { \mathrm { m a x } }$ steps. The dependence of $s _ { i }$ on the distance $d _ { j , i }$ , implies a weighing of 1 for the winning node (i.e. distance equals zero), and lower than 1 for neighbour nodes. Note that other neighbourhood structures have been proposed as well, for example using the four closest neighbours on the grid, which results in a kernel with a plus-shape (Fortuin et al., 2019). + +# 2.2 DEEP-SOM MODELS + +All deep-SOM research has focused on combining autoencoders (Ferles et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest et al., 2021) with a SOM. These models can broadly be summarized as a vector-quantized (VQ) VAE (van den Oord et al., 2017), with a topological organization of the vectors in the quantization codebook: the SOM. The models are trained end-to-end using error backpropagation of both a reconstruction task loss $\mathcal { L } _ { \mathrm { t a s k } }$ and a loss $\mathcal { L } _ { \mathrm { t o p o } }$ that encourages topological ordering in the SOM. In general, a deep-SOM training objective takes the following form: + +$$ +\mathcal { L } _ { \mathrm { d e e p . 5 o M } } = \mathcal { L } _ { \mathrm { t a k } } + \alpha \mathcal { L } _ { \mathrm { t o p o } } , \qquad ( 4 ) \quad \mathrm { ~ w i t h ~ } \quad \mathcal { L } _ { \mathrm { t o p o } } ( z ^ { ( n ) } ) = \mathbb { E } _ { \mathcal { Z } } \Big [ \sum _ { i = 1 } ^ { k } S _ { i } \big ( \phi _ { i = j } ^ { ( n ) } \big ) \big | | z ^ { ( n ) } - \phi _ { i } ^ { ( n ) } | \big | _ { 2 } ^ { 2 } \Big ] . +$$ + +Hyperparameter $\alpha$ controls the trade-off. The topological loss thus replaces the original update rule of the SOM algorithm (see eq. (1)). The features $z \in { \mathcal { Z } }$ are jointly optimized, and thus also depend on $n$ now. To prevent clutter we will, however, omit the (n)-superscript in the following. + +Fortuin et al. (2019) propose the SOM-VAE model. As opposed to VQ-VAE, SOM-VAE has two decoders, as it also decodes the continuous latents. Topological organization of the codebook vectors is enforced by using a plus-shaped neighbourhood kernel, which affects the codebook vectors of the direct neighbours of the winning node (i.e. up, down, left, and right on the grid). The encoder parameters are, however, unaffected by the quantization error of these neighbour nodes. To facilitate the latter, the topological loss was split in a commitment loss (committing the winning codebook vector to $_ z$ and vice versa) and a $S O M$ loss (pulling the codebook vectors of the neighbours to $z$ ): $\begin{array} { r } { \mathcal { L } _ { \mathrm { t o p o } } = \mathcal { L } _ { \mathrm { c o m m i t m e n t } } + \frac { \beta } { \alpha } \mathcal { L } _ { \mathrm { S O M } } } \end{array}$ . Formally: + +$$ +\mathcal { L } _ { \mathrm { c o m m i n e n t } } = \mathbb { E } _ { \mathcal { Z } } \Big [ \big \| | \boldsymbol { z } - \phi _ { i } | \big \| _ { 2 } ^ { 2 } \Big ] , \quad ( 6 ) \qquad \mathrm { a n d } \quad \mathcal { L } _ { \mathrm { s o m } } = \mathbb { E } _ { \mathcal { Z } } \Big [ \sum _ { \substack { i = 1 , i \neq j } } ^ { k } S _ { i } \big ( \phi \quad \big ) | | \operatorname { s g } [ \boldsymbol { z } ] - \phi \quad | | _ { 2 } ^ { 2 } \Big ] , +$$ + +with $\mathrm { s g } [ \cdot ]$ a gradient blocker that impedes gradient updates to the encoder. Note that for $0 < \beta / \alpha < 1$ the proposed neighbourhood plus-kernel is a coarse approximation of the Gaussian kernel. The sum of the reconstruction losses $\mathcal { L } _ { \mathrm { r e c o n , c o n t } }$ and $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n , d i s c } }$ from the continuous and discrete decoder, respectively, yield the total task loss $\mathcal { L } _ { \mathrm { t a s k } }$ , which is combined with the topological loss to create the training objective of the SOM-VAE model. + +SOM-VAE-prob (Fortuin et al., 2019) and (T)-DPSOM (Manduchi et al., 2021) are extensions of SOM-VAE. SOM-VAE-prob enforces smoothness over time by adding a transition loss (multiplied by $\gamma$ ) to optimize a first-order Markov model to learn the node transition probabilities, and a smoothness loss (multiplied by $\tau$ in there work, we will use $\zeta$ here) to minimize the quantization error of highly probable transitions. DPSOM is a probabilistic model, based on a variational autoencoder with a non-degenerate approximate posterior (Kingma & Welling, 2013) with soft cluster assignment and a cluster assignment hardening (CAH) loss (Xie et al., 2016). T-DPSOM additionally incorporates a temporal smoothness loss, and an LSTM, which aims to predict the future latent space. This latter functionality is similar to the future-prediction task that is already naturally embedded in the + +![](images/e3dea705ac52dbf29bfc0d9f7bf4a4a97bc89c2d7e1aa6bf5466407d9678da7a.jpg) +Figure 1: Architectures of different deep-SOM models including the gradient paths in green. a) SOM-VAE (Fortuin et al., 2019) b) DESOM (Forest et al., 2021) c) SOM-CPC (ours). The two decoders in the SOM-VAE model are independent and have their own trainable parameters, while the visualized encoders in the SOMCPC model are all the same (i.e. parameters are shared). The $g _ { \psi }$ block in the SOM-CPC model indicates an autoregressive component (e.g. a GRU), and $\mathcal { Z } _ { p } ^ { \prime }$ refers to a set of drawn negative embeddings. + +CPC objective that we propose as a task loss (see section 3.2). The probabilistic additions in the (T-)DPSOM model, with respect to SOM-VAE, are orthogonal to the developments in this work. + +Forest et al. (2021) propose Deep embedded SOM (DESOM). Compared to SOM-VAE, the decoder on the discrete space is omitted (therewith also $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n , d i s c } } )$ , gradients from $\mathcal { L } _ { \mathrm { S O M } }$ to the encoder are not being blocked (i.e. $\mathrm { s g } [ \cdot ]$ is removed from eq. (7)), and the topological loss $\mathcal { L } _ { \mathrm { t o p o } }$ is given by eq. (2), i.e. with a Gaussian neighbourhood function with decaying variance. In a short work, Forest et al. (2019) speculate about adding an LSTM in the latent space to train a SOM on sequential data, and refer to this model as LSTM-DESOM. Figure 1a-b visualizes the SOM-VAE and DESOM architecture. + +# 3 SOM-CPC + +# 3.1 MOTIVATION + +In this work, we propose the SOM-CPC model, a representation learning model that learns to map windows of time series data to a structured 2D grid. The model jointly optimizes a temporal contrastive learning objective to extract features, and a topological loss that organizes the SOM space. + +In order to learn features that are both suitable for SOM organization and accurately reflect the data, the model should ideally invert the original data generating process (which is in general unknown and implicit). Assuming that this generative process has been highly non-linear, feature learning can be formulated as a non-linear independent component analysis (ICA) problem, which has proven to be non-identifiable (Hyvärinen & Pajunen, 1999). However, recent advances showed that the problem becomes identifiable under the assumed presence of an auxiliary variable (Hyvärinen et al., 2018). Such an auxiliary variable (e.g. a temporal component) is not present in plain autoencoders, but the contrastive learning paradigm was shown to conform to this assumption (Hyvärinen et al., 2018; Zimmermann et al., 2021). This theory is in line with the hypothesis stated by Mrabah et al. (2020) that a reconstruction objective may hamper clustering performance in the latent space. + +# 3.2 ALGORITHMIC DETAILS + +We introduce $\mathcal { X } = \{ \ldots , \substack { x ( t ) , x ( t + 1 ) , \ldots \} }$ , a set of non-overlapping data windows $\pmb { x } ( t ) \in \mathbb { R } ^ { c h \times T }$ , with $c h$ the number of channels, and $T$ the number of samples in the window. For brevity we omit the time index when possible. An encoder, parameterized by $\theta$ , maps each data window $_ { \textbf { \em x } }$ to a latent representation $\bar { \boldsymbol { z } } = f _ { \boldsymbol { \theta } } ( \pmb { x } ) \in \mathbb { R } ^ { F }$ , with $F$ the number of features. The set $\mathcal { Z }$ includes the embeddings of all windows in $\mathcal { X }$ . A causal auto-regressive (AR) module $g _ { \psi }$ parameterized by $\psi$ , e.g. a gated-recurrent unit (GRU), subsequently aggregates the current and $L$ previous embeddings, to generate a (current) context vector $\pmb { c } ( \dot { t } ) \in \dot { \mathbb { R } } ^ { F }$ . Given this context, the pretext task in our SOM-CPC model aims to minimize the prediction error for $P$ future (or ‘positive’) embeddings $z ( t + p )$ , for $p \in \{ 1 , \ldots , P \}$ , compared to this error for $N$ ‘negative’ embeddings. These negatives may be sampled across the dataset, or within the same time-series, and are on the fly encoded to their latent representation during training. The task objective, being the InfoNCE loss (Oord et al., 2019), is defined as: + +$$ +\mathcal { L } _ { \mathrm { t a s k } } : = \mathcal { L } _ { \mathrm { I n f o N C E } } = \frac { 1 } { P } \sum _ { p = 1 } ^ { P } \mathcal { L } _ { p } , \quad \mathrm { w i t h } \quad \mathcal { L } _ { p } = - \frac { \mathbb { E } } { \chi } \bigg [ \log \frac { \exp \Big ( z ( t + p ) \mathbf { W } _ { p } c ( t ) \Big ) } { \sum _ { z ^ { \prime } \in \mathcal { Z } _ { p } ^ { \prime } \cup \{ z ( t + p ) \} } \exp \Big ( z ^ { \prime } \mathbf { W } _ { p } c ( t ) \Big ) } \bigg ] , +$$ + +with $\mathcal { Z } _ { p } ^ { \prime } \subset \mathcal { Z }$ a set of embeddings of drawn negative samples $\left. \mathcal { Z } _ { p } ^ { \prime } \right| = N )$ , and $\mathbf { W } _ { p } \in \mathbb { R } ^ { F \times F }$ a trainable mapping between the current context vector and the $p ^ { \mathrm { t h } }$ future embedding. + +The context vector is not only used to predict future embeddings, it is also the input to the SOM module that selects the winning node. The SOM is optimized using the topological loss $\mathcal { L } _ { \mathrm { t o p o } }$ , as defined in eq. (5), with a Gaussian neighbourhood kernel $s$ , as defined in eq. (2). Depending on the use case, it was found to not always be necessary, or even beneficial (due to higher risk of overfitting), to use an AR module $g _ { \psi }$ to aggregate causal context into the current embedding. If the AR module is not used, the future predictions are made directly from the current (continuous) latent space $z ( t )$ instead of $\mathbf { } c ( t )$ . Likewise, $z ( t )$ rather than $\mathbf { } c ( t )$ is being quantized by the SOM module. Depending on the presence of this AR module, both $\mathcal { L } _ { \mathrm { t o p o } }$ and $\mathcal { L } _ { \mathrm { t a s k } }$ are thus computed on either $z ( t )$ or $\mathbf { } c ( t )$ . + +All model elements are optimized jointly, with the training objective being: $\mathcal { L } _ { \mathrm { S O M - C P C } } = \mathcal { L } _ { \mathrm { t a s k } } +$ $\alpha \mathcal { L } _ { \mathrm { t o p o } }$ , which adheres to the general objective of a deep-SOM model as formulated in eq. (4). Figure 1c provides an overview of the SOM-CPC model, and its gradient paths in green. The initial standard deviation $\boldsymbol { \sigma } ^ { ( 0 ) }$ of the Gaussian kernel (from eq. (2)) was set to half the squared-root of the number of SOM nodes $k$ . Given the square topology of the SOM grid, this setting of $\sigma _ { 0 }$ ensures that the full grid is captured by the neighbourhood kernel at the start of training. Algorithm 1 in appendix A.1 provides pseudocode of the full SOM-CPC algorithm. + +# 3.3 PERFORMANCE EVALUATION + +Forest et al. (2020) provide a taxology of SOM metrics that distinguishes external vs internal and topological vs clustering metrics. External metrics are related to labels (which are not used during unsupervised training), while internal metrics do not depend on such information. Topological metrics assess the topological ordering (i.e. neighbourhood relations) of the SOM, while clustering metrics are more related to, for example, pureness of nodes. + +To evaluate clustering performance, linked to external labels, we leverage purity and the normalized mutual information (NMI). The latter corrects for a high number of clusters (i.e. nodes), which could easily lead to high pureness, but leaves the NMI more conservative. + +Even though scoring high on external metrics is not the main goal of a representation learning model like SOM-CPC, we do report it as it provides an indication of how well information was preserved. To compute regression/classification performance, we first ‘color’ (or label) each node with the most occurring (for discrete labels) or median (for continuous labels) label from the training set. The test set predictions are then converted from node indices to label predictions by using these colorings. Regression performance is expressed as the average squared regression error with the target: $\mathrm { S E } _ { \mathrm { t a r g e t } }$ Classification performance is reported with Cohen’s kappa (Cohen, 1960), a commonly used metric that corrects for correctness by chance. + +Topographic performance is measured using the (internal) topographic error (TE) (Kiviluoto, 1996), which reports the fraction of windows (between 0 and 1) for which the winning and second-best winning node are not neighbours in the SOM (lower is better). Finally, to measure whether a time series conveys a smooth trajectory through SOM space, we measure the average Euclidean distance (denoted $\ell _ { \mathrm { { 2 , s m o o t h } } } )$ between all subsequent windows in each time series. The lower this value, the less frequently large jumps in the 2D map occur. Note that in extreme cases where many windows collapsed to the same node, both the TE and the average $\ell _ { 2 , \mathrm { s m o o t h } }$ metric are artificially pushed down. We can thus only interpret these metrics in conjunction with earlier-mentioned clustering and classification metrics. + +![](images/095d5981ef212a3ff5ed70dacb4a9ae875fce08aa8ac819333ff01b60320f3cd.jpg) +Figure 2: SOMs and regression plots for SOM-VAE (a), DESOM (b) and SOM-CPC (c). Both DESOM and SOM-CPC show a gradual change of frequency over the grid, but the regression error $\mathrm { S E } _ { \mathrm { t a r g e t } }$ is lower for SOM-CPC, which can also be seen from the regression plot, where the predicted window frequencies are plotted against the target frequencies (i.e. training set median label) for the node on which the window was mapped. d) Task loss versus the topological loss for SOM-VAE (with Gaussian neighbourhood), DESOM and SOM-CPC (both with and without $\mathrm { s g } [ \cdot ]$ ). The different curves display various values of $\alpha$ , for which the DESOM model seems most sensitive. The SOM-CPC models follow a smooth optimization curve, minimizing both the task and topological loss, while these losses seem to be more conflicting in SOM-VAE and DESOM training. + +# 4 EXPERIMENTS + +We compare SOM-CPC to several other 2D representation learning methods. First, deep-SOM models with a reconstruction task loss (i.e. SOM-VAE, SOM-VAE-prob, and (GRU-)DESOM). Second, vanilla CPC with a multi-dimensional latent space $F \gg 2$ ) (Oord et al., 2019), followed by PCA for additional dimensionality reduction to 2D. Third, CPC with a 2D latent space ( $F = 2$ ). For the latter two CPC-based models, linear and non-linear read-out is, respectively, tested using a linear neural classifier, and K-means clustering with the same number of clusters as the number of nodes used in SOM-CPC. High-dimensional vanilla CPC ( $F \gg 2$ ) without additional dimensionality reduction is, moreover, tested as well as it sets a baseline for the amount of information that can be preserved given the encoder architecture, while not providing an interpretable 2D representation. The same encoder architecture is used for all models that are compared in a single application domain, and all models are run with the same seed for randomization. Details on model architectures and training settings for the different applications can be found in appendix A.3.1, A.4.2, and A.5.1. + +# 4.1 SYNTHETIC DATA + +Data generation: A synthetic dataset was created, consisting of sinusoids with an initial frequency sampled from a uniform distribution between 20 and $4 0 \ : \mathrm { H z }$ . The frequency of the signals was altered over time according to a random walk process with a step size of $0 . 1 \ : \mathrm { H z }$ . As such, at each time step (i.e. sample), the signal’s frequency either increased with $0 . 1 \ : \mathrm { H z }$ (with probability $p _ { \mathrm { u p } } = 0 . 1$ ), decreased with $0 . 1 \ \mathrm { H z }$ $( p _ { \mathrm { d o w n } } = 0 . 1 )$ , or remained constant $\gamma _ { \mathrm { c } } = 0 . 8 )$ . In case the random walk crossed either 1 or $6 0 \mathrm { H z }$ , the probabilities were (temporarily) altered to $[ p _ { \mathrm { u p } } , p _ { \mathrm { c } } , p _ { \mathrm { d o w n } } ] = [ 0 . 5 , 0 . 5 , 0 ]$ or $[ p _ { \mathrm { u p } } , p _ { \mathrm { c } } , p _ { \mathrm { d o w n } } ] = [ 0 . 0 , 0 . 5 , 0 . 5 ]$ , respectively. All series were finally corrupted with an additive white Gaussian noise vector $\epsilon \sim \mathcal { N } ( 0 , 0 . 0 \dot { 1 } )$ . Formally, each generated signal took the form: $\begin{array} { r } { \pmb { x } [ n ] = \mathrm { s i n } \left( 2 \pi \frac { f [ n - 1 ] + \Delta f } { f _ { s } } n \right) + \epsilon } \end{array}$ , with $f [ n = 0 ] \sim U [ 2 0 , 4 0 ] , \Delta f \sim \mathrm { C a t e g o r i c a l } ( [ p _ { \mathrm { u p } } , p _ { \mathrm { c } } , p _ { \mathrm { d o w n } } ] ) .$ and $f _ { s } = 1 2 8 \mathrm { { H z } }$ the sampling frequency. A total of 200 of such time-series, each of 5 minutes, were generated, and labels were defined per 1-second window by taking the median frequency. The set was randomly divided into a training $\mathit { n } = 1 0 0 $ ), validation $\mathrm { \Delta } n = 5 0 $ ), and test split $\mathit { n } = 5 0$ ). + +![](images/9c64afdd535a9684c986a72e86f2d42657c2e53f3627425cdb320431daaf6b9c.jpg) +Figure 3: Progression of SOM training (nodes are indicated in black) in the SOM-CPC model, using either a Gaussian (top) or plus neighbourhood (bottom) kernel. The PCA projection of the test set latent space is plotted behind the nodes. It can clearly be seen that the Gaussian kernel enforces a more strict organization of SOM nodes, where nodes are non-uniformly quantizing the latent space, placing more nodes at higher density areas. + +Results: Table 1 shows that SOM-CPC outperforms all deep-SOM baselines on all metrics. Figure 6 in appendix A.3.2 shows the PCA projections of the (continuous) latent spaces of the three deep-SOM models indicated with $^ { \textrm { a * } }$ in table 1. It reveals that the latent space disentanglement of the SOM-CPC model is much better than that of the SOM-VAE and DESOM models. Figure 2a-c displays the resulting SOMs (colored with the median test set labels) for the same three models. Uncolored nodes in the SOM were not assigned in the test set. Interestingly, although the SOM for the DESOM and SOM-CPC model look similar, the $\mathrm { S E } _ { \mathrm { t a r g e t } }$ is higher for the DESOM model, which can also be seen from the regression plots below the SOMs. + +The addition of two temporal losses in the SOM-VAE-prob model, as compared to SOM-VAE, did deteriorate the given metrics, even though a range of values for multipliers $\alpha , \gamma$ and $\zeta$ was tested (see table 3 in appendix A.3.2 for the full sweep). The deterioration of the results can be explained by the difficulty of finding the correct scaling factors for these additional losses. Note that the SOM-CPC model automatically incorporates smoothness over time thanks to the nature of the CPC task loss, therewith preventing additional hyperparameter tuning. + +Additionally, we study the optimization behavior of SOM-VAE, DESOM, and SOM-CPC by plotting the progression of the task versus the topological loss during training (see fig. 2d). To make a fair comparison, we plot the SOM-VAE models that are trained with a Gaussian neighbourhood kernel. Different curves in the graphs indicate runs with varying values for $\alpha$ , and the line color’s gradient denotes the training iteration. The SOM-CPC graphs include the models run with and without gradient detachment of $\mathcal { L } _ { \mathrm { S O M } }$ to the encoder. It can be seen that both losses jointly minimize in SOM-CPC training, while there is a counteracting effect visible for SOM-VAE, and a high influence of the value of $\alpha$ for DESOM training. + +Comparing to non-deep-SOM baselines, it can be seen from table 1 that CPC (with $F = 2$ ), and CPC followed by PCA, resulted in a much higher regression error $\mathrm { S E } _ { \mathrm { t a r g e t } }$ than SOM-CPC when using linear read-out. Non-linear K-means clustering improved performance for both cases, but only for CPC with $F = 2$ , performance nearly reached SOM-CPC performance. Later we will see that optimizing CPC with $F = 2$ can hamper optimization for more intricate data spaces (see section 4.3). Interestingly, regression performance of SOM-CPC was found to be even slightly better than that of the vanilla multi-dimensional CPC model (with $F = 1 2 8$ ), both for linear classification and K-means. This could be explained by the additional regularization that the SOM provides in SOM-CPC training. + +We perform several ablation experiments on SOM-CPC, which are reported below the dashed line in table 1. Blocking the gradients of the neighbour nodes with respect to the encoder during training $( \mathcal { L } _ { \mathrm { { S O M } S g [ \cdot ] } }$ column) slightly improved the regression error and temporal smoothness, but decreased the topographic error. Looking at the models with various values for $\alpha$ (reported in table 3, appendix A.3.2), the effect of gradient blocking can be considered small and ambiguous when considering different metrics. Disjoint training $( \mathbf { C P C } + \mathbf { S O M } )$ resulted in a less smooth trajectory over time through the 2D SOM space, seen from the higher $\ell _ { 2 , \mathrm { s m o o t h } }$ . Using a plus neighbourhood kernel instead of a Gaussian kernel decreased performance on all three metrics. The increase in TE, is well explainable by the fact that a plus kernel takes into account fewer neighbours (at least at the start of training) and therefore has more difficulty to find a good topological mapping. Figure 3 shows the development of the SOM node spread (projected on top of a PCA projection of the continuous test set latents), during training for SOM-CPC with the two type of kernels. It can indeed be seen that the Gaussian kernel enforces a more strict topological organization. Interestingly, the kernel does not only influence the codebook vectors, but also seems to influence the organization of the latent space, seen from the differently-shaped PCA projections in the background. + +# 4.2 SLEEP + +We analyse SOM-CPC on subset 3 of the Montreal Archive of Sleep Studies (MASS) database (O’Reilly et al., 2014), consisting of whole-night polysomnography recordings, for which every 30-second window is labelled with a sleep stage label from $\{ \mathrm { N } 1 $ , N2, N3, REM, $\mathrm { W a k e } \}$ . The 62 recordings (from 62 unique subjects) were randomly split into a training $n = 4 8$ ), validation $( n = 8 )$ and hold-out test set $( n = 7$ ). Details on the data preprocessing can be found in appendix A.4.1. + +Table 5 in appendix A.4.3 shows that SOM-CPC again clearly outperformed SOM-VAE and DESOM on all metrics. Whether or not the gradients of the SOM loss were stopped towards the encoder did not greatly influence SOM-CPC performance. Topological ordering, measured by TE, and temporal smoothness $\cdot \ell _ { 2 , \mathrm { s m o o t h } } )$ deteriorated when changing the Gaussian kernel to a plus kernel, or training high-dimensional CPC and SOM disjointly. SOMCPC’s classification performance was higher than that of CPC with $F = 2$ , and CPC followed by PCA. + +![](images/acf92fe50c3eea2c8f7684ee051c86ae9cb1a1f4a96133c190f8934736bc0b6a.jpg) +Figure 4: Deep sleep N3 is isolated from light sleep N1, Wake and REM sleep with a cluster of medium-deep sleep N2. + +Figure 4 shows the test set PCA projection of the latent space of CPC $F = 1 2 8 )$ ), with the K-means nodes as black stars (left), and the SOM (right) trained by the SOM-CPC model (nodes are colored with the most-occurring label in the test set). Both visualizations show similar clustering patterns: deep sleep N3 is isolated from lighter forms of sleep (i.e. N1, Wake and REM sleep) by a thick cluster of medium-deep sleep N2. However, the higher performance of SOM-CPC (see table 5) indicates that more information is preserved in the 2D space resulting from the SOM-CPC model. The size of the nodes in the SOM map of SOM-CPC indicates the average time in the night of windows on that node. A difference is visible in node sizes within the Wake, N2 and N3 clusters, suggesting a possible existence of different sub-categories of sleep within the pre-defined sleep stages. + +# 4.3 AUDIO + +For the audio experiments, we use a subset of the publicly available LibriSpeech dataset (Panayotov et al., 2015). The dataset contains multiple minute-long English voice recordings of 251 different speakers, sampled at $1 6 ~ \mathrm { K H z }$ . We used the publicly available train-test split, as provided by Oord et al. (2019), and created an additional validation set by randomly selecting $2 5 \%$ of the training set. Recordings of the ten speakers with the longest recording time were selected to alleviate computational burden. This resulted in a total of 150.9, 54.6, and 46.5 minutes in the training, validation, respectively test set. The full model and training details can be found in appendix A.5.1. + +Table 7 in appendix A.5.2 shows the results of SOM-CPC (which includes a GRU for this dataset), compared to different variants of the DESOM model. SOM-CPC outperforms all DESOM variants by a wide margin and for all choices of the $\alpha$ parameter. The difference in performance between DESOM and SOM-CPC is also visible in fig. 5. SOM-CPC has clustered the SOM nodes belonging to the same speaker, and seems to group male and female speakers (denoted with the node’s shape), while these effects are not present in the SOM of the GRU-DESOM model. Minimizing the InfoNCE training objective of CPC with $F = 2$ was found challenging for this dataset, which resulted in non-competitive performance of the linear classifier and K-means clustering trained on the resulted 2D latent space. Using PCA for dimensionality reduction of the high-dimemnsional CPC latent space (with $F = 5 1 2$ ) performed better, but still inferior to SOM-CPC. + +The SOM of the SOM-CPC model (fig. 5-right) reveals two separate clusters both for speaker 2 (green) and 3 (red). The LibriSpeech corpus contains multiple recordings from each speaker, grouped by (book) chapters from which the speaker was reading. Interestingly, additional analyses revealed that the red and green sub-clusters represented recordings belonging to different chapters: $9 9 . 9 9 \%$ of the test-set windows mapped to the upper red sub-cluster belong to the same chapter, while $9 9 . 9 7 \%$ of the windows in the lower red sub-cluster belong to another chapter read by this speaker. Similarly, $1 0 0 . 0 \%$ of the test set windows in the right green sub-cluster belong to two chapters read by speaker 2, while $9 8 . 9 1 \%$ of the windows in the left green sub-cluster belong to another chapter. An auditory inspection revealed that the room acoustics of the recordings belonging to the chapters in different clusters were different, causing changes in the signals which the SOM-CPC model has picked upon. This division between recordings of the same speaker is not visible in the 2D PCA projection of the CPC (with $\cdot$ ) features, as seen from fig. 8 in appendix A.5.2. + +![](images/12d5cb12915dbf4c542c3627bd53f66f59cbcd84d2ce303a58fa90adcdd92341.jpg) +Figure 5: SOM-CPC is able to better cluster different speakers than GRU-DESOM. Stars denote women and dots are men. + +# 5 DISCUSSION + +We proposed a new member of the deep-SOM family: SOM-CPC, suitable for interpretable 2D representation learning of high-rate data streams. Earlier proposed deep-SOM models mainly used reconstruction objectives. In general, SOM-CPC outperformed these models with a wide gap on a variety of metrics. Moreover, it implicitly enforces temporal smoothness, while autoencoder-based models require additional losses and hyperparameter tuning to achieve this. SOM-CPC’s task loss was found to align better with the topological SOM objective than a reconstruction loss, as already hypothesized by Mrabah et al. (2020). While for some applications CPC could succesfully be trained with a 2D latent space directly, optimization was found to be hampered in case of more intricate data spaces. Compared to vanilla CPC with a multi-dimensional latent space, SOM-CPC enables pattern recognition and knowledge discovery. The SOM objective did not hamper CPC optimization. Even better, in the synthetic setup it had a regularizing effect, resulting in lower regression error than vanilla CPC. The use of a Gaussian neighbourhood kernel, as opposed to a plus kernel, was found to improve the topological ordering in the SOM. No decisive conclusions could be made regarding gradient blocking from the SOM loss towards the encoder parameters. Allowing these gradients to flow did not hurt performance, so for coding simplicity, we would advice to not detach the SOM loss. + +Setting an appropriate stopping criterion for self-supervised (SSL) models is debatable. In the SSL literature models with the best test set performance are sometimes reported (He et al., 2019; Fortuin et al., 2019). This is, however, questionable as it may artificially boost reported performance. As such, we created a validation set to apply early stopping in all experiments. Another challenge arises when dealing with aggregated loss functions, since not all losses may smoothly decay and the weighted summation of losses may result in a different optimal epoch than the sub-losses separately. Besides, classification performance (often used as a proxy for information preservation) does not necessarily align with SOM performance or information preservation (see fig. 7 in appendix A.4.3). + +We believe that SOM-CPC will facilitate knowledge discovery in real-life time series and opens up new research directions for representation learning of time series. Directions include investigation to whether additions like the soft-cluster assignment, cluster hardening loss or a Gaussian latent prior - which have shown to improve the SOM-VAE model (Manduchi et al., 2021) - improve SOM-CPC performance as well. Moreover, the CPC objective assumes slowly (or non-changing) data characteristics within the time frame in which positive samples are drawn. A multi-modal variational future prediction could possibly improve performance for data that do not meet this assumption. + +# REPRODUCIBILITY STATEMENT + +All code used to train and evaluate the models as presented in this paper can be found at https: //anonymous.4open.science/r/SOM-CPC. The details regarding model architectures and training settings for each of the application domains are also presented in appendix A.3.1, A.4.2, and A.5.1. Pseudocode of the proposed SOM-CPC algorithm is given in algorithm 1 in appendix A.1. + +# REFERENCES + +Richard B. Berry, Rita Brooks, Charlene E. Gamaldo, Susan M. Harding, Robin M. Lloyd, Carole L. Marcus, and Bradley V. Vaughn. The AASM manual for the scoring of sleep and associated events. Rules, Terminology and Technical Specifications, Darien, Illinois, American Academy of Sleep Medicine, 176:2012, 2012. + +Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. 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Contrastive Learning Inverts the Data Generating Process. arxiv, 2021. URL http://arxiv. org/abs/2102.08850. + +# A EXPERIMENTAL DETAILS + +This appendix contains all information for full reproducability of the experiments. Domainindependent details on SOM-CPC and its evaluation are provided in appendix A.1, while benchmark implementations are discussed in appendix A.2. Domain-specific settings for the synthetic, sleep and audio experiments are discussed in sections A.3, A.4, and A.5, respectively. + +# A.1 GENERAL DETAILS + +Algorithm 1 provides pseudocode of SOM-CPC, when considering the presence of an AR module. For reproducibility, the full code base can be found at https://anonymous.4open.science/ r/SOM-CPC. + +# Algorithm 1 SOM-CPC + +Input: Dataset $\mathcal { X }$ , model comprising $f _ { \theta }$ , $g _ { \psi }$ , $\{ \mathbf { W } _ { p } \} _ { p = 1 } ^ { P }$ , number of past windows $L$ , # positive samples $P$ , # negative samples $N$ , # SOM nodes $k$ , Gaussian neighbourhood function $s$ with $\sigma ^ { ( n _ { \mathrm { m a x } } ) }$ , $n _ { \mathrm { m a x } }$ , loss trade-off parameter $\alpha$ . +Output: A trained SOM: topologically ordered codebook $\Phi$ that represents data √ $\mathcal { X }$ in 2D. - Initialize the Gaussian kernel according to eq. (2) with: $\begin{array} { r } { \sigma ^ { ( 0 ) } = \frac { 1 } { 2 } \sqrt { k } } \end{array}$ and $\lambda = - n _ { \mathrm { m a x } } / \log \left( \sigma ^ { ( n _ { \mathrm { m a x } } ) } / \sigma ^ { ( 0 ) } \right)$ for $n$ in $n _ { \mathrm { m a x } }$ do - Sample a sequence of datapoints: $[ { \pmb x } ( t - L ) , \dots , { \pmb x } ( t ) ] \sim { \pmb \chi }$ - Define $P$ positive samples: $\{ \pmb { x } ( t + p ) \} _ { p = 1 } ^ { P }$ - Sample $P \times N$ negative samples ${ { \mathcal { X } } ^ { \prime } } \subset { { \mathcal { X } } }$ , with $| { \mathcal { X } } ^ { \prime } | = P \times N$ - Encode - data sequence: $\pmb { c } ( t ) = g _ { \psi } \Big ( f _ { \theta } \big ( [ \pmb { x } ( t - L ) , \ldots , \pmb { x } ( t ) ] \big ) \Big )$ - positive samples: $\{ z ( t + p ) \} _ { p = 1 } ^ { P } = f _ { \theta } \left( \{ \pmb { x } ( t + p ) \} _ { p = 1 } ^ { P } \right)$ - negative samples: $\mathcal { Z } ^ { \prime } = f _ { \theta } \left( \mathcal { X } ^ { \prime } \right)$ - Predict future: $\hat { \mathbf { z } } ( t + p ) = \mathbf { W } _ { p } \dot { \mathbf { c } } ( t )$ , with $1 \leq p \leq P$ - Quantize: $\begin{array} { r l } { \phi } & { { } = \mathrm { S O M } _ { \Phi } \left( \begin{array} { l l l } \end{array} \right. } \end{array}$ $\mathbf { \Psi } ( \mathbf { c } ( t ) )$ - Update $\boldsymbol { \sigma } ^ { ( n ) }$ according to eq. (3) - Compute $\mathcal { L } _ { \mathrm { t o p o } } \big ( \mathbf { c } ( t ) \big )$ and $\mathcal { L } _ { \mathrm { i n f o N C E } }$ according to eq. (5) and eq. (8) - Update trainable parameters $\propto \alpha \mathcal { L } _ { \mathrm { t o p o } }$ and $\mathcal { L } _ { \mathrm { { i n f o N C E } } }$ + +end for + +Several metrics were used to quantify SOM performance, as explained in section 3.3. The purity implementation is taken from the Github implementation of Fortuin et al. (2019), while NMI was computed using the sklearn library (Pedregosa et al., 2011). The topographic error implementation comes from the SOMperf python library (Forest et al., 2020). Finally, the $\ell _ { \mathrm { { 2 , s m o o t h } } }$ distance is computed using the norm function in the linalg library of numpy. + +Section 3.3 already shortly elucidated upon the way in which Cohen’s kappa and $\mathrm { S E } _ { \mathrm { t a r g e t } }$ were computed. For both metrics, each SOM node was labeled/colored with the most-occuring (in case of Cohen’s kappa) or median (in case of $\mathrm { S E } _ { \mathrm { t a r g e t } } )$ label in the training set. Note that it can be questioned whether this node labelling should be done on a training or validation set, or directly on the test set for which performance is reported. In earlier times when more conventional clustering approaches (e.g. K-means) were used, a training/validation/test split was typically not made. As a result, clustering was directly performed on the one and only (test) set, that was also used to label the clusters/nodes. Moving towards deep learning based approaches where more hyperparameters need to be set and overfitting can become a larger problem, we found it necessary to report in this work on a test set that was not used for labelling the nodes and/or setting hyperparameters. It should thus be taken into account, that direct comparison to results in other deep-clustering works may need a critical eye to see whether similar procedures were used or not. + +# A.2 BENCHMARKS AND ABLATIONS + +To benchmark our implementation of the SOM-VAE model (and the very similar DESOM model), we replicated the results on MNIST. MNIST was not used for further experimentation with SOM-CPC in the main body of this paper since this work focuses on high-rate time series. Using $k = 1 6$ SOM nodes, Fortuin et al. (2019) report purity $= 0 . 7 3 1 \pm 0 . 0 0 4$ and $\mathbf { N M I } = 0 . 5 9 4 \pm 0 . 0 0 4$ in table 1, and purity $= 0 . 7 2 1 \pm 0 . 0 0 6$ and $\mathbf { N M I } = 0 . 5 8 7 \pm 0 . 0 0 3$ in table S1. All numbers are averages and standard errors over 10 runs. + +Settings that were not provided in the paper, were taken from the hard-coded settings that we found in the provided code base. Instead of splitting the standard MNIST training set in a training and test split, as done by Fortuin et al. (2019), we used the available train/test split that comes with the standard MNIST dataloader from Pytorch. From the first ten runs, one run fully collapsed and resulted in extremely poor performance. Considering this run as an outlier, we run an $1 1 ^ { \mathrm { t h } }$ run and here report the average and standard errors of the 10 non-collapsed runs: purity $= 0 . 7 0 5 \pm 0 . 0 0 2$ and $\mathbf { N M I } = 0 . 5 8 4 \pm 0 . 0 0 1$ . Our performance does reasonably well match the reported performance by Fortuin et al. (2019), given the fact that a different test set of MNIST was used for our experiments. + +To restrict the search space of hyperparameters in the SOM-VAE(-prob) model, which has multiple loss multipliers, we fixed some of the these multipliers for further experiments in this paper. Multiplier $\beta$ scales the SOM loss that sums the quantization error of the four neighbour nodes in the plus kernel. It is, therefore, expected to be at least 4 times larger than the commitment loss, which only reflects the quantization error of the winning node. Choosing $\beta = \alpha / 4$ would imply that the weighing of the summed quantization error of the four neighbours is equal to the weighing of this error for the winning node. To give the winning node a slightly higher importance, we set $\beta = \alpha / 5 = 0 . 2 \alpha$ . The authors of SOM-VAE (Fortuin et al., 2019) used, instead, a search strategy to find the optimal setting. Their code base1 shows that the SOM loss was multiplied with 0.9, while $\alpha = 1$ . Taking into account that their implementation of the commitment loss averaged the quantization error of the four neighbour nodes, while we summed the contribution, their effective setting was thus set to $\beta = \textstyle { \frac { 0 . 9 } { 4 } } \alpha = 0 . 2 2 5 \alpha$ , which is close to what we used in our experiments. + +We compared SOM-CPC also against vanilla CPC training followed by a linear classifier or Kmeans clustering, while freezing the encoder parameters. The supervised linear classifier took in all experiments the form of one fully-connected layer, including biases, that was followed by a log-softmax activation for the sleep and audio cases. It was trained using the mean squared error for the synthetic data set, and cross-entropy loss for sleep and audio experiments. K-means clustering was run from the sklearn library, with the default settings. The number of clusters was chosen to be equal to the number of nodes in the SOM-CPC models against which the performance was compared. Also the disjointly-trained SOM had exactly the same settings as the SOM in the SOM-CPC models with which it was compared. + +Several ablation are performed on the SOM-CPC model. We test the effect of propagating gradients of ${ \mathcal { L } } _ { \mathrm { S O M } }$ to the encoder parameters, the difference between using a Gaussian neighbourhoood kernel versus a plus kernel, and the effect of jointly training CPC and the SOM. Moreover, the effect of certain settings that are typically used in the CPC objective are investigated. CPC’s InfoNCE objective for one window $\cdot$ , given in eq. (8), can as follows be generalized to a more general contrastive learning objective: + +$$ +\mathcal { L } _ { p } = - \frac { \mathbb { E } } { \chi } \Big [ \log \frac { \exp \Big ( \sin ( z _ { a } , z _ { p } ) / \tau \Big ) } { \sum _ { z ^ { \prime } \in \mathcal { Z } _ { p } ^ { \prime } \cup \{ z _ { p } \} } \exp \Big ( \sin ( z _ { a } , z _ { p } ^ { \prime } ) / \tau \Big ) } \Big ] , +$$ + +where $z _ { a }$ is the latent space of the current (or anchor) window, $\cdot$ a similarity metric, and the other symbols are equivalent to eq. (8). CPC typically uses $\tau = 1$ , and the dot product as the similarity metric. However, other related contrastive learning objectives, e.g. in SimCLR (Chen et al., 2020), use a temperature value that is often set to 0.07 (Chen et al., 2020; Woo et al., 2022), and a cosine similarity instead of the (unnormalized) dot product. As such, we add ablations where we set $\cdot$ at 1 or 0.07, and use either the dot-product or the cosine similarity as the similarity metric. + +# A.3 SYNTHETIC EXPERIMENTS + +# A.3.1 TRAINING DETAILS + +Table 2 summarizes the encoder and decoder architectures used in this experiment. The output size column in the table uses channels-first notation. The SOM-VAE and DESOM models were found to benefit from a convolutional part of the encoder that did not fully reduce the temporal dimension to size 1. As such, the last convolutional layer of the SOM-CPC encoder was changed for a fully connected layer preceded by a flattening operation for the autoencoder-based models. The SOM-CPC and CPC model are run without an AR module, to make the fairest comparison to the SOM-VAE(prob) and DESOM models, which also do not incorporate such a component. + +Table 2: Model details for the synthetic data experiments in section 4.1. + +
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU(0.01)911same
MaxPool1Dbs ×16× 32-44=·
Dropout (0.1)bs ×16×32--=-
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8----
Conv1Dbs ×64×864Leaky ReLU (0.01)311same
MaxPool1Dbs ×64×2-44-
Dropout (0.1)bs ×64×2----
Conv1Dbs ×128×2128Leaky ReLU (0.01)311same
MaxPool1Dbs ×128 ×1-22=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)911same
MaxPool1Dbs ×16× 32-44-
Dropout (0.1)bs ×16×32----
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)31same
Flattenbs ×512-==
Fully Connectedbs ×128128Leaky ReLU (0.01)=
DecoderforSOM-VAEandDESOM
Fully Connectedbs×512512Leaky ReLU (0.01)
Unflattenbs ×64×8-
Conv1Dbs ×32×832Leaky ReLU (0.01)311same
ConvTranspose1Dbs ×32 × 3232None4410
Conv1Dbs ×16× 3216Leaky ReLU(0.01)711same
ConvTranspose1Dbs ×16× 12816None4410
Conv1Dbs ×1 × 1281Tanh911same
+ +For SOM-CPC, $P = 3$ future predictions (i.e. positive samples) were used, and $N = 3$ negative samples were drawn for each positive sample. The latter were drawn randomly from the entire training set. The standard deviation of the Gaussian neighbourhood kernel was exponentially decayed until $\sigma ^ { ( n _ { \mathrm { m a x } } ) } = 2$ . Choosing a lower value at the end of training induced instable optimization behavior. + +All models (including the benchmarks) were trained using the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 0.001 and a batch size of 128. Each model was trained for maximally 1000 epochs. The best model was selected based on the lowest task loss on the validation set, being $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n } }$ for SOM-VAE and DESOM and $\mathcal { L } _ { \mathrm { { I n f o N C E } } }$ for SOM-CPC and CPC. We did not use the full training objective $\mathcal { L } _ { \mathrm { d e e p - S O M } }$ as model selection criterion, as both the commitment and SOM loss showed to be low initially (possibly due to low values of the random initialization of the model), while both increased and reached a steady-state later in training. The linear classifier and the disjointly-trained SOM on the CPC embeddings were trained until convergence, for maximally 1000 epochs. + +# A.3.2 EXTENDED RESULTS + +Table 3 extends table 1 with additional sweeps of hyperparameters $\alpha , \gamma _ { ; }$ , and $\zeta$ , and ablations with different settings of the temperature value $\tau$ and the used similarity metric. Compared to SOM-VAE, SOM-VAE-prob is expected to show more smooth trajectories over time, captured in the $\ell _ { 2 , \mathrm { s m o o t h } }$ distance metric, thanks to the additional transition and smoothness loss (multiplied by $\gamma$ and $\cdot$ , respectively). It can be seen that tuning these hyperparameters is a complex process, and a sweep did not result in one SOM-VAE-prob run that performed better than the best SOM-VAE model. In contrary, the addition of the extra losses possibly interfered with the optimization process, and only very delicate settings of $\gamma$ and $\cdot$ might improve model performance eventually. A sweep over the topological loss multiplier $\alpha$ , revealed a low sensitivity of SOM-CPC to this value. It can be seen that changing the temperature value and/or the similarity metric did not significantly alter the performance consistently on all reported metrics. + +Table 3: This is the extended version of table 1 on the synthetic data results, including a sweep of hyperparameters $\alpha , \gamma$ and $\zeta$ . Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). The models indicated with $^ { \textrm { a * } }$ were used to depict trained SOMs in fig. 2 and PCA projections in fig. 6. + +
ModelαSLsom sg[-]SEtargetl2.smoothTE
CPC+linearclassifier==2.62±2.37==
CPC + K-means===1.09±.62=
CPC(F=2) + linear classifier25.01±42.94
CPC(F = 2)+K-means.76±1.31
CPC+PCA+ linear classifier42.81±58.12
CPC + PCA+K-means-=4.42±9.01
SOM-VAE1e-3Plus11.59±13.692.60±.46.38±.042
*1e-2Plus14.57±44.102.98±.84.38±.052
.1 1Plus Plus8.02±4.582.41±.68.28±.070
SOM-VAE1e-5Gaussian13.43±3.66 11.12±17.052.75±.45.33±.048
1e-4Gaussian1.93±.29.069±.029
1e-31.52±26.611.95±.36.075±.028
SOM-VAE-prob1e-2Gaussian Gaussian11.60±25.431.92±.34.056±.023
(γ=5e-5,S= 1e-3)1e-318.13±48.662.03±.42.086±.030
(γ = 4e-5,= 1e-3)Plus21.26±55.003.78±.52.93±.042
1e-3Plus21.30±37.374.23±.57.88±.051
(γ=3.3e-5,S= 1e-3) (γ= 5e-4,S= 1e-2)1e-3Plus22.52±48.973.81±.51.98±.015
(γ=4e-4,= 1e-2)1e-2Plus14.65±19.583.70±.51.86±.083
1e-2Plus27.25±72.823.54±.67.94±.022
(γ= 3.3e-4,S= 1e-2)1e-2Plus21.62±38.783.71±.76.97±.014
(γ= 5e-5,S= 1e-2).1Plus26.82±68.843.23±.80.68±.067
(γ=4e-5,= 1e-2).1Plus22.34±64.923.11±.73.74±.072
(γ=3.3e-5,S=1e-2) (γ= 5e-4,=.1).1Plus Plus2.10±48.803.15±.73.63±.050
.14.85±75.963.06±.55.84±.050
(γ= 4e-4,S=.1) (γ=3.3e-4,S=.1).1Plus29.96±46.962.81±.34.82±.13
DESOM.1 1e-5Plus GaussianX3.96±56.943.95±.83.85±.11
1e-4GaussianX14.00±3.10 19.09±44.041.99±.36 1.95±.28.13±.069
1e-3GaussianX12.58±27.101.92±.31.11±.046 .077±.033
1e-2GaussianX13.66±44.711.89±.33.065±.028
.1GaussianX10.77±10.852.20±.44.061±.019
SOM-CPC (ours)1GaussianX22.86±55.712.26±.43.092±.034
*1e-5GaussianX.95±2.401.24±.31.048±.020
1e-4GaussianX.72±1.081.37±.37.022±.011
1e-3GaussianX.81±.751.06±.28.028±.012
1e-2Gaussian Gaussian_X.62±.711.08±.28.059±.035
1 1e-5GaussianX √1.90±4.07 .64±.711.18±.30.039±.016
1e-4Gaussian√.68±.671.04±.26 1.19±.43.039±.020
1e-3Gaussian.75±1.021.12±.32.057±.033
1e-2Gaussian.47±.48.059±.027
.1.99±.24.069±.040
Gaussian√.89±1.501.16±.32.069±.031
1e-4PlusX1.71±1.152.46±0.51.30±0.062
Gaussian1e-2 1e-4Plus√1.16±.61 1.47±2.601.85±.26 1.15±.34.12±.037 .014±.015
SOM-CPC(T = O.07,sim = cosine sim.) SOM-CPC(τ = 1,sim = cosine sim.) 1e-4 Gaussian
+ +![](images/348104e47be032cc7d8471ede902b7d2ad2ea136cd2b21a97db4e35e8521f89c.jpg) +Figure 6: PCA projections of the continuous latent spaces of the full test set for the SOM-VAE, DESOM, and SOM-CPC models of which the SOMs were visualized in fig. 2. Disentanglement of the signal frequencies is much better in the SOM-CPC model. + +Figure 6 shows PCA projections of the latent space of the SOM-VAE, DESOM, and SOM-CPC models that are indicated with a \* in tables 1 and 3, and for which the SOMs were visualized in fig. 2. Disentanglement of the signal frequencies is much better for the SOM-CPC model, providing an explanation for the better (i.e. lower) $\mathrm { S E } _ { \mathrm { t a r g e t } }$ of this model, as compared to SOM-VAE and DESOM. + +# A.4 SLEEP EXPERIMENTS + +# A.4.1 DATA PROCESSING + +For the experiments on sleep data, we used subset 3 of the publicly available Montreal Archive of Sleep Studies (MASS) database (O’Reilly et al., 2014), consisting of 62 whole-night polysomnography recordings. Each recording contains, among others, electroencephalography (EEG), chin electromyography (EMG), and electrooculography (EOG) data. We refer the reader to O’Reilly et al. (2014) for more details regarding this dataset. We selected the channels that are typically used in clinical practice, comprising three EEG channels (F4, C4, O2), the two EOG channels, and one chin EMG derivation, and downsampled the data to $1 2 8 \ : \mathrm { H z }$ . Sleep stage labels that follow the guidelines of the American Academy of Sleep Medicine (AASM) Berry et al. (2012) (being wakefulness (Wake), rapid-eye movement (REM) sleep, or non-REM1 till non-REM3 (N1, N2, N3)) were available for every non-overlapping 30-second window, the common label resolution in clinical practice. + +Some processing had already been done by the distributors of the MASS dataset (O’Reilly et al., 2014). The $6 0 \mathrm { H z }$ powerline interference was, however, not fully suppressed, and we wanted to down sample each signal to $1 2 8 \ : \mathrm { H z }$ to reduce computational complexity. As such, before downsampling, all derivations were additionally filtered with a zero-phase (i.e. two-directional) $5 ^ { \mathrm { t h } }$ order Butterworth band-pass filter $( 0 . 3 - 5 9 \ : \mathrm { H z } )$ , followed by another zero-phase $5 ^ { \mathrm { t h } }$ order Butterworth notch filter $( 5 9 - 6 1 \ : \mathrm { H z } )$ . Channels were normalized within-patient and per channel, yielding mean subtraction, followed by normalization such that amplitudes of $9 5 \%$ of the samples were mapped between $^ { - 1 }$ and $+ 1$ . The 62 recordings (numbered $1 - 6 4$ , with number 43 and 49 missing) were split into a training set including patients $1 - 4 8$ ${ \mathrm { ' } n = 4 7 }$ ), a validation set including patients $5 0 - 5 7$ ${ \mathrm { \Delta } n = 8 }$ ), and hold-out test set that included patients $5 8 - 6 4$ ( ${ \mathrm { ~ \it ~ n ~ } } = { \mathrm { ~ \it ~ 7 ~ } }$ ). + +# A.4.2 TRAINING DETAILS + +Dimensionality reduction of polysomnography data was done by encoding all selected channels in each non-overlapping 30-second window using standard convolutional encoder. Table 4 summarizes the used encoder and decoder (for SOM-VAE and DESOM) architectures. The latent space for decoding in the SOM-VAE and DESOM benchmark models was not fully reduced to a 1D vector to enhance training. Nevertheless, the last adaptive average pooling layer that was used in the encoder of SOM-CPC, was applied in the bottleneck of SOM-VAE and DESOM before SOM quantization took place. As a result the feature vectors in all models were of size $F = 1 2 8$ . The decoder architecture (see table 4) was used both for the continuous and discrete decoding in the SOM-VAE model (without weight tying). No AR-component was used in the SOM-CPC model to make a fair comparison to the SOM-VAE and DESOM model that also did not include such a component. + +For the SOM-CPC model, $P = 3$ future predictions (i.e. positive samples) were used, and $N = 3$ negative samples were drawn for each positive sample. The latter were drawn from the same subject as the positive sample. The $\sigma$ of the Gaussian neighbourhood kernel was exponentially annealed to σ(nmax) = 0.5 during training. + +All models were trained with the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 1e-4 and a batch size of 128. Each model was trained for maximally 500 epochs, and the best model was selected based on the lowest $\scriptstyle { \mathcal { L } } _ { \mathrm { r e c o n } }$ (for SOM-VAE and DESOM) or $\mathcal { L } _ { \mathrm { { I n f o N C E } } }$ (for SOM-CPC and CPC) on the validation set. + +# A.4.3 EXTENDED RESULTS + +Table 5 shows the quantitative results on sleep data, comparing SOM-CPC against deep-SOM models (SOM-VAE and DESOM) and disjoint training of CPC, followed by either a supervised linear classifier, K-means or a SOM. Discussion of the main results in this table can be found in section 4.2. The ablation experiments in which the value of $\cdot$ and/or the similarity metric was altered show that classification and clustering performance slightly dropped when using the cosine similarity with a temperature value of 1, while the topographic organization slightly improved (i.e. lower TE). These effects vanished when using a temperature of $\cdot$ . Both runs showed worse temporal smoothness (i.e. higher $\cdot$ ). As expected, only changing the temperature value to 0.07 did almost not affect results, suggesting that the linear projector heads were able to adjust for this scaling factor. + +Table 4: Model details for the sleep experiments in section 4.2. + +
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU (0.01)15110
MaxPool1Dbs ×16×32-55=
Dropout (0.1)bs ×16× 32---=
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55
Dropout (0.1)bs ×32×8-=-==
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2-=
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
AdaptiveAvgPool1Dbs ×128×1=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)1511(18,17)
MaxPool1Dbs ×16×32-55
Dropout (0.1)bs ×16×32--==
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55=
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2--==
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
DecoderforSOM-VAEandDESOM
Conv1Dbs×64×264Leaky ReLU(0.01)3110
ConvTranspose1Dbs ×64×264None5510
Conv1Dbs ×32×232Leaky ReLU (0.01)5110
ConvTranspose1Dbs ×32×232None5510
Conv1Dbs ×16×216Leaky ReLU (0.01)9110
ConvTranspose1Dbs ×16×216None5510
Conv1Dbs ×6×26None15110
+ +We also tested the performance when using the SimCLR (Chen et al., 2020) objective for the task loss, instead of the CPC objective. SimCLR is also a contrastive learning framework, but instead of drawing positive samples from the future latent space, these samples are created by applying augmentations on the anchor window. Inspired by Um et al. (2017) we used the following augmentations: independent and identically distributed Gaussian noise $\mathcal { N } ( 0 , 0 . 0 5 )$ was added (called jitter in their implementation), each channel was scaled with a value drawn from $\cdot$ , windows were split in 4 sub-windows of minimal 2 seconds and randomly permuted, and lastly time series were both time warped and magnitude warped. The latter two augmentations make use of smooth curves that smoothly vary the positions of time stamps or magnitude values, respectively. + +Besides the difference on how to create positive samples, the originally proposed SimCLR model has some other slight differences with respect to the CPC model: + +• The SimCLR loss uses the cosine similarity, while CPC uses the (unnormalized) dot product as the similarity metric (see eq. (9)). +• SimCLR uses an additional temperature $\tau$ in its loss function (see eq. (9)), for which the value is often set to 0.07 (Chen et al., 2020; Woo et al., 2022). CPC does not incorporate such a temperature, which effectively means that it uses a value of 1. +• SimCLR uses a non-linear MLP projection head, while CPC uses linear projection heads. +• SimCLR uses negative samples from within the batch, while this is not specified in the CPC paper. This specified design choice makes SimCLR typically very sensitive to the batch size. +SimCLR was not proposed to include an auto-regressive component, and can not straightforwardly be extended to do so, while CPC can be implemented with or without such a module. + +For the most fair comparison, the procedure for drawing negative samples in SimCLR is done equivalently as for SOM-CPC, i.e. within the recording, instead of within the batch. However, in the SOM-SimCLR model (i.e. the joint training of SOM with SimCLR), each drawn negative sample is added to the set of negative samples both in its raw form, and with a random augmentation, which effectively doubles the number of negative samples. Table 3 reports the performance of the baseline SOM-SimCLR model (i.e. with settings $\tau = 0 . 0 7$ and the cosine similarity), and variants using a temperature value of 1 and/or the dot product as the similarity metric. All settings regarding training procedure and the SOM were set equivalently as in the SOM-CPC training. Table 5 shows that SOM-SimCLR results for $\cdot$ are better than those with $\tau = 1$ , which is in line with findings from Chen et al. (2020); Woo et al. (2022). However, even with $\cdot$ , performance of SOM-SimCLR is lower on all metrics compared to the SOM-CPC model with the same value for $\alpha$ . The higher $\ell _ { \mathrm { { 2 , \mathrm { { s m o o t h } } } } }$ metric of SOM-SimCLR indicates on average larger jumps over the SOM map through time, which might be caused by the fact that the SimCLR task objective does not incorporate temporal information, while InfoNCE does exploit this. Training time of SOM-SimCLR was, moverover, considerably longer than SOM-CPC with the same settings due to the additional augmentations that need to be computed for every data window and its negative samples. + +Table 5: Test set performance of various models trained on sleep recordings. SOMs of models with a \* are visualized in fig. 4. Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). + +
ModelCPC + linear classifierLsom sg[-]PurityNMICohen's kappa
CPC + K-means CPC(F= 2) + linear classifier- - =- =- - --.68±.10
-=.79 -.29.61±.11 .52±.10 .55±.090-- =
CPC(F= 2)+K-means CPC + PCA+ linear classifier *=.74.24.54±.086
CPC +PCA +K-means - 1e-3 Plus 1e-2 Plus-.77.26 .57±.082
SOM-VAE.71 .23.51±.042.36±.26.24±.031
√ √.71.23 .51±.042.67±.17 2.60±.34.30±.037 .28±.042
DESOM√ √.72 .71.23 .23.52±.03 .53±.033.08±.32 .31±.054
1e-6 Gaussian 1e-5 Gaussian 1e-4 Gaussian 1e-3 Gaussian 1e-2 GaussianX X.70 .70.27 .53±.05 .23 .50±.042.14±.32 2.10±.26.095±.020 .11±.028
X.71.22 .51±.042.35±.24 2.40±.16.17±.035
SOM-SimCLR(T = 0.07)X X.71 .71.22 .51±.05 .22 .50±.042.30±.26.22±.0085 .23±.021
1e-3 Gaussian SOM-SimCLR(T = 1) 1e-3 SOM-CPC (ours)X X.73 .70.23 .20.53±.13 .48±.162.21±.35 1.87±.30.29±.026 .50±.068
Gaussian 1e-5 Gaussian 1e-4 Gaussian * 1e-3 GaussianX X X.78 .27 .78 .27 .78.59±.11 .61±.101.03±.11 .041±.014 1.01±.10 .062±.018
1e-2 Gaussian .1 Gaussian 1e-3 Gaussian 1e-2 Gaussian GaussianX X √.27 .79 .28 .79 .28 .78 .27.61±.12 .60±.11 .65±.07 .62王.10 .60±.101.02±.09 .032±.0096 1.08±.11 .19±.042 1.09±.09 .19±.04 1.02±.12 .067±.025
.1 Vaaeais 1 Gaussian √ 1e-3 Plus X 1e-3 Plus √ SOM-CPC(T = 0.07,sim = cosine sim.) 1e-3 Gaussian X SOM-CPC (τ = 1,sim = cosine sim.) X.78 .78 .78 .79 .79 .78 .73 .79.27 .27 .27 .28 .28 .27 .27
1e-3 Gaussian SOM-CPC(τ = 0.07,sim = dot prod.) 1e-3 Gaussian CPC + SOM (disjoint) Gaussian =X .79 = 0.80 Mwv1.43±.15 .025±.0094 1.06±.11 .059±.020 1.21±.11 .52±.042 Cohen's kappa
InfoNCE 1.2 0.10 1.0 0.08Commitment lossSOM loss 5Purity bsl00.30NMI
0.840.75wAb众
0.063 20.70 0.650.25
0.60.0410.600.200.2
0.40.0200.550.150.0Train Validation
0 2000 200400200 4000.50 02004000.10 0 200
400 Epoch0 EpochEpochEpochEpoch400 0200 400 Epoch
+ +Figure 7: Training curves of SOM-CPC (the model indicated with a \* in table 5) for both the training and validation set. The green dashed line indicates the epoch of the used model, i.e. the one with the lowest validation InfoNCE loss. It can be seen that the epoch with the best clustering and classification performance does not necessarily align with the epoch that has the lowest loss commitment and/or SOM loss. + +Figure 7 shows training curves of the training and validation set for the SOM-CPC model that is indicated with a \* in table 5. The green line indicates the epoch with the lowest InfoNCE validation loss. These graphs show that the performance of InfoNCE, the commitment and SOM loss, and classification metrics do not necessarily align, making it dependent on your final goal with the SOM-CPC model what is the most appropriate stopping-criterion. + +# A.5 AUDIO EXPERIMENTS + +# A.5.1 TRAINING DETAILS + +Audio streams were encoded in windows of 0.01 seconds $( = 1 6 0$ samples). For CPC, GRU-DESOM, and SOM-CPC the contextual information of $L = 1 2 7$ previous windows was aggregated using a GRU, equivalent as proposed by Oord et al. (2019). The InfoNCE objective for (SOM-)CPC was computed on top of the last context vector of the GRU. To test different settings for the GRU-desom model, we distinguished GRU-DESOM that decodes only the last window, given the last context vector, and GRU-DESOM that decodes the full sequence of 128 windows from the respective context vectors. + +Table 6 provides the model details of the encoder, and the decoder for the DESOM benchmarks. The reconstruction loss of the DESOM model was found to be hampered in its optimization when using the encoder architecture, as adopted for the SOM-CPC model. As such, the downsampling factor of the encoder was reduced for the DESOM model to enable minimization of the task loss during training. + +Table 6: Model details for the audio experiments in section 4.3. + +
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
Encoder forSOM-CPCand CPC
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512512ReLU4211
GRUbs × 512512--==-
EncoderforSOM-VAEandDESOM
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512 ×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512×2512ReLU411same
Flattenbs ×1024-===
GRUbs ×10241024
DecoderforSOM-VAEandDESOM
Unflattenbs×512×2==
Conv1Dbs × 512×2512ReLU411same
ConvTranspose1Dbs × 512×4512ReLU4211
ConvTranspose1Dbs × 512×8512ReLU4211
ConvTranspose1Dbs × 512× 32512ReLU8412
ConvTranspose1Dbs ×1×160512ReLU10513 (+ output pad = 1)
+ +For training of SOM-CPC, we followed the settings from Oord et al. (2019) and set $P = 1 2$ . The number of negative samples was set to $N = 1 0$ , which were drawn randomly from the entire training set. All deep-SOM models were trained for maximally 3000 epochs, using the Adam optimizer Kingma & Ba (2014) with a learning rate of 1e-4 and a batch size of 8. One epoch was defined as a push through of one sequence of 128 windows (or 1 window for DESOM) from each recording. The best model was selected based on the lowest validation task loss. + +The supervised linear classifier and disjoint SOM training on top of the frozen CPC embeddings were trained with a batch size of 128, and a learning rate of 1e-4 and 1e-2, respectively. Both models were stopped upon convergence of the validation loss (which was after 200 and 250 epochs, respectively). + +# A.5.2 EXTENDED RESULTS + +Quantitative results of the audio experiments can be found in table 7. For this application, Cohen’s kappa is computed as the average over all data windows in the test set, which is different from the synthetic and sleep case, where it was computed as the average and one standard deviation across recordings. In these audio experiments, all windows from one recording contain the same speaker id label. Computing Cohen’s kappa per-recording, i.e. with having the same label for all windows in that recording, is therefore inappropriate as the computation can not correct for correctness by chance. Table 7 show that SOM-CPC clearly outperformed all variants of the (GRU-)DESOM model, and feature extraction using CPC, followed by PCA and linear or non-linear classification. + +Ablations with respect to the temperature $^ { \prime }$ and the similarity metric in the loss function indicated, similarly as in the sleep case, that simply changing the temperature value did hardly affect SOM-CPC performance. However, changing the similarity metric to be the cosine similarity caused a drop in clustering and classification performance, only when using a temperature value of 1. This result overlaps with the experimental findings in the sleep case (see appendix A.4.3), and suggests that a low temperature value - which was found beneficial in the SimCLR objective (Chen et al., 2020; Woo et al., 2022) - does not equivalently improve SOM-CPC performance. + +Figure 8 compares the test set projection on the 2D PCA space, created on the CPC features (with $F = 1 2 8$ ), to the SOM from the SOM-CPC model that is denoted with a $\cdot$ in table 7 (and also visualized in fig. 5-right). + +Table 7: Test set performance of various models trained on audio recordings. SOMs of models with a \* are visualized in fig. 5. Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). + +
ModelαSLsom sg[·]PurityNMICohen'skappaTE
CPC+linearclassifier----1.00-
CPC + K-means==1.00.601.00
CPC(F=2)+ linear classifier==-.00=
CPC(F= 2)+K-means.13.013.025
CPC+PCA+ linear classifier=--.86
CPC + PCA + K-means-=-.89.54..88
DESOM1e-5GaussianX.18.03.06.14±.035
1e-4GaussianX.23.08.11.34±.045
1e-3GaussianX.31.13.20.68±.050
1e-2GaussianX.13-.00.001.0±.00
GRU-DESOM (reconstructing last window)1e-5GaussianX.19.04.08.13±.028
1e-4GaussianX.26.09.15.20±.036
1e-3GaussianX.31.13.21.42±.078
1e-2GaussianX.32.14.22.46±.057
GRU-DESOM (reconstructing full sequence)1e-5GaussianX.19.05.08.34±.048
1e-4GaussianX.30.12.20.59±.072
1e-3GaussianX.33.14.22.78±.048
1e-2GaussianX.29.12.19.57±.069
SOM-CPC(ours)1e-5GaussianX.99.73.99.14±.081
1e-4GaussianX1.00.631.00.24±.12
*1e-3GaussianX1.00.611.00.33±.098
1e-2GaussianX1.00.61.99.33±.099
1e-3Gaussian√1.00.61.99.28±.087
1e-2Gaussian√1.00.61.99.35±0.10
.1Gaussian√1.00.611.00.35±.095
1Gaussian√1.00.611.00.38±.11
SOM-CPC (T = 0.07,sim = cosine sim.)1e-3GaussianX.99.61.99.42±0.12
SOM-CPC(τ = 1,sim = cosine sim.)1e-3GaussianX.88.55.86.17±.063
SOM-CPC(τ = 0.07,sim = dot prod.)1e-3GaussianX1.00.61.99.38±.098
CPC + SOM(disjoint)-Gaussian-1.00.621.00.28±.11
+ +![](images/25f87c5f96cd1f71916e3973abf24aec8831127866c3d6e679b873b8ffb4c103.jpg) +Figure 8: Projecting the test set on the 2D PCA space shows no division of the green and the red clusters in two sub-clusters, something that is visible in the SOM of the SOM-CPC model. These sub-clusters were found to relate to recordings that were made with different room acoustics. \ No newline at end of file diff --git a/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF_content_list.json b/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f4ffbf08bcfae87553201739df997a6a984cf49c --- /dev/null +++ b/parse/dev/DAxQXzdq8SF/DAxQXzdq8SF_content_list.json @@ -0,0 +1,2015 @@ +[ + { + "type": "text", + "text": "SOM-CPC: UNSUPERVISED CONTRASTIVE LEARNING WITH SELF-ORGANIZING MAPS FOR STRUCTURED REPRESENTATIONS OF HIGH-RATE TIME SERIES ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 195, + 398, + 223 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 260, + 544, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Continuous monitoring with an ever-increasing number of sensors has become ubiquitous across many application domains. Acquired data are typically highdimensional and difficult to interpret, but they are also hypothesized to lie on a lowdimensional manifold. Dimensionality reduction techniques have, therefore, been sought for. Recently, expressive non-linear deep learning (DL) models have gained popularity over more conventional methods like Principle Component Analysis (PCA) and Self-Organizing Maps (SOMs). However, the resulting latent space of a DL model often remains difficult to interpret. In this work we propose SOM-CPC, a model that jointly optimizes Contrastive Predictive Coding and a SOM to find an organized 2D manifold, while preserving higher-dimensional information. We address a largely unexplored and challenging set of scenarios comprising highrate time series, and show on both synthetic and real-life data (medical sleep data and audio recordings) that SOM-CPC outperforms both DL-based feature extraction, followed by PCA, K-means or a SOM, and strong deep-SOM baselines that jointly optimize a DL model and a SOM. SOM-CPC has great potential to expose latent patterns in high-rate data streams and may therefore contribute to a better understanding of many different processes and systems. ", + "bbox": [ + 232, + 291, + 766, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 558, + 336, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The improvement and abundance of sensor technology has led to large amounts of high-dimensional, information-rich continuous data streams. However, gaining actionable insights from these data is challenging due to their low interpretability. The main objective of this study is, therefore, to develop an algorithm for acquiring a structured and interpretable representation of (high-rate) time series. We define such an interpretable representation as one that has the ability to be informative and to facilitate exploration of the underlying structure (Lipton, 2018). ", + "bbox": [ + 173, + 589, + 825, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "According to the manifold hypothesis, high-dimensional real-world data lies on a low-dimensional manifold, comprising disentangled latent factors of variation. The area of unsupervised representation learning is concerned with models that learn this manifold from a set of training data, without the bias of human annotations. Dimensionality reduction techniques like Principle Component Analysis (PCA), possibly in combination with clustering methods like K-means clustering, have conventionally been used for this purpose. Acquiring an interpretable representation with PCA requires omitting many principle components in order to achieve an interpretable number of components. This, however, may discard important information that can not linearly be projected on these few dimensions. A Self-Organizing Map (Kohonen, 1990), on the other hand, is an extension of K-means clustering that creates a low-dimensional interpretable visualization, while still representing the data in multiple dimensions. However, SOMs typically act on features, which need to be selected heuristically and may, therefore, strongly depend on the use case and/or data modality. ", + "bbox": [ + 173, + 680, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep learning (DL) models have become popular alternatives for non-linear dimensionality reduction that can be applied directly on raw data. Such models have been combined with joint clustering objectives in the latent space (Xie et al., 2016; Yang et al., 2017; Madiraju, 2018; Lee & Schaar, 2020). These methods, however, do typically not create a (visually) interpretable representation, and sometimes make use of label information during training (Lee & Schaar, 2020). To enhance interpretability, latent space representations of DL models are often visualized using a t-distributed stochastic neighbor embedding (t-SNE) (Hinton & Roweis, 2002). Albeit its frequent use, t-SNE does not allow a direct deployment on unseen data as it does not learn a reusable mapping between the multi-dimensional and the low-dimensional space. ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To acquire visually interpretable data representations from raw data, without assuming that data must live in two or three dimensions only, non-linear DL encoders have been combined with SOMs (Ferles et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest et al., 2021). In the resulting joint training strategy of these deep-SOM models, the SOM objective can be seen as a regularizer on the encoding procedure, as it promotes a cluster-friendly feature space. Most of these models have focused on autoencoders as feature extractors. However, similar to Mrabah et al. (2020), we hypothesize that their reconstruction objective may hamper the clustering or structured representation learning objective: while within-cluster similarities should remain preserved for latent clustering, reconstruction demands a preservation of all factors of similarity. Moreover, in the context of time series representation learning, other self-supervised models - that take the temporal nature of the data into account during training - might be more suitable. ", + "bbox": [ + 173, + 166, + 825, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contrastive self-supervised learning approaches have quickly become popular thanks to their superior representation learning performance in many domains (see Le-Khac et al. (2020) for a review). While many of these models rely on data augmentations during training in order to construct pairs of similar data points, Contrastive Predictive Coding (CPC) (Oord et al., 2019) leverages the temporal dimension for this purposes, making it a natural choice for self-supervised representation learning of time series. In CPC, the temporal dimension not only serves as a pretext task, but simultaneously enforces latent smoothness over time. The contributions of this work are as follows: ", + "bbox": [ + 174, + 325, + 825, + 424 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a new model in the deep-SOM family: SOM-CPC, which is suitable for learning structured and interpretable 2D representations of (high-rate) time series by encoding subsequent data windows to a topologically ordered set of quantization vectors. • Using regression and classification probing tasks, we show that SOM-CPC preserves more information in its 2D representation than CPC that is followed by PCA, and a linear classifier or K-means, or directly encoding CPC’s latent space to two dimensions. SOM-CPC’s joint optimization, moreover, facilitates a smooth temporal trajectory through 2D space. • We show that SOM-CPC quantitatively and qualitatively outperforms deep-SOM models with a reconstruction objective in terms of both clustering and topological ordering. It, moreover, requires less auxiliary loss functions (and associated hyperparameter tuning) thanks to its natural tendency to incorporate temporal smoothness. Lastly, SOM-CPC’s training behavior shows that the SOM clustering objective better aligns with the CPC objective than with a reconstruction loss. ", + "bbox": [ + 217, + 434, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 642, + 339, + 659 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 KOHONEN SELF-ORGANIZING MAPS ", + "text_level": 1, + "bbox": [ + 176, + 671, + 470, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Kohonen’s Self-Organizing Map (SOM) (Kohonen, 1990) is an algorithm to find a visually interpretable topological data representation. It has been found useful to reveal intricate patterns and structure in a plethora of applications. The algorithm’s output, the low-dimensional visualization, is often referred to as a SOM as well. We choose to use a use a 2D visualization to enhance interpretability. ", + "bbox": [ + 174, + 696, + 825, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We define a set of data points $\\mathcal { Z }$ , and quantized counterparts $q _ { \\Phi } ( z ) \\in \\Phi$ for $z \\in { \\mathcal { Z } }$ . The set $\\Phi : \\{ \\phi _ { 1 } , \\ldots , \\phi _ { k } \\}$ is a trainable quantization codebook containing $k$ vectors or prototypes $\\phi _ { i } \\in$ $\\mathbb { R } ^ { F } , 1 \\le i \\le k$ . The jth prototype $\\boldsymbol \\phi ^ { ( n ) } = q _ { \\Phi } ( z )$ is the ‘winning vector’ for data point $_ { z }$ , at iteration $\\cdot$ of the training procedure. The learned codebook vectors are placed on a pre-defined 2D grid by assigning an xy-coordinate to each vector at initialization. Note that this creates a 2D representation, while each data point $_ z$ still lives in $\\mathbb { R } ^ { F }$ , with $F \\gg 2$ . This is conceptually different than the way in which PCA achieves dimensionality reduction to 2D, where all information in the $3 ^ { \\mathrm { r d } }$ and higher principle components is strictly omitted. During training of a SOM, each $\\phi _ { i }$ is updated as follows (Kohonen, 1990), with $z \\in { \\mathcal { Z } }$ : ", + "bbox": [ + 173, + 775, + 825, + 904 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/561ca698c22d04afb1648da2b7bbc46a6dd5d0f4ba9754b6dc952592531533ed.jpg", + "text": "$$\n\\phi _ { i } ^ { ( n + 1 ) } = \\phi _ { i } ^ { ( n ) } + \\eta ^ { ( n ) } S _ { i } \\big ( \\phi ^ { ( n ) } \\big ) \\big ( z - \\phi _ { i } ^ { ( n ) } \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 356, + 905, + 642, + 926 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\eta ^ { ( n ) }$ is a time-decreasing learning rate. Topological neighborhood structure is promoted via a neighbourhood kernel $s$ that weighs nodes inversely proportional to their distance with the winning node. A Gaussian kernel is often used which weighs node $i$ according to: ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/276860374c3d4568a561e9a0ef57351f397e5a75ec8410db28b87c465366004f.jpg", + "text": "$$\n\\begin{array} { r l } & { S _ { i } \\big ( \\phi ^ { ( n ) } \\big ) = \\exp \\Big ( - \\frac { d _ { j , i } ^ { ( n ) } } { 2 ( \\sigma ^ { ( n ) } ) ^ { 2 } } \\Big ) , \\quad \\mathrm { ~ w i t h ~ } } \\\\ & { d _ { j , i } ^ { ( n ) } = | | \\mathcal { P } \\{ \\phi ^ { ( n ) } \\} , \\mathcal { P } \\{ \\phi _ { i } ^ { ( n ) } \\} | | _ { 2 } ^ { 2 } \\quad \\quad \\mathrm { ~ a n d ~ } \\quad \\quad \\sigma ^ { ( n ) } = \\sigma ^ { ( 0 ) } \\exp ( - n / \\lambda ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 151, + 740, + 218 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { P }$ projects a codebook vector to its corresponding coordinate on the grid, $\\boldsymbol { \\sigma } ^ { ( 0 ) }$ denotes the initial standard deviation, and $\\lambda$ the decay factor. Setting $\\lambda = - n _ { \\mathrm { m a x } } / \\log \\left( \\sigma ^ { ( n _ { \\mathrm { m a x } } ) } / \\sigma ^ { ( 0 ) } \\right)$ sweeps $\\sigma$ between $\\boldsymbol { \\sigma } ^ { ( 0 ) }$ and $\\sigma ^ { n _ { \\mathrm { m a x } } }$ in $n _ { \\mathrm { m a x } }$ steps. The dependence of $s _ { i }$ on the distance $d _ { j , i }$ , implies a weighing of 1 for the winning node (i.e. distance equals zero), and lower than 1 for neighbour nodes. Note that other neighbourhood structures have been proposed as well, for example using the four closest neighbours on the grid, which results in a kernel with a plus-shape (Fortuin et al., 2019). ", + "bbox": [ + 173, + 223, + 825, + 311 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 DEEP-SOM MODELS ", + "text_level": 1, + "bbox": [ + 174, + 328, + 359, + 342 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "All deep-SOM research has focused on combining autoencoders (Ferles et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest et al., 2021) with a SOM. These models can broadly be summarized as a vector-quantized (VQ) VAE (van den Oord et al., 2017), with a topological organization of the vectors in the quantization codebook: the SOM. The models are trained end-to-end using error backpropagation of both a reconstruction task loss $\\mathcal { L } _ { \\mathrm { t a s k } }$ and a loss $\\mathcal { L } _ { \\mathrm { t o p o } }$ that encourages topological ordering in the SOM. In general, a deep-SOM training objective takes the following form: ", + "bbox": [ + 173, + 353, + 826, + 452 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1ab7a8d7952e75436d34f53af92f92f8ce9bb07960874c7bc6b3d0683e841b33.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { d e e p . 5 o M } } = \\mathcal { L } _ { \\mathrm { t a k } } + \\alpha \\mathcal { L } _ { \\mathrm { t o p o } } , \\qquad ( 4 ) \\quad \\mathrm { ~ w i t h ~ } \\quad \\mathcal { L } _ { \\mathrm { t o p o } } ( z ^ { ( n ) } ) = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\sum _ { i = 1 } ^ { k } S _ { i } \\big ( \\phi _ { i = j } ^ { ( n ) } \\big ) \\big | | z ^ { ( n ) } - \\phi _ { i } ^ { ( n ) } | \\big | _ { 2 } ^ { 2 } \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 459, + 776, + 500 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Hyperparameter $\\alpha$ controls the trade-off. The topological loss thus replaces the original update rule of the SOM algorithm (see eq. (1)). The features $z \\in { \\mathcal { Z } }$ are jointly optimized, and thus also depend on $n$ now. To prevent clutter we will, however, omit the (n)-superscript in the following. ", + "bbox": [ + 174, + 503, + 825, + 546 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Fortuin et al. (2019) propose the SOM-VAE model. As opposed to VQ-VAE, SOM-VAE has two decoders, as it also decodes the continuous latents. Topological organization of the codebook vectors is enforced by using a plus-shaped neighbourhood kernel, which affects the codebook vectors of the direct neighbours of the winning node (i.e. up, down, left, and right on the grid). The encoder parameters are, however, unaffected by the quantization error of these neighbour nodes. To facilitate the latter, the topological loss was split in a commitment loss (committing the winning codebook vector to $_ z$ and vice versa) and a $S O M$ loss (pulling the codebook vectors of the neighbours to $z$ ): $\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { t o p o } } = \\mathcal { L } _ { \\mathrm { c o m m i t m e n t } } + \\frac { \\beta } { \\alpha } \\mathcal { L } _ { \\mathrm { S O M } } } \\end{array}$ . Formally: ", + "bbox": [ + 173, + 551, + 825, + 667 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b3cebab8223d9c59800ea0a551a1fdb898a275b19b30304ee2b06a13beed48d6.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { c o m m i n e n t } } = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\big \\| | \\boldsymbol { z } - \\phi _ { i } | \\big \\| _ { 2 } ^ { 2 } \\Big ] , \\quad ( 6 ) \\qquad \\mathrm { a n d } \\quad \\mathcal { L } _ { \\mathrm { s o m } } = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\sum _ { \\substack { i = 1 , i \\neq j } } ^ { k } S _ { i } \\big ( \\phi \\quad \\big ) | | \\operatorname { s g } [ \\boldsymbol { z } ] - \\phi \\quad | | _ { 2 } ^ { 2 } \\Big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 179, + 674, + 767, + 717 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "with $\\mathrm { s g } [ \\cdot ]$ a gradient blocker that impedes gradient updates to the encoder. Note that for $0 < \\beta / \\alpha < 1$ the proposed neighbourhood plus-kernel is a coarse approximation of the Gaussian kernel. The sum of the reconstruction losses $\\mathcal { L } _ { \\mathrm { r e c o n , c o n t } }$ and $\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n , d i s c } }$ from the continuous and discrete decoder, respectively, yield the total task loss $\\mathcal { L } _ { \\mathrm { t a s k } }$ , which is combined with the topological loss to create the training objective of the SOM-VAE model. ", + "bbox": [ + 174, + 722, + 825, + 791 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "SOM-VAE-prob (Fortuin et al., 2019) and (T)-DPSOM (Manduchi et al., 2021) are extensions of SOM-VAE. SOM-VAE-prob enforces smoothness over time by adding a transition loss (multiplied by $\\gamma$ ) to optimize a first-order Markov model to learn the node transition probabilities, and a smoothness loss (multiplied by $\\tau$ in there work, we will use $\\zeta$ here) to minimize the quantization error of highly probable transitions. DPSOM is a probabilistic model, based on a variational autoencoder with a non-degenerate approximate posterior (Kingma & Welling, 2013) with soft cluster assignment and a cluster assignment hardening (CAH) loss (Xie et al., 2016). T-DPSOM additionally incorporates a temporal smoothness loss, and an LSTM, which aims to predict the future latent space. This latter functionality is similar to the future-prediction task that is already naturally embedded in the ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/e3dea705ac52dbf29bfc0d9f7bf4a4a97bc89c2d7e1aa6bf5466407d9678da7a.jpg", + "image_caption": [ + "Figure 1: Architectures of different deep-SOM models including the gradient paths in green. a) SOM-VAE (Fortuin et al., 2019) b) DESOM (Forest et al., 2021) c) SOM-CPC (ours). The two decoders in the SOM-VAE model are independent and have their own trainable parameters, while the visualized encoders in the SOMCPC model are all the same (i.e. parameters are shared). The $g _ { \\psi }$ block in the SOM-CPC model indicates an autoregressive component (e.g. a GRU), and $\\mathcal { Z } _ { p } ^ { \\prime }$ refers to a set of drawn negative embeddings. " + ], + "image_footnote": [], + "bbox": [ + 207, + 98, + 790, + 290 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "CPC objective that we propose as a task loss (see section 3.2). The probabilistic additions in the (T-)DPSOM model, with respect to SOM-VAE, are orthogonal to the developments in this work. ", + "bbox": [ + 176, + 395, + 821, + 422 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Forest et al. (2021) propose Deep embedded SOM (DESOM). Compared to SOM-VAE, the decoder on the discrete space is omitted (therewith also $\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n , d i s c } } )$ , gradients from $\\mathcal { L } _ { \\mathrm { S O M } }$ to the encoder are not being blocked (i.e. $\\mathrm { s g } [ \\cdot ]$ is removed from eq. (7)), and the topological loss $\\mathcal { L } _ { \\mathrm { t o p o } }$ is given by eq. (2), i.e. with a Gaussian neighbourhood function with decaying variance. In a short work, Forest et al. (2019) speculate about adding an LSTM in the latent space to train a SOM on sequential data, and refer to this model as LSTM-DESOM. Figure 1a-b visualizes the SOM-VAE and DESOM architecture. ", + "bbox": [ + 174, + 429, + 825, + 513 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 SOM-CPC ", + "text_level": 1, + "bbox": [ + 176, + 534, + 299, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 MOTIVATION ", + "text_level": 1, + "bbox": [ + 174, + 566, + 305, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this work, we propose the SOM-CPC model, a representation learning model that learns to map windows of time series data to a structured 2D grid. The model jointly optimizes a temporal contrastive learning objective to extract features, and a topological loss that organizes the SOM space. ", + "bbox": [ + 174, + 593, + 825, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to learn features that are both suitable for SOM organization and accurately reflect the data, the model should ideally invert the original data generating process (which is in general unknown and implicit). Assuming that this generative process has been highly non-linear, feature learning can be formulated as a non-linear independent component analysis (ICA) problem, which has proven to be non-identifiable (Hyvärinen & Pajunen, 1999). However, recent advances showed that the problem becomes identifiable under the assumed presence of an auxiliary variable (Hyvärinen et al., 2018). Such an auxiliary variable (e.g. a temporal component) is not present in plain autoencoders, but the contrastive learning paradigm was shown to conform to this assumption (Hyvärinen et al., 2018; Zimmermann et al., 2021). This theory is in line with the hypothesis stated by Mrabah et al. (2020) that a reconstruction objective may hamper clustering performance in the latent space. ", + "bbox": [ + 174, + 642, + 825, + 781 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 ALGORITHMIC DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 800, + 380, + 814 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce $\\mathcal { X } = \\{ \\ldots , \\substack { x ( t ) , x ( t + 1 ) , \\ldots \\} }$ , a set of non-overlapping data windows $\\pmb { x } ( t ) \\in \\mathbb { R } ^ { c h \\times T }$ , with $c h$ the number of channels, and $T$ the number of samples in the window. For brevity we omit the time index when possible. An encoder, parameterized by $\\theta$ , maps each data window $_ { \\textbf { \\em x } }$ to a latent representation $\\bar { \\boldsymbol { z } } = f _ { \\boldsymbol { \\theta } } ( \\pmb { x } ) \\in \\mathbb { R } ^ { F }$ , with $F$ the number of features. The set $\\mathcal { Z }$ includes the embeddings of all windows in $\\mathcal { X }$ . A causal auto-regressive (AR) module $g _ { \\psi }$ parameterized by $\\psi$ , e.g. a gated-recurrent unit (GRU), subsequently aggregates the current and $L$ previous embeddings, to generate a (current) context vector $\\pmb { c } ( \\dot { t } ) \\in \\dot { \\mathbb { R } } ^ { F }$ . Given this context, the pretext task in our SOM-CPC model aims to minimize the prediction error for $P$ future (or ‘positive’) embeddings $z ( t + p )$ , for $p \\in \\{ 1 , \\ldots , P \\}$ , compared to this error for $N$ ‘negative’ embeddings. These negatives may be sampled across the dataset, or within the same time-series, and are on the fly encoded to their latent representation during training. The task objective, being the InfoNCE loss (Oord et al., 2019), is defined as: ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 102, + 823, + 174 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a4c31def364e3e72c823897d1f9ca5a4940ced9037a15dae88ff0e35cef4ecac.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { t a s k } } : = \\mathcal { L } _ { \\mathrm { I n f o N C E } } = \\frac { 1 } { P } \\sum _ { p = 1 } ^ { P } \\mathcal { L } _ { p } , \\quad \\mathrm { w i t h } \\quad \\mathcal { L } _ { p } = - \\frac { \\mathbb { E } } { \\chi } \\bigg [ \\log \\frac { \\exp \\Big ( z ( t + p ) \\mathbf { W } _ { p } c ( t ) \\Big ) } { \\sum _ { z ^ { \\prime } \\in \\mathcal { Z } _ { p } ^ { \\prime } \\cup \\{ z ( t + p ) \\} } \\exp \\Big ( z ^ { \\prime } \\mathbf { W } _ { p } c ( t ) \\Big ) } \\bigg ] ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 181, + 797, + 236 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "with $\\mathcal { Z } _ { p } ^ { \\prime } \\subset \\mathcal { Z }$ a set of embeddings of drawn negative samples $\\left. \\mathcal { Z } _ { p } ^ { \\prime } \\right| = N )$ , and $\\mathbf { W } _ { p } \\in \\mathbb { R } ^ { F \\times F }$ a trainable mapping between the current context vector and the $p ^ { \\mathrm { t h } }$ future embedding. ", + "bbox": [ + 176, + 247, + 825, + 280 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The context vector is not only used to predict future embeddings, it is also the input to the SOM module that selects the winning node. The SOM is optimized using the topological loss $\\mathcal { L } _ { \\mathrm { t o p o } }$ , as defined in eq. (5), with a Gaussian neighbourhood kernel $s$ , as defined in eq. (2). Depending on the use case, it was found to not always be necessary, or even beneficial (due to higher risk of overfitting), to use an AR module $g _ { \\psi }$ to aggregate causal context into the current embedding. If the AR module is not used, the future predictions are made directly from the current (continuous) latent space $z ( t )$ instead of $\\mathbf { } c ( t )$ . Likewise, $z ( t )$ rather than $\\mathbf { } c ( t )$ is being quantized by the SOM module. Depending on the presence of this AR module, both $\\mathcal { L } _ { \\mathrm { t o p o } }$ and $\\mathcal { L } _ { \\mathrm { t a s k } }$ are thus computed on either $z ( t )$ or $\\mathbf { } c ( t )$ . ", + "bbox": [ + 173, + 286, + 825, + 398 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "All model elements are optimized jointly, with the training objective being: $\\mathcal { L } _ { \\mathrm { S O M - C P C } } = \\mathcal { L } _ { \\mathrm { t a s k } } +$ $\\alpha \\mathcal { L } _ { \\mathrm { t o p o } }$ , which adheres to the general objective of a deep-SOM model as formulated in eq. (4). Figure 1c provides an overview of the SOM-CPC model, and its gradient paths in green. The initial standard deviation $\\boldsymbol { \\sigma } ^ { ( 0 ) }$ of the Gaussian kernel (from eq. (2)) was set to half the squared-root of the number of SOM nodes $k$ . Given the square topology of the SOM grid, this setting of $\\sigma _ { 0 }$ ensures that the full grid is captured by the neighbourhood kernel at the start of training. Algorithm 1 in appendix A.1 provides pseudocode of the full SOM-CPC algorithm. ", + "bbox": [ + 174, + 405, + 825, + 503 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 PERFORMANCE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 526, + 415, + 540 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Forest et al. (2020) provide a taxology of SOM metrics that distinguishes external vs internal and topological vs clustering metrics. External metrics are related to labels (which are not used during unsupervised training), while internal metrics do not depend on such information. Topological metrics assess the topological ordering (i.e. neighbourhood relations) of the SOM, while clustering metrics are more related to, for example, pureness of nodes. ", + "bbox": [ + 174, + 554, + 825, + 625 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To evaluate clustering performance, linked to external labels, we leverage purity and the normalized mutual information (NMI). The latter corrects for a high number of clusters (i.e. nodes), which could easily lead to high pureness, but leaves the NMI more conservative. ", + "bbox": [ + 176, + 631, + 825, + 674 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Even though scoring high on external metrics is not the main goal of a representation learning model like SOM-CPC, we do report it as it provides an indication of how well information was preserved. To compute regression/classification performance, we first ‘color’ (or label) each node with the most occurring (for discrete labels) or median (for continuous labels) label from the training set. The test set predictions are then converted from node indices to label predictions by using these colorings. Regression performance is expressed as the average squared regression error with the target: $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ Classification performance is reported with Cohen’s kappa (Cohen, 1960), a commonly used metric that corrects for correctness by chance. ", + "bbox": [ + 174, + 680, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Topographic performance is measured using the (internal) topographic error (TE) (Kiviluoto, 1996), which reports the fraction of windows (between 0 and 1) for which the winning and second-best winning node are not neighbours in the SOM (lower is better). Finally, to measure whether a time series conveys a smooth trajectory through SOM space, we measure the average Euclidean distance (denoted $\\ell _ { \\mathrm { { 2 , s m o o t h } } } )$ between all subsequent windows in each time series. The lower this value, the less frequently large jumps in the 2D map occur. Note that in extreme cases where many windows collapsed to the same node, both the TE and the average $\\ell _ { 2 , \\mathrm { s m o o t h } }$ metric are artificially pushed down. We can thus only interpret these metrics in conjunction with earlier-mentioned clustering and classification metrics. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/095d5981ef212a3ff5ed70dacb4a9ae875fce08aa8ac819333ff01b60320f3cd.jpg", + "image_caption": [ + "Figure 2: SOMs and regression plots for SOM-VAE (a), DESOM (b) and SOM-CPC (c). Both DESOM and SOM-CPC show a gradual change of frequency over the grid, but the regression error $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ is lower for SOM-CPC, which can also be seen from the regression plot, where the predicted window frequencies are plotted against the target frequencies (i.e. training set median label) for the node on which the window was mapped. d) Task loss versus the topological loss for SOM-VAE (with Gaussian neighbourhood), DESOM and SOM-CPC (both with and without $\\mathrm { s g } [ \\cdot ]$ ). The different curves display various values of $\\alpha$ , for which the DESOM model seems most sensitive. The SOM-CPC models follow a smooth optimization curve, minimizing both the task and topological loss, while these losses seem to be more conflicting in SOM-VAE and DESOM training. " + ], + "image_footnote": [], + "bbox": [ + 184, + 152, + 795, + 482 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 627, + 326, + 642 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare SOM-CPC to several other 2D representation learning methods. First, deep-SOM models with a reconstruction task loss (i.e. SOM-VAE, SOM-VAE-prob, and (GRU-)DESOM). Second, vanilla CPC with a multi-dimensional latent space $F \\gg 2$ ) (Oord et al., 2019), followed by PCA for additional dimensionality reduction to 2D. Third, CPC with a 2D latent space ( $F = 2$ ). For the latter two CPC-based models, linear and non-linear read-out is, respectively, tested using a linear neural classifier, and K-means clustering with the same number of clusters as the number of nodes used in SOM-CPC. High-dimensional vanilla CPC ( $F \\gg 2$ ) without additional dimensionality reduction is, moreover, tested as well as it sets a baseline for the amount of information that can be preserved given the encoder architecture, while not providing an interpretable 2D representation. The same encoder architecture is used for all models that are compared in a single application domain, and all models are run with the same seed for randomization. Details on model architectures and training settings for the different applications can be found in appendix A.3.1, A.4.2, and A.5.1. ", + "bbox": [ + 174, + 657, + 825, + 824 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 SYNTHETIC DATA ", + "text_level": 1, + "bbox": [ + 176, + 842, + 336, + 856 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Data generation: A synthetic dataset was created, consisting of sinusoids with an initial frequency sampled from a uniform distribution between 20 and $4 0 \\ : \\mathrm { H z }$ . The frequency of the signals was altered over time according to a random walk process with a step size of $0 . 1 \\ : \\mathrm { H z }$ . As such, at each time step (i.e. sample), the signal’s frequency either increased with $0 . 1 \\ : \\mathrm { H z }$ (with probability $p _ { \\mathrm { u p } } = 0 . 1$ ), decreased with $0 . 1 \\ \\mathrm { H z }$ $( p _ { \\mathrm { d o w n } } = 0 . 1 )$ , or remained constant $\\gamma _ { \\mathrm { c } } = 0 . 8 )$ . In case the random walk crossed either 1 or $6 0 \\mathrm { H z }$ , the probabilities were (temporarily) altered to $[ p _ { \\mathrm { u p } } , p _ { \\mathrm { c } } , p _ { \\mathrm { d o w n } } ] = [ 0 . 5 , 0 . 5 , 0 ]$ or $[ p _ { \\mathrm { u p } } , p _ { \\mathrm { c } } , p _ { \\mathrm { d o w n } } ] = [ 0 . 0 , 0 . 5 , 0 . 5 ]$ , respectively. All series were finally corrupted with an additive white Gaussian noise vector $\\epsilon \\sim \\mathcal { N } ( 0 , 0 . 0 \\dot { 1 } )$ . Formally, each generated signal took the form: $\\begin{array} { r } { \\pmb { x } [ n ] = \\mathrm { s i n } \\left( 2 \\pi \\frac { f [ n - 1 ] + \\Delta f } { f _ { s } } n \\right) + \\epsilon } \\end{array}$ , with $f [ n = 0 ] \\sim U [ 2 0 , 4 0 ] , \\Delta f \\sim \\mathrm { C a t e g o r i c a l } ( [ p _ { \\mathrm { u p } } , p _ { \\mathrm { c } } , p _ { \\mathrm { d o w n } } ] ) .$ and $f _ { s } = 1 2 8 \\mathrm { { H z } }$ the sampling frequency. A total of 200 of such time-series, each of 5 minutes, were generated, and labels were defined per 1-second window by taking the median frequency. The set was randomly divided into a training $\\mathit { n } = 1 0 0 $ ), validation $\\mathrm { \\Delta } n = 5 0 $ ), and test split $\\mathit { n } = 5 0$ ). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/9c64afdd535a9684c986a72e86f2d42657c2e53f3627425cdb320431daaf6b9c.jpg", + "image_caption": [ + "Figure 3: Progression of SOM training (nodes are indicated in black) in the SOM-CPC model, using either a Gaussian (top) or plus neighbourhood (bottom) kernel. The PCA projection of the test set latent space is plotted behind the nodes. It can clearly be seen that the Gaussian kernel enforces a more strict organization of SOM nodes, where nodes are non-uniformly quantizing the latent space, placing more nodes at higher density areas. " + ], + "image_footnote": [], + "bbox": [ + 176, + 118, + 813, + 263 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 357, + 825, + 479 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results: Table 1 shows that SOM-CPC outperforms all deep-SOM baselines on all metrics. Figure 6 in appendix A.3.2 shows the PCA projections of the (continuous) latent spaces of the three deep-SOM models indicated with $^ { \\textrm { a * } }$ in table 1. It reveals that the latent space disentanglement of the SOM-CPC model is much better than that of the SOM-VAE and DESOM models. Figure 2a-c displays the resulting SOMs (colored with the median test set labels) for the same three models. Uncolored nodes in the SOM were not assigned in the test set. Interestingly, although the SOM for the DESOM and SOM-CPC model look similar, the $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ is higher for the DESOM model, which can also be seen from the regression plots below the SOMs. ", + "bbox": [ + 173, + 484, + 825, + 597 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The addition of two temporal losses in the SOM-VAE-prob model, as compared to SOM-VAE, did deteriorate the given metrics, even though a range of values for multipliers $\\alpha , \\gamma$ and $\\zeta$ was tested (see table 3 in appendix A.3.2 for the full sweep). The deterioration of the results can be explained by the difficulty of finding the correct scaling factors for these additional losses. Note that the SOM-CPC model automatically incorporates smoothness over time thanks to the nature of the CPC task loss, therewith preventing additional hyperparameter tuning. ", + "bbox": [ + 174, + 603, + 825, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Additionally, we study the optimization behavior of SOM-VAE, DESOM, and SOM-CPC by plotting the progression of the task versus the topological loss during training (see fig. 2d). To make a fair comparison, we plot the SOM-VAE models that are trained with a Gaussian neighbourhood kernel. Different curves in the graphs indicate runs with varying values for $\\alpha$ , and the line color’s gradient denotes the training iteration. The SOM-CPC graphs include the models run with and without gradient detachment of $\\mathcal { L } _ { \\mathrm { S O M } }$ to the encoder. It can be seen that both losses jointly minimize in SOM-CPC training, while there is a counteracting effect visible for SOM-VAE, and a high influence of the value of $\\alpha$ for DESOM training. ", + "bbox": [ + 174, + 694, + 825, + 806 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparing to non-deep-SOM baselines, it can be seen from table 1 that CPC (with $F = 2$ ), and CPC followed by PCA, resulted in a much higher regression error $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ than SOM-CPC when using linear read-out. Non-linear K-means clustering improved performance for both cases, but only for CPC with $F = 2$ , performance nearly reached SOM-CPC performance. Later we will see that optimizing CPC with $F = 2$ can hamper optimization for more intricate data spaces (see section 4.3). Interestingly, regression performance of SOM-CPC was found to be even slightly better than that of the vanilla multi-dimensional CPC model (with $F = 1 2 8$ ), both for linear classification and K-means. This could be explained by the additional regularization that the SOM provides in SOM-CPC training. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We perform several ablation experiments on SOM-CPC, which are reported below the dashed line in table 1. Blocking the gradients of the neighbour nodes with respect to the encoder during training $( \\mathcal { L } _ { \\mathrm { { S O M } S g [ \\cdot ] } }$ column) slightly improved the regression error and temporal smoothness, but decreased the topographic error. Looking at the models with various values for $\\alpha$ (reported in table 3, appendix A.3.2), the effect of gradient blocking can be considered small and ambiguous when considering different metrics. Disjoint training $( \\mathbf { C P C } + \\mathbf { S O M } )$ resulted in a less smooth trajectory over time through the 2D SOM space, seen from the higher $\\ell _ { 2 , \\mathrm { s m o o t h } }$ . Using a plus neighbourhood kernel instead of a Gaussian kernel decreased performance on all three metrics. The increase in TE, is well explainable by the fact that a plus kernel takes into account fewer neighbours (at least at the start of training) and therefore has more difficulty to find a good topological mapping. Figure 3 shows the development of the SOM node spread (projected on top of a PCA projection of the continuous test set latents), during training for SOM-CPC with the two type of kernels. It can indeed be seen that the Gaussian kernel enforces a more strict topological organization. Interestingly, the kernel does not only influence the codebook vectors, but also seems to influence the organization of the latent space, seen from the differently-shaped PCA projections in the background. ", + "bbox": [ + 174, + 104, + 825, + 311 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 SLEEP ", + "text_level": 1, + "bbox": [ + 174, + 330, + 259, + 344 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We analyse SOM-CPC on subset 3 of the Montreal Archive of Sleep Studies (MASS) database (O’Reilly et al., 2014), consisting of whole-night polysomnography recordings, for which every 30-second window is labelled with a sleep stage label from $\\{ \\mathrm { N } 1 $ , N2, N3, REM, $\\mathrm { W a k e } \\}$ . The 62 recordings (from 62 unique subjects) were randomly split into a training $n = 4 8$ ), validation $( n = 8 )$ and hold-out test set $( n = 7$ ). Details on the data preprocessing can be found in appendix A.4.1. ", + "bbox": [ + 173, + 356, + 823, + 426 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 5 in appendix A.4.3 shows that SOM-CPC again clearly outperformed SOM-VAE and DESOM on all metrics. Whether or not the gradients of the SOM loss were stopped towards the encoder did not greatly influence SOM-CPC performance. Topological ordering, measured by TE, and temporal smoothness $\\cdot \\ell _ { 2 , \\mathrm { s m o o t h } } )$ deteriorated when changing the Gaussian kernel to a plus kernel, or training high-dimensional CPC and SOM disjointly. SOMCPC’s classification performance was higher than that of CPC with $F = 2$ , and CPC followed by PCA. ", + "bbox": [ + 176, + 434, + 549, + 571 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/acf92fe50c3eea2c8f7684ee051c86ae9cb1a1f4a96133c190f8934736bc0b6a.jpg", + "image_caption": [ + "Figure 4: Deep sleep N3 is isolated from light sleep N1, Wake and REM sleep with a cluster of medium-deep sleep N2. " + ], + "image_footnote": [], + "bbox": [ + 566, + 434, + 821, + 531 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 4 shows the test set PCA projection of the latent space of CPC $F = 1 2 8 )$ ), with the K-means nodes as black stars (left), and the SOM (right) trained by the SOM-CPC model (nodes are colored with the most-occurring label in the test set). Both visualizations show similar clustering patterns: deep sleep N3 is isolated from lighter forms of sleep (i.e. N1, Wake and REM sleep) by a thick cluster of medium-deep sleep N2. However, the higher performance of SOM-CPC (see table 5) indicates that more information is preserved in the 2D space resulting from the SOM-CPC model. The size of the nodes in the SOM map of SOM-CPC indicates the average time in the night of windows on that node. A difference is visible in node sizes within the Wake, N2 and N3 clusters, suggesting a possible existence of different sub-categories of sleep within the pre-defined sleep stages. ", + "bbox": [ + 173, + 593, + 825, + 718 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 AUDIO ", + "text_level": 1, + "bbox": [ + 174, + 737, + 261, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For the audio experiments, we use a subset of the publicly available LibriSpeech dataset (Panayotov et al., 2015). The dataset contains multiple minute-long English voice recordings of 251 different speakers, sampled at $1 6 ~ \\mathrm { K H z }$ . We used the publicly available train-test split, as provided by Oord et al. (2019), and created an additional validation set by randomly selecting $2 5 \\%$ of the training set. Recordings of the ten speakers with the longest recording time were selected to alleviate computational burden. This resulted in a total of 150.9, 54.6, and 46.5 minutes in the training, validation, respectively test set. The full model and training details can be found in appendix A.5.1. ", + "bbox": [ + 174, + 763, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 7 in appendix A.5.2 shows the results of SOM-CPC (which includes a GRU for this dataset), compared to different variants of the DESOM model. SOM-CPC outperforms all DESOM variants by a wide margin and for all choices of the $\\alpha$ parameter. The difference in performance between DESOM and SOM-CPC is also visible in fig. 5. SOM-CPC has clustered the SOM nodes belonging to the same speaker, and seems to group male and female speakers (denoted with the node’s shape), while these effects are not present in the SOM of the GRU-DESOM model. Minimizing the InfoNCE training objective of CPC with $F = 2$ was found challenging for this dataset, which resulted in non-competitive performance of the linear classifier and K-means clustering trained on the resulted 2D latent space. Using PCA for dimensionality reduction of the high-dimemnsional CPC latent space (with $F = 5 1 2$ ) performed better, but still inferior to SOM-CPC. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The SOM of the SOM-CPC model (fig. 5-right) reveals two separate clusters both for speaker 2 (green) and 3 (red). The LibriSpeech corpus contains multiple recordings from each speaker, grouped by (book) chapters from which the speaker was reading. Interestingly, additional analyses revealed that the red and green sub-clusters represented recordings belonging to different chapters: $9 9 . 9 9 \\%$ of the test-set windows mapped to the upper red sub-cluster belong to the same chapter, while $9 9 . 9 7 \\%$ of the windows in the lower red sub-cluster belong to another chapter read by this speaker. Similarly, $1 0 0 . 0 \\%$ of the test set windows in the right green sub-cluster belong to two chapters read by speaker 2, while $9 8 . 9 1 \\%$ of the windows in the left green sub-cluster belong to another chapter. An auditory inspection revealed that the room acoustics of the recordings belonging to the chapters in different clusters were different, causing changes in the signals which the SOM-CPC model has picked upon. This division between recordings of the same speaker is not visible in the 2D PCA projection of the CPC (with $\\cdot$ ) features, as seen from fig. 8 in appendix A.5.2. ", + "bbox": [ + 174, + 194, + 549, + 347 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/12d5cb12915dbf4c542c3627bd53f66f59cbcd84d2ce303a58fa90adcdd92341.jpg", + "image_caption": [ + "Figure 5: SOM-CPC is able to better cluster different speakers than GRU-DESOM. Stars denote women and dots are men. " + ], + "image_footnote": [], + "bbox": [ + 563, + 193, + 823, + 284 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 347, + 825, + 430 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 457, + 310, + 473 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We proposed a new member of the deep-SOM family: SOM-CPC, suitable for interpretable 2D representation learning of high-rate data streams. Earlier proposed deep-SOM models mainly used reconstruction objectives. In general, SOM-CPC outperformed these models with a wide gap on a variety of metrics. Moreover, it implicitly enforces temporal smoothness, while autoencoder-based models require additional losses and hyperparameter tuning to achieve this. SOM-CPC’s task loss was found to align better with the topological SOM objective than a reconstruction loss, as already hypothesized by Mrabah et al. (2020). While for some applications CPC could succesfully be trained with a 2D latent space directly, optimization was found to be hampered in case of more intricate data spaces. Compared to vanilla CPC with a multi-dimensional latent space, SOM-CPC enables pattern recognition and knowledge discovery. The SOM objective did not hamper CPC optimization. Even better, in the synthetic setup it had a regularizing effect, resulting in lower regression error than vanilla CPC. The use of a Gaussian neighbourhood kernel, as opposed to a plus kernel, was found to improve the topological ordering in the SOM. No decisive conclusions could be made regarding gradient blocking from the SOM loss towards the encoder parameters. Allowing these gradients to flow did not hurt performance, so for coding simplicity, we would advice to not detach the SOM loss. ", + "bbox": [ + 174, + 492, + 825, + 700 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Setting an appropriate stopping criterion for self-supervised (SSL) models is debatable. In the SSL literature models with the best test set performance are sometimes reported (He et al., 2019; Fortuin et al., 2019). This is, however, questionable as it may artificially boost reported performance. As such, we created a validation set to apply early stopping in all experiments. Another challenge arises when dealing with aggregated loss functions, since not all losses may smoothly decay and the weighted summation of losses may result in a different optimal epoch than the sub-losses separately. Besides, classification performance (often used as a proxy for information preservation) does not necessarily align with SOM performance or information preservation (see fig. 7 in appendix A.4.3). ", + "bbox": [ + 174, + 708, + 825, + 819 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We believe that SOM-CPC will facilitate knowledge discovery in real-life time series and opens up new research directions for representation learning of time series. Directions include investigation to whether additions like the soft-cluster assignment, cluster hardening loss or a Gaussian latent prior - which have shown to improve the SOM-VAE model (Manduchi et al., 2021) - improve SOM-CPC performance as well. Moreover, the CPC objective assumes slowly (or non-changing) data characteristics within the time frame in which positive samples are drawn. A multi-modal variational future prediction could possibly improve performance for data that do not meet this assumption. ", + "bbox": [ + 173, + 827, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REPRODUCIBILITY STATEMENT ", + "text_level": 1, + "bbox": [ + 176, + 102, + 434, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "All code used to train and evaluate the models as presented in this paper can be found at https: //anonymous.4open.science/r/SOM-CPC. The details regarding model architectures and training settings for each of the application domains are also presented in appendix A.3.1, A.4.2, and A.5.1. Pseudocode of the proposed SOM-CPC algorithm is given in algorithm 1 in appendix A.1. ", + "bbox": [ + 174, + 133, + 826, + 190 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 212, + 285, + 227 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Richard B. Berry, Rita Brooks, Charlene E. Gamaldo, Susan M. Harding, Robin M. Lloyd, Carole L. Marcus, and Bradley V. Vaughn. 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In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 3861–3870. PMLR, 06–11 Aug 2017. ", + "bbox": [ + 174, + 103, + 826, + 160 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Roland S. Zimmermann, Yash Sharma, Steffen Schneider, Matthias Bethge, and Wieland Brendel. Contrastive Learning Inverts the Data Generating Process. arxiv, 2021. URL http://arxiv. org/abs/2102.08850. ", + "bbox": [ + 174, + 169, + 828, + 210 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 178, + 102, + 418, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "This appendix contains all information for full reproducability of the experiments. Domainindependent details on SOM-CPC and its evaluation are provided in appendix A.1, while benchmark implementations are discussed in appendix A.2. Domain-specific settings for the synthetic, sleep and audio experiments are discussed in sections A.3, A.4, and A.5, respectively. ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 GENERAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 207, + 349, + 220 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 1 provides pseudocode of SOM-CPC, when considering the presence of an AR module. For reproducibility, the full code base can be found at https://anonymous.4open.science/ r/SOM-CPC. ", + "bbox": [ + 174, + 232, + 826, + 273 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 1 SOM-CPC ", + "text_level": 1, + "bbox": [ + 174, + 292, + 336, + 306 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Input: Dataset $\\mathcal { X }$ , model comprising $f _ { \\theta }$ , $g _ { \\psi }$ , $\\{ \\mathbf { W } _ { p } \\} _ { p = 1 } ^ { P }$ , number of past windows $L$ , # positive samples $P$ , # negative samples $N$ , # SOM nodes $k$ , Gaussian neighbourhood function $s$ with $\\sigma ^ { ( n _ { \\mathrm { m a x } } ) }$ , $n _ { \\mathrm { m a x } }$ , loss trade-off parameter $\\alpha$ . \nOutput: A trained SOM: topologically ordered codebook $\\Phi$ that represents data √ $\\mathcal { X }$ in 2D. - Initialize the Gaussian kernel according to eq. (2) with: $\\begin{array} { r } { \\sigma ^ { ( 0 ) } = \\frac { 1 } { 2 } \\sqrt { k } } \\end{array}$ and $\\lambda = - n _ { \\mathrm { m a x } } / \\log \\left( \\sigma ^ { ( n _ { \\mathrm { m a x } } ) } / \\sigma ^ { ( 0 ) } \\right)$ for $n$ in $n _ { \\mathrm { m a x } }$ do - Sample a sequence of datapoints: $[ { \\pmb x } ( t - L ) , \\dots , { \\pmb x } ( t ) ] \\sim { \\pmb \\chi }$ - Define $P$ positive samples: $\\{ \\pmb { x } ( t + p ) \\} _ { p = 1 } ^ { P }$ - Sample $P \\times N$ negative samples ${ { \\mathcal { X } } ^ { \\prime } } \\subset { { \\mathcal { X } } }$ , with $| { \\mathcal { X } } ^ { \\prime } | = P \\times N$ - Encode - data sequence: $\\pmb { c } ( t ) = g _ { \\psi } \\Big ( f _ { \\theta } \\big ( [ \\pmb { x } ( t - L ) , \\ldots , \\pmb { x } ( t ) ] \\big ) \\Big )$ - positive samples: $\\{ z ( t + p ) \\} _ { p = 1 } ^ { P } = f _ { \\theta } \\left( \\{ \\pmb { x } ( t + p ) \\} _ { p = 1 } ^ { P } \\right)$ - negative samples: $\\mathcal { Z } ^ { \\prime } = f _ { \\theta } \\left( \\mathcal { X } ^ { \\prime } \\right)$ - Predict future: $\\hat { \\mathbf { z } } ( t + p ) = \\mathbf { W } _ { p } \\dot { \\mathbf { c } } ( t )$ , with $1 \\leq p \\leq P$ - Quantize: $\\begin{array} { r l } { \\phi } & { { } = \\mathrm { S O M } _ { \\Phi } \\left( \\begin{array} { l l l } \\end{array} \\right. } \\end{array}$ $\\mathbf { \\Psi } ( \\mathbf { c } ( t ) )$ - Update $\\boldsymbol { \\sigma } ^ { ( n ) }$ according to eq. (3) - Compute $\\mathcal { L } _ { \\mathrm { t o p o } } \\big ( \\mathbf { c } ( t ) \\big )$ and $\\mathcal { L } _ { \\mathrm { i n f o N C E } }$ according to eq. (5) and eq. (8) - Update trainable parameters $\\propto \\alpha \\mathcal { L } _ { \\mathrm { t o p o } }$ and $\\mathcal { L } _ { \\mathrm { { i n f o N C E } } }$ ", + "bbox": [ + 171, + 310, + 826, + 568 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "end for ", + "bbox": [ + 189, + 563, + 236, + 574 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Several metrics were used to quantify SOM performance, as explained in section 3.3. The purity implementation is taken from the Github implementation of Fortuin et al. (2019), while NMI was computed using the sklearn library (Pedregosa et al., 2011). The topographic error implementation comes from the SOMperf python library (Forest et al., 2020). Finally, the $\\ell _ { \\mathrm { { 2 , s m o o t h } } }$ distance is computed using the norm function in the linalg library of numpy. ", + "bbox": [ + 174, + 594, + 825, + 665 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Section 3.3 already shortly elucidated upon the way in which Cohen’s kappa and $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ were computed. For both metrics, each SOM node was labeled/colored with the most-occuring (in case of Cohen’s kappa) or median (in case of $\\mathrm { S E } _ { \\mathrm { t a r g e t } } )$ label in the training set. Note that it can be questioned whether this node labelling should be done on a training or validation set, or directly on the test set for which performance is reported. In earlier times when more conventional clustering approaches (e.g. K-means) were used, a training/validation/test split was typically not made. As a result, clustering was directly performed on the one and only (test) set, that was also used to label the clusters/nodes. Moving towards deep learning based approaches where more hyperparameters need to be set and overfitting can become a larger problem, we found it necessary to report in this work on a test set that was not used for labelling the nodes and/or setting hyperparameters. It should thus be taken into account, that direct comparison to results in other deep-clustering works may need a critical eye to see whether similar procedures were used or not. ", + "bbox": [ + 173, + 671, + 825, + 838 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 BENCHMARKS AND ABLATIONS ", + "text_level": 1, + "bbox": [ + 176, + 856, + 436, + 869 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To benchmark our implementation of the SOM-VAE model (and the very similar DESOM model), we replicated the results on MNIST. MNIST was not used for further experimentation with SOM-CPC in the main body of this paper since this work focuses on high-rate time series. Using $k = 1 6$ SOM nodes, Fortuin et al. (2019) report purity $= 0 . 7 3 1 \\pm 0 . 0 0 4$ and $\\mathbf { N M I } = 0 . 5 9 4 \\pm 0 . 0 0 4$ in table 1, and purity $= 0 . 7 2 1 \\pm 0 . 0 0 6$ and $\\mathbf { N M I } = 0 . 5 8 7 \\pm 0 . 0 0 3$ in table S1. All numbers are averages and standard errors over 10 runs. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 145 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Settings that were not provided in the paper, were taken from the hard-coded settings that we found in the provided code base. Instead of splitting the standard MNIST training set in a training and test split, as done by Fortuin et al. (2019), we used the available train/test split that comes with the standard MNIST dataloader from Pytorch. From the first ten runs, one run fully collapsed and resulted in extremely poor performance. Considering this run as an outlier, we run an $1 1 ^ { \\mathrm { t h } }$ run and here report the average and standard errors of the 10 non-collapsed runs: purity $= 0 . 7 0 5 \\pm 0 . 0 0 2$ and $\\mathbf { N M I } = 0 . 5 8 4 \\pm 0 . 0 0 1$ . Our performance does reasonably well match the reported performance by Fortuin et al. (2019), given the fact that a different test set of MNIST was used for our experiments. ", + "bbox": [ + 174, + 152, + 825, + 263 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "To restrict the search space of hyperparameters in the SOM-VAE(-prob) model, which has multiple loss multipliers, we fixed some of the these multipliers for further experiments in this paper. Multiplier $\\beta$ scales the SOM loss that sums the quantization error of the four neighbour nodes in the plus kernel. It is, therefore, expected to be at least 4 times larger than the commitment loss, which only reflects the quantization error of the winning node. Choosing $\\beta = \\alpha / 4$ would imply that the weighing of the summed quantization error of the four neighbours is equal to the weighing of this error for the winning node. To give the winning node a slightly higher importance, we set $\\beta = \\alpha / 5 = 0 . 2 \\alpha$ . The authors of SOM-VAE (Fortuin et al., 2019) used, instead, a search strategy to find the optimal setting. Their code base1 shows that the SOM loss was multiplied with 0.9, while $\\alpha = 1$ . Taking into account that their implementation of the commitment loss averaged the quantization error of the four neighbour nodes, while we summed the contribution, their effective setting was thus set to $\\beta = \\textstyle { \\frac { 0 . 9 } { 4 } } \\alpha = 0 . 2 2 5 \\alpha$ , which is close to what we used in our experiments. ", + "bbox": [ + 173, + 270, + 826, + 438 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We compared SOM-CPC also against vanilla CPC training followed by a linear classifier or Kmeans clustering, while freezing the encoder parameters. The supervised linear classifier took in all experiments the form of one fully-connected layer, including biases, that was followed by a log-softmax activation for the sleep and audio cases. It was trained using the mean squared error for the synthetic data set, and cross-entropy loss for sleep and audio experiments. K-means clustering was run from the sklearn library, with the default settings. The number of clusters was chosen to be equal to the number of nodes in the SOM-CPC models against which the performance was compared. Also the disjointly-trained SOM had exactly the same settings as the SOM in the SOM-CPC models with which it was compared. ", + "bbox": [ + 173, + 444, + 825, + 570 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Several ablation are performed on the SOM-CPC model. We test the effect of propagating gradients of ${ \\mathcal { L } } _ { \\mathrm { S O M } }$ to the encoder parameters, the difference between using a Gaussian neighbourhoood kernel versus a plus kernel, and the effect of jointly training CPC and the SOM. Moreover, the effect of certain settings that are typically used in the CPC objective are investigated. CPC’s InfoNCE objective for one window $\\cdot$ , given in eq. (8), can as follows be generalized to a more general contrastive learning objective: ", + "bbox": [ + 173, + 577, + 825, + 660 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/acf4c414c2a0b7ce238ef796130a6b683a8a17f956ce8c987bdff576aaf4cfd5.jpg", + "text": "$$\n\\mathcal { L } _ { p } = - \\frac { \\mathbb { E } } { \\chi } \\Big [ \\log \\frac { \\exp \\Big ( \\sin ( z _ { a } , z _ { p } ) / \\tau \\Big ) } { \\sum _ { z ^ { \\prime } \\in \\mathcal { Z } _ { p } ^ { \\prime } \\cup \\{ z _ { p } \\} } \\exp \\Big ( \\sin ( z _ { a } , z _ { p } ^ { \\prime } ) / \\tau \\Big ) } \\Big ] ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 659, + 663, + 707 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where $z _ { a }$ is the latent space of the current (or anchor) window, $\\cdot$ a similarity metric, and the other symbols are equivalent to eq. (8). CPC typically uses $\\tau = 1$ , and the dot product as the similarity metric. However, other related contrastive learning objectives, e.g. in SimCLR (Chen et al., 2020), use a temperature value that is often set to 0.07 (Chen et al., 2020; Woo et al., 2022), and a cosine similarity instead of the (unnormalized) dot product. As such, we add ablations where we set $\\cdot$ at 1 or 0.07, and use either the dot-product or the cosine similarity as the similarity metric. ", + "bbox": [ + 173, + 710, + 825, + 794 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3 SYNTHETIC EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 810, + 403, + 824 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.3.1 TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 837, + 367, + 851 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Table 2 summarizes the encoder and decoder architectures used in this experiment. The output size column in the table uses channels-first notation. The SOM-VAE and DESOM models were found to benefit from a convolutional part of the encoder that did not fully reduce the temporal dimension to size 1. As such, the last convolutional layer of the SOM-CPC encoder was changed for a fully connected layer preceded by a flattening operation for the autoencoder-based models. The SOM-CPC and CPC model are run without an AR module, to make the fairest comparison to the SOM-VAE(prob) and DESOM models, which also do not incorporate such a component. ", + "bbox": [ + 176, + 861, + 821, + 888 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/4d8ba43b619be7f612f06a08926c66cf07733d4956185b39fca9b545a587495a.jpg", + "table_caption": [ + "Table 2: Model details for the synthetic data experiments in section 4.1. " + ], + "table_footnote": [], + "table_body": "
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU(0.01)911same
MaxPool1Dbs ×16× 32-44=·
Dropout (0.1)bs ×16×32--=-
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8----
Conv1Dbs ×64×864Leaky ReLU (0.01)311same
MaxPool1Dbs ×64×2-44-
Dropout (0.1)bs ×64×2----
Conv1Dbs ×128×2128Leaky ReLU (0.01)311same
MaxPool1Dbs ×128 ×1-22=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)911same
MaxPool1Dbs ×16× 32-44-
Dropout (0.1)bs ×16×32----
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)31same
Flattenbs ×512-==
Fully Connectedbs ×128128Leaky ReLU (0.01)=
DecoderforSOM-VAEandDESOM
Fully Connectedbs×512512Leaky ReLU (0.01)
Unflattenbs ×64×8-
Conv1Dbs ×32×832Leaky ReLU (0.01)311same
ConvTranspose1Dbs ×32 × 3232None4410
Conv1Dbs ×16× 3216Leaky ReLU(0.01)711same
ConvTranspose1Dbs ×16× 12816None4410
Conv1Dbs ×1 × 1281Tanh911same
", + "bbox": [ + 218, + 126, + 777, + 429 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 464, + 823, + 534 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For SOM-CPC, $P = 3$ future predictions (i.e. positive samples) were used, and $N = 3$ negative samples were drawn for each positive sample. The latter were drawn randomly from the entire training set. The standard deviation of the Gaussian neighbourhood kernel was exponentially decayed until $\\sigma ^ { ( n _ { \\mathrm { m a x } } ) } = 2$ . Choosing a lower value at the end of training induced instable optimization behavior. ", + "bbox": [ + 174, + 540, + 825, + 598 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "All models (including the benchmarks) were trained using the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 0.001 and a batch size of 128. Each model was trained for maximally 1000 epochs. The best model was selected based on the lowest task loss on the validation set, being $\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n } }$ for SOM-VAE and DESOM and $\\mathcal { L } _ { \\mathrm { { I n f o N C E } } }$ for SOM-CPC and CPC. We did not use the full training objective $\\mathcal { L } _ { \\mathrm { d e e p - S O M } }$ as model selection criterion, as both the commitment and SOM loss showed to be low initially (possibly due to low values of the random initialization of the model), while both increased and reached a steady-state later in training. The linear classifier and the disjointly-trained SOM on the CPC embeddings were trained until convergence, for maximally 1000 epochs. ", + "bbox": [ + 174, + 604, + 825, + 717 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.3.2 EXTENDED RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 742, + 375, + 756 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 3 extends table 1 with additional sweeps of hyperparameters $\\alpha , \\gamma _ { ; }$ , and $\\zeta$ , and ablations with different settings of the temperature value $\\tau$ and the used similarity metric. Compared to SOM-VAE, SOM-VAE-prob is expected to show more smooth trajectories over time, captured in the $\\ell _ { 2 , \\mathrm { s m o o t h } }$ distance metric, thanks to the additional transition and smoothness loss (multiplied by $\\gamma$ and $\\cdot$ , respectively). It can be seen that tuning these hyperparameters is a complex process, and a sweep did not result in one SOM-VAE-prob run that performed better than the best SOM-VAE model. In contrary, the addition of the extra losses possibly interfered with the optimization process, and only very delicate settings of $\\gamma$ and $\\cdot$ might improve model performance eventually. A sweep over the topological loss multiplier $\\alpha$ , revealed a low sensitivity of SOM-CPC to this value. It can be seen that changing the temperature value and/or the similarity metric did not significantly alter the performance consistently on all reported metrics. ", + "bbox": [ + 173, + 770, + 826, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/2a5041fedb49be4c6e76f1a5f3832a183f4a102e831206af4e48df134c2fabf1.jpg", + "table_caption": [ + "Table 3: This is the extended version of table 1 on the synthetic data results, including a sweep of hyperparameters $\\alpha , \\gamma$ and $\\zeta$ . Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). The models indicated with $^ { \\textrm { a * } }$ were used to depict trained SOMs in fig. 2 and PCA projections in fig. 6. " + ], + "table_footnote": [], + "table_body": "
ModelαSLsom sg[-]SEtargetl2.smoothTE
CPC+linearclassifier==2.62±2.37==
CPC + K-means===1.09±.62=
CPC(F=2) + linear classifier25.01±42.94
CPC(F = 2)+K-means.76±1.31
CPC+PCA+ linear classifier42.81±58.12
CPC + PCA+K-means-=4.42±9.01
SOM-VAE1e-3Plus11.59±13.692.60±.46.38±.042
*1e-2Plus14.57±44.102.98±.84.38±.052
.1 1Plus Plus8.02±4.582.41±.68.28±.070
SOM-VAE1e-5Gaussian13.43±3.66 11.12±17.052.75±.45.33±.048
1e-4Gaussian1.93±.29.069±.029
1e-31.52±26.611.95±.36.075±.028
SOM-VAE-prob1e-2Gaussian Gaussian11.60±25.431.92±.34.056±.023
(γ=5e-5,S= 1e-3)1e-318.13±48.662.03±.42.086±.030
(γ = 4e-5,= 1e-3)Plus21.26±55.003.78±.52.93±.042
1e-3Plus21.30±37.374.23±.57.88±.051
(γ=3.3e-5,S= 1e-3) (γ= 5e-4,S= 1e-2)1e-3Plus22.52±48.973.81±.51.98±.015
(γ=4e-4,= 1e-2)1e-2Plus14.65±19.583.70±.51.86±.083
1e-2Plus27.25±72.823.54±.67.94±.022
(γ= 3.3e-4,S= 1e-2)1e-2Plus21.62±38.783.71±.76.97±.014
(γ= 5e-5,S= 1e-2).1Plus26.82±68.843.23±.80.68±.067
(γ=4e-5,= 1e-2).1Plus22.34±64.923.11±.73.74±.072
(γ=3.3e-5,S=1e-2) (γ= 5e-4,=.1).1Plus Plus2.10±48.803.15±.73.63±.050
.14.85±75.963.06±.55.84±.050
(γ= 4e-4,S=.1) (γ=3.3e-4,S=.1).1Plus29.96±46.962.81±.34.82±.13
DESOM.1 1e-5Plus GaussianX3.96±56.943.95±.83.85±.11
1e-4GaussianX14.00±3.10 19.09±44.041.99±.36 1.95±.28.13±.069
1e-3GaussianX12.58±27.101.92±.31.11±.046 .077±.033
1e-2GaussianX13.66±44.711.89±.33.065±.028
.1GaussianX10.77±10.852.20±.44.061±.019
SOM-CPC (ours)1GaussianX22.86±55.712.26±.43.092±.034
*1e-5GaussianX.95±2.401.24±.31.048±.020
1e-4GaussianX.72±1.081.37±.37.022±.011
1e-3GaussianX.81±.751.06±.28.028±.012
1e-2Gaussian Gaussian_X.62±.711.08±.28.059±.035
1 1e-5GaussianX √1.90±4.07 .64±.711.18±.30.039±.016
1e-4Gaussian√.68±.671.04±.26 1.19±.43.039±.020
1e-3Gaussian.75±1.021.12±.32.057±.033
1e-2Gaussian.47±.48.059±.027
.1.99±.24.069±.040
Gaussian√.89±1.501.16±.32.069±.031
1e-4PlusX1.71±1.152.46±0.51.30±0.062
Gaussian1e-2 1e-4Plus√1.16±.61 1.47±2.601.85±.26 1.15±.34.12±.037 .014±.015
SOM-CPC(T = O.07,sim = cosine sim.) SOM-CPC(τ = 1,sim = cosine sim.) 1e-4 Gaussian
", + "bbox": [ + 196, + 165, + 799, + 608 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/348104e47be032cc7d8471ede902b7d2ad2ea136cd2b21a97db4e35e8521f89c.jpg", + "image_caption": [ + "Figure 6: PCA projections of the continuous latent spaces of the full test set for the SOM-VAE, DESOM, and SOM-CPC models of which the SOMs were visualized in fig. 2. Disentanglement of the signal frequencies is much better in the SOM-CPC model. " + ], + "image_footnote": [], + "bbox": [ + 178, + 640, + 820, + 776 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Figure 6 shows PCA projections of the latent space of the SOM-VAE, DESOM, and SOM-CPC models that are indicated with a \\* in tables 1 and 3, and for which the SOMs were visualized in fig. 2. Disentanglement of the signal frequencies is much better for the SOM-CPC model, providing an explanation for the better (i.e. lower) $\\mathrm { S E } _ { \\mathrm { t a r g e t } }$ of this model, as compared to SOM-VAE and DESOM. ", + "bbox": [ + 174, + 857, + 826, + 915 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 SLEEP EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 364, + 117 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4.1 DATA PROCESSING ", + "text_level": 1, + "bbox": [ + 176, + 128, + 362, + 143 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the experiments on sleep data, we used subset 3 of the publicly available Montreal Archive of Sleep Studies (MASS) database (O’Reilly et al., 2014), consisting of 62 whole-night polysomnography recordings. Each recording contains, among others, electroencephalography (EEG), chin electromyography (EMG), and electrooculography (EOG) data. We refer the reader to O’Reilly et al. (2014) for more details regarding this dataset. We selected the channels that are typically used in clinical practice, comprising three EEG channels (F4, C4, O2), the two EOG channels, and one chin EMG derivation, and downsampled the data to $1 2 8 \\ : \\mathrm { H z }$ . Sleep stage labels that follow the guidelines of the American Academy of Sleep Medicine (AASM) Berry et al. (2012) (being wakefulness (Wake), rapid-eye movement (REM) sleep, or non-REM1 till non-REM3 (N1, N2, N3)) were available for every non-overlapping 30-second window, the common label resolution in clinical practice. ", + "bbox": [ + 173, + 154, + 825, + 292 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Some processing had already been done by the distributors of the MASS dataset (O’Reilly et al., 2014). The $6 0 \\mathrm { H z }$ powerline interference was, however, not fully suppressed, and we wanted to down sample each signal to $1 2 8 \\ : \\mathrm { H z }$ to reduce computational complexity. As such, before downsampling, all derivations were additionally filtered with a zero-phase (i.e. two-directional) $5 ^ { \\mathrm { t h } }$ order Butterworth band-pass filter $( 0 . 3 - 5 9 \\ : \\mathrm { H z } )$ , followed by another zero-phase $5 ^ { \\mathrm { t h } }$ order Butterworth notch filter $( 5 9 - 6 1 \\ : \\mathrm { H z } )$ . Channels were normalized within-patient and per channel, yielding mean subtraction, followed by normalization such that amplitudes of $9 5 \\%$ of the samples were mapped between $^ { - 1 }$ and $+ 1$ . The 62 recordings (numbered $1 - 6 4$ , with number 43 and 49 missing) were split into a training set including patients $1 - 4 8$ ${ \\mathrm { ' } n = 4 7 }$ ), a validation set including patients $5 0 - 5 7$ ${ \\mathrm { \\Delta } n = 8 }$ ), and hold-out test set that included patients $5 8 - 6 4$ ( ${ \\mathrm { ~ \\it ~ n ~ } } = { \\mathrm { ~ \\it ~ 7 ~ } }$ ). ", + "bbox": [ + 174, + 299, + 826, + 439 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4.2 TRAINING DETAILS", + "text_level": 1, + "bbox": [ + 176, + 454, + 366, + 469 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Dimensionality reduction of polysomnography data was done by encoding all selected channels in each non-overlapping 30-second window using standard convolutional encoder. Table 4 summarizes the used encoder and decoder (for SOM-VAE and DESOM) architectures. The latent space for decoding in the SOM-VAE and DESOM benchmark models was not fully reduced to a 1D vector to enhance training. Nevertheless, the last adaptive average pooling layer that was used in the encoder of SOM-CPC, was applied in the bottleneck of SOM-VAE and DESOM before SOM quantization took place. As a result the feature vectors in all models were of size $F = 1 2 8$ . The decoder architecture (see table 4) was used both for the continuous and discrete decoding in the SOM-VAE model (without weight tying). No AR-component was used in the SOM-CPC model to make a fair comparison to the SOM-VAE and DESOM model that also did not include such a component. ", + "bbox": [ + 173, + 478, + 825, + 617 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For the SOM-CPC model, $P = 3$ future predictions (i.e. positive samples) were used, and $N = 3$ negative samples were drawn for each positive sample. The latter were drawn from the same subject as the positive sample. The $\\sigma$ of the Gaussian neighbourhood kernel was exponentially annealed to σ(nmax) = 0.5 during training. ", + "bbox": [ + 174, + 625, + 825, + 681 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "All models were trained with the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 1e-4 and a batch size of 128. Each model was trained for maximally 500 epochs, and the best model was selected based on the lowest $\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n } }$ (for SOM-VAE and DESOM) or $\\mathcal { L } _ { \\mathrm { { I n f o N C E } } }$ (for SOM-CPC and CPC) on the validation set. ", + "bbox": [ + 174, + 689, + 825, + 744 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4.3 EXTENDED RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 760, + 377, + 775 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 5 shows the quantitative results on sleep data, comparing SOM-CPC against deep-SOM models (SOM-VAE and DESOM) and disjoint training of CPC, followed by either a supervised linear classifier, K-means or a SOM. Discussion of the main results in this table can be found in section 4.2. The ablation experiments in which the value of $\\cdot$ and/or the similarity metric was altered show that classification and clustering performance slightly dropped when using the cosine similarity with a temperature value of 1, while the topographic organization slightly improved (i.e. lower TE). These effects vanished when using a temperature of $\\cdot$ . Both runs showed worse temporal smoothness (i.e. higher $\\cdot$ ). As expected, only changing the temperature value to 0.07 did almost not affect results, suggesting that the linear projector heads were able to adjust for this scaling factor. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/f5ff7724dbf4120d91f844b492b2e9256e143e42203e9ef222849bea260cd042.jpg", + "table_caption": [ + "Table 4: Model details for the sleep experiments in section 4.2. " + ], + "table_footnote": [], + "table_body": "
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU (0.01)15110
MaxPool1Dbs ×16×32-55=
Dropout (0.1)bs ×16× 32---=
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55
Dropout (0.1)bs ×32×8-=-==
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2-=
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
AdaptiveAvgPool1Dbs ×128×1=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)1511(18,17)
MaxPool1Dbs ×16×32-55
Dropout (0.1)bs ×16×32--==
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55=
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2--==
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
DecoderforSOM-VAEandDESOM
Conv1Dbs×64×264Leaky ReLU(0.01)3110
ConvTranspose1Dbs ×64×264None5510
Conv1Dbs ×32×232Leaky ReLU (0.01)5110
ConvTranspose1Dbs ×32×232None5510
Conv1Dbs ×16×216Leaky ReLU (0.01)9110
ConvTranspose1Dbs ×16×216None5510
Conv1Dbs ×6×26None15110
", + "bbox": [ + 210, + 73, + 782, + 385 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We also tested the performance when using the SimCLR (Chen et al., 2020) objective for the task loss, instead of the CPC objective. SimCLR is also a contrastive learning framework, but instead of drawing positive samples from the future latent space, these samples are created by applying augmentations on the anchor window. Inspired by Um et al. (2017) we used the following augmentations: independent and identically distributed Gaussian noise $\\mathcal { N } ( 0 , 0 . 0 5 )$ was added (called jitter in their implementation), each channel was scaled with a value drawn from $\\cdot$ , windows were split in 4 sub-windows of minimal 2 seconds and randomly permuted, and lastly time series were both time warped and magnitude warped. The latter two augmentations make use of smooth curves that smoothly vary the positions of time stamps or magnitude values, respectively. ", + "bbox": [ + 173, + 393, + 826, + 518 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Besides the difference on how to create positive samples, the originally proposed SimCLR model has some other slight differences with respect to the CPC model: ", + "bbox": [ + 173, + 526, + 825, + 554 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• The SimCLR loss uses the cosine similarity, while CPC uses the (unnormalized) dot product as the similarity metric (see eq. (9)). \n• SimCLR uses an additional temperature $\\tau$ in its loss function (see eq. (9)), for which the value is often set to 0.07 (Chen et al., 2020; Woo et al., 2022). CPC does not incorporate such a temperature, which effectively means that it uses a value of 1. \n• SimCLR uses a non-linear MLP projection head, while CPC uses linear projection heads. \n• SimCLR uses negative samples from within the batch, while this is not specified in the CPC paper. This specified design choice makes SimCLR typically very sensitive to the batch size. \nSimCLR was not proposed to include an auto-regressive component, and can not straightforwardly be extended to do so, while CPC can be implemented with or without such a module. ", + "bbox": [ + 217, + 566, + 825, + 743 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "For the most fair comparison, the procedure for drawing negative samples in SimCLR is done equivalently as for SOM-CPC, i.e. within the recording, instead of within the batch. However, in the SOM-SimCLR model (i.e. the joint training of SOM with SimCLR), each drawn negative sample is added to the set of negative samples both in its raw form, and with a random augmentation, which effectively doubles the number of negative samples. Table 3 reports the performance of the baseline SOM-SimCLR model (i.e. with settings $\\tau = 0 . 0 7$ and the cosine similarity), and variants using a temperature value of 1 and/or the dot product as the similarity metric. All settings regarding training procedure and the SOM were set equivalently as in the SOM-CPC training. Table 5 shows that SOM-SimCLR results for $\\cdot$ are better than those with $\\tau = 1$ , which is in line with findings from Chen et al. (2020); Woo et al. (2022). However, even with $\\cdot$ , performance of SOM-SimCLR is lower on all metrics compared to the SOM-CPC model with the same value for $\\alpha$ . The higher $\\ell _ { \\mathrm { { 2 , \\mathrm { { s m o o t h } } } } }$ metric of SOM-SimCLR indicates on average larger jumps over the SOM map through time, which might be caused by the fact that the SimCLR task objective does not incorporate temporal information, while InfoNCE does exploit this. Training time of SOM-SimCLR was, moverover, considerably longer than SOM-CPC with the same settings due to the additional augmentations that need to be computed for every data window and its negative samples. ", + "bbox": [ + 173, + 757, + 826, + 924 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/62440512ae2346c7d6d057665f01c364e6fa9ee44267d170c2f0e712c16f79e0.jpg", + "table_caption": [ + "Table 5: Test set performance of various models trained on sleep recordings. SOMs of models with a \\* are visualized in fig. 4. Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). " + ], + "table_footnote": [], + "table_body": "
ModelCPC + linear classifierLsom sg[-]PurityNMICohen's kappa
CPC + K-means CPC(F= 2) + linear classifier- - =- =- - --.68±.10
-=.79 -.29.61±.11 .52±.10 .55±.090-- =
CPC(F= 2)+K-means CPC + PCA+ linear classifier *=.74.24.54±.086
CPC +PCA +K-means - 1e-3 Plus 1e-2 Plus-.77.26 .57±.082
SOM-VAE.71 .23.51±.042.36±.26.24±.031
√ √.71.23 .51±.042.67±.17 2.60±.34.30±.037 .28±.042
DESOM√ √.72 .71.23 .23.52±.03 .53±.033.08±.32 .31±.054
1e-6 Gaussian 1e-5 Gaussian 1e-4 Gaussian 1e-3 Gaussian 1e-2 GaussianX X.70 .70.27 .53±.05 .23 .50±.042.14±.32 2.10±.26.095±.020 .11±.028
X.71.22 .51±.042.35±.24 2.40±.16.17±.035
SOM-SimCLR(T = 0.07)X X.71 .71.22 .51±.05 .22 .50±.042.30±.26.22±.0085 .23±.021
1e-3 Gaussian SOM-SimCLR(T = 1) 1e-3 SOM-CPC (ours)X X.73 .70.23 .20.53±.13 .48±.162.21±.35 1.87±.30.29±.026 .50±.068
Gaussian 1e-5 Gaussian 1e-4 Gaussian * 1e-3 GaussianX X X.78 .27 .78 .27 .78.59±.11 .61±.101.03±.11 .041±.014 1.01±.10 .062±.018
1e-2 Gaussian .1 Gaussian 1e-3 Gaussian 1e-2 Gaussian GaussianX X √.27 .79 .28 .79 .28 .78 .27.61±.12 .60±.11 .65±.07 .62王.10 .60±.101.02±.09 .032±.0096 1.08±.11 .19±.042 1.09±.09 .19±.04 1.02±.12 .067±.025
.1 Vaaeais 1 Gaussian √ 1e-3 Plus X 1e-3 Plus √ SOM-CPC(T = 0.07,sim = cosine sim.) 1e-3 Gaussian X SOM-CPC (τ = 1,sim = cosine sim.) X.78 .78 .78 .79 .79 .78 .73 .79.27 .27 .27 .28 .28 .27 .27
1e-3 Gaussian SOM-CPC(τ = 0.07,sim = dot prod.) 1e-3 Gaussian CPC + SOM (disjoint) Gaussian =X .79 = 0.80 Mwv1.43±.15 .025±.0094 1.06±.11 .059±.020 1.21±.11 .52±.042 Cohen's kappa
InfoNCE 1.2 0.10 1.0 0.08Commitment lossSOM loss 5Purity bsl00.30NMI
0.840.75wAb众
0.063 20.70 0.650.25
0.60.0410.600.200.2
0.40.0200.550.150.0Train Validation
0 2000 200400200 4000.50 02004000.10 0 200
400 Epoch0 EpochEpochEpochEpoch400 0200 400 Epoch
", + "bbox": [ + 160, + 152, + 861, + 574 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "", + "image_caption": [ + "Figure 7: Training curves of SOM-CPC (the model indicated with a \\* in table 5) for both the training and validation set. The green dashed line indicates the epoch of the used model, i.e. the one with the lowest validation InfoNCE loss. It can be seen that the epoch with the best clustering and classification performance does not necessarily align with the epoch that has the lowest loss commitment and/or SOM loss. " + ], + "image_footnote": [], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 659, + 825, + 714 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 7 shows training curves of the training and validation set for the SOM-CPC model that is indicated with a \\* in table 5. The green line indicates the epoch with the lowest InfoNCE validation loss. These graphs show that the performance of InfoNCE, the commitment and SOM loss, and classification metrics do not necessarily align, making it dependent on your final goal with the SOM-CPC model what is the most appropriate stopping-criterion. ", + "bbox": [ + 173, + 722, + 825, + 791 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.5 AUDIO EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 367, + 117 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A.5.1 TRAINING DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 128, + 366, + 143 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Audio streams were encoded in windows of 0.01 seconds $( = 1 6 0$ samples). For CPC, GRU-DESOM, and SOM-CPC the contextual information of $L = 1 2 7$ previous windows was aggregated using a GRU, equivalent as proposed by Oord et al. (2019). The InfoNCE objective for (SOM-)CPC was computed on top of the last context vector of the GRU. To test different settings for the GRU-desom model, we distinguished GRU-DESOM that decodes only the last window, given the last context vector, and GRU-DESOM that decodes the full sequence of 128 windows from the respective context vectors. ", + "bbox": [ + 173, + 154, + 825, + 251 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Table 6 provides the model details of the encoder, and the decoder for the DESOM benchmarks. The reconstruction loss of the DESOM model was found to be hampered in its optimization when using the encoder architecture, as adopted for the SOM-CPC model. As such, the downsampling factor of the encoder was reduced for the DESOM model to enable minimization of the task loss during training. ", + "bbox": [ + 174, + 257, + 825, + 328 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/c9ef71e01b5d1e49ca6eec7826d5397ed9f0335aeb0912d193ad13ef39e84ede.jpg", + "table_caption": [ + "Table 6: Model details for the audio experiments in section 4.3. " + ], + "table_footnote": [], + "table_body": "
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
Encoder forSOM-CPCand CPC
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512512ReLU4211
GRUbs × 512512--==-
EncoderforSOM-VAEandDESOM
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512 ×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512×2512ReLU411same
Flattenbs ×1024-===
GRUbs ×10241024
DecoderforSOM-VAEandDESOM
Unflattenbs×512×2==
Conv1Dbs × 512×2512ReLU411same
ConvTranspose1Dbs × 512×4512ReLU4211
ConvTranspose1Dbs × 512×8512ReLU4211
ConvTranspose1Dbs × 512× 32512ReLU8412
ConvTranspose1Dbs ×1×160512ReLU10513 (+ output pad = 1)
", + "bbox": [ + 214, + 366, + 790, + 592 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For training of SOM-CPC, we followed the settings from Oord et al. (2019) and set $P = 1 2$ . The number of negative samples was set to $N = 1 0$ , which were drawn randomly from the entire training set. All deep-SOM models were trained for maximally 3000 epochs, using the Adam optimizer Kingma & Ba (2014) with a learning rate of 1e-4 and a batch size of 8. One epoch was defined as a push through of one sequence of 128 windows (or 1 window for DESOM) from each recording. The best model was selected based on the lowest validation task loss. ", + "bbox": [ + 174, + 604, + 825, + 689 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The supervised linear classifier and disjoint SOM training on top of the frozen CPC embeddings were trained with a batch size of 128, and a learning rate of 1e-4 and 1e-2, respectively. Both models were stopped upon convergence of the validation loss (which was after 200 and 250 epochs, respectively). ", + "bbox": [ + 174, + 695, + 825, + 738 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "A.5.2 EXTENDED RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 752, + 379, + 767 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Quantitative results of the audio experiments can be found in table 7. For this application, Cohen’s kappa is computed as the average over all data windows in the test set, which is different from the synthetic and sleep case, where it was computed as the average and one standard deviation across recordings. In these audio experiments, all windows from one recording contain the same speaker id label. Computing Cohen’s kappa per-recording, i.e. with having the same label for all windows in that recording, is therefore inappropriate as the computation can not correct for correctness by chance. Table 7 show that SOM-CPC clearly outperformed all variants of the (GRU-)DESOM model, and feature extraction using CPC, followed by PCA and linear or non-linear classification. ", + "bbox": [ + 174, + 776, + 825, + 888 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Ablations with respect to the temperature $^ { \\prime }$ and the similarity metric in the loss function indicated, similarly as in the sleep case, that simply changing the temperature value did hardly affect SOM-CPC performance. However, changing the similarity metric to be the cosine similarity caused a drop in clustering and classification performance, only when using a temperature value of 1. This result overlaps with the experimental findings in the sleep case (see appendix A.4.3), and suggests that a low temperature value - which was found beneficial in the SimCLR objective (Chen et al., 2020; Woo et al., 2022) - does not equivalently improve SOM-CPC performance. ", + "bbox": [ + 171, + 896, + 828, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Figure 8 compares the test set projection on the 2D PCA space, created on the CPC features (with $F = 1 2 8$ ), to the SOM from the SOM-CPC model that is denoted with a $\\cdot$ in table 7 (and also visualized in fig. 5-right). ", + "bbox": [ + 174, + 181, + 825, + 223 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/604121c4109407dc274605e72e54e43c43391ea726b9eba5124e495e129d8868.jpg", + "table_caption": [ + "Table 7: Test set performance of various models trained on audio recordings. SOMs of models with a \\* are visualized in fig. 5. Bold values indicate the best performance per column (excluding the upper bound of the vanilla CPC model, which does not result in a 2D representation). " + ], + "table_footnote": [], + "table_body": "
ModelαSLsom sg[·]PurityNMICohen'skappaTE
CPC+linearclassifier----1.00-
CPC + K-means==1.00.601.00
CPC(F=2)+ linear classifier==-.00=
CPC(F= 2)+K-means.13.013.025
CPC+PCA+ linear classifier=--.86
CPC + PCA + K-means-=-.89.54..88
DESOM1e-5GaussianX.18.03.06.14±.035
1e-4GaussianX.23.08.11.34±.045
1e-3GaussianX.31.13.20.68±.050
1e-2GaussianX.13-.00.001.0±.00
GRU-DESOM (reconstructing last window)1e-5GaussianX.19.04.08.13±.028
1e-4GaussianX.26.09.15.20±.036
1e-3GaussianX.31.13.21.42±.078
1e-2GaussianX.32.14.22.46±.057
GRU-DESOM (reconstructing full sequence)1e-5GaussianX.19.05.08.34±.048
1e-4GaussianX.30.12.20.59±.072
1e-3GaussianX.33.14.22.78±.048
1e-2GaussianX.29.12.19.57±.069
SOM-CPC(ours)1e-5GaussianX.99.73.99.14±.081
1e-4GaussianX1.00.631.00.24±.12
*1e-3GaussianX1.00.611.00.33±.098
1e-2GaussianX1.00.61.99.33±.099
1e-3Gaussian√1.00.61.99.28±.087
1e-2Gaussian√1.00.61.99.35±0.10
.1Gaussian√1.00.611.00.35±.095
1Gaussian√1.00.611.00.38±.11
SOM-CPC (T = 0.07,sim = cosine sim.)1e-3GaussianX.99.61.99.42±0.12
SOM-CPC(τ = 1,sim = cosine sim.)1e-3GaussianX.88.55.86.17±.063
SOM-CPC(τ = 0.07,sim = dot prod.)1e-3GaussianX1.00.61.99.38±.098
CPC + SOM(disjoint)-Gaussian-1.00.621.00.28±.11
", + "bbox": [ + 187, + 285, + 807, + 569 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/25f87c5f96cd1f71916e3973abf24aec8831127866c3d6e679b873b8ffb4c103.jpg", + "image_caption": [ + "Figure 8: Projecting the test set on the 2D PCA space shows no division of the green and the red clusters in two sub-clusters, something that is visible in the SOM of the SOM-CPC model. These sub-clusters were found to relate to recordings that were made with different room acoustics. 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This, however,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "may discard important information that can not linearly be projected on these few dimensions. A", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "Self-Organizing Map (Kohonen, 1990), on the other hand, is an extension of K-means clustering that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "creates a low-dimensional interpretable visualization, while still representing the data in multiple", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "dimensions. However, SOMs typically act on features, which need to be selected heuristically and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 659, + 385, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 385, + 673 + ], + "score": 1.0, + "content": "may, therefore, strongly depend on the use case and/or data modality.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 539, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Deep learning (DL) models have become popular alternatives for non-linear dimensionality reduction", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 685, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 505, + 702 + ], + "score": 1.0, + "content": "that can be applied directly on raw data. Such models have been combined with joint clustering", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 697, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 507, + 712 + ], + "score": 1.0, + "content": "objectives in the latent space (Xie et al., 2016; Yang et al., 2017; Madiraju, 2018; Lee & Schaar,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "2020). These methods, however, do typically not create a (visually) interpretable representation,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and sometimes make use of label information during training (Lee & Schaar, 2020). To enhance", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "interpretability, latent space representations of DL models are often visualized using a t-distributed", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "stochastic neighbor embedding (t-SNE) (Hinton & Roweis, 2002). Albeit its frequent use, t-SNE", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "does not allow a direct deployment on unseen data as it does not learn a reusable mapping between", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 325, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 325, + 128 + ], + "score": 1.0, + "content": "the multi-dimensional and the low-dimensional space.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 676, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "interpretability, latent space representations of DL models are often visualized using a t-distributed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "stochastic neighbor embedding (t-SNE) (Hinton & Roweis, 2002). Albeit its frequent use, t-SNE", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "does not allow a direct deployment on unseen data as it does not learn a reusable mapping between", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 325, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 325, + 128 + ], + "score": 1.0, + "content": "the multi-dimensional and the low-dimensional space.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "To acquire visually interpretable data representations from raw data, without assuming that data must", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "live in two or three dimensions only, non-linear DL encoders have been combined with SOMs (Ferles", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 166 + ], + "score": 1.0, + "content": "et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "et al., 2021). In the resulting joint training strategy of these deep-SOM models, the SOM objective", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "can be seen as a regularizer on the encoding procedure, as it promotes a cluster-friendly feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "space. Most of these models have focused on autoencoders as feature extractors. However, similar to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "score": 1.0, + "content": "Mrabah et al. (2020), we hypothesize that their reconstruction objective may hamper the clustering or", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "structured representation learning objective: while within-cluster similarities should remain preserved", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "for latent clustering, reconstruction demands a preservation of all factors of similarity. Moreover,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 242 + ], + "score": 1.0, + "content": "in the context of time series representation learning, other self-supervised models - that take the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 432, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 432, + 254 + ], + "score": 1.0, + "content": "temporal nature of the data into account during training - might be more suitable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "Contrastive self-supervised learning approaches have quickly become popular thanks to their superior", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "representation learning performance in many domains (see Le-Khac et al. (2020) for a review). While", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "many of these models rely on data augmentations during training in order to construct pairs of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "similar data points, Contrastive Predictive Coding (CPC) (Oord et al., 2019) leverages the temporal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "dimension for this purposes, making it a natural choice for self-supervised representation learning of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "time series. In CPC, the temporal dimension not only serves as a pretext task, but simultaneously", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 443, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 443, + 336 + ], + "score": 1.0, + "content": "enforces latent smoothness over time. The contributions of this work are as follows:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 133, + 344, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 132, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 132, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "• We propose a new model in the deep-SOM family: SOM-CPC, which is suitable for learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "structured and interpretable 2D representations of (high-rate) time series by encoding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 366, + 461, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 461, + 380 + ], + "score": 1.0, + "content": "subsequent data windows to a topologically ordered set of quantization vectors.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 136, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 136, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "• Using regression and classification probing tasks, we show that SOM-CPC preserves more", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "information in its 2D representation than CPC that is followed by PCA, and a linear classifier", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "or K-means, or directly encoding CPC’s latent space to two dimensions. SOM-CPC’s joint", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 412, + 474, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 474, + 427 + ], + "score": 1.0, + "content": "optimization, moreover, facilitates a smooth temporal trajectory through 2D space.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 136, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 136, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "• We show that SOM-CPC quantitatively and qualitatively outperforms deep-SOM models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 141, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "with a reconstruction objective in terms of both clustering and topological ordering. It,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "moreover, requires less auxiliary loss functions (and associated hyperparameter tuning)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 141, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "thanks to its natural tendency to incorporate temporal smoothness. Lastly, SOM-CPC’s", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 142, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "training behavior shows that the SOM clustering objective better aligns with the CPC", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 483, + 308, + 494 + ], + "spans": [ + { + "bbox": [ + 142, + 483, + 308, + 494 + ], + "score": 1.0, + "content": "objective than with a reconstruction loss.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 509, + 208, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 507, + 210, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 210, + 524 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 288, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 289, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 289, + 545 + ], + "score": 1.0, + "content": "2.1 KOHONEN SELF-ORGANIZING MAPS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "Kohonen’s Self-Organizing Map (SOM) (Kohonen, 1990) is an algorithm to find a visually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "interpretable topological data representation. It has been found useful to reveal intricate patterns and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 574, + 507, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 507, + 588 + ], + "score": 1.0, + "content": "structure in a plethora of applications. The algorithm’s output, the low-dimensional visualization,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "is often referred to as a SOM as well. We choose to use a use a 2D visualization to enhance", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 595, + 171, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 171, + 611 + ], + "score": 1.0, + "content": "interpretability.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 235, + 627 + ], + "score": 1.0, + "content": "We define a set of data points", + "type": "text" + }, + { + "bbox": [ + 236, + 615, + 245, + 624 + ], + "score": 0.8, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 613, + 365, + 627 + ], + "score": 1.0, + "content": ", and quantized counterparts", + "type": "text" + }, + { + "bbox": [ + 365, + 614, + 415, + 626 + ], + "score": 0.92, + "content": "q _ { \\Phi } ( z ) \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 613, + 432, + 627 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 432, + 615, + 464, + 625 + ], + "score": 0.88, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 613, + 506, + 627 + ], + "score": 1.0, + "content": ". The set", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 180, + 638 + ], + "score": 0.92, + "content": "\\Phi : \\{ \\phi _ { 1 } , \\ldots , \\phi _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 624, + 382, + 638 + ], + "score": 1.0, + "content": "is a trainable quantization codebook containing", + "type": "text" + }, + { + "bbox": [ + 382, + 627, + 389, + 635 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 624, + 482, + 638 + ], + "score": 1.0, + "content": "vectors or prototypes", + "type": "text" + }, + { + "bbox": [ + 482, + 626, + 505, + 637 + ], + "score": 0.88, + "content": "\\phi _ { i } \\in", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 636, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 167, + 651 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { F } , 1 \\le i \\le k", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 636, + 241, + 653 + ], + "score": 1.0, + "content": ". The jth prototype", + "type": "text" + }, + { + "bbox": [ + 241, + 637, + 298, + 653 + ], + "score": 0.93, + "content": "\\boldsymbol \\phi ^ { ( n ) } = q _ { \\Phi } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 637, + 448, + 652 + ], + "score": 1.0, + "content": "is the ‘winning vector’ for data point", + "type": "text" + }, + { + "bbox": [ + 449, + 641, + 456, + 649 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ", at iteration", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 114, + 660 + ], + "score": 0.68, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "of the training procedure. The learned codebook vectors are placed on a pre-defined 2D grid by", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 673 + ], + "score": 1.0, + "content": "assigning an xy-coordinate to each vector at initialization. Note that this creates a 2D representation,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 195, + 685 + ], + "score": 1.0, + "content": "while each data point", + "type": "text" + }, + { + "bbox": [ + 196, + 673, + 203, + 682 + ], + "score": 0.76, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 670, + 254, + 685 + ], + "score": 1.0, + "content": "still lives in", + "type": "text" + }, + { + "bbox": [ + 255, + 673, + 269, + 682 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 670, + 294, + 685 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 294, + 674, + 324, + 682 + ], + "score": 0.9, + "content": "F \\gg 2", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 670, + 506, + 685 + ], + "score": 1.0, + "content": ". This is conceptually different than the way", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 682, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 682, + 446, + 695 + ], + "score": 1.0, + "content": "in which PCA achieves dimensionality reduction to 2D, where all information in the", + "type": "text" + }, + { + "bbox": [ + 446, + 683, + 459, + 693 + ], + "score": 0.85, + "content": "3 ^ { \\mathrm { r d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 682, + 506, + 695 + ], + "score": 1.0, + "content": "and higher", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 694, + 505, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 405, + 706 + ], + "score": 1.0, + "content": "principle components is strictly omitted. During training of a SOM, each", + "type": "text" + }, + { + "bbox": [ + 406, + 694, + 416, + 705 + ], + "score": 0.88, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 694, + 505, + 706 + ], + "score": 1.0, + "content": "is updated as follows", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 705, + 230, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 199, + 716 + ], + "score": 1.0, + "content": "(Kohonen, 1990), with", + "type": "text" + }, + { + "bbox": [ + 199, + 706, + 226, + 715 + ], + "score": 0.88, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 705, + 230, + 716 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 717, + 393, + 734 + ], + "lines": [ + { + "bbox": [ + 218, + 717, + 393, + 734 + ], + "spans": [ + { + "bbox": [ + 218, + 717, + 393, + 734 + ], + "score": 0.93, + "content": "\\phi _ { i } ^ { ( n + 1 ) } = \\phi _ { i } ^ { ( n ) } + \\eta ^ { ( n ) } S _ { i } \\big ( \\phi ^ { ( n ) } \\big ) \\big ( z - \\phi _ { i } ^ { ( n ) } \\big ) ,", + "type": "interline_equation", + "image_path": "561ca698c22d04afb1648da2b7bbc46a6dd5d0f4ba9754b6dc952592531533ed.jpg" + } + ] + } + ], + "index": 51, + "virtual_lines": [ + { + "bbox": [ + 218, + 717, + 393, + 734 + ], + "spans": [], + "index": 51 + } + ] + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 505, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 506, + 144 + ], + "score": 1.0, + "content": "To acquire visually interpretable data representations from raw data, without assuming that data must", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "live in two or three dimensions only, non-linear DL encoders have been combined with SOMs (Ferles", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 166 + ], + "score": 1.0, + "content": "et al., 2018; Pesteie et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "et al., 2021). In the resulting joint training strategy of these deep-SOM models, the SOM objective", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "can be seen as a regularizer on the encoding procedure, as it promotes a cluster-friendly feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "space. Most of these models have focused on autoencoders as feature extractors. However, similar to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 210 + ], + "score": 1.0, + "content": "Mrabah et al. 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Moreover,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 242 + ], + "score": 1.0, + "content": "in the context of time series representation learning, other self-supervised models - that take the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 432, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 432, + 254 + ], + "score": 1.0, + "content": "temporal nature of the data into account during training - might be more suitable.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 132, + 506, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "Contrastive self-supervised learning approaches have quickly become popular thanks to their superior", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "representation learning performance in many domains (see Le-Khac et al. (2020) for a review). While", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "many of these models rely on data augmentations during training in order to construct pairs of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "similar data points, Contrastive Predictive Coding (CPC) (Oord et al., 2019) leverages the temporal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "dimension for this purposes, making it a natural choice for self-supervised representation learning of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "time series. In CPC, the temporal dimension not only serves as a pretext task, but simultaneously", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 443, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 443, + 336 + ], + "score": 1.0, + "content": "enforces latent smoothness over time. The contributions of this work are as follows:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 259, + 506, + 336 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 344, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 132, + 343, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 132, + 343, + 505, + 357 + ], + "score": 1.0, + "content": "• We propose a new model in the deep-SOM family: SOM-CPC, which is suitable for learning", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 505, + 369 + ], + "score": 1.0, + "content": "structured and interpretable 2D representations of (high-rate) time series by encoding", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 366, + 461, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 461, + 380 + ], + "score": 1.0, + "content": "subsequent data windows to a topologically ordered set of quantization vectors.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 136, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 136, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "• Using regression and classification probing tasks, we show that SOM-CPC preserves more", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 392, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 505, + 403 + ], + "score": 1.0, + "content": "information in its 2D representation than CPC that is followed by PCA, and a linear classifier", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 142, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "or K-means, or directly encoding CPC’s latent space to two dimensions. SOM-CPC’s joint", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 412, + 474, + 427 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 474, + 427 + ], + "score": 1.0, + "content": "optimization, moreover, facilitates a smooth temporal trajectory through 2D space.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 136, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 136, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "• We show that SOM-CPC quantitatively and qualitatively outperforms deep-SOM models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 141, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "with a reconstruction objective in terms of both clustering and topological ordering. It,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 141, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "moreover, requires less auxiliary loss functions (and associated hyperparameter tuning)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 141, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "thanks to its natural tendency to incorporate temporal smoothness. Lastly, SOM-CPC’s", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 142, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 142, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "training behavior shows that the SOM clustering objective better aligns with the CPC", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 483, + 308, + 494 + ], + "spans": [ + { + "bbox": [ + 142, + 483, + 308, + 494 + ], + "score": 1.0, + "content": "objective than with a reconstruction loss.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 28, + "bbox_fs": [ + 132, + 343, + 506, + 494 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 509, + 208, + 522 + ], + "lines": [ + { + "bbox": [ + 104, + 507, + 210, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 507, + 210, + 524 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 288, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 289, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 289, + 545 + ], + "score": 1.0, + "content": "2.1 KOHONEN SELF-ORGANIZING MAPS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 566 + ], + "score": 1.0, + "content": "Kohonen’s Self-Organizing Map (SOM) (Kohonen, 1990) is an algorithm to find a visually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "interpretable topological data representation. It has been found useful to reveal intricate patterns and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 574, + 507, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 507, + 588 + ], + "score": 1.0, + "content": "structure in a plethora of applications. The algorithm’s output, the low-dimensional visualization,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "is often referred to as a SOM as well. 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The set", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 180, + 638 + ], + "score": 0.92, + "content": "\\Phi : \\{ \\phi _ { 1 } , \\ldots , \\phi _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 624, + 382, + 638 + ], + "score": 1.0, + "content": "is a trainable quantization codebook containing", + "type": "text" + }, + { + "bbox": [ + 382, + 627, + 389, + 635 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 624, + 482, + 638 + ], + "score": 1.0, + "content": "vectors or prototypes", + "type": "text" + }, + { + "bbox": [ + 482, + 626, + 505, + 637 + ], + "score": 0.88, + "content": "\\phi _ { i } \\in", + "type": "inline_equation" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 636, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 167, + 651 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { F } , 1 \\le i \\le k", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 636, + 241, + 653 + ], + "score": 1.0, + "content": ". The jth prototype", + "type": "text" + }, + { + "bbox": [ + 241, + 637, + 298, + 653 + ], + "score": 0.93, + "content": "\\boldsymbol \\phi ^ { ( n ) } = q _ { \\Phi } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 637, + 448, + 652 + ], + "score": 1.0, + "content": "is the ‘winning vector’ for data point", + "type": "text" + }, + { + "bbox": [ + 449, + 641, + 456, + 649 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ", at iteration", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 114, + 660 + ], + "score": 0.68, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "of the training procedure. 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A Gaussian kernel is often used which weighs node", + "type": "text" + }, + { + "bbox": [ + 340, + 106, + 344, + 114 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 105, + 400, + 117 + ], + "score": 1.0, + "content": "according to:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 120, + 453, + 173 + ], + "lines": [ + { + "bbox": [ + 157, + 120, + 453, + 173 + ], + "spans": [ + { + "bbox": [ + 157, + 120, + 453, + 173 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { S _ { i } \\big ( \\phi ^ { ( n ) } \\big ) = \\exp \\Big ( - \\frac { d _ { j , i } ^ { ( n ) } } { 2 ( \\sigma ^ { ( n ) } ) ^ { 2 } } \\Big ) , \\quad \\mathrm { ~ w i t h ~ } } \\\\ & { d _ { j , i } ^ { ( n ) } = | | \\mathcal { P } \\{ \\phi ^ { ( n ) } \\} , \\mathcal { P } \\{ \\phi _ { i } ^ { ( n ) } \\} | | _ { 2 } ^ { 2 } \\quad \\quad \\mathrm { ~ a n d ~ } \\quad \\quad \\sigma ^ { ( n ) } = \\sigma ^ { ( 0 ) } \\exp ( - n / \\lambda ) , } \\end{array}", + "type": "interline_equation", + "image_path": "276860374c3d4568a561e9a0ef57351f397e5a75ec8410db28b87c465366004f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 157, + 120, + 453, + 137.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 157, + 137.66666666666666, + 453, + 155.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 157, + 155.33333333333331, + 453, + 172.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 133, + 190 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 178, + 143, + 188 + ], + "score": 0.81, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 177, + 437, + 190 + ], + "score": 1.0, + "content": "projects a codebook vector to its corresponding coordinate on the grid,", + "type": "text" + }, + { + "bbox": [ + 437, + 177, + 454, + 188 + ], + "score": 0.88, + "content": "\\boldsymbol { \\sigma } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 188, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 230, + 204 + ], + "score": 1.0, + "content": "initial standard deviation, and", + "type": "text" + }, + { + "bbox": [ + 230, + 190, + 237, + 200 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 188, + 340, + 204 + ], + "score": 1.0, + "content": "the decay factor. Setting", + "type": "text" + }, + { + "bbox": [ + 340, + 189, + 463, + 202 + ], + "score": 0.9, + "content": "\\lambda = - n _ { \\mathrm { m a x } } / \\log \\left( \\sigma ^ { ( n _ { \\mathrm { m a x } } ) } / \\sigma ^ { ( 0 ) } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 188, + 496, + 204 + ], + "score": 1.0, + "content": "sweeps", + "type": "text" + }, + { + "bbox": [ + 497, + 193, + 504, + 200 + ], + "score": 0.7, + "content": "\\sigma", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 141, + 216 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 142, + 201, + 159, + 213 + ], + "score": 0.89, + "content": "\\boldsymbol { \\sigma } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 200, + 178, + 216 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 178, + 202, + 199, + 213 + ], + "score": 0.88, + "content": "\\sigma ^ { n _ { \\mathrm { m a x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 200, + 210, + 216 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 211, + 204, + 231, + 213 + ], + "score": 0.85, + "content": "n _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 200, + 335, + 216 + ], + "score": 1.0, + "content": "steps. The dependence of", + "type": "text" + }, + { + "bbox": [ + 335, + 203, + 345, + 213 + ], + "score": 0.87, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 200, + 408, + 216 + ], + "score": 1.0, + "content": "on the distance", + "type": "text" + }, + { + "bbox": [ + 408, + 203, + 424, + 215 + ], + "score": 0.89, + "content": "d _ { j , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 200, + 505, + 216 + ], + "score": 1.0, + "content": ", implies a weighing", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 226 + ], + "score": 1.0, + "content": "of 1 for the winning node (i.e. distance equals zero), and lower than 1 for neighbour nodes. Note", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 224, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 506, + 237 + ], + "score": 1.0, + "content": "that other neighbourhood structures have been proposed as well, for example using the four closest", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 461, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 461, + 247 + ], + "score": 1.0, + "content": "neighbours on the grid, which results in a kernel with a plus-shape (Fortuin et al., 2019).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 260, + 220, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 221, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 221, + 273 + ], + "score": 1.0, + "content": "2.2 DEEP-SOM MODELS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 506, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "All deep-SOM research has focused on combining autoencoders (Ferles et al., 2018; Pesteie", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "et al., 2018; Fortuin et al., 2019; Forest et al., 2019; Manduchi et al., 2021; Forest", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "et al., 2021) with a SOM. These models can broadly be summarized as a vector-quantized", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "(VQ) VAE (van den Oord et al., 2017), with a topological organization of the vectors", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "in the quantization codebook: the SOM. The models are trained end-to-end using error", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 308, + 349 + ], + "score": 1.0, + "content": "backpropagation of both a reconstruction task loss", + "type": "text" + }, + { + "bbox": [ + 308, + 336, + 328, + 347 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { t a s k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 335, + 371, + 349 + ], + "score": 1.0, + "content": "and a loss", + "type": "text" + }, + { + "bbox": [ + 371, + 336, + 392, + 348 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { t o p o } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "that encourages topological", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 347, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 359 + ], + "score": 1.0, + "content": "ordering in the SOM. In general, a deep-SOM training objective takes the following form:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 364, + 475, + 396 + ], + "lines": [ + { + "bbox": [ + 117, + 364, + 475, + 396 + ], + "spans": [ + { + "bbox": [ + 117, + 364, + 475, + 396 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { \\mathrm { d e e p . 5 o M } } = \\mathcal { L } _ { \\mathrm { t a k } } + \\alpha \\mathcal { L } _ { \\mathrm { t o p o } } , \\qquad ( 4 ) \\quad \\mathrm { ~ w i t h ~ } \\quad \\mathcal { L } _ { \\mathrm { t o p o } } ( z ^ { ( n ) } ) = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\sum _ { i = 1 } ^ { k } S _ { i } \\big ( \\phi _ { i = j } ^ { ( n ) } \\big ) \\big | | z ^ { ( n ) } - \\phi _ { i } ^ { ( n ) } | \\big | _ { 2 } ^ { 2 } \\Big ] .", + "type": "interline_equation", + "image_path": "1ab7a8d7952e75436d34f53af92f92f8ce9bb07960874c7bc6b3d0683e841b33.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 117, + 364, + 475, + 374.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 117, + 374.6666666666667, + 475, + 385.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 117, + 385.33333333333337, + 475, + 396.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 173, + 412 + ], + "score": 1.0, + "content": "Hyperparameter", + "type": "text" + }, + { + "bbox": [ + 174, + 402, + 182, + 409 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "controls the trade-off. The topological loss thus replaces the original update rule", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 302, + 422 + ], + "score": 1.0, + "content": "of the SOM algorithm (see eq. (1)). The features", + "type": "text" + }, + { + "bbox": [ + 303, + 411, + 330, + 420 + ], + "score": 0.9, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "are jointly optimized, and thus also depend", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 460, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 119, + 434 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 423, + 126, + 431 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 420, + 460, + 434 + ], + "score": 1.0, + "content": "now. To prevent clutter we will, however, omit the (n)-superscript in the following.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "Fortuin et al. (2019) propose the SOM-VAE model. As opposed to VQ-VAE, SOM-VAE", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "has two decoders, as it also decodes the continuous latents. Topological organization of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "codebook vectors is enforced by using a plus-shaped neighbourhood kernel, which affects the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "codebook vectors of the direct neighbours of the winning node (i.e. up, down, left, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "right on the grid). 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In general, a deep-SOM training objective takes the following form:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 280, + 506, + 359 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 364, + 475, + 396 + ], + "lines": [ + { + "bbox": [ + 117, + 364, + 475, + 396 + ], + "spans": [ + { + "bbox": [ + 117, + 364, + 475, + 396 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { \\mathrm { d e e p . 5 o M } } = \\mathcal { L } _ { \\mathrm { t a k } } + \\alpha \\mathcal { L } _ { \\mathrm { t o p o } } , \\qquad ( 4 ) \\quad \\mathrm { ~ w i t h ~ } \\quad \\mathcal { L } _ { \\mathrm { t o p o } } ( z ^ { ( n ) } ) = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\sum _ { i = 1 } ^ { k } S _ { i } \\big ( \\phi _ { i = j } ^ { ( n ) } \\big ) \\big | | z ^ { ( n ) } - \\phi _ { i } ^ { ( n ) } | \\big | _ { 2 } ^ { 2 } \\Big ] .", + "type": "interline_equation", + "image_path": "1ab7a8d7952e75436d34f53af92f92f8ce9bb07960874c7bc6b3d0683e841b33.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 117, + 364, + 475, + 374.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 117, + 374.6666666666667, + 475, + 385.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 117, + 385.33333333333337, + 475, + 396.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 433 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 173, + 412 + ], + "score": 1.0, + "content": "Hyperparameter", + "type": "text" + }, + { + "bbox": [ + 174, + 402, + 182, + 409 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "controls the trade-off. The topological loss thus replaces the original update rule", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 302, + 422 + ], + "score": 1.0, + "content": "of the SOM algorithm (see eq. (1)). The features", + "type": "text" + }, + { + "bbox": [ + 303, + 411, + 330, + 420 + ], + "score": 0.9, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "are jointly optimized, and thus also depend", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 460, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 119, + 434 + ], + "score": 1.0, + "content": "on", + "type": "text" + }, + { + "bbox": [ + 119, + 423, + 126, + 431 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 420, + 460, + 434 + ], + "score": 1.0, + "content": "now. To prevent clutter we will, however, omit the (n)-superscript in the following.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 399, + 505, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 451 + ], + "score": 1.0, + "content": "Fortuin et al. (2019) propose the SOM-VAE model. As opposed to VQ-VAE, SOM-VAE", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "has two decoders, as it also decodes the continuous latents. Topological organization of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "codebook vectors is enforced by using a plus-shaped neighbourhood kernel, which affects the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "codebook vectors of the direct neighbours of the winning node (i.e. up, down, left, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "score": 1.0, + "content": "right on the grid). The encoder parameters are, however, unaffected by the quantization", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "error of these neighbour nodes. To facilitate the latter, the topological loss was split in a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 363, + 516 + ], + "score": 1.0, + "content": "commitment loss (committing the winning codebook vector to", + "type": "text" + }, + { + "bbox": [ + 364, + 506, + 371, + 514 + ], + "score": 0.75, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 504, + 462, + 516 + ], + "score": 1.0, + "content": "and vice versa) and a", + "type": "text" + }, + { + "bbox": [ + 462, + 504, + 486, + 514 + ], + "score": 0.35, + "content": "S O M", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "loss", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 515, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 321, + 530 + ], + "score": 1.0, + "content": "(pulling the codebook vectors of the neighbours to", + "type": "text" + }, + { + "bbox": [ + 322, + 518, + 330, + 527 + ], + "score": 0.67, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 515, + 339, + 530 + ], + "score": 1.0, + "content": "):", + "type": "text" + }, + { + "bbox": [ + 339, + 515, + 459, + 529 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { t o p o } } = \\mathcal { L } _ { \\mathrm { c o m m i t m e n t } } + \\frac { \\beta } { \\alpha } \\mathcal { L } _ { \\mathrm { S O M } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 515, + 506, + 530 + ], + "score": 1.0, + "content": ". Formally:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 437, + 506, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 110, + 534, + 470, + 568 + ], + "lines": [ + { + "bbox": [ + 110, + 534, + 470, + 568 + ], + "spans": [ + { + "bbox": [ + 110, + 534, + 470, + 568 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { \\mathrm { c o m m i n e n t } } = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\big \\| | \\boldsymbol { z } - \\phi _ { i } | \\big \\| _ { 2 } ^ { 2 } \\Big ] , \\quad ( 6 ) \\qquad \\mathrm { a n d } \\quad \\mathcal { L } _ { \\mathrm { s o m } } = \\mathbb { E } _ { \\mathcal { Z } } \\Big [ \\sum _ { \\substack { i = 1 , i \\neq j } } ^ { k } S _ { i } \\big ( \\phi \\quad \\big ) | | \\operatorname { s g } [ \\boldsymbol { z } ] - \\phi \\quad | | _ { 2 } ^ { 2 } \\Big ] ,", + "type": "interline_equation", + "image_path": "b3cebab8223d9c59800ea0a551a1fdb898a275b19b30304ee2b06a13beed48d6.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 110, + 534, + 470, + 545.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 110, + 545.3333333333334, + 470, + 556.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 110, + 556.6666666666667, + 470, + 568.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 502, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 125, + 585 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 572, + 144, + 584 + ], + "score": 0.55, + "content": "\\mathrm { s g } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 571, + 447, + 585 + ], + "score": 1.0, + "content": "a gradient blocker that impedes gradient updates to the encoder. Note that for", + "type": "text" + }, + { + "bbox": [ + 448, + 572, + 502, + 584 + ], + "score": 0.92, + "content": "0 < \\beta / \\alpha < 1", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "the proposed neighbourhood plus-kernel is a coarse approximation of the Gaussian kernel. 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(2019)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 384, + 507, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 507, + 398 + ], + "score": 1.0, + "content": "speculate about adding an LSTM in the latent space to train a SOM on sequential data, and refer to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 395, + 489, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 489, + 408 + ], + "score": 1.0, + "content": "this model as LSTM-DESOM. Figure 1a-b visualizes the SOM-VAE and DESOM architecture.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 423, + 183, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 422, + 185, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 185, + 441 + ], + "score": 1.0, + "content": "3 SOM-CPC", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 449, + 187, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 189, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 189, + 463 + ], + "score": 1.0, + "content": "3.1 MOTIVATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "In this work, we propose the SOM-CPC model, a representation learning model that learns to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "map windows of time series data to a structured 2D grid. The model jointly optimizes a temporal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "contrastive learning objective to extract features, and a topological loss that organizes the SOM space.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 507, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 507, + 522 + ], + "score": 1.0, + "content": "In order to learn features that are both suitable for SOM organization and accurately reflect the data,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "the model should ideally invert the original data generating process (which is in general unknown and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "implicit). Assuming that this generative process has been highly non-linear, feature learning can be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "formulated as a non-linear independent component analysis (ICA) problem, which has proven to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "non-identifiable (Hyvärinen & Pajunen, 1999). However, recent advances showed that the problem", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "becomes identifiable under the assumed presence of an auxiliary variable (Hyvärinen et al., 2018).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "Such an auxiliary variable (e.g. a temporal component) is not present in plain autoencoders, but the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "contrastive learning paradigm was shown to conform to this assumption (Hyvärinen et al., 2018;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "score": 1.0, + "content": "Zimmermann et al., 2021). This theory is in line with the hypothesis stated by Mrabah et al. (2020)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 608, + 452, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 452, + 621 + ], + "score": 1.0, + "content": "that a reconstruction objective may hamper clustering performance in the latent space.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 233, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 234, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 234, + 646 + ], + "score": 1.0, + "content": "3.2 ALGORITHMIC DETAILS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 103, + 652, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 103, + 652, + 161, + 668 + ], + "score": 1.0, + "content": "We introduce", + "type": "text" + }, + { + "bbox": [ + 161, + 655, + 285, + 667 + ], + "score": 0.92, + "content": "\\mathcal { X } = \\{ \\ldots , \\substack { x ( t ) , x ( t + 1 ) , \\ldots \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 652, + 443, + 668 + ], + "score": 1.0, + "content": ", a set of non-overlapping data windows", + "type": "text" + }, + { + "bbox": [ + 443, + 654, + 502, + 667 + ], + "score": 0.93, + "content": "\\pmb { x } ( t ) \\in \\mathbb { R } ^ { c h \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 652, + 506, + 668 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 126, + 678 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 666, + 138, + 676 + ], + "score": 0.43, + "content": "c h", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 666, + 255, + 678 + ], + "score": 1.0, + "content": "the number of channels, and", + "type": "text" + }, + { + "bbox": [ + 256, + 667, + 264, + 676 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "the number of samples in the window. 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(7)), and the topological loss", + "type": "text" + }, + { + "bbox": [ + 396, + 363, + 416, + 375 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { t o p o } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 361, + 507, + 376 + ], + "score": 1.0, + "content": "is given by eq. (2), i.e.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 386 + ], + "score": 1.0, + "content": "with a Gaussian neighbourhood function with decaying variance. In a short work, Forest et al. 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Figure 1a-b visualizes the SOM-VAE and DESOM architecture.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 341, + 507, + 408 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 423, + 183, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 422, + 185, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 185, + 441 + ], + "score": 1.0, + "content": "3 SOM-CPC", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 449, + 187, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 189, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 189, + 463 + ], + "score": 1.0, + "content": "3.1 MOTIVATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "In this work, we propose the SOM-CPC model, a representation learning model that learns to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "map windows of time series data to a structured 2D grid. The model jointly optimizes a temporal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "contrastive learning objective to extract features, and a topological loss that organizes the SOM space.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 470, + 506, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 507, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 507, + 522 + ], + "score": 1.0, + "content": "In order to learn features that are both suitable for SOM organization and accurately reflect the data,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "the model should ideally invert the original data generating process (which is in general unknown and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 505, + 543 + ], + "score": 1.0, + "content": "implicit). Assuming that this generative process has been highly non-linear, feature learning can be", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "formulated as a non-linear independent component analysis (ICA) problem, which has proven to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "non-identifiable (Hyvärinen & Pajunen, 1999). However, recent advances showed that the problem", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "becomes identifiable under the assumed presence of an auxiliary variable (Hyvärinen et al., 2018).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "Such an auxiliary variable (e.g. a temporal component) is not present in plain autoencoders, but the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "contrastive learning paradigm was shown to conform to this assumption (Hyvärinen et al., 2018;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 609 + ], + "score": 1.0, + "content": "Zimmermann et al., 2021). This theory is in line with the hypothesis stated by Mrabah et al. (2020)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 608, + 452, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 452, + 621 + ], + "score": 1.0, + "content": "that a reconstruction objective may hamper clustering performance in the latent space.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 508, + 507, + 621 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 233, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 234, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 234, + 646 + ], + "score": 1.0, + "content": "3.2 ALGORITHMIC DETAILS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 103, + 652, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 103, + 652, + 161, + 668 + ], + "score": 1.0, + "content": "We introduce", + "type": "text" + }, + { + "bbox": [ + 161, + 655, + 285, + 667 + ], + "score": 0.92, + "content": "\\mathcal { X } = \\{ \\ldots , \\substack { x ( t ) , x ( t + 1 ) , \\ldots \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 652, + 443, + 668 + ], + "score": 1.0, + "content": ", a set of non-overlapping data windows", + "type": "text" + }, + { + "bbox": [ + 443, + 654, + 502, + 667 + ], + "score": 0.93, + "content": "\\pmb { x } ( t ) \\in \\mathbb { R } ^ { c h \\times T }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 652, + 506, + 668 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 126, + 678 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 666, + 138, + 676 + ], + "score": 0.43, + "content": "c h", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 666, + 255, + 678 + ], + "score": 1.0, + "content": "the number of channels, and", + "type": "text" + }, + { + "bbox": [ + 256, + 667, + 264, + 676 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "the number of samples in the window. For brevity we omit", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 363, + 689 + ], + "score": 1.0, + "content": "the time index when possible. An encoder, parameterized by", + "type": "text" + }, + { + "bbox": [ + 363, + 677, + 370, + 687 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 677, + 477, + 689 + ], + "score": 1.0, + "content": ", maps each data window", + "type": "text" + }, + { + "bbox": [ + 477, + 679, + 485, + 687 + ], + "score": 0.75, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 193, + 700 + ], + "score": 1.0, + "content": "latent representation", + "type": "text" + }, + { + "bbox": [ + 194, + 687, + 269, + 700 + ], + "score": 0.92, + "content": "\\bar { \\boldsymbol { z } } = f _ { \\boldsymbol { \\theta } } ( \\pmb { x } ) \\in \\mathbb { R } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 687, + 295, + 700 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 295, + 688, + 304, + 698 + ], + "score": 0.83, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 687, + 442, + 700 + ], + "score": 1.0, + "content": "the number of features. The set", + "type": "text" + }, + { + "bbox": [ + 442, + 688, + 451, + 698 + ], + "score": 0.8, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "includes the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 697, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 227, + 713 + ], + "score": 1.0, + "content": "embeddings of all windows in", + "type": "text" + }, + { + "bbox": [ + 227, + 700, + 237, + 708 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 697, + 394, + 713 + ], + "score": 1.0, + "content": ". A causal auto-regressive (AR) module", + "type": "text" + }, + { + "bbox": [ + 394, + 700, + 406, + 711 + ], + "score": 0.84, + "content": "g _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 697, + 478, + 713 + ], + "score": 1.0, + "content": "parameterized by", + "type": "text" + }, + { + "bbox": [ + 478, + 699, + 486, + 710 + ], + "score": 0.83, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 697, + 507, + 713 + ], + "score": 1.0, + "content": ", e.g.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 394, + 723 + ], + "score": 1.0, + "content": "a gated-recurrent unit (GRU), subsequently aggregates the current and", + "type": "text" + }, + { + "bbox": [ + 394, + 711, + 402, + 720 + ], + "score": 0.78, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "previous embeddings, to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 245, + 733 + ], + "score": 1.0, + "content": "generate a (current) context vector", + "type": "text" + }, + { + "bbox": [ + 246, + 720, + 289, + 732 + ], + "score": 0.93, + "content": "\\pmb { c } ( \\dot { t } ) \\in \\dot { \\mathbb { R } } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ". Given this context, the pretext task in our SOM-CPC", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 303, + 94 + ], + "score": 1.0, + "content": "model aims to minimize the prediction error for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 303, + 83, + 312, + 92 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 313, + 82, + 451, + 94 + ], + "score": 1.0, + "content": "future (or ‘positive’) embeddings", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 451, + 82, + 487, + 95 + ], + "score": 0.93, + "content": "z ( t + p )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 487, + 82, + 505, + 94 + ], + "score": 1.0, + "content": ", for", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 173, + 106 + ], + "score": 0.92, + "content": "p \\in \\{ 1 , \\ldots , P \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 173, + 93, + 289, + 107 + ], + "score": 1.0, + "content": ", compared to this error for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 289, + 94, + 300, + 104 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 300, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "‘negative’ embeddings. 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The task objective, being the InfoNCE loss (Oord et al., 2019), is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 152, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 152, + 139 + ], + "score": 1.0, + "content": "defined as:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 144, + 488, + 187 + ], + "lines": [ + { + "bbox": [ + 111, + 144, + 488, + 187 + ], + "spans": [ + { + "bbox": [ + 111, + 144, + 488, + 187 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { \\mathrm { t a s k } } : = \\mathcal { L } _ { \\mathrm { I n f o N C E } } = \\frac { 1 } { P } \\sum _ { p = 1 } ^ { P } \\mathcal { L } _ { p } , \\quad \\mathrm { w i t h } \\quad \\mathcal { L } _ { p } = - \\frac { \\mathbb { E } } { \\chi } \\bigg [ \\log \\frac { \\exp \\Big ( z ( t + p ) \\mathbf { W } _ { p } c ( t ) \\Big ) } { \\sum _ { z ^ { \\prime } \\in \\mathcal { Z } _ { p } ^ { \\prime } \\cup \\{ z ( t + p ) \\} } \\exp \\Big ( z ^ { \\prime } \\mathbf { W } _ { p } c ( t ) \\Big ) } \\bigg ] ,", + "type": "interline_equation", + "image_path": "a4c31def364e3e72c823897d1f9ca5a4940ced9037a15dae88ff0e35cef4ecac.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 144, + 488, + 158.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 158.33333333333334, + 488, + 172.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 172.66666666666669, + 488, + 187.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 196, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 104, + 195, + 504, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 127, + 211 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 197, + 164, + 211 + ], + "score": 0.93, + "content": "\\mathcal { Z } _ { p } ^ { \\prime } \\subset \\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 195, + 367, + 211 + ], + "score": 1.0, + "content": "a set of embeddings of drawn negative samples", + "type": "text" + }, + { + "bbox": [ + 368, + 197, + 414, + 210 + ], + "score": 0.89, + "content": "\\left. \\mathcal { Z } _ { p } ^ { \\prime } \\right| = N )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 195, + 438, + 211 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 438, + 196, + 496, + 210 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { p } \\in \\mathbb { R } ^ { F \\times F }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 195, + 504, + 211 + ], + "score": 1.0, + "content": "a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 442, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 352, + 226 + ], + "score": 1.0, + "content": "trainable mapping between the current context vector and the", + "type": "text" + }, + { + "bbox": [ + 353, + 210, + 365, + 222 + ], + "score": 0.9, + "content": "p ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 208, + 442, + 226 + ], + "score": 1.0, + "content": "future embedding.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "The context vector is not only used to predict future embeddings, it is also the input to the SOM", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 469, + 252 + ], + "score": 1.0, + "content": "module that selects the winning node. 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(4). Figure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "1c provides an overview of the SOM-CPC model, and its gradient paths in green. The initial standard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 145, + 369 + ], + "score": 1.0, + "content": "deviation", + "type": "text" + }, + { + "bbox": [ + 145, + 354, + 163, + 365 + ], + "score": 0.9, + "content": "\\boldsymbol { \\sigma } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 353, + 506, + 369 + ], + "score": 1.0, + "content": "of the Gaussian kernel (from eq. 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Given the square topology of the SOM grid, this setting of", + "type": "text" + }, + { + "bbox": [ + 410, + 367, + 421, + 377 + ], + "score": 0.86, + "content": "\\sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "ensures that the full", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "grid is captured by the neighbourhood kernel at the start of training. Algorithm 1 in appendix A.1", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 324, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 324, + 401 + ], + "score": 1.0, + "content": "provides pseudocode of the full SOM-CPC algorithm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 417, + 254, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 255, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 255, + 430 + ], + "score": 1.0, + "content": "3.3 PERFORMANCE EVALUATION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Forest et al. (2020) provide a taxology of SOM metrics that distinguishes external vs internal and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "topological vs clustering metrics. External metrics are related to labels (which are not used during", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "unsupervised training), while internal metrics do not depend on such information. Topological metrics", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "assess the topological ordering (i.e. neighbourhood relations) of the SOM, while clustering metrics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "score": 1.0, + "content": "are more related to, for example, pureness of nodes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 108, + 500, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "To evaluate clustering performance, linked to external labels, we leverage purity and the normalized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "mutual information (NMI). The latter corrects for a high number of clusters (i.e. nodes), which could", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 523, + 378, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 378, + 535 + ], + "score": 1.0, + "content": "easily lead to high pureness, but leaves the NMI more conservative.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Even though scoring high on external metrics is not the main goal of a representation learning model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "like SOM-CPC, we do report it as it provides an indication of how well information was preserved.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "To compute regression/classification performance, we first ‘color’ (or label) each node with the most", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "occurring (for discrete labels) or median (for continuous labels) label from the training set. The test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "set predictions are then converted from node indices to label predictions by using these colorings.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 592, + 503, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 473, + 609 + ], + "score": 1.0, + "content": "Regression performance is expressed as the average squared regression error with the target:", + "type": "text" + }, + { + "bbox": [ + 473, + 594, + 503, + 606 + ], + "score": 0.73, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "Classification performance is reported with Cohen’s kappa (Cohen, 1960), a commonly used metric", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 264, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 264, + 628 + ], + "score": 1.0, + "content": "that corrects for correctness by chance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "Topographic performance is measured using the (internal) topographic error (TE) (Kiviluoto, 1996),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "which reports the fraction of windows (between 0 and 1) for which the winning and second-best", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "winning node are not neighbours in the SOM (lower is better). Finally, to measure whether a time", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "series conveys a smooth trajectory through SOM space, we measure the average Euclidean distance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 144, + 691 + ], + "score": 1.0, + "content": "(denoted", + "type": "text" + }, + { + "bbox": [ + 144, + 677, + 180, + 689 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { { 2 , s m o o t h } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "between all subsequent windows in each time series. The lower this value, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "less frequently large jumps in the 2D map occur. Note that in extreme cases where many windows", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 348, + 713 + ], + "score": 1.0, + "content": "collapsed to the same node, both the TE and the average", + "type": "text" + }, + { + "bbox": [ + 348, + 699, + 381, + 711 + ], + "score": 0.87, + "content": "\\ell _ { 2 , \\mathrm { s m o o t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "metric are artificially pushed", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "down. We can thus only interpret these metrics in conjunction with earlier-mentioned clustering and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 194, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 194, + 732 + ], + "score": 1.0, + "content": "classification metrics.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 504, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 139 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 144, + 488, + 187 + ], + "lines": [ + { + "bbox": [ + 111, + 144, + 488, + 187 + ], + "spans": [ + { + "bbox": [ + 111, + 144, + 488, + 187 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { \\mathrm { t a s k } } : = \\mathcal { L } _ { \\mathrm { I n f o N C E } } = \\frac { 1 } { P } \\sum _ { p = 1 } ^ { P } \\mathcal { L } _ { p } , \\quad \\mathrm { w i t h } \\quad \\mathcal { L } _ { p } = - \\frac { \\mathbb { E } } { \\chi } \\bigg [ \\log \\frac { \\exp \\Big ( z ( t + p ) \\mathbf { W } _ { p } c ( t ) \\Big ) } { \\sum _ { z ^ { \\prime } \\in \\mathcal { Z } _ { p } ^ { \\prime } \\cup \\{ z ( t + p ) \\} } \\exp \\Big ( z ^ { \\prime } \\mathbf { W } _ { p } c ( t ) \\Big ) } \\bigg ] ,", + "type": "interline_equation", + "image_path": "a4c31def364e3e72c823897d1f9ca5a4940ced9037a15dae88ff0e35cef4ecac.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 144, + 488, + 158.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 158.33333333333334, + 488, + 172.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 172.66666666666669, + 488, + 187.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 196, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 104, + 195, + 504, + 211 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 127, + 211 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 197, + 164, + 211 + ], + "score": 0.93, + "content": "\\mathcal { Z } _ { p } ^ { \\prime } \\subset \\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 195, + 367, + 211 + ], + "score": 1.0, + "content": "a set of embeddings of drawn negative samples", + "type": "text" + }, + { + "bbox": [ + 368, + 197, + 414, + 210 + ], + "score": 0.89, + "content": "\\left. \\mathcal { Z } _ { p } ^ { \\prime } \\right| = N )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 195, + 438, + 211 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 438, + 196, + 496, + 210 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { p } \\in \\mathbb { R } ^ { F \\times F }", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 195, + 504, + 211 + ], + "score": 1.0, + "content": "a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 442, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 352, + 226 + ], + "score": 1.0, + "content": "trainable mapping between the current context vector and the", + "type": "text" + }, + { + "bbox": [ + 353, + 210, + 365, + 222 + ], + "score": 0.9, + "content": "p ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 208, + 442, + 226 + ], + "score": 1.0, + "content": "future embedding.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 195, + 504, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "The context vector is not only used to predict future embeddings, it is also the input to the SOM", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 469, + 252 + ], + "score": 1.0, + "content": "module that selects the winning node. 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Depending on the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "use case, it was found to not always be necessary, or even beneficial (due to higher risk of overfitting),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 270, + 193, + 284 + ], + "score": 1.0, + "content": "to use an AR module", + "type": "text" + }, + { + "bbox": [ + 194, + 273, + 205, + 284 + ], + "score": 0.86, + "content": "g _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 270, + 506, + 284 + ], + "score": 1.0, + "content": "to aggregate causal context into the current embedding. 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Likewise,", + "type": "text" + }, + { + "bbox": [ + 211, + 293, + 230, + 305 + ], + "score": 0.91, + "content": "z ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 292, + 277, + 307 + ], + "score": 1.0, + "content": "rather than", + "type": "text" + }, + { + "bbox": [ + 277, + 293, + 294, + 305 + ], + "score": 0.9, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 292, + 506, + 307 + ], + "score": 1.0, + "content": "is being quantized by the SOM module. 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(4). Figure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "1c provides an overview of the SOM-CPC model, and its gradient paths in green. The initial standard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 145, + 369 + ], + "score": 1.0, + "content": "deviation", + "type": "text" + }, + { + "bbox": [ + 145, + 354, + 163, + 365 + ], + "score": 0.9, + "content": "\\boldsymbol { \\sigma } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 353, + 506, + 369 + ], + "score": 1.0, + "content": "of the Gaussian kernel (from eq. (2)) was set to half the squared-root of the number of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 157, + 379 + ], + "score": 1.0, + "content": "SOM nodes", + "type": "text" + }, + { + "bbox": [ + 157, + 366, + 164, + 376 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 365, + 409, + 379 + ], + "score": 1.0, + "content": ". Given the square topology of the SOM grid, this setting of", + "type": "text" + }, + { + "bbox": [ + 410, + 367, + 421, + 377 + ], + "score": 0.86, + "content": "\\sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "ensures that the full", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 390 + ], + "score": 1.0, + "content": "grid is captured by the neighbourhood kernel at the start of training. Algorithm 1 in appendix A.1", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 324, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 324, + 401 + ], + "score": 1.0, + "content": "provides pseudocode of the full SOM-CPC algorithm.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 320, + 506, + 401 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 417, + 254, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 255, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 255, + 430 + ], + "score": 1.0, + "content": "3.3 PERFORMANCE EVALUATION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Forest et al. (2020) provide a taxology of SOM metrics that distinguishes external vs internal and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 464 + ], + "score": 1.0, + "content": "topological vs clustering metrics. External metrics are related to labels (which are not used during", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "unsupervised training), while internal metrics do not depend on such information. Topological metrics", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "assess the topological ordering (i.e. neighbourhood relations) of the SOM, while clustering metrics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 314, + 496 + ], + "score": 1.0, + "content": "are more related to, for example, pureness of nodes.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 439, + 505, + 496 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 500, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "To evaluate clustering performance, linked to external labels, we leverage purity and the normalized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "mutual information (NMI). The latter corrects for a high number of clusters (i.e. nodes), which could", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 523, + 378, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 378, + 535 + ], + "score": 1.0, + "content": "easily lead to high pureness, but leaves the NMI more conservative.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 500, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Even though scoring high on external metrics is not the main goal of a representation learning model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "like SOM-CPC, we do report it as it provides an indication of how well information was preserved.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "To compute regression/classification performance, we first ‘color’ (or label) each node with the most", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "occurring (for discrete labels) or median (for continuous labels) label from the training set. The test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "set predictions are then converted from node indices to label predictions by using these colorings.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 592, + 503, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 473, + 609 + ], + "score": 1.0, + "content": "Regression performance is expressed as the average squared regression error with the target:", + "type": "text" + }, + { + "bbox": [ + 473, + 594, + 503, + 606 + ], + "score": 0.73, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "Classification performance is reported with Cohen’s kappa (Cohen, 1960), a commonly used metric", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 264, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 264, + 628 + ], + "score": 1.0, + "content": "that corrects for correctness by chance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 540, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "Topographic performance is measured using the (internal) topographic error (TE) (Kiviluoto, 1996),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "which reports the fraction of windows (between 0 and 1) for which the winning and second-best", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "winning node are not neighbours in the SOM (lower is better). Finally, to measure whether a time", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 679 + ], + "score": 1.0, + "content": "series conveys a smooth trajectory through SOM space, we measure the average Euclidean distance", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 144, + 691 + ], + "score": 1.0, + "content": "(denoted", + "type": "text" + }, + { + "bbox": [ + 144, + 677, + 180, + 689 + ], + "score": 0.9, + "content": "\\ell _ { \\mathrm { { 2 , s m o o t h } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "between all subsequent windows in each time series. The lower this value, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "less frequently large jumps in the 2D map occur. Note that in extreme cases where many windows", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 348, + 713 + ], + "score": 1.0, + "content": "collapsed to the same node, both the TE and the average", + "type": "text" + }, + { + "bbox": [ + 348, + 699, + 381, + 711 + ], + "score": 0.87, + "content": "\\ell _ { 2 , \\mathrm { s m o o t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "metric are artificially pushed", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "down. We can thus only interpret these metrics in conjunction with earlier-mentioned clustering and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 194, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 194, + 732 + ], + "score": 1.0, + "content": "classification metrics.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 633, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 113, + 121, + 487, + 382 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 121, + 487, + 382 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 121, + 487, + 382 + ], + "spans": [ + { + "bbox": [ + 113, + 121, + 487, + 382 + ], + "score": 0.41, + "type": "image", + "image_path": "095d5981ef212a3ff5ed70dacb4a9ae875fce08aa8ac819333ff01b60320f3cd.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 113, + 121, + 487, + 208.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 113, + 208.0, + 487, + 295.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 113, + 295.0, + 487, + 382.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 396, + 505, + 476 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "Figure 2: SOMs and regression plots for SOM-VAE (a), DESOM (b) and SOM-CPC (c). Both DESOM and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 431, + 418 + ], + "score": 1.0, + "content": "SOM-CPC show a gradual change of frequency over the grid, but the regression error", + "type": "text" + }, + { + "bbox": [ + 431, + 406, + 458, + 416 + ], + "score": 0.89, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "is lower for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "SOM-CPC, which can also be seen from the regression plot, where the predicted window frequencies are plotted", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "against the target frequencies (i.e. training set median label) for the node on which the window was mapped. d)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Task loss versus the topological loss for SOM-VAE (with Gaussian neighbourhood), DESOM and SOM-CPC", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 191, + 457 + ], + "score": 1.0, + "content": "(both with and without", + "type": "text" + }, + { + "bbox": [ + 192, + 446, + 209, + 457 + ], + "score": 0.69, + "content": "\\mathrm { s g } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 446, + 384, + 457 + ], + "score": 1.0, + "content": "). The different curves display various values of", + "type": "text" + }, + { + "bbox": [ + 384, + 448, + 392, + 455 + ], + "score": 0.62, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 446, + 505, + 457 + ], + "score": 1.0, + "content": ", for which the DESOM model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "seems most sensitive. The SOM-CPC models follow a smooth optimization curve, minimizing both the task and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 465, + 468, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 468, + 478 + ], + "score": 1.0, + "content": "topological loss, while these losses seem to be more conflicting in SOM-VAE and DESOM training.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "title", + "bbox": [ + 108, + 497, + 200, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 201, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 201, + 511 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "We compare SOM-CPC to several other 2D representation learning methods. First, deep-SOM models", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "with a reconstruction task loss (i.e. SOM-VAE, SOM-VAE-prob, and (GRU-)DESOM). Second,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 308, + 556 + ], + "score": 1.0, + "content": "vanilla CPC with a multi-dimensional latent space", + "type": "text" + }, + { + "bbox": [ + 308, + 544, + 339, + 554 + ], + "score": 0.85, + "content": "F \\gg 2", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 544, + 506, + 556 + ], + "score": 1.0, + "content": ") (Oord et al., 2019), followed by PCA for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 417, + 568 + ], + "score": 1.0, + "content": "additional dimensionality reduction to 2D. Third, CPC with a 2D latent space (", + "type": "text" + }, + { + "bbox": [ + 417, + 555, + 445, + 565 + ], + "score": 0.86, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "). For the latter", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "two CPC-based models, linear and non-linear read-out is, respectively, tested using a linear neural", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 589 + ], + "score": 1.0, + "content": "classifier, and K-means clustering with the same number of clusters as the number of nodes used in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 288, + 600 + ], + "score": 1.0, + "content": "SOM-CPC. High-dimensional vanilla CPC (", + "type": "text" + }, + { + "bbox": [ + 288, + 587, + 320, + 598 + ], + "score": 0.84, + "content": "F \\gg 2", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ") without additional dimensionality reduction", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 599, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 611 + ], + "score": 1.0, + "content": "is, moreover, tested as well as it sets a baseline for the amount of information that can be preserved", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "given the encoder architecture, while not providing an interpretable 2D representation. The same", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "encoder architecture is used for all models that are compared in a single application domain, and all", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 645 + ], + "score": 1.0, + "content": "models are run with the same seed for randomization. Details on model architectures and training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 642, + 457, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 457, + 654 + ], + "score": 1.0, + "content": "settings for the different applications can be found in appendix A.3.1, A.4.2, and A.5.1.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 206, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 207, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 207, + 680 + ], + "score": 1.0, + "content": "4.1 SYNTHETIC DATA", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "Data generation: A synthetic dataset was created, consisting of sinusoids with an initial frequency", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 315, + 711 + ], + "score": 1.0, + "content": "sampled from a uniform distribution between 20 and", + "type": "text" + }, + { + "bbox": [ + 315, + 699, + 340, + 709 + ], + "score": 0.68, + "content": "4 0 \\ : \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". The frequency of the signals was altered", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 379, + 721 + ], + "score": 1.0, + "content": "over time according to a random walk process with a step size of", + "type": "text" + }, + { + "bbox": [ + 379, + 710, + 408, + 720 + ], + "score": 0.73, + "content": "0 . 1 \\ : \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 710, + 505, + 721 + ], + "score": 1.0, + "content": ". 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Both DESOM and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 431, + 418 + ], + "score": 1.0, + "content": "SOM-CPC show a gradual change of frequency over the grid, but the regression error", + "type": "text" + }, + { + "bbox": [ + 431, + 406, + 458, + 416 + ], + "score": 0.89, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "is lower for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "SOM-CPC, which can also be seen from the regression plot, where the predicted window frequencies are plotted", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "against the target frequencies (i.e. training set median label) for the node on which the window was mapped. d)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Task loss versus the topological loss for SOM-VAE (with Gaussian neighbourhood), DESOM and SOM-CPC", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 191, + 457 + ], + "score": 1.0, + "content": "(both with and without", + "type": "text" + }, + { + "bbox": [ + 192, + 446, + 209, + 457 + ], + "score": 0.69, + "content": "\\mathrm { s g } [ \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 446, + 384, + 457 + ], + "score": 1.0, + "content": "). The different curves display various values of", + "type": "text" + }, + { + "bbox": [ + 384, + 448, + 392, + 455 + ], + "score": 0.62, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 446, + 505, + 457 + ], + "score": 1.0, + "content": ", for which the DESOM model", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 468 + ], + "score": 1.0, + "content": "seems most sensitive. The SOM-CPC models follow a smooth optimization curve, minimizing both the task and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 465, + 468, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 468, + 478 + ], + "score": 1.0, + "content": "topological loss, while these losses seem to be more conflicting in SOM-VAE and DESOM training.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "title", + "bbox": [ + 108, + 497, + 200, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 201, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 201, + 511 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "We compare SOM-CPC to several other 2D representation learning methods. First, deep-SOM models", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 507, + 546 + ], + "score": 1.0, + "content": "with a reconstruction task loss (i.e. SOM-VAE, SOM-VAE-prob, and (GRU-)DESOM). Second,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 308, + 556 + ], + "score": 1.0, + "content": "vanilla CPC with a multi-dimensional latent space", + "type": "text" + }, + { + "bbox": [ + 308, + 544, + 339, + 554 + ], + "score": 0.85, + "content": "F \\gg 2", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 544, + 506, + 556 + ], + "score": 1.0, + "content": ") (Oord et al., 2019), followed by PCA for", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 417, + 568 + ], + "score": 1.0, + "content": "additional dimensionality reduction to 2D. 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Interestingly, although the SOM for the DESOM and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 247, + 464 + ], + "score": 1.0, + "content": "SOM-CPC model look similar, the", + "type": "text" + }, + { + "bbox": [ + 247, + 451, + 277, + 462 + ], + "score": 0.87, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "is higher for the DESOM model, which can also be seen", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 461, + 280, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 280, + 474 + ], + "score": 1.0, + "content": "from the regression plots below the SOMs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "The addition of two temporal losses in the SOM-VAE-prob model, as compared to SOM-VAE, did", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 400, + 502 + ], + "score": 1.0, + "content": "deteriorate the given metrics, even though a range of values for multipliers", + "type": "text" + }, + { + "bbox": [ + 400, + 491, + 419, + 500 + ], + "score": 0.27, + "content": "\\alpha , \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 488, + 436, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 490, + 443, + 501 + ], + "score": 0.85, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "was tested (see", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "table 3 in appendix A.3.2 for the full sweep). The deterioration of the results can be explained by the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "difficulty of finding the correct scaling factors for these additional losses. Note that the SOM-CPC", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 523, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 507, + 535 + ], + "score": 1.0, + "content": "model automatically incorporates smoothness over time thanks to the nature of the CPC task loss,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 329, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 329, + 547 + ], + "score": 1.0, + "content": "therewith preventing additional hyperparameter tuning.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 104, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "Additionally, we study the optimization behavior of SOM-VAE, DESOM, and SOM-CPC by plotting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "the progression of the task versus the topological loss during training (see fig. 2d). To make a fair", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "comparison, we plot the SOM-VAE models that are trained with a Gaussian neighbourhood kernel.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 379, + 595 + ], + "score": 1.0, + "content": "Different curves in the graphs indicate runs with varying values for", + "type": "text" + }, + { + "bbox": [ + 379, + 585, + 387, + 593 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 583, + 506, + 595 + ], + "score": 1.0, + "content": ", and the line color’s gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "denotes the training iteration. The SOM-CPC graphs include the models run with and without", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 204, + 618 + ], + "score": 1.0, + "content": "gradient detachment of", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 228, + 616 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { S O M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "to the encoder. 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Non-linear K-means clustering improved performance for both cases, but only", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 163, + 689 + ], + "score": 1.0, + "content": "for CPC with", + "type": "text" + }, + { + "bbox": [ + 163, + 677, + 191, + 687 + ], + "score": 0.91, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 677, + 506, + 689 + ], + "score": 1.0, + "content": ", performance nearly reached SOM-CPC performance. Later we will see that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 192, + 700 + ], + "score": 1.0, + "content": "optimizing CPC with", + "type": "text" + }, + { + "bbox": [ + 192, + 688, + 219, + 698 + ], + "score": 0.92, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "can hamper optimization for more intricate data spaces (see section 4.3).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "Interestingly, regression performance of SOM-CPC was found to be even slightly better than that of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 294, + 721 + ], + "score": 1.0, + "content": "the vanilla multi-dimensional CPC model (with", + "type": "text" + }, + { + "bbox": [ + 294, + 710, + 332, + 720 + ], + "score": 0.86, + "content": "F = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "), both for linear classification and K-means.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 507, + 734 + ], + "score": 1.0, + "content": "This could be explained by the additional regularization that the SOM provides in SOM-CPC training.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 94, + 498, + 209 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 94, + 498, + 209 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 94, + 498, + 209 + ], + "spans": [ + { + "bbox": [ + 108, + 94, + 498, + 209 + ], + "score": 0.967, + "type": "image", + "image_path": "9c64afdd535a9684c986a72e86f2d42657c2e53f3627425cdb320431daaf6b9c.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 94, + 498, + 132.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 132.33333333333334, + 498, + 170.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 170.66666666666669, + 498, + 209.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 216, + 505, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "Figure 3: Progression of SOM training (nodes are indicated in black) in the SOM-CPC model, using either a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 227, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 237 + ], + "score": 1.0, + "content": "Gaussian (top) or plus neighbourhood (bottom) kernel. The PCA projection of the test set latent space is plotted", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 236, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 505, + 247 + ], + "score": 1.0, + "content": "behind the nodes. It can clearly be seen that the Gaussian kernel enforces a more strict organization of SOM", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 246, + 504, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 504, + 258 + ], + "score": 1.0, + "content": "nodes, where nodes are non-uniformly quantizing the latent space, placing more nodes at higher density areas.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 283, + 505, + 380 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 105, + 283, + 506, + 380 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "Results: Table 1 shows that SOM-CPC outperforms all deep-SOM baselines on all metrics. Figure 6", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "in appendix A.3.2 shows the PCA projections of the (continuous) latent spaces of the three deep-SOM", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 194, + 418 + ], + "score": 1.0, + "content": "models indicated with", + "type": "text" + }, + { + "bbox": [ + 195, + 408, + 209, + 417 + ], + "score": 0.26, + "content": "^ { \\textrm { a * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 407, + 505, + 418 + ], + "score": 1.0, + "content": "in table 1. It reveals that the latent space disentanglement of the SOM-CPC", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 504, + 429 + ], + "score": 1.0, + "content": "model is much better than that of the SOM-VAE and DESOM models. Figure 2a-c displays the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "resulting SOMs (colored with the median test set labels) for the same three models. Uncolored nodes", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "in the SOM were not assigned in the test set. Interestingly, although the SOM for the DESOM and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 247, + 464 + ], + "score": 1.0, + "content": "SOM-CPC model look similar, the", + "type": "text" + }, + { + "bbox": [ + 247, + 451, + 277, + 462 + ], + "score": 0.87, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "is higher for the DESOM model, which can also be seen", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 461, + 280, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 280, + 474 + ], + "score": 1.0, + "content": "from the regression plots below the SOMs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 384, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "The addition of two temporal losses in the SOM-VAE-prob model, as compared to SOM-VAE, did", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 400, + 502 + ], + "score": 1.0, + "content": "deteriorate the given metrics, even though a range of values for multipliers", + "type": "text" + }, + { + "bbox": [ + 400, + 491, + 419, + 500 + ], + "score": 0.27, + "content": "\\alpha , \\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 488, + 436, + 502 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 490, + 443, + 501 + ], + "score": 0.85, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "was tested (see", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 513 + ], + "score": 1.0, + "content": "table 3 in appendix A.3.2 for the full sweep). The deterioration of the results can be explained by the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "difficulty of finding the correct scaling factors for these additional losses. Note that the SOM-CPC", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 523, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 507, + 535 + ], + "score": 1.0, + "content": "model automatically incorporates smoothness over time thanks to the nature of the CPC task loss,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 329, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 329, + 547 + ], + "score": 1.0, + "content": "therewith preventing additional hyperparameter tuning.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 478, + 507, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 639 + ], + "lines": [ + { + "bbox": [ + 104, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "Additionally, we study the optimization behavior of SOM-VAE, DESOM, and SOM-CPC by plotting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "the progression of the task versus the topological loss during training (see fig. 2d). To make a fair", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "comparison, we plot the SOM-VAE models that are trained with a Gaussian neighbourhood kernel.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 379, + 595 + ], + "score": 1.0, + "content": "Different curves in the graphs indicate runs with varying values for", + "type": "text" + }, + { + "bbox": [ + 379, + 585, + 387, + 593 + ], + "score": 0.72, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 583, + 506, + 595 + ], + "score": 1.0, + "content": ", and the line color’s gradient", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "denotes the training iteration. The SOM-CPC graphs include the models run with and without", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 204, + 618 + ], + "score": 1.0, + "content": "gradient detachment of", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 228, + 616 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { S O M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "to the encoder. It can be seen that both losses jointly minimize in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "SOM-CPC training, while there is a counteracting effect visible for SOM-VAE, and a high influence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 263, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 166, + 640 + ], + "score": 1.0, + "content": "of the value of", + "type": "text" + }, + { + "bbox": [ + 167, + 628, + 174, + 637 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 626, + 263, + 640 + ], + "score": 1.0, + "content": "for DESOM training.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 548, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 451, + 655 + ], + "score": 1.0, + "content": "Comparing to non-deep-SOM baselines, it can be seen from table 1 that CPC (with", + "type": "text" + }, + { + "bbox": [ + 451, + 644, + 480, + 654 + ], + "score": 0.88, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 644, + 506, + 655 + ], + "score": 1.0, + "content": "), and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 381, + 668 + ], + "score": 1.0, + "content": "CPC followed by PCA, resulted in a much higher regression error", + "type": "text" + }, + { + "bbox": [ + 381, + 655, + 411, + 667 + ], + "score": 0.81, + "content": "\\mathrm { S E } _ { \\mathrm { t a r g e t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "than SOM-CPC when", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "using linear read-out. Non-linear K-means clustering improved performance for both cases, but only", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 163, + 689 + ], + "score": 1.0, + "content": "for CPC with", + "type": "text" + }, + { + "bbox": [ + 163, + 677, + 191, + 687 + ], + "score": 0.91, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 677, + 506, + 689 + ], + "score": 1.0, + "content": ", performance nearly reached SOM-CPC performance. Later we will see that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 192, + 700 + ], + "score": 1.0, + "content": "optimizing CPC with", + "type": "text" + }, + { + "bbox": [ + 192, + 688, + 219, + 698 + ], + "score": 0.92, + "content": "F = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "can hamper optimization for more intricate data spaces (see section 4.3).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "Interestingly, regression performance of SOM-CPC was found to be even slightly better than that of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 294, + 721 + ], + "score": 1.0, + "content": "the vanilla multi-dimensional CPC model (with", + "type": "text" + }, + { + "bbox": [ + 294, + 710, + 332, + 720 + ], + "score": 0.86, + "content": "F = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 710, + 506, + 721 + ], + "score": 1.0, + "content": "), both for linear classification and K-means.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 507, + 734 + ], + "score": 1.0, + "content": "This could be explained by the additional regularization that the SOM provides in SOM-CPC training.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 644, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We perform several ablation experiments on SOM-CPC, which are reported below the dashed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 108 + ], + "score": 1.0, + "content": "line in table 1. Blocking the gradients of the neighbour nodes with respect to the encoder during", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 141, + 117 + ], + "score": 1.0, + "content": "training", + "type": "text" + }, + { + "bbox": [ + 141, + 104, + 187, + 117 + ], + "score": 0.89, + "content": "( \\mathcal { L } _ { \\mathrm { { S O M } S g [ \\cdot ] } }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "column) slightly improved the regression error and temporal smoothness, but", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 507, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 417, + 128 + ], + "score": 1.0, + "content": "decreased the topographic error. Looking at the models with various values for", + "type": "text" + }, + { + "bbox": [ + 417, + 117, + 425, + 125 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 114, + 507, + 128 + ], + "score": 1.0, + "content": "(reported in table 3,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "appendix A.3.2), the effect of gradient blocking can be considered small and ambiguous when", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 301, + 151 + ], + "score": 1.0, + "content": "considering different metrics. Disjoint training", + "type": "text" + }, + { + "bbox": [ + 301, + 137, + 358, + 148 + ], + "score": 0.86, + "content": "( \\mathbf { C P C } + \\mathbf { S O M } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "resulted in a less smooth trajectory", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 350, + 162 + ], + "score": 1.0, + "content": "over time through the 2D SOM space, seen from the higher", + "type": "text" + }, + { + "bbox": [ + 351, + 149, + 383, + 160 + ], + "score": 0.89, + "content": "\\ell _ { 2 , \\mathrm { s m o o t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 147, + 506, + 162 + ], + "score": 1.0, + "content": ". Using a plus neighbourhood", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 171 + ], + "score": 1.0, + "content": "kernel instead of a Gaussian kernel decreased performance on all three metrics. The increase in TE, is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "well explainable by the fact that a plus kernel takes into account fewer neighbours (at least at the start", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "of training) and therefore has more difficulty to find a good topological mapping. Figure 3 shows the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "development of the SOM node spread (projected on top of a PCA projection of the continuous test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "set latents), during training for SOM-CPC with the two type of kernels. It can indeed be seen that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "Gaussian kernel enforces a more strict topological organization. Interestingly, the kernel does not", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 507, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 507, + 238 + ], + "score": 1.0, + "content": "only influence the codebook vectors, but also seems to influence the organization of the latent space,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 237, + 383, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 383, + 248 + ], + "score": 1.0, + "content": "seen from the differently-shaped PCA projections in the background.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 159, + 273 + ], + "lines": [ + { + "bbox": [ + 105, + 260, + 161, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 161, + 275 + ], + "score": 1.0, + "content": "4.2 SLEEP", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 282, + 504, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 506, + 295 + ], + "score": 1.0, + "content": "We analyse SOM-CPC on subset 3 of the Montreal Archive of Sleep Studies (MASS) database", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "(O’Reilly et al., 2014), consisting of whole-night polysomnography recordings, for which every", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 355, + 317 + ], + "score": 1.0, + "content": "30-second window is labelled with a sleep stage label from", + "type": "text" + }, + { + "bbox": [ + 356, + 305, + 373, + 317 + ], + "score": 0.68, + "content": "\\{ \\mathrm { N } 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 305, + 438, + 317 + ], + "score": 1.0, + "content": ", N2, N3, REM,", + "type": "text" + }, + { + "bbox": [ + 438, + 305, + 466, + 317 + ], + "score": 0.52, + "content": "\\mathrm { W a k e } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 305, + 505, + 317 + ], + "score": 1.0, + "content": ". The 62", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 316, + 502, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 393, + 329 + ], + "score": 1.0, + "content": "recordings (from 62 unique subjects) were randomly split into a training", + "type": "text" + }, + { + "bbox": [ + 393, + 316, + 425, + 326 + ], + "score": 0.86, + "content": "n = 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 316, + 475, + 329 + ], + "score": 1.0, + "content": "), validation", + "type": "text" + }, + { + "bbox": [ + 475, + 316, + 502, + 326 + ], + "score": 0.84, + "content": "( n = 8 )", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 492, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 192, + 340 + ], + "score": 1.0, + "content": "and hold-out test set", + "type": "text" + }, + { + "bbox": [ + 192, + 327, + 219, + 338 + ], + "score": 0.85, + "content": "( n = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 326, + 492, + 340 + ], + "score": 1.0, + "content": "). Details on the data preprocessing can be found in appendix A.4.1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 344, + 336, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 336, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 336, + 356 + ], + "score": 1.0, + "content": "Table 5 in appendix A.4.3 shows that SOM-CPC again", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 354, + 337, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 337, + 366 + ], + "score": 1.0, + "content": "clearly outperformed SOM-VAE and DESOM on all", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 366, + 337, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 337, + 378 + ], + "score": 1.0, + "content": "metrics. Whether or not the gradients of the SOM loss", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 337, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 337, + 389 + ], + "score": 1.0, + "content": "were stopped towards the encoder did not greatly influence", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 337, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 337, + 399 + ], + "score": 1.0, + "content": "SOM-CPC performance. Topological ordering, measured", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 337, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 247, + 411 + ], + "score": 1.0, + "content": "by TE, and temporal smoothness", + "type": "text" + }, + { + "bbox": [ + 247, + 399, + 284, + 411 + ], + "score": 0.79, + "content": "\\cdot \\ell _ { 2 , \\mathrm { s m o o t h } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 398, + 337, + 411 + ], + "score": 1.0, + "content": "deteriorated", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 409, + 337, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 337, + 421 + ], + "score": 1.0, + "content": "when changing the Gaussian kernel to a plus kernel, or", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 420, + 338, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 338, + 432 + ], + "score": 1.0, + "content": "training high-dimensional CPC and SOM disjointly. 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Both visualizations show similar clustering patterns:", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "deep sleep N3 is isolated from lighter forms of sleep (i.e. N1, Wake and REM sleep) by a thick cluster", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 513, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 528 + ], + "score": 1.0, + "content": "of medium-deep sleep N2. However, the higher performance of SOM-CPC (see table 5) indicates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "that more information is preserved in the 2D space resulting from the SOM-CPC model. The size", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "score": 1.0, + "content": "of the nodes in the SOM map of SOM-CPC indicates the average time in the night of windows on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 545, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 507, + 561 + ], + "score": 1.0, + "content": "that node. 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The dataset contains multiple minute-long English voice recordings of 251 different", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 193, + 640 + ], + "score": 1.0, + "content": "speakers, sampled at", + "type": "text" + }, + { + "bbox": [ + 193, + 627, + 226, + 637 + ], + "score": 0.49, + "content": "1 6 ~ \\mathrm { K H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 627, + 505, + 640 + ], + "score": 1.0, + "content": ". We used the publicly available train-test split, as provided by Oord", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 422, + 652 + ], + "score": 1.0, + "content": "et al. (2019), and created an additional validation set by randomly selecting", + "type": "text" + }, + { + "bbox": [ + 422, + 638, + 442, + 649 + ], + "score": 0.88, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "of the training", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "set. Recordings of the ten speakers with the longest recording time were selected to alleviate", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "score": 1.0, + "content": "computational burden. 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Figure 3 shows the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "development of the SOM node spread (projected on top of a PCA projection of the continuous test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "set latents), during training for SOM-CPC with the two type of kernels. It can indeed be seen that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "Gaussian kernel enforces a more strict topological organization. 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Both visualizations show similar clustering patterns:", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "deep sleep N3 is isolated from lighter forms of sleep (i.e. N1, Wake and REM sleep) by a thick cluster", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 513, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 528 + ], + "score": 1.0, + "content": "of medium-deep sleep N2. However, the higher performance of SOM-CPC (see table 5) indicates", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "that more information is preserved in the 2D space resulting from the SOM-CPC model. The size", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 548 + ], + "score": 1.0, + "content": "of the nodes in the SOM map of SOM-CPC indicates the average time in the night of windows on", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 545, + 507, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 507, + 561 + ], + "score": 1.0, + "content": "that node. 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The dataset contains multiple minute-long English voice recordings of 251 different", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 193, + 640 + ], + "score": 1.0, + "content": "speakers, sampled at", + "type": "text" + }, + { + "bbox": [ + 193, + 627, + 226, + 637 + ], + "score": 0.49, + "content": "1 6 ~ \\mathrm { K H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 627, + 505, + 640 + ], + "score": 1.0, + "content": ". We used the publicly available train-test split, as provided by Oord", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 104, + 637, + 422, + 652 + ], + "score": 1.0, + "content": "et al. (2019), and created an additional validation set by randomly selecting", + "type": "text" + }, + { + "bbox": [ + 422, + 638, + 442, + 649 + ], + "score": 0.88, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "of the training", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "set. Recordings of the ten speakers with the longest recording time were selected to alleviate", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 673 + ], + "score": 1.0, + "content": "computational burden. This resulted in a total of 150.9, 54.6, and 46.5 minutes in the training,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 505, + 683 + ], + "score": 1.0, + "content": "validation, respectively test set. The full model and training details can be found in appendix A.5.1.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52, + "bbox_fs": [ + 104, + 605, + 507, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Table 7 in appendix A.5.2 shows the results of SOM-CPC (which includes a GRU for this dataset),", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "compared to different variants of the DESOM model. 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SOM-CPC has clustered the SOM nodes belonging", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "to the same speaker, and seems to group male and female speakers (denoted with the node’s shape),", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "while these effects are not present in the SOM of the GRU-DESOM model. 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Stars", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 344, + 256, + 465, + 267 + ], + "spans": [ + { + "bbox": [ + 344, + 256, + 465, + 267 + ], + "score": 1.0, + "content": "denote women and dots are men.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 21.75 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 409, + 287 + ], + "score": 1.0, + "content": "in the right green sub-cluster belong to two chapters read by speaker 2, while", + "type": "text" + }, + { + "bbox": [ + 409, + 274, + 442, + 285 + ], + "score": 0.89, + "content": "9 8 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "of the windows", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "in the left green sub-cluster belong to another chapter. An auditory inspection revealed that the room", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "acoustics of the recordings belonging to the chapters in different clusters were different, causing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "changes in the signals which the SOM-CPC model has picked upon. This division between recordings", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 415, + 331 + ], + "score": 1.0, + "content": "of the same speaker is not visible in the 2D PCA projection of the CPC (with", + "type": "text" + }, + { + "bbox": [ + 416, + 319, + 453, + 329 + ], + "score": 0.86, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 318, + 506, + 331 + ], + "score": 1.0, + "content": ") features, as", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 329, + 247, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 247, + 341 + ], + "score": 1.0, + "content": "seen from fig. 8 in appendix A.5.2.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 190, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 192, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 192, + 378 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "We proposed a new member of the deep-SOM family: SOM-CPC, suitable for interpretable 2D", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "representation learning of high-rate data streams. Earlier proposed deep-SOM models mainly used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "reconstruction objectives. In general, SOM-CPC outperformed these models with a wide gap on a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "variety of metrics. Moreover, it implicitly enforces temporal smoothness, while autoencoder-based", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "models require additional losses and hyperparameter tuning to achieve this. SOM-CPC’s task loss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "was found to align better with the topological SOM objective than a reconstruction loss, as already", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "hypothesized by Mrabah et al. (2020). While for some applications CPC could succesfully be trained", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "with a 2D latent space directly, optimization was found to be hampered in case of more intricate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "data spaces. Compared to vanilla CPC with a multi-dimensional latent space, SOM-CPC enables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "pattern recognition and knowledge discovery. The SOM objective did not hamper CPC optimization.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Even better, in the synthetic setup it had a regularizing effect, resulting in lower regression error than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "vanilla CPC. The use of a Gaussian neighbourhood kernel, as opposed to a plus kernel, was found", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "to improve the topological ordering in the SOM. No decisive conclusions could be made regarding", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "gradient blocking from the SOM loss towards the encoder parameters. Allowing these gradients to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "flow did not hurt performance, so for coding simplicity, we would advice to not detach the SOM loss.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Setting an appropriate stopping criterion for self-supervised (SSL) models is debatable. In the SSL", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "literature models with the best test set performance are sometimes reported (He et al., 2019; Fortuin", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "et al., 2019). This is, however, questionable as it may artificially boost reported performance. As such,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "we created a validation set to apply early stopping in all experiments. Another challenge arises when", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "dealing with aggregated loss functions, since not all losses may smoothly decay and the weighted", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "summation of losses may result in a different optimal epoch than the sub-losses separately. Besides,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "classification performance (often used as a proxy for information preservation) does not necessarily", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 639, + 459, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 459, + 650 + ], + "score": 1.0, + "content": "align with SOM performance or information preservation (see fig. 7 in appendix A.4.3).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5 + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "We believe that SOM-CPC will facilitate knowledge discovery in real-life time series and opens up", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "new research directions for representation learning of time series. Directions include investigation", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "to whether additions like the soft-cluster assignment, cluster hardening loss or a Gaussian latent", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "prior - which have shown to improve the SOM-VAE model (Manduchi et al., 2021) - improve", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "SOM-CPC performance as well. 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An auditory inspection revealed that the room", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "acoustics of the recordings belonging to the chapters in different clusters were different, causing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 504, + 320 + ], + "score": 1.0, + "content": "changes in the signals which the SOM-CPC model has picked upon. 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Stars", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 344, + 256, + 465, + 267 + ], + "spans": [ + { + "bbox": [ + 344, + 256, + 465, + 267 + ], + "score": 1.0, + "content": "denote women and dots are men.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + } + ], + "index": 21.75 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 341 + ], + "lines": [], + "index": 28.5, + "bbox_fs": [ + 105, + 274, + 506, + 341 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 190, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 192, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 192, + 378 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "We proposed a new member of the deep-SOM family: SOM-CPC, suitable for interpretable 2D", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "representation learning of high-rate data streams. Earlier proposed deep-SOM models mainly used", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 506, + 425 + ], + "score": 1.0, + "content": "reconstruction objectives. In general, SOM-CPC outperformed these models with a wide gap on a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "variety of metrics. Moreover, it implicitly enforces temporal smoothness, while autoencoder-based", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "models require additional losses and hyperparameter tuning to achieve this. SOM-CPC’s task loss", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "was found to align better with the topological SOM objective than a reconstruction loss, as already", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "hypothesized by Mrabah et al. (2020). While for some applications CPC could succesfully be trained", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "with a 2D latent space directly, optimization was found to be hampered in case of more intricate", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "data spaces. Compared to vanilla CPC with a multi-dimensional latent space, SOM-CPC enables", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "pattern recognition and knowledge discovery. The SOM objective did not hamper CPC optimization.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "Even better, in the synthetic setup it had a regularizing effect, resulting in lower regression error than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "vanilla CPC. The use of a Gaussian neighbourhood kernel, as opposed to a plus kernel, was found", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "to improve the topological ordering in the SOM. No decisive conclusions could be made regarding", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "gradient blocking from the SOM loss towards the encoder parameters. Allowing these gradients to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "flow did not hurt performance, so for coding simplicity, we would advice to not detach the SOM loss.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 390, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "Setting an appropriate stopping criterion for self-supervised (SSL) models is debatable. In the SSL", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "literature models with the best test set performance are sometimes reported (He et al., 2019; Fortuin", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "et al., 2019). This is, however, questionable as it may artificially boost reported performance. As such,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "we created a validation set to apply early stopping in all experiments. Another challenge arises when", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "dealing with aggregated loss functions, since not all losses may smoothly decay and the weighted", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "summation of losses may result in a different optimal epoch than the sub-losses separately. Besides,", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "classification performance (often used as a proxy for information preservation) does not necessarily", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 639, + 459, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 459, + 650 + ], + "score": 1.0, + "content": "align with SOM performance or information preservation (see fig. 7 in appendix A.4.3).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 561, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 668 + ], + "score": 1.0, + "content": "We believe that SOM-CPC will facilitate knowledge discovery in real-life time series and opens up", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 677 + ], + "score": 1.0, + "content": "new research directions for representation learning of time series. Directions include investigation", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "to whether additions like the soft-cluster assignment, cluster hardening loss or a Gaussian latent", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "prior - which have shown to improve the SOM-VAE model (Manduchi et al., 2021) - improve", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "SOM-CPC performance as well. 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A multi-modal variational", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 719, + 491, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 491, + 733 + ], + "score": 1.0, + "content": "future prediction could possibly improve performance for data that do not meet this assumption.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 59, + "bbox_fs": [ + 105, + 653, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 266, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 267, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 267, + 96 + ], + "score": 1.0, + "content": "REPRODUCIBILITY STATEMENT", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 506, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 507, + 119 + ], + "score": 1.0, + "content": "All code used to train and evaluate the models as presented in this paper can be found at https:", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "score": 1.0, + "content": "//anonymous.4open.science/r/SOM-CPC. The details regarding model architectures and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "score": 1.0, + "content": "training settings for each of the application domains are also presented in appendix A.3.1, A.4.2, and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 498, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 498, + 152 + ], + "score": 1.0, + "content": "A.5.1. 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As a result, clustering", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 598, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 610 + ], + "score": 1.0, + "content": "was directly performed on the one and only (test) set, that was also used to label the clusters/nodes.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 610, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 506, + 622 + ], + "score": 1.0, + "content": "Moving towards deep learning based approaches where more hyperparameters need to be set and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "overfitting can become a larger problem, we found it necessary to report in this work on a test set", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "that was not used for labelling the nodes and/or setting hyperparameters. It should thus be taken into", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "account, that direct comparison to results in other deep-clustering works may need a critical eye to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 304, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 304, + 666 + ], + "score": 1.0, + "content": "see whether similar procedures were used or not.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 531, + 506, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 267, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 269, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 269, + 690 + ], + "score": 1.0, + "content": "A.2 BENCHMARKS AND ABLATIONS", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "To benchmark our implementation of the SOM-VAE model (and the very similar DESOM model), we", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "replicated the results on MNIST. MNIST was not used for further experimentation with SOM-CPC", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 449, + 734 + ], + "score": 1.0, + "content": "in the main body of this paper since this work focuses on high-rate time series. Using", + "type": "text" + }, + { + "bbox": [ + 450, + 721, + 480, + 731 + ], + "score": 0.85, + "content": "k = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "SOM", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 275, + 95 + ], + "score": 1.0, + "content": "nodes, Fortuin et al. (2019) report purity", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 275, + 82, + 348, + 93 + ], + "score": 0.88, + "content": "= 0 . 7 3 1 \\pm 0 . 0 0 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 348, + 82, + 366, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 367, + 82, + 461, + 93 + ], + "score": 0.89, + "content": "\\mathbf { N M I } = 0 . 5 9 4 \\pm 0 . 0 0 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 461, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "in table 1,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 149, + 105 + ], + "score": 1.0, + "content": "and purity", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 149, + 94, + 221, + 104 + ], + "score": 0.86, + "content": "= 0 . 7 2 1 \\pm 0 . 0 0 6", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 221, + 93, + 239, + 105 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 239, + 94, + 332, + 105 + ], + "score": 0.89, + "content": "\\mathbf { N M I } = 0 . 5 8 7 \\pm 0 . 0 0 3", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 332, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "in table S1. All numbers are averages and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 223, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 223, + 116 + ], + "score": 1.0, + "content": "standard errors over 10 runs.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 275, + 95 + ], + "score": 1.0, + "content": "nodes, Fortuin et al. 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All numbers are averages and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 223, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 223, + 116 + ], + "score": 1.0, + "content": "standard errors over 10 runs.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Settings that were not provided in the paper, were taken from the hard-coded settings that we found", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "in the provided code base. Instead of splitting the standard MNIST training set in a training and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "test split, as done by Fortuin et al. (2019), we used the available train/test split that comes with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "the standard MNIST dataloader from Pytorch. From the first ten runs, one run fully collapsed and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 453, + 177 + ], + "score": 1.0, + "content": "resulted in extremely poor performance. Considering this run as an outlier, we run an", + "type": "text" + }, + { + "bbox": [ + 454, + 165, + 471, + 176 + ], + "score": 0.87, + "content": "1 1 ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "run and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 417, + 189 + ], + "score": 1.0, + "content": "here report the average and standard errors of the 10 non-collapsed runs: purity", + "type": "text" + }, + { + "bbox": [ + 417, + 176, + 487, + 187 + ], + "score": 0.85, + "content": "= 0 . 7 0 5 \\pm 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 198, + 198 + ], + "score": 0.9, + "content": "\\mathbf { N M I } = 0 . 5 8 4 \\pm 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 187, + 505, + 200 + ], + "score": 1.0, + "content": ". Our performance does reasonably well match the reported performance by", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "Fortuin et al. (2019), given the fact that a different test set of MNIST was used for our experiments.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 506, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "To restrict the search space of hyperparameters in the SOM-VAE(-prob) model, which has multiple", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "loss multipliers, we fixed some of the these multipliers for further experiments in this paper. Multiplier", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 114, + 248 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "scales the SOM loss that sums the quantization error of the four neighbour nodes in the plus kernel.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "It is, therefore, expected to be at least 4 times larger than the commitment loss, which only reflects", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 326, + 272 + ], + "score": 1.0, + "content": "the quantization error of the winning node. Choosing", + "type": "text" + }, + { + "bbox": [ + 327, + 259, + 365, + 271 + ], + "score": 0.92, + "content": "\\beta = \\alpha / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "would imply that the weighing of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "the summed quantization error of the four neighbours is equal to the weighing of this error for the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 429, + 294 + ], + "score": 1.0, + "content": "winning node. To give the winning node a slightly higher importance, we set", + "type": "text" + }, + { + "bbox": [ + 429, + 281, + 503, + 293 + ], + "score": 0.92, + "content": "\\beta = \\alpha / 5 = 0 . 2 \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 280, + 506, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "The authors of SOM-VAE (Fortuin et al., 2019) used, instead, a search strategy to find the optimal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 301, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 442, + 317 + ], + "score": 1.0, + "content": "setting. Their code base1 shows that the SOM loss was multiplied with 0.9, while", + "type": "text" + }, + { + "bbox": [ + 443, + 303, + 469, + 313 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 301, + 506, + 317 + ], + "score": 1.0, + "content": ". Taking", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "into account that their implementation of the commitment loss averaged the quantization error of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "the four neighbour nodes, while we summed the contribution, their effective setting was thus set to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 335, + 398, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 335, + 189, + 349 + ], + "score": 0.93, + "content": "\\beta = \\textstyle { \\frac { 0 . 9 } { 4 } } \\alpha = 0 . 2 2 5 \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 335, + 398, + 349 + ], + "score": 1.0, + "content": ", which is close to what we used in our experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "We compared SOM-CPC also against vanilla CPC training followed by a linear classifier or K-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "means clustering, while freezing the encoder parameters. The supervised linear classifier took in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "all experiments the form of one fully-connected layer, including biases, that was followed by a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "log-softmax activation for the sleep and audio cases. It was trained using the mean squared error for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 395, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 411 + ], + "score": 1.0, + "content": "the synthetic data set, and cross-entropy loss for sleep and audio experiments. K-means clustering", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "was run from the sklearn library, with the default settings. The number of clusters was chosen to be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "equal to the number of nodes in the SOM-CPC models against which the performance was compared.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "Also the disjointly-trained SOM had exactly the same settings as the SOM in the SOM-CPC models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 223, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 223, + 453 + ], + "score": 1.0, + "content": "with which it was compared.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "Several ablation are performed on the SOM-CPC model. We test the effect of propagating gradients", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 117, + 480 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 470, + 141, + 479 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { S O M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "to the encoder parameters, the difference between using a Gaussian neighbourhoood kernel", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "versus a plus kernel, and the effect of jointly training CPC and the SOM. Moreover, the effect of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 504, + 501 + ], + "score": 1.0, + "content": "certain settings that are typically used in the CPC objective are investigated. CPC’s InfoNCE objective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 169, + 514 + ], + "score": 1.0, + "content": "for one window", + "type": "text" + }, + { + "bbox": [ + 170, + 506, + 175, + 513 + ], + "score": 0.7, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 500, + 505, + 514 + ], + "score": 1.0, + "content": ", given in eq. 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(8). CPC typically uses", + "type": "text" + }, + { + "bbox": [ + 336, + 576, + 361, + 583 + ], + "score": 0.88, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 572, + 505, + 586 + ], + "score": 1.0, + "content": ", and the dot product as the similarity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "metric. However, other related contrastive learning objectives, e.g. in SimCLR (Chen et al., 2020),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "use a temperature value that is often set to 0.07 (Chen et al., 2020; Woo et al., 2022), and a cosine", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 480, + 618 + ], + "score": 1.0, + "content": "similarity instead of the (unnormalized) dot product. 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The output", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 692, + 504, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 504, + 705 + ], + "score": 1.0, + "content": "size column in the table uses channels-first notation. The SOM-VAE and DESOM models were", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 712, + 499, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 501, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 501, + 723 + ], + "score": 1.0, + "content": "1https://github.com/ratschlab/SOM-VAE/blob/master/som_vae/somvae_train.", + "type": "text" + } + ] + }, + { + "bbox": [ + 104, + 721, + 151, + 732 + ], + "spans": [ + { + "bbox": [ + 104, + 721, + 151, + 732 + ], + "score": 1.0, + "content": "py, line 79.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 115 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 106, + 82, + 506, + 116 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "Settings that were not provided in the paper, were taken from the hard-coded settings that we found", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "in the provided code base. Instead of splitting the standard MNIST training set in a training and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "test split, as done by Fortuin et al. (2019), we used the available train/test split that comes with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "the standard MNIST dataloader from Pytorch. From the first ten runs, one run fully collapsed and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 453, + 177 + ], + "score": 1.0, + "content": "resulted in extremely poor performance. Considering this run as an outlier, we run an", + "type": "text" + }, + { + "bbox": [ + 454, + 165, + 471, + 176 + ], + "score": 0.87, + "content": "1 1 ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "run and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 417, + 189 + ], + "score": 1.0, + "content": "here report the average and standard errors of the 10 non-collapsed runs: purity", + "type": "text" + }, + { + "bbox": [ + 417, + 176, + 487, + 187 + ], + "score": 0.85, + "content": "= 0 . 7 0 5 \\pm 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 198, + 198 + ], + "score": 0.9, + "content": "\\mathbf { N M I } = 0 . 5 8 4 \\pm 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 187, + 505, + 200 + ], + "score": 1.0, + "content": ". Our performance does reasonably well match the reported performance by", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 211 + ], + "score": 1.0, + "content": "Fortuin et al. (2019), given the fact that a different test set of MNIST was used for our experiments.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 122, + 506, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 506, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "To restrict the search space of hyperparameters in the SOM-VAE(-prob) model, which has multiple", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "loss multipliers, we fixed some of the these multipliers for further experiments in this paper. Multiplier", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 114, + 248 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "scales the SOM loss that sums the quantization error of the four neighbour nodes in the plus kernel.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "It is, therefore, expected to be at least 4 times larger than the commitment loss, which only reflects", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 326, + 272 + ], + "score": 1.0, + "content": "the quantization error of the winning node. Choosing", + "type": "text" + }, + { + "bbox": [ + 327, + 259, + 365, + 271 + ], + "score": 0.92, + "content": "\\beta = \\alpha / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "would imply that the weighing of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "the summed quantization error of the four neighbours is equal to the weighing of this error for the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 280, + 429, + 294 + ], + "score": 1.0, + "content": "winning node. To give the winning node a slightly higher importance, we set", + "type": "text" + }, + { + "bbox": [ + 429, + 281, + 503, + 293 + ], + "score": 0.92, + "content": "\\beta = \\alpha / 5 = 0 . 2 \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 280, + 506, + 294 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "The authors of SOM-VAE (Fortuin et al., 2019) used, instead, a search strategy to find the optimal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 301, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 442, + 317 + ], + "score": 1.0, + "content": "setting. Their code base1 shows that the SOM loss was multiplied with 0.9, while", + "type": "text" + }, + { + "bbox": [ + 443, + 303, + 469, + 313 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 301, + 506, + 317 + ], + "score": 1.0, + "content": ". Taking", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "into account that their implementation of the commitment loss averaged the quantization error of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "the four neighbour nodes, while we summed the contribution, their effective setting was thus set to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 335, + 398, + 349 + ], + "spans": [ + { + "bbox": [ + 107, + 335, + 189, + 349 + ], + "score": 0.93, + "content": "\\beta = \\textstyle { \\frac { 0 . 9 } { 4 } } \\alpha = 0 . 2 2 5 \\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 335, + 398, + 349 + ], + "score": 1.0, + "content": ", which is close to what we used in our experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 214, + 506, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "We compared SOM-CPC also against vanilla CPC training followed by a linear classifier or K-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 364, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 506, + 376 + ], + "score": 1.0, + "content": "means clustering, while freezing the encoder parameters. The supervised linear classifier took in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "all experiments the form of one fully-connected layer, including biases, that was followed by a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "log-softmax activation for the sleep and audio cases. It was trained using the mean squared error for", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 395, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 411 + ], + "score": 1.0, + "content": "the synthetic data set, and cross-entropy loss for sleep and audio experiments. K-means clustering", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "was run from the sklearn library, with the default settings. The number of clusters was chosen to be", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "equal to the number of nodes in the SOM-CPC models against which the performance was compared.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "Also the disjointly-trained SOM had exactly the same settings as the SOM in the SOM-CPC models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 223, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 223, + 453 + ], + "score": 1.0, + "content": "with which it was compared.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 352, + 506, + 453 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 470 + ], + "score": 1.0, + "content": "Several ablation are performed on the SOM-CPC model. 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Moreover, the effect of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 490, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 504, + 501 + ], + "score": 1.0, + "content": "certain settings that are typically used in the CPC objective are investigated. CPC’s InfoNCE objective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 169, + 514 + ], + "score": 1.0, + "content": "for one window", + "type": "text" + }, + { + "bbox": [ + 170, + 506, + 175, + 513 + ], + "score": 0.7, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 500, + 505, + 514 + ], + "score": 1.0, + "content": ", given in eq. 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However, other related contrastive learning objectives, e.g. in SimCLR (Chen et al., 2020),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "use a temperature value that is often set to 0.07 (Chen et al., 2020; Woo et al., 2022), and a cosine", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 606, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 480, + 618 + ], + "score": 1.0, + "content": "similarity instead of the (unnormalized) dot product. 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The output", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 692, + 504, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 504, + 705 + ], + "score": 1.0, + "content": "size column in the table uses channels-first notation. The SOM-VAE and DESOM models were", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "found to benefit from a convolutional part of the encoder that did not fully reduce the temporal", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "dimension to size 1. 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The", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "SOM-CPC and CPC model are run without an AR module, to make the fairest comparison to the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 411, + 462, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 462, + 424 + ], + "score": 1.0, + "content": "SOM-VAE(prob) and DESOM models, which also do not incorporate such a component.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 680, + 505, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 134, + 100, + 476, + 340 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 174, + 90, + 434, + 100 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 176, + 90, + 435, + 102 + ], + "spans": [ + { + "bbox": [ + 176, + 90, + 435, + 102 + ], + "score": 1.0, + "content": "Table 2: Model details for the synthetic data experiments in section 4.1.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 134, + 100, + 476, + 340 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 134, + 100, + 476, + 340 + ], + "spans": [ + { + "bbox": [ + 134, + 100, + 476, + 340 + ], + "score": 0.984, + "html": "
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU(0.01)911same
MaxPool1Dbs ×16× 32-44=·
Dropout (0.1)bs ×16×32--=-
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8----
Conv1Dbs ×64×864Leaky ReLU (0.01)311same
MaxPool1Dbs ×64×2-44-
Dropout (0.1)bs ×64×2----
Conv1Dbs ×128×2128Leaky ReLU (0.01)311same
MaxPool1Dbs ×128 ×1-22=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)911same
MaxPool1Dbs ×16× 32-44-
Dropout (0.1)bs ×16×32----
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)31same
Flattenbs ×512-==
Fully Connectedbs ×128128Leaky ReLU (0.01)=
DecoderforSOM-VAEandDESOM
Fully Connectedbs×512512Leaky ReLU (0.01)
Unflattenbs ×64×8-
Conv1Dbs ×32×832Leaky ReLU (0.01)311same
ConvTranspose1Dbs ×32 × 3232None4410
Conv1Dbs ×16× 3216Leaky ReLU(0.01)711same
ConvTranspose1Dbs ×16× 12816None4410
Conv1Dbs ×1 × 1281Tanh911same
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Choosing a lower value at the end of training induced instable optimization behavior.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "All models (including the benchmarks) were trained using the Adam optimizer (Kingma & Ba, 2014),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "with a learning rate of 0.001 and a batch size of 128. Each model was trained for maximally 1000", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 500, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 479, + 516 + ], + "score": 1.0, + "content": "epochs. The best model was selected based on the lowest task loss on the validation set, being", + "type": "text" + }, + { + "bbox": [ + 480, + 502, + 504, + 513 + ], + "score": 0.7, + "content": "\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 240, + 525 + ], + "score": 1.0, + "content": "for SOM-VAE and DESOM and", + "type": "text" + }, + { + "bbox": [ + 240, + 513, + 274, + 524 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { { I n f o N C E } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "for SOM-CPC and CPC. We did not use the full training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 145, + 536 + ], + "score": 1.0, + "content": "objective", + "type": "text" + }, + { + "bbox": [ + 146, + 524, + 185, + 535 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { d e e p - S O M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "as model selection criterion, as both the commitment and SOM loss showed to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "be low initially (possibly due to low values of the random initialization of the model), while both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "increased and reached a steady-state later in training. The linear classifier and the disjointly-trained", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 556, + 471, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 471, + 569 + ], + "score": 1.0, + "content": "SOM on the CPC embeddings were trained until convergence, for maximally 1000 epochs.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 588, + 230, + 599 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 231, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 231, + 601 + ], + "score": 1.0, + "content": "A.3.2 EXTENDED RESULTS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 378, + 624 + ], + "score": 1.0, + "content": "Table 3 extends table 1 with additional sweeps of hyperparameters", + "type": "text" + }, + { + "bbox": [ + 379, + 613, + 398, + 623 + ], + "score": 0.34, + "content": "\\alpha , \\gamma _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 610, + 418, + 624 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 418, + 612, + 424, + 623 + ], + "score": 0.75, + "content": "\\zeta", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 610, + 506, + 624 + ], + "score": 1.0, + "content": ", and ablations with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 273, + 634 + ], + "score": 1.0, + "content": "different settings of the temperature value", + "type": "text" + }, + { + "bbox": [ + 273, + 624, + 281, + 632 + ], + "score": 0.77, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "and the used similarity metric. 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Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU(0.01)911same
MaxPool1Dbs ×16× 32-44=·
Dropout (0.1)bs ×16×32--=-
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8----
Conv1Dbs ×64×864Leaky ReLU (0.01)311same
MaxPool1Dbs ×64×2-44-
Dropout (0.1)bs ×64×2----
Conv1Dbs ×128×2128Leaky ReLU (0.01)311same
MaxPool1Dbs ×128 ×1-22=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)911same
MaxPool1Dbs ×16× 32-44-
Dropout (0.1)bs ×16×32----
Conv1Dbs ×32×3232Leaky ReLU (0.01)711same
MaxPool1Dbs ×32×8-44-
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)31same
Flattenbs ×512-==
Fully Connectedbs ×128128Leaky ReLU (0.01)=
DecoderforSOM-VAEandDESOM
Fully Connectedbs×512512Leaky ReLU (0.01)
Unflattenbs ×64×8-
Conv1Dbs ×32×832Leaky ReLU (0.01)311same
ConvTranspose1Dbs ×32 × 3232None4410
Conv1Dbs ×16× 3216Leaky ReLU(0.01)711same
ConvTranspose1Dbs ×16× 12816None4410
Conv1Dbs ×1 × 1281Tanh911same
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The latter were drawn randomly from the entire training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 462 + ], + "score": 1.0, + "content": "set. The standard deviation of the Gaussian neighbourhood kernel was exponentially decayed until", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 460, + 499, + 477 + ], + "spans": [ + { + "bbox": [ + 107, + 461, + 153, + 473 + ], + "score": 0.9, + "content": "\\sigma ^ { ( n _ { \\mathrm { m a x } } ) } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 460, + 499, + 477 + ], + "score": 1.0, + "content": ". Choosing a lower value at the end of training induced instable optimization behavior.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 428, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "All models (including the benchmarks) were trained using the Adam optimizer (Kingma & Ba, 2014),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "with a learning rate of 0.001 and a batch size of 128. Each model was trained for maximally 1000", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 500, + 504, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 479, + 516 + ], + "score": 1.0, + "content": "epochs. The best model was selected based on the lowest task loss on the validation set, being", + "type": "text" + }, + { + "bbox": [ + 480, + 502, + 504, + 513 + ], + "score": 0.7, + "content": "\\scriptstyle { \\mathcal { L } } _ { \\mathrm { r e c o n } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 240, + 525 + ], + "score": 1.0, + "content": "for SOM-VAE and DESOM and", + "type": "text" + }, + { + "bbox": [ + 240, + 513, + 274, + 524 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { { I n f o N C E } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "for SOM-CPC and CPC. We did not use the full training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 145, + 536 + ], + "score": 1.0, + "content": "objective", + "type": "text" + }, + { + "bbox": [ + 146, + 524, + 185, + 535 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { d e e p - S O M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "as model selection criterion, as both the commitment and SOM loss showed to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "be low initially (possibly due to low values of the random initialization of the model), while both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "increased and reached a steady-state later in training. 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ModelαSLsom sg[-]SEtargetl2.smoothTE
CPC+linearclassifier==2.62±2.37==
CPC + K-means===1.09±.62=
CPC(F=2) + linear classifier25.01±42.94
CPC(F = 2)+K-means.76±1.31
CPC+PCA+ linear classifier42.81±58.12
CPC + PCA+K-means-=4.42±9.01
SOM-VAE1e-3Plus11.59±13.692.60±.46.38±.042
*1e-2Plus14.57±44.102.98±.84.38±.052
.1 1Plus Plus8.02±4.582.41±.68.28±.070
SOM-VAE1e-5Gaussian13.43±3.66 11.12±17.052.75±.45.33±.048
1e-4Gaussian1.93±.29.069±.029
1e-31.52±26.611.95±.36.075±.028
SOM-VAE-prob1e-2Gaussian Gaussian11.60±25.431.92±.34.056±.023
(γ=5e-5,S= 1e-3)1e-318.13±48.662.03±.42.086±.030
(γ = 4e-5,= 1e-3)Plus21.26±55.003.78±.52.93±.042
1e-3Plus21.30±37.374.23±.57.88±.051
(γ=3.3e-5,S= 1e-3) (γ= 5e-4,S= 1e-2)1e-3Plus22.52±48.973.81±.51.98±.015
(γ=4e-4,= 1e-2)1e-2Plus14.65±19.583.70±.51.86±.083
1e-2Plus27.25±72.823.54±.67.94±.022
(γ= 3.3e-4,S= 1e-2)1e-2Plus21.62±38.783.71±.76.97±.014
(γ= 5e-5,S= 1e-2).1Plus26.82±68.843.23±.80.68±.067
(γ=4e-5,= 1e-2).1Plus22.34±64.923.11±.73.74±.072
(γ=3.3e-5,S=1e-2) (γ= 5e-4,=.1).1Plus Plus2.10±48.803.15±.73.63±.050
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(γ= 4e-4,S=.1) (γ=3.3e-4,S=.1).1Plus29.96±46.962.81±.34.82±.13
DESOM.1 1e-5Plus GaussianX3.96±56.943.95±.83.85±.11
1e-4GaussianX14.00±3.10 19.09±44.041.99±.36 1.95±.28.13±.069
1e-3GaussianX12.58±27.101.92±.31.11±.046 .077±.033
1e-2GaussianX13.66±44.711.89±.33.065±.028
.1GaussianX10.77±10.852.20±.44.061±.019
SOM-CPC (ours)1GaussianX22.86±55.712.26±.43.092±.034
*1e-5GaussianX.95±2.401.24±.31.048±.020
1e-4GaussianX.72±1.081.37±.37.022±.011
1e-3GaussianX.81±.751.06±.28.028±.012
1e-2Gaussian Gaussian_X.62±.711.08±.28.059±.035
1 1e-5GaussianX √1.90±4.07 .64±.711.18±.30.039±.016
1e-4Gaussian√.68±.671.04±.26 1.19±.43.039±.020
1e-3Gaussian.75±1.021.12±.32.057±.033
1e-2Gaussian.47±.48.059±.027
.1.99±.24.069±.040
Gaussian√.89±1.501.16±.32.069±.031
1e-4PlusX1.71±1.152.46±0.51.30±0.062
Gaussian1e-2 1e-4Plus√1.16±.61 1.47±2.601.85±.26 1.15±.34.12±.037 .014±.015
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ModelαSLsom sg[-]SEtargetl2.smoothTE
CPC+linearclassifier==2.62±2.37==
CPC + K-means===1.09±.62=
CPC(F=2) + linear classifier25.01±42.94
CPC(F = 2)+K-means.76±1.31
CPC+PCA+ linear classifier42.81±58.12
CPC + PCA+K-means-=4.42±9.01
SOM-VAE1e-3Plus11.59±13.692.60±.46.38±.042
*1e-2Plus14.57±44.102.98±.84.38±.052
.1 1Plus Plus8.02±4.582.41±.68.28±.070
SOM-VAE1e-5Gaussian13.43±3.66 11.12±17.052.75±.45.33±.048
1e-4Gaussian1.93±.29.069±.029
1e-31.52±26.611.95±.36.075±.028
SOM-VAE-prob1e-2Gaussian Gaussian11.60±25.431.92±.34.056±.023
(γ=5e-5,S= 1e-3)1e-318.13±48.662.03±.42.086±.030
(γ = 4e-5,= 1e-3)Plus21.26±55.003.78±.52.93±.042
1e-3Plus21.30±37.374.23±.57.88±.051
(γ=3.3e-5,S= 1e-3) (γ= 5e-4,S= 1e-2)1e-3Plus22.52±48.973.81±.51.98±.015
(γ=4e-4,= 1e-2)1e-2Plus14.65±19.583.70±.51.86±.083
1e-2Plus27.25±72.823.54±.67.94±.022
(γ= 3.3e-4,S= 1e-2)1e-2Plus21.62±38.783.71±.76.97±.014
(γ= 5e-5,S= 1e-2).1Plus26.82±68.843.23±.80.68±.067
(γ=4e-5,= 1e-2).1Plus22.34±64.923.11±.73.74±.072
(γ=3.3e-5,S=1e-2) (γ= 5e-4,=.1).1Plus Plus2.10±48.803.15±.73.63±.050
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DESOM.1 1e-5Plus GaussianX3.96±56.943.95±.83.85±.11
1e-4GaussianX14.00±3.10 19.09±44.041.99±.36 1.95±.28.13±.069
1e-3GaussianX12.58±27.101.92±.31.11±.046 .077±.033
1e-2GaussianX13.66±44.711.89±.33.065±.028
.1GaussianX10.77±10.852.20±.44.061±.019
SOM-CPC (ours)1GaussianX22.86±55.712.26±.43.092±.034
*1e-5GaussianX.95±2.401.24±.31.048±.020
1e-4GaussianX.72±1.081.37±.37.022±.011
1e-3GaussianX.81±.751.06±.28.028±.012
1e-2Gaussian Gaussian_X.62±.711.08±.28.059±.035
1 1e-5GaussianX √1.90±4.07 .64±.711.18±.30.039±.016
1e-4Gaussian√.68±.671.04±.26 1.19±.43.039±.020
1e-3Gaussian.75±1.021.12±.32.057±.033
1e-2Gaussian.47±.48.059±.027
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Gaussian√.89±1.501.16±.32.069±.031
1e-4PlusX1.71±1.152.46±0.51.30±0.062
Gaussian1e-2 1e-4Plus√1.16±.61 1.47±2.601.85±.26 1.15±.34.12±.037 .014±.015
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Each recording contains, among others, electroencephalography (EEG),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "chin electromyography (EMG), and electrooculography (EOG) data. We refer the reader to O’Reilly", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 506, + 179 + ], + "score": 1.0, + "content": "et al. (2014) for more details regarding this dataset. We selected the channels that are typically used in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "clinical practice, comprising three EEG channels (F4, C4, O2), the two EOG channels, and one chin", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 294, + 201 + ], + "score": 1.0, + "content": "EMG derivation, and downsampled the data to", + "type": "text" + }, + { + "bbox": [ + 294, + 188, + 324, + 199 + ], + "score": 0.57, + "content": "1 2 8 \\ : \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 188, + 505, + 201 + ], + "score": 1.0, + "content": ". 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The", + "type": "text" + }, + { + "bbox": [ + 152, + 249, + 178, + 259 + ], + "score": 0.7, + "content": "6 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "powerline interference was, however, not fully suppressed, and we wanted to down", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 507, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 195, + 273 + ], + "score": 1.0, + "content": "sample each signal to", + "type": "text" + }, + { + "bbox": [ + 196, + 260, + 226, + 271 + ], + "score": 0.71, + "content": "1 2 8 \\ : \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 259, + 507, + 273 + ], + "score": 1.0, + "content": "to reduce computational complexity. 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The decoder architecture", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "(see table 4) was used both for the continuous and discrete decoding in the SOM-VAE model (without", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "weight tying). No AR-component was used in the SOM-CPC model to make a fair comparison to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 409, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 409, + 490 + ], + "score": 1.0, + "content": "SOM-VAE and DESOM model that also did not include such a component.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 379, + 506, + 490 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 216, + 507 + ], + "score": 1.0, + "content": "For the SOM-CPC model,", + "type": "text" + }, + { + "bbox": [ + 216, + 495, + 244, + 505 + ], + "score": 0.9, + "content": "P = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 495, + 475, + 507 + ], + "score": 1.0, + "content": "future predictions (i.e. positive samples) were used, and", + "type": "text" + }, + { + "bbox": [ + 475, + 495, + 505, + 505 + ], + "score": 0.88, + "content": "N = 3", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "negative samples were drawn for each positive sample. 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The", + "type": "text" + }, + { + "bbox": [ + 218, + 519, + 225, + 527 + ], + "score": 0.79, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "of the Gaussian neighbourhood kernel was exponentially annealed to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 528, + 226, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 226, + 542 + ], + "score": 1.0, + "content": "σ(nmax) = 0.5 during training.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 495, + 505, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "All models were trained with the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 1e-4", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "and a batch size of 128. 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Discussion of the main results in this table can be found in section 4.2.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 297, + 667 + ], + "score": 1.0, + "content": "The ablation experiments in which the value of", + "type": "text" + }, + { + "bbox": [ + 297, + 657, + 304, + 665 + ], + "score": 0.74, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "and/or the similarity metric was altered show that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "classification and clustering performance slightly dropped when using the cosine similarity with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "a temperature value of 1, while the topographic organization slightly improved (i.e. lower TE).", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 320, + 700 + ], + "score": 1.0, + "content": "These effects vanished when using a temperature of", + "type": "text" + }, + { + "bbox": [ + 321, + 690, + 358, + 698 + ], + "score": 0.83, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ". Both runs showed worse temporal", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 206, + 711 + ], + "score": 1.0, + "content": "smoothness (i.e. higher", + "type": "text" + }, + { + "bbox": [ + 207, + 701, + 239, + 711 + ], + "score": 0.89, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "). As expected, only changing the temperature value to 0.07 did", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "almost not affect results, suggesting that the linear projector heads were able to adjust for this scaling", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 719, + 135, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 135, + 734 + ], + "score": 1.0, + "content": "factor.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 621, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 129, + 58, + 479, + 305 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 191, + 48, + 419, + 58 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 190, + 47, + 419, + 59 + ], + "spans": [ + { + "bbox": [ + 190, + 47, + 419, + 59 + ], + "score": 1.0, + "content": "Table 4: Model details for the sleep experiments in section 4.2.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 129, + 58, + 479, + 305 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 58, + 479, + 305 + ], + "spans": [ + { + "bbox": [ + 129, + 58, + 479, + 305 + ], + "score": 0.983, + "html": "
Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU (0.01)15110
MaxPool1Dbs ×16×32-55=
Dropout (0.1)bs ×16× 32---=
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55
Dropout (0.1)bs ×32×8-=-==
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2-=
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
AdaptiveAvgPool1Dbs ×128×1=
EncoderforSOM-VAEandDESOM
Conv1Dbs×16×12816Leaky ReLU (0.01)1511(18,17)
MaxPool1Dbs ×16×32-55
Dropout (0.1)bs ×16×32--==
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55=
Dropout (0.1)bs ×32×8---=
Conv1Dbs ×64×864Leaky ReLU (0.01)5110
MaxPool1Dbs ×64×2-55
Dropout (0.1)bs ×64×2--==
Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
DecoderforSOM-VAEandDESOM
Conv1Dbs×64×264Leaky ReLU(0.01)3110
ConvTranspose1Dbs ×64×264None5510
Conv1Dbs ×32×232Leaky ReLU (0.01)5110
ConvTranspose1Dbs ×32×232None5510
Conv1Dbs ×16×216Leaky ReLU (0.01)9110
ConvTranspose1Dbs ×16×216None5510
Conv1Dbs ×6×26None15110
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SimCLR is also a contrastive learning framework, but instead of drawing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "score": 1.0, + "content": "positive samples from the future latent space, these samples are created by applying augmentations on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "the anchor window. Inspired by Um et al. (2017) we used the following augmentations: independent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 269, + 368 + ], + "score": 1.0, + "content": "and identically distributed Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 270, + 356, + 315, + 368 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , 0 . 0 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "was added (called jitter in their implementation),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 307, + 379 + ], + "score": 1.0, + "content": "each channel was scaled with a value drawn from", + "type": "text" + }, + { + "bbox": [ + 307, + 367, + 347, + 379 + ], + "score": 0.91, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 365, + 505, + 379 + ], + "score": 1.0, + "content": ", windows were split in 4 sub-windows", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "of minimal 2 seconds and randomly permuted, and lastly time series were both time warped and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "magnitude warped. The latter two augmentations make use of smooth curves that smoothly vary the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 399, + 344, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 344, + 412 + ], + "score": 1.0, + "content": "positions of time stamps or magnitude values, respectively.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Besides the difference on how to create positive samples, the originally proposed SimCLR model has", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 428, + 351, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 351, + 439 + ], + "score": 1.0, + "content": "some other slight differences with respect to the CPC model:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 133, + 449, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 132, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 132, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "• The SimCLR loss uses the cosine similarity, while CPC uses the (unnormalized) dot product", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 460, + 289, + 472 + ], + "spans": [ + { + "bbox": [ + 141, + 460, + 289, + 472 + ], + "score": 1.0, + "content": "as the similarity metric (see eq. (9)).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 132, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 132, + 475, + 308, + 488 + ], + "score": 1.0, + "content": "• SimCLR uses an additional temperature", + "type": "text" + }, + { + "bbox": [ + 308, + 478, + 315, + 486 + ], + "score": 0.72, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "in its loss function (see eq. (9)), for which the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 142, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "value is often set to 0.07 (Chen et al., 2020; Woo et al., 2022). CPC does not incorporate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 497, + 419, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 419, + 510 + ], + "score": 1.0, + "content": "such a temperature, which effectively means that it uses a value of 1.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 133, + 513, + 501, + 526 + ], + "spans": [ + { + "bbox": [ + 133, + 513, + 501, + 526 + ], + "score": 1.0, + "content": "• SimCLR uses a non-linear MLP projection head, while CPC uses linear projection heads.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 133, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 133, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "• SimCLR uses negative samples from within the batch, while this is not specified in the CPC", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 541, + 506, + 553 + ], + "score": 1.0, + "content": "paper. This specified design choice makes SimCLR typically very sensitive to the batch size.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 138, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 138, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "SimCLR was not proposed to include an auto-regressive component, and can not", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 141, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "straightforwardly be extended to do so, while CPC can be implemented with or without such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 578, + 184, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 578, + 184, + 590 + ], + "score": 1.0, + "content": "a module.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "For the most fair comparison, the procedure for drawing negative samples in SimCLR is done", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "score": 1.0, + "content": "equivalently as for SOM-CPC, i.e. within the recording, instead of within the batch. However,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "in the SOM-SimCLR model (i.e. the joint training of SOM with SimCLR), each drawn negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "sample is added to the set of negative samples both in its raw form, and with a random augmentation,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "which effectively doubles the number of negative samples. Table 3 reports the performance of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 307, + 667 + ], + "score": 1.0, + "content": "baseline SOM-SimCLR model (i.e. with settings", + "type": "text" + }, + { + "bbox": [ + 307, + 657, + 344, + 665 + ], + "score": 0.82, + "content": "\\tau = 0 . 0 7", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "and the cosine similarity), and variants", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "using a temperature value of 1 and/or the dot product as the similarity metric. All settings regarding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "training procedure and the SOM were set equivalently as in the SOM-CPC training. 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Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
EncoderforSOM-CPCandCPC
Conv1Dbs×16×12816Leaky ReLU (0.01)15110
MaxPool1Dbs ×16×32-55=
Dropout (0.1)bs ×16× 32---=
Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
MaxPool1Dbs ×32×8-55
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AdaptiveAvgPool1Dbs ×128×1=
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Conv1Dbs ×32×3232Leaky ReLU (0.01)9110
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Conv1Dbs ×64×864Leaky ReLU (0.01)5110
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Conv1Dbs ×128×2128Leaky ReLU (0.01)3110
DecoderforSOM-VAEandDESOM
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ConvTranspose1Dbs ×64×264None5510
Conv1Dbs ×32×232Leaky ReLU (0.01)5110
ConvTranspose1Dbs ×32×232None5510
Conv1Dbs ×16×216Leaky ReLU (0.01)9110
ConvTranspose1Dbs ×16×216None5510
Conv1Dbs ×6×26None15110
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SimCLR is also a contrastive learning framework, but instead of drawing", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "score": 1.0, + "content": "positive samples from the future latent space, these samples are created by applying augmentations on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "the anchor window. Inspired by Um et al. (2017) we used the following augmentations: independent", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 269, + 368 + ], + "score": 1.0, + "content": "and identically distributed Gaussian noise", + "type": "text" + }, + { + "bbox": [ + 270, + 356, + 315, + 368 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , 0 . 0 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "was added (called jitter in their implementation),", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 307, + 379 + ], + "score": 1.0, + "content": "each channel was scaled with a value drawn from", + "type": "text" + }, + { + "bbox": [ + 307, + 367, + 347, + 379 + ], + "score": 0.91, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 365, + 505, + 379 + ], + "score": 1.0, + "content": ", windows were split in 4 sub-windows", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "of minimal 2 seconds and randomly permuted, and lastly time series were both time warped and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "magnitude warped. The latter two augmentations make use of smooth curves that smoothly vary the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 399, + 344, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 344, + 412 + ], + "score": 1.0, + "content": "positions of time stamps or magnitude values, respectively.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 312, + 506, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Besides the difference on how to create positive samples, the originally proposed SimCLR model has", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 428, + 351, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 351, + 439 + ], + "score": 1.0, + "content": "some other slight differences with respect to the CPC model:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 415, + 505, + 439 + ] + }, + { + "type": "list", + "bbox": [ + 133, + 449, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 132, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 132, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "• The SimCLR loss uses the cosine similarity, while CPC uses the (unnormalized) dot product", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 460, + 289, + 472 + ], + "spans": [ + { + "bbox": [ + 141, + 460, + 289, + 472 + ], + "score": 1.0, + "content": "as the similarity metric (see eq. (9)).", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 132, + 475, + 308, + 488 + ], + "score": 1.0, + "content": "• SimCLR uses an additional temperature", + "type": "text" + }, + { + "bbox": [ + 308, + 478, + 315, + 486 + ], + "score": 0.72, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "in its loss function (see eq. (9)), for which the", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 142, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "value is often set to 0.07 (Chen et al., 2020; Woo et al., 2022). CPC does not incorporate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 497, + 419, + 510 + ], + "spans": [ + { + "bbox": [ + 141, + 497, + 419, + 510 + ], + "score": 1.0, + "content": "such a temperature, which effectively means that it uses a value of 1.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 513, + 501, + 526 + ], + "spans": [ + { + "bbox": [ + 133, + 513, + 501, + 526 + ], + "score": 1.0, + "content": "• SimCLR uses a non-linear MLP projection head, while CPC uses linear projection heads.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 133, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 133, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "• SimCLR uses negative samples from within the batch, while this is not specified in the CPC", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 541, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 141, + 541, + 506, + 553 + ], + "score": 1.0, + "content": "paper. This specified design choice makes SimCLR typically very sensitive to the batch size.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 138, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 138, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "SimCLR was not proposed to include an auto-regressive component, and can not", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 141, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "straightforwardly be extended to do so, while CPC can be implemented with or without such", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 578, + 184, + 590 + ], + "spans": [ + { + "bbox": [ + 141, + 578, + 184, + 590 + ], + "score": 1.0, + "content": "a module.", + "type": "text" + } + ], + "index": 25, + "is_list_end_line": true + } + ], + "index": 20, + "bbox_fs": [ + 132, + 447, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "For the most fair comparison, the procedure for drawing negative samples in SimCLR is done", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "score": 1.0, + "content": "equivalently as for SOM-CPC, i.e. within the recording, instead of within the batch. However,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "in the SOM-SimCLR model (i.e. the joint training of SOM with SimCLR), each drawn negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "sample is added to the set of negative samples both in its raw form, and with a random augmentation,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "which effectively doubles the number of negative samples. Table 3 reports the performance of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 307, + 667 + ], + "score": 1.0, + "content": "baseline SOM-SimCLR model (i.e. with settings", + "type": "text" + }, + { + "bbox": [ + 307, + 657, + 344, + 665 + ], + "score": 0.82, + "content": "\\tau = 0 . 0 7", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "and the cosine similarity), and variants", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "using a temperature value of 1 and/or the dot product as the similarity metric. All settings regarding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "training procedure and the SOM were set equivalently as in the SOM-CPC training. Table 5 shows", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 235, + 699 + ], + "score": 1.0, + "content": "that SOM-SimCLR results for", + "type": "text" + }, + { + "bbox": [ + 235, + 690, + 274, + 698 + ], + "score": 0.86, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 688, + 385, + 699 + ], + "score": 1.0, + "content": "are better than those with", + "type": "text" + }, + { + "bbox": [ + 385, + 690, + 412, + 698 + ], + "score": 0.84, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 688, + 505, + 699 + ], + "score": 1.0, + "content": ", which is in line with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 399, + 711 + ], + "score": 1.0, + "content": "findings from Chen et al. (2020); Woo et al. (2022). However, even with", + "type": "text" + }, + { + "bbox": [ + 399, + 699, + 437, + 709 + ], + "score": 0.84, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 698, + 506, + 711 + ], + "score": 1.0, + "content": ", performance of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "SOM-SimCLR is lower on all metrics compared to the SOM-CPC model with the same value for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 723, + 114, + 731 + ], + "score": 0.45, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 719, + 166, + 734 + ], + "score": 1.0, + "content": ". The higher", + "type": "text" + }, + { + "bbox": [ + 167, + 721, + 199, + 733 + ], + "score": 0.88, + "content": "\\ell _ { \\mathrm { { 2 , \\mathrm { { s m o o t h } } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "metric of SOM-SimCLR indicates on average larger jumps over the SOM", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "map through time, which might be caused by the fact that the SimCLR task objective does not", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "incorporate temporal information, while InfoNCE does exploit this. 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ModelCPC + linear classifierLsom sg[-]PurityNMICohen's kappa
CPC + K-means CPC(F= 2) + linear classifier- - =- =- - --.68±.10
-=.79 -.29.61±.11 .52±.10 .55±.090-- =
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CPC +PCA +K-means - 1e-3 Plus 1e-2 Plus-.77.26 .57±.082
SOM-VAE.71 .23.51±.042.36±.26.24±.031
√ √.71.23 .51±.042.67±.17 2.60±.34.30±.037 .28±.042
DESOM√ √.72 .71.23 .23.52±.03 .53±.033.08±.32 .31±.054
1e-6 Gaussian 1e-5 Gaussian 1e-4 Gaussian 1e-3 Gaussian 1e-2 GaussianX X.70 .70.27 .53±.05 .23 .50±.042.14±.32 2.10±.26.095±.020 .11±.028
X.71.22 .51±.042.35±.24 2.40±.16.17±.035
SOM-SimCLR(T = 0.07)X X.71 .71.22 .51±.05 .22 .50±.042.30±.26.22±.0085 .23±.021
1e-3 Gaussian SOM-SimCLR(T = 1) 1e-3 SOM-CPC (ours)X X.73 .70.23 .20.53±.13 .48±.162.21±.35 1.87±.30.29±.026 .50±.068
Gaussian 1e-5 Gaussian 1e-4 Gaussian * 1e-3 GaussianX X X.78 .27 .78 .27 .78.59±.11 .61±.101.03±.11 .041±.014 1.01±.10 .062±.018
1e-2 Gaussian .1 Gaussian 1e-3 Gaussian 1e-2 Gaussian GaussianX X √.27 .79 .28 .79 .28 .78 .27.61±.12 .60±.11 .65±.07 .62王.10 .60±.101.02±.09 .032±.0096 1.08±.11 .19±.042 1.09±.09 .19±.04 1.02±.12 .067±.025
.1 Vaaeais 1 Gaussian √ 1e-3 Plus X 1e-3 Plus √ SOM-CPC(T = 0.07,sim = cosine sim.) 1e-3 Gaussian X SOM-CPC (τ = 1,sim = cosine sim.) X.78 .78 .78 .79 .79 .78 .73 .79.27 .27 .27 .28 .28 .27 .27
1e-3 Gaussian SOM-CPC(τ = 0.07,sim = dot prod.) 1e-3 Gaussian CPC + SOM (disjoint) Gaussian =X .79 = 0.80 Mwv1.43±.15 .025±.0094 1.06±.11 .059±.020 1.21±.11 .52±.042 Cohen's kappa
InfoNCE 1.2 0.10 1.0 0.08Commitment lossSOM loss 5Purity bsl00.30NMI
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ModelCPC + linear classifierLsom sg[-]PurityNMICohen's kappa
CPC + K-means CPC(F= 2) + linear classifier- - =- =- - --.68±.10
-=.79 -.29.61±.11 .52±.10 .55±.090-- =
CPC(F= 2)+K-means CPC + PCA+ linear classifier *=.74.24.54±.086
CPC +PCA +K-means - 1e-3 Plus 1e-2 Plus-.77.26 .57±.082
SOM-VAE.71 .23.51±.042.36±.26.24±.031
√ √.71.23 .51±.042.67±.17 2.60±.34.30±.037 .28±.042
DESOM√ √.72 .71.23 .23.52±.03 .53±.033.08±.32 .31±.054
1e-6 Gaussian 1e-5 Gaussian 1e-4 Gaussian 1e-3 Gaussian 1e-2 GaussianX X.70 .70.27 .53±.05 .23 .50±.042.14±.32 2.10±.26.095±.020 .11±.028
X.71.22 .51±.042.35±.24 2.40±.16.17±.035
SOM-SimCLR(T = 0.07)X X.71 .71.22 .51±.05 .22 .50±.042.30±.26.22±.0085 .23±.021
1e-3 Gaussian SOM-SimCLR(T = 1) 1e-3 SOM-CPC (ours)X X.73 .70.23 .20.53±.13 .48±.162.21±.35 1.87±.30.29±.026 .50±.068
Gaussian 1e-5 Gaussian 1e-4 Gaussian * 1e-3 GaussianX X X.78 .27 .78 .27 .78.59±.11 .61±.101.03±.11 .041±.014 1.01±.10 .062±.018
1e-2 Gaussian .1 Gaussian 1e-3 Gaussian 1e-2 Gaussian GaussianX X √.27 .79 .28 .79 .28 .78 .27.61±.12 .60±.11 .65±.07 .62王.10 .60±.101.02±.09 .032±.0096 1.08±.11 .19±.042 1.09±.09 .19±.04 1.02±.12 .067±.025
.1 Vaaeais 1 Gaussian √ 1e-3 Plus X 1e-3 Plus √ SOM-CPC(T = 0.07,sim = cosine sim.) 1e-3 Gaussian X SOM-CPC (τ = 1,sim = cosine sim.) X.78 .78 .78 .79 .79 .78 .73 .79.27 .27 .27 .28 .28 .27 .27
1e-3 Gaussian SOM-CPC(τ = 0.07,sim = dot prod.) 1e-3 Gaussian CPC + SOM (disjoint) Gaussian =X .79 = 0.80 Mwv1.43±.15 .025±.0094 1.06±.11 .059±.020 1.21±.11 .52±.042 Cohen's kappa
InfoNCE 1.2 0.10 1.0 0.08Commitment lossSOM loss 5Purity bsl00.30NMI
0.840.75wAb众
0.063 20.70 0.650.25
0.60.0410.600.200.2
0.40.0200.550.150.0Train Validation
0 2000 200400200 4000.50 02004000.10 0 200
400 Epoch0 EpochEpochEpochEpoch400 0200 400 Epoch
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Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
Encoder forSOM-CPCand CPC
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512512ReLU4211
GRUbs × 512512--==-
EncoderforSOM-VAEandDESOM
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512 ×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512×2512ReLU411same
Flattenbs ×1024-===
GRUbs ×10241024
DecoderforSOM-VAEandDESOM
Unflattenbs×512×2==
Conv1Dbs × 512×2512ReLU411same
ConvTranspose1Dbs × 512×4512ReLU4211
ConvTranspose1Dbs × 512×8512ReLU4211
ConvTranspose1Dbs × 512× 32512ReLU8412
ConvTranspose1Dbs ×1×160512ReLU10513 (+ output pad = 1)
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Layer typeOutput sizeChannelsActivationKernel sizeStridesDilationPadding
Encoder forSOM-CPCand CPC
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512512ReLU4211
GRUbs × 512512--==-
EncoderforSOM-VAEandDESOM
Conv1Dbs×512×32512ReLU10513
Conv1Dbs × 512 ×8512ReLU8412
Conv1Dbs × 512×4512ReLU4211
Conv1Dbs × 512×2512ReLU4211
Conv1Dbs × 512×2512ReLU411same
Flattenbs ×1024-===
GRUbs ×10241024
DecoderforSOM-VAEandDESOM
Unflattenbs×512×2==
Conv1Dbs × 512×2512ReLU411same
ConvTranspose1Dbs × 512×4512ReLU4211
ConvTranspose1Dbs × 512×8512ReLU4211
ConvTranspose1Dbs × 512× 32512ReLU8412
ConvTranspose1Dbs ×1×160512ReLU10513 (+ output pad = 1)
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ModelαSLsom sg[·]PurityNMICohen'skappaTE
CPC+linearclassifier----1.00-
CPC + K-means==1.00.601.00
CPC(F=2)+ linear classifier==-.00=
CPC(F= 2)+K-means.13.013.025
CPC+PCA+ linear classifier=--.86
CPC + PCA + K-means-=-.89.54..88
DESOM1e-5GaussianX.18.03.06.14±.035
1e-4GaussianX.23.08.11.34±.045
1e-3GaussianX.31.13.20.68±.050
1e-2GaussianX.13-.00.001.0±.00
GRU-DESOM (reconstructing last window)1e-5GaussianX.19.04.08.13±.028
1e-4GaussianX.26.09.15.20±.036
1e-3GaussianX.31.13.21.42±.078
1e-2GaussianX.32.14.22.46±.057
GRU-DESOM (reconstructing full sequence)1e-5GaussianX.19.05.08.34±.048
1e-4GaussianX.30.12.20.59±.072
1e-3GaussianX.33.14.22.78±.048
1e-2GaussianX.29.12.19.57±.069
SOM-CPC(ours)1e-5GaussianX.99.73.99.14±.081
1e-4GaussianX1.00.631.00.24±.12
*1e-3GaussianX1.00.611.00.33±.098
1e-2GaussianX1.00.61.99.33±.099
1e-3Gaussian√1.00.61.99.28±.087
1e-2Gaussian√1.00.61.99.35±0.10
.1Gaussian√1.00.611.00.35±.095
1Gaussian√1.00.611.00.38±.11
SOM-CPC (T = 0.07,sim = cosine sim.)1e-3GaussianX.99.61.99.42±0.12
SOM-CPC(τ = 1,sim = cosine sim.)1e-3GaussianX.88.55.86.17±.063
SOM-CPC(τ = 0.07,sim = dot prod.)1e-3GaussianX1.00.61.99.38±.098
CPC + SOM(disjoint)-Gaussian-1.00.621.00.28±.11
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ModelαSLsom sg[·]PurityNMICohen'skappaTE
CPC+linearclassifier----1.00-
CPC + K-means==1.00.601.00
CPC(F=2)+ linear classifier==-.00=
CPC(F= 2)+K-means.13.013.025
CPC+PCA+ linear classifier=--.86
CPC + PCA + K-means-=-.89.54..88
DESOM1e-5GaussianX.18.03.06.14±.035
1e-4GaussianX.23.08.11.34±.045
1e-3GaussianX.31.13.20.68±.050
1e-2GaussianX.13-.00.001.0±.00
GRU-DESOM (reconstructing last window)1e-5GaussianX.19.04.08.13±.028
1e-4GaussianX.26.09.15.20±.036
1e-3GaussianX.31.13.21.42±.078
1e-2GaussianX.32.14.22.46±.057
GRU-DESOM (reconstructing full sequence)1e-5GaussianX.19.05.08.34±.048
1e-4GaussianX.30.12.20.59±.072
1e-3GaussianX.33.14.22.78±.048
1e-2GaussianX.29.12.19.57±.069
SOM-CPC(ours)1e-5GaussianX.99.73.99.14±.081
1e-4GaussianX1.00.631.00.24±.12
*1e-3GaussianX1.00.611.00.33±.098
1e-2GaussianX1.00.61.99.33±.099
1e-3Gaussian√1.00.61.99.28±.087
1e-2Gaussian√1.00.61.99.35±0.10
.1Gaussian√1.00.611.00.35±.095
1Gaussian√1.00.611.00.38±.11
SOM-CPC (T = 0.07,sim = cosine sim.)1e-3GaussianX.99.61.99.42±0.12
SOM-CPC(τ = 1,sim = cosine sim.)1e-3GaussianX.88.55.86.17±.063
SOM-CPC(τ = 0.07,sim = dot prod.)1e-3GaussianX1.00.61.99.38±.098
CPC + SOM(disjoint)-Gaussian-1.00.621.00.28±.11
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These sub-clusters were found to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 654, + 344, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 344, + 664 + ], + "score": 1.0, + "content": "relate to recordings that were made with different room acoustics.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 16.5 + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/dev/FKXVK9dyMM/FKXVK9dyMM.md b/parse/dev/FKXVK9dyMM/FKXVK9dyMM.md new file mode 100644 index 0000000000000000000000000000000000000000..d99044ce5e081f7e5757329b9e0cd3b1a2ef3254 --- /dev/null +++ b/parse/dev/FKXVK9dyMM/FKXVK9dyMM.md @@ -0,0 +1,378 @@ +# LIGHTGCL: SIMPLE YET EFFECTIVE GRAPH CON-TRASTIVE LEARNING FOR RECOMMENDATION + +Xuheng Cai Chao Huang∗ Lianghao Xia Xubin Ren Department of Computer Science, University of Hong Kong {rickcai, lhaoxia}@hku.hk chaohuang75gmail.com + +xubinrencs@gmail.com + +# ABSTRACT + +Graph neural network (GNN) is a powerful learning approach for graph-based recommender systems. Recently, GNNs integrated with contrastive learning have shown superior performance in recommendation with their data augmentation schemes, aiming at dealing with highly sparse data. Despite their success, most existing graph contrastive learning methods either perform stochastic augmentation (e.g., node/edge perturbation) on the user-item interaction graph, or rely on the heuristic-based augmentation techniques (e.g., user clustering) for generating contrastive views. We argue that these methods cannot well preserve the intrinsic semantic structures and are easily biased by the noise perturbation. In this paper, we propose a simple yet effective graph contrastive learning paradigm LightGCL that mitigates these issues impairing the generality and robustness of CL-based recommenders. Our model exclusively utilizes singular value decomposition for contrastive augmentation, which enables the unconstrained structural refinement with global collaborative relation modeling. Experiments conducted on several benchmark datasets demonstrate the significant improvement in performance of our model over the state-of-the-arts. Further analyses demonstrate the superiority of LightGCL’s robustness against data sparsity and popularity bias. The source code of our model is available at https://github.com/HKUDS/LightGCL. + +# 1 INTRODUCTION + +Graph neural networks (GNNs) have shown effectiveness in graph-based recommender systems by extracting local collaborative signals via neighborhood representation aggregation (Wang et al., 2019; Chen et al., 2020b). In general, to learn user and item representations, GNN-based recommenders perform embedding propagation on the user-item interaction graph by stacking multiple message passing layers for exploring high-order connectivity (He et al., 2020; Zhang et al., 2019; Liu et al., 2021a). Most GNN-based collaborative filtering models adhere to the supervised learning paradigm, requiring sufficient quality labelled data for model training. However, many practical recommendation scenarios struggle with the data sparsity issue in learning high-quality user and item representations from limited interaction data (Liu et al., 2021b; Lin et al., 2021). To address the label scarcity issue, the benefits of contrastive learning have been brought into the recommendation for data augmentation (Wu et al., 2021). The main idea of contrastive learning in enhancing the user and item representation is to research the agreement between the generated embedding views by contrasting the defined positive pairs with negative instance counterparts (Xie et al., 2022). + +While contrastive learning has been shown to be effective in improving the performance of graphbased recommendation methods, the view generators serve as the core part of data augmentation through identifying accurate contrasting samples. Most of current graph contrastive learning (GCL) approaches employ heuristic-based contrastive view generators to maximize the mutual information between the input positive pairs and push apart negative instances(Wu et al., 2021; Yu et al., 2022a; Xia et al., 2022b). To construct perturbed views, SGL (Wu et al., 2021) has been proposed to generate node pairs of positive view by corrupting the structural information of user-item interaction graph using stochastic augmentation strategies, e.g., node dropping and edge perturbation. To improve the graph contrastive learning in recommendation, SimGCL (Yu et al., 2022a) offers embedding augmentation with random noise perturbation. To work on identifying semantic neighbors of nodes (users and items), HCCF (Xia et al., 2022b) and NCL (Lin et al., 2022) are introduced to pursue consistent representations between the structurally adjacent nodes and semantic neighbors. Despite their effectiveness, state-of-the-art contrastive recommender systems suffer from several inherent limitations: i) Graph augmentation with random perturbation may lose useful structural information, which misleads the representation learning. ii) The success of heuristic-guided representation contrasting schemes is largely built upon the view generator, which limits the model generality and is vulnerable to the noisy user behaviors. iii) Most of current GNN-based contrastive recommenders are limited by the over-smoothing issue which leads to indistinguishable representations. + +In light of the above limitations and challenges, we revisit the graph contrastive learning paradigm for recommendation with a proposed simple yet effective augmentation method LightGCL. In our model, the graph augmentation is guided by singular value decomposition (SVD) to not only distill the useful information of user-item interactions but also inject the global collaborative context into the representation alignment of contrastive learning. Instead of generating two handcrafted augmented views, important semantic of user-item interactions can be well preserved with our robust graph contrastive learning paradigm. This enables our self-augmented representations to be reflective of both user-specific preferences and cross-user global dependencies. + +Our contributions are highlighted as follows: + +• In this paper, we enhance the recommender systems by designing a lightweight and robust graph contrastive learning framework to address the identified key challenges pertaining to this task. • We propose an effective and efficient contrastive learning paradigm LightGCL for graph augmentation. With the injection of global collaborative relations, our model can mitigate the issues brought by inaccurate contrastive signals. • Our method exhibits improved training efficiency compared to existing GCL-based approaches. • Extensive experiments on several real-world datasets justify the performance superiority of our LightGCL. In-depth analyzes demonstrate the rationality and robustness of LightGCL. + +# 2 RELATED WORK + +Graph Contrastive Learning for Recommendation. A promising line of recent studies has incorporated contrastive learning (CL) into graph-based recommenders, to address the label sparsity issue with self-supervision signals. Particularly, SGL (Wu et al., 2021) and SimGCL (Yu et al., 2022a) perform data augmentation over graph structure and embeddings with random dropout operations. However, such stochastic augmentation may drop important information, which may make the sparsity issue of inactive users even worse. Furthermore, some recent alternative CL-based recommenders, such as HCCF (Xia et al., 2022b) and NCL (Lin et al., 2022), design heuristic-based strategies to construct view for embedding contrasting. Despite their effectiveness, their success heavily relies on their incorporated heuristics (e.g., the number of hyperedges or user clusters) for contrastive view generation, which can hardly be adaptive to different recommendation tasks. + +Self-Supervised Learning on Graphs. Recently, self-supervised learning (SSL) has advanced the graph learning paradigm by enhancing node representation from unlabeled graph data (Zhu et al., 2021a;b; Velickovic et al., 2019; Hassani & Khasahmadi, 2020; Peng et al., 2020; Zhu et al., 2020; Wu et al., 2022). For example, to improve the predictive SSL paradigm, AutoSSL (Jin et al., 2022) automatically combines multiple pretext tasks for augmentation. Towards the line of contrastive SSL over graph structures, recent efforts focus on designing various graph contrastive learning methods (Yu et al., 2022b; Yin et al., 2022; Zhang et al., 2022; Xia et al., 2022a; Suresh et al., 2021). For instance, SimGRACE Xia et al. (2022a) proposes to generate contrastive views with the GNN encoder perturbations. In AutoGCL Yin et al. (2022), graph view generators are designed to be jointly trained with the graph encoder in an end-to-end way. Additionally, GCA (Zhu et al., 2021b) performs both topology-level and attribute-level data augmentation for contrastive view generation. In this method, important edges and features will be identified for adaptive augmentation. GraphCL (You et al., 2020) generates correlated graph representation views using various augmentation strategies, such as node/edge perturbation and attribute masking. + +![](images/0feaf6d22694bb9f83b4e1a65bef3f8fa8b0bec94a5309d7d6e04c5aec214e73.jpg) +Figure 1: Overall structure of LightGCL. + +# 3 METHODOLOGY + +In this section, we describe our proposed LightGCL framework in detail. LightGCL is a lightweight graph contrastive learning paradigm as illustrated in Fig. 1. Complementary to the GCN backbone (the upper half of the figure) extracting the local graph dependency, the SVD-guided augmentation (the lower half of the figure) empowers the graph contrastive learning with global collaborative relation analysis for learning effective user and item representations. + +# 3.1 LOCAL GRAPH DEPENDENCY MODELING + +As a common practice of collaborative filtering, we assign each user $u _ { i }$ and item $v _ { j }$ with an embedding vector $e _ { i } ^ { ( u ) } , e _ { j } ^ { ( v ) } \in \mathbb { R } ^ { d }$ , where $d$ is the embedding size. The collections of all user and item embeddings are defined as $\pmb { { E } } ^ { ( u ) } \in \mathbb { R } ^ { I \times d }$ and $\pmb { { \cal E } } ^ { ( v ) } \in \mathbb { R } ^ { J \times d }$ , where $I$ and $J$ are the number of users and items, respectively. Following Xia et al. (2022b), we adopt a two-layer GCN to aggregate the neighboring information for each node. In layer $l$ , the aggregation process is expressed as follows: + +$$ +\begin{array} { r } { \boldsymbol { z } _ { i , l } ^ { ( u ) } = \sigma ( p ( \tilde { \boldsymbol { A } } _ { i , : } ) \cdot \boldsymbol { E } _ { l - 1 } ^ { ( v ) } ) , \quad \boldsymbol { z } _ { j , l } ^ { ( v ) } = \sigma ( p ( \tilde { \boldsymbol { A } } _ { : , j } ) \cdot \boldsymbol { E } _ { l - 1 } ^ { ( u ) } ) } \end{array} +$$ + +where z(u)i,l and z(v)j,l denote the $l$ -th layer aggregated embedding for user $u _ { i }$ and item $v _ { j }$ . $\sigma ( \cdot )$ represents the LeakyReLU with a negative slope of 0.5. $\tilde { \boldsymbol { \mathcal { A } } }$ is the normalized adjacency matrix, on which we perform the edge dropout denoted as $p ( \cdot )$ , to mitigate the overfitting issue. We implement the residual connections in each layer to retain the original information of the nodes as follows: + +$$ +\pmb { e } _ { i , l } ^ { ( u ) } = \pmb { z } _ { i , l } ^ { ( u ) } + \pmb { e } _ { i , l - 1 } ^ { ( u ) } , \quad \pmb { e } _ { j , l } ^ { ( v ) } = \pmb { z } _ { j , l } ^ { ( v ) } + \pmb { e } _ { j , l - 1 } ^ { ( v ) } +$$ + +The final embedding for a node is the sum of its embeddings across all layers, and the inner product between the final embedding of a user $u _ { i }$ and an item $v _ { j }$ predicts $u _ { i }$ ’s preference towards $v _ { j }$ : + +$$ +\pmb { e } _ { i } ^ { ( u ) } = \sum _ { l = 0 } ^ { L } \pmb { e } _ { i , l } ^ { ( u ) } , \quad \pmb { e } _ { j } ^ { ( v ) } = \sum _ { l = 0 } ^ { L } \pmb { e } _ { j , l } ^ { ( v ) } , \quad \hat { y } _ { i , j } = e _ { i } ^ { ( u ) \top } \pmb { e } _ { j } ^ { ( v ) } +$$ + +# 3.2 EFFICIENT GLOBAL COLLABORATIVE RELATION LEARNING + +To empower graph contrastive learning for recommendation with global structure learning, we equip our LightGCL with the SVD scheme (Rajwade et al., 2012; Rangarajan, 2001) to efficiently distill important collaborative signals from the global perspective. Specifically, we first perform SVD on the adjacency matrix $\mathcal { A }$ as $\mathbf { \mathcal { A } } = U S V ^ { \top }$ . Here, $U / V$ is an $I \times I / J \times J$ orthonormal matrix with columns being the eigenvectors of $\mathcal { A }$ ’s row-row $/$ column-column correlation matrix. $_ { s }$ is an $I \times J$ diagonal matrix storing the singular values of $\mathcal { A }$ . The largest singular values are usually associated with the principal components of the matrix. Thus, we truncate the list of singular values to keep the largest q values, and reconstruct the adjacency matrix with the truncated matrices as $\hat { \ b { A } } = \ b { U } _ { q } \ b { S } _ { q } \ b { V } _ { q } ^ { \top }$ , where $U _ { q } \in \mathbb { R } ^ { I \times q }$ and $V _ { q } \in \mathbb { R } ^ { J \times q }$ contain the first $q$ columns of $U$ and $V$ respectively. $S _ { q } \in \mathbb { R } ^ { q \times q }$ is the diagonal matrix of the $q$ largest singular values. + +The reconstructed matrix $\hat { A }$ is a low-rank approximation of the adjacency matrix $\mathcal { A }$ , for it holds that $r a n k ( { \hat { A } } ) = q$ . The advantages of SVD-based graph structure learning are two-folds. Firstly, it emphasizes the principal components of the graph by identifying the user-item interactions that are important and reliable to user preference representations. Secondly, the generated new graph structures preserve the global collaborative signals by considering each user-item pair. Given the $\hat { A }$ , we perform message propagation on the reconstructed user-item relation graph in each layer: + +$$ +\pmb { g } _ { i , l } ^ { ( u ) } = \sigma ( \hat { \mathcal { A } } _ { i , : } \cdot \pmb { E } _ { l - 1 } ^ { ( v ) } ) , \quad \pmb { g } _ { j , l } ^ { ( v ) } = \sigma ( \hat { \mathcal { A } } _ { : , j } \cdot \pmb { E } _ { l - 1 } ^ { ( u ) } ) +$$ + +However, performing the exact SVD on large matrices is highly expensive, making it impractical for handling large-scale user-item matrix. Therefore, we adopt the randomized SVD algorithm proposed by Halko et al. (2011), whose key idea is to first approximate the range of the input matrix with a low-rank orthonormal matrix, and then perform SVD on this smaller matrix. + +$$ +\hat { U } _ { q } , \hat { S } _ { q } , \hat { V } _ { q } ^ { \top } = \mathrm { A p p r o x } { \mathrm { S V D } } ( { \cal A } , q ) , \quad \hat { A } _ { S V D } = \hat { U } _ { q } \hat { S } _ { q } \hat { V } _ { q } ^ { \top } +$$ + +where $q$ is the required rank for the decomposed matrices, and $\hat { { \cal U } } _ { q } \in \mathbb { R } ^ { I \times q } , \hat { { \cal S } } _ { q } \in \mathbb { R } ^ { q \times q } , \hat { { \cal V } } _ { q } \in \mathbb { R } ^ { J \times q }$ are the approximated versions of $U _ { q }$ , $S _ { q }$ , $V _ { q }$ . Thus, we rewrite the message propagation rules in Eq. 4 with the approximated matrices and the collective representations of the embeddings as follows: + +$$ +\pmb { G } _ { l } ^ { ( u ) } = \sigma ( \hat { A } _ { S V D } \pmb { E } _ { l - 1 } ^ { ( v ) } ) = \sigma ( \hat { U } _ { q } \hat { S } _ { q } \hat { V } _ { q } ^ { \top } \pmb { E } _ { l - 1 } ^ { ( v ) } ) ; \quad \pmb { G } _ { l } ^ { ( v ) } = \sigma ( \hat { A } _ { S V D } ^ { \top } \pmb { E } _ { l - 1 } ^ { ( u ) } ) = \sigma ( \hat { V } _ { q } \hat { S } _ { q } \hat { U } _ { q } ^ { \top } \pmb { E } _ { l - 1 } ^ { ( u ) } ) +$$ + +where $G _ { l } ^ { ( u ) }$ and $G _ { l } ^ { ( v ) }$ are the collections of user and item embeddings encoded from the new generated graph structure view. Note that we do not need to compute and store the large dense matrix $\hat { \boldsymbol { \mathcal { A } } } _ { S V D }$ . Instead, we can store $\hat { U } _ { q } , \hat { S } _ { q }$ and $\hat { V } _ { q }$ , which are of low dimensions. By pre-calculating $( \hat { U } _ { q } \hat { S } _ { q } )$ and $( \hat { V } _ { q } \hat { S } _ { q } )$ during the preprocessing stage with SVD, the model efficiency is improved. + +# 3.3 SIMPLIFIED LOCAL-GLOBAL CONTRASTIVE LEARNING + +The conventional GCL methods such as SGL and SimGCL contrast node embeddings by constructing two extra views, while the embeddings generated from the original graph (the main-view) are not directly involved in the InfoNCE loss. The reason for adopting such a cumbersome three-view paradigm may be that the random perturbation used to augment the graph may provide misleading signals to the main-view embeddings. In our proposed method, however, the augmented graph view is created with global collaborative relations, which can enhance the main-view representations. Therefore, we simplify the CL framework by directly contrasting the SVD-augmented view embeddings g(u)i,l with the main-view embeddings $\boldsymbol { z } _ { i , l } ^ { ( u ) }$ in the InfoNCE loss (Oord et al., 2018): + +$$ +\mathcal { L } _ { s } ^ { ( u ) } = \sum _ { i = 0 } ^ { I } \sum _ { l = 0 } ^ { L } - \log \frac { \exp ( s ( z _ { i , l } ^ { ( u ) } , \pmb { g } _ { i , l } ^ { ( u ) } / \tau ) ) } { \sum _ { i ^ { \prime } = 0 } ^ { I } \exp ( s ( z _ { i , l } ^ { ( u ) } , \pmb { g } _ { i ^ { \prime } , l } ^ { ( u ) } ) / \tau ) } +$$ + +where $s ( \cdot )$ and $\tau$ stand for the cosine similarity and the temperature respectively. The InfoNCE loss $\mathcal { L } _ { s } ^ { ( v ) }$ for the items are defined in the same way. To prevent overfitting, we implement a random node dropout in each batch to exclude some nodes from participating in the contrastive learning. As shown in Eq. 8, the contrastive loss is jointly optimized with our main objective function for the recommendation task (where $\hat { y } _ { i , p _ { s } }$ and $\hat { y } _ { i , n _ { s } }$ denote the predicted scores for a pair of positive and negative items of user $\romannumeral 1$ ): + +$$ +\mathcal { L } = \mathcal { L } _ { r } + \lambda _ { 1 } \cdot ( \mathcal { L } _ { s } ^ { ( u ) } + \mathcal { L } _ { s } ^ { ( v ) } ) + \lambda _ { 2 } \cdot \Vert \Theta \Vert _ { 2 } ^ { 2 } ; \quad \mathcal { L } _ { r } = \sum _ { i = 0 } ^ { I } \sum _ { s = 1 } ^ { S } \operatorname* { m a x } ( 0 , 1 - \hat { y } _ { i , p _ { s } } + \hat { y } _ { i , n _ { s } } ) +$$ + +# 4 EVALUATION + +To verify the superiority and effectiveness of the proposed LightGCL method, we perform extensive experiments to answer the following research questions: + +• RQ1: How does LightGCL perform on different datasets compared to various SOTA baselines? • RQ2: How does the lightweight graph contrastive learning improve the model efficiency? • RQ3: How does our model perform against data sparsity, popularity bias and over-smoothing? • RQ4: How does the local-global contrastive learning contribute to the performance of our model? • RQ5: How do different parameter settings affect our model performance? + +# 4.1 EXPERIMENTAL SETTINGS + +# 4.1.1 DATASETS AND EVALUATION PROTOCOLS + +We evaluate our model and the baselines on five real-world datasets: Yelp (29,601 users, 24,734 items, 1,517,326 interactions): a dataset collected from the rating interactions on Yelp platform; Gowalla (50,821 users, 57,440 items, 1,172,425 interactions): a dataset containing users’ check-in records collected from Gowalla platform; ML-10M (69,878 users, 10,195 items, 9,988,816 interactions): a well-known movie-rating dataset for collaborative filtering; Amazon-book (78,578 users, 77,801 items, 2,240,156 interactions): a dataset composed of users’ ratings on books collected from Amazon; and Tmall (47,939 users, 41,390 items, 2,357,450 interactions): a E-commerce dataset containing users’ purchase records on different products in Tmall platform. + +In accordance with He et al. (2020) and Wu et al. (2021), we split the datasets into training, validation and testing sets with a ratio of 7:2:1. We adopt the Recall $@ \mathbf { N }$ and Normalized Discounted Cumulative Gain $( \mathrm { N D C G } ) @ \mathrm { N }$ , where $\Nu = \{ 2 0 , 4 0 \}$ , as the evaluation metrics. + +# 4.1.2 BASELINE METHODS + +We compare our model against 16 state-of-the-art baselines with different learning paradigms: + +• MLP-enhanced Collaborative Filtering: NCF (He et al., 2017). +• GNN-based Collaborative Filtering: GCCF (Chen et al., 2020c), LightGCN (He et al., 2020). +• Disentangled Graph Collaborative Filtering: DGCF (Wang et al., 2020b). +• Hypergraph-based Collaborative Filtering: HyRec (Wang et al., 2020a). +• Self-Supervised Learning Recommender Systems: GraphCL (You et al., 2020), GRACE (Zhu et al., 2020), GCA (Zhu et al., 2021b), MHCN (Yu et al., 2021), SAIL (Yu et al., 2022b), AutoGCL (Yin et al., 2022), SimGRACE (Xia et al., 2022a), SGL (Wu et al., 2021), HCCF (Xia et al., 2022b), SHT (Xia et al., 2022c), SimGCL (Yu et al., 2022a). + +Due to space limit, the detailed descriptions of baselines are presented in Appendix A. + +# 4.1.3 HYPERPARAMETER SETTINGS + +To ensure a fair comparison, we tune the hyperparameters of all the baselines within the ranges suggested in the original papers, except the following fixed settings for all the models: the embedding size is set as 32; the batch size is 256; two convolutional layers are used for GCN models. + +For our LightGCL, the regularization weights $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are tuned from $\{ 1 \mathrm { e } { - } 5 , 1 \mathrm { e } { - } 6 , 1 \mathrm { e } { - } 7 \}$ and {1e4, 1e- $\{ 5 \}$ , respectively. The temperature $\tau$ is searched from $\{ 0 . 3 , 0 . 5 , 1 , \dot { 3 } , 1 0 \}$ . The dropout rate is chosen from $\{ 0 , 0 . 2 5 \}$ . The rank (i.e., $\grave { q } ,$ ) for SVD, is set as 5. We use the Adam optimizer with a learning rate of 0.001 decaying at the rate of 0.98 until the rate reaches 0.0005.\* + +# 4.2 PERFORMANCE VALIDATION (RQ1) + +We summarize the experimental result in Table $1 ^ { \dagger }$ , with the following observations and conclusions: + +Table 1: Performance comparison with baselines on five datasets. + +
DataMetricDGCFHyRecLightGCNMHCNSGLSimGRACEGCAHCCFSHTSimGCLLightGCLp-val.impr.
oR@200.04660.04720.04820.05030.05260.06030.06210.06260.06510.07180.07937e-910%
N@200.03950.03950.04090.04240.04440.04350.05300.05270.05460.06150.06688e-98%
R@400.07740.07910.08030.08260.08690.09890.10210.10400.10910.11660.12922e-910%
N@400.05110.05220.05270.05440.05710.06560.06770.06810.07090.07780.08522e-99%
GoeaalR@200.09440.09010.09850.09550.10300.08690.08960.10700.12320.13570.15781e-616%
N@200.05220.04980.05930.05740.06230.05280.05370.06440.07310.08180.09352e-614%
R@400.14010.13560.14310.13930.15000.12760.13220.15350.18040.19560.22453e-614%
N@400.06710.06600.07100.06890.07460.06370.06510.07670.08810.09750.11083e-613%
WOI-TNR@200.17630.18010.17890.14970.18330.22540.21450.22190.21730.22650.26131e-915%
N@200.21010.21780.21280.18140.22050.26860.26130.26290.25730.26130.31063e-918%
R@400.26810.26850.26500.22500.27680.32950.32310.32650.32110.33450.37997e-1013%
N@400.23400.23400.23220.19620.24260.29390.28710.28800.33180.28800.33871e-917%
VAzaaoR@200.02110.03020.03190.02960.03270.03810.03090.03220.04410.04740.05852e-723%
N@200.01540.02250.02360.02190.02490.02910.02380.02470.03280.03600.04362e-621%
R@400.03510.04320.04990.04890.05310.06210.04980.05250.07190.07500.09331e-724%
N@400.02010.02460.02900.02840.03120.03710.03010.03140.04200.04510.05519e-722%
[igR@200.02350.02330.02250.02030.02680.02220.03730.03140.03870.04730.05283e-511%
N@200.01630.01600.01540.01390.01830.01520.02520.02130.02620.03280.03611e-410%
R@400.03940.03500.03780.03400.04460.03670.06160.05190.06450.07660.08521e-511%
N@400.02180.01990.02080.01880.02460.02030.03370.02840.03520.04290.04737e-510%
+ +• Contrastive Learning Dominates. As can be seen from the table, recent methods implementing contrastive learning (SGL, HCCF, SimGCL) exhibit consistent superiority as compared to traditional graph-based (GCCF, LightGCN) or hypergraph-based (HyRec) models. They also perform better than some of other self-supervised learning approaches (MHCN). This could be attributed to the effectiveness of CL to learn evenly distributed embeddings (Yu et al., 2022a). + +• Contrastive Learning Enhancement. Our method consistently outperforms all the contrastive learning baselines. We attribute such performance improvement to the effective augmentation of graph contrastive learning via injecting global collaborative contextual signals. Other compared contrastive learning-based recommenders (e.g., SGL, SimGCL, and HCCF) are easily biased by noisy interaction information and generate misleading self-supervised signals. + +# 4.3 EFFICIENCY STUDY (RQ2) + +GCL models often suffer from a high computational cost due to the construction of extra views and the convolution operations performed on them during training. However, the low-rank nature of the SVD-reconstructed graph and the simplified CL structure enable the training of our LightGCL to be highly efficient. We analyze the pre-processing and per-batch training complexity of our model in comparison to three competitive baselines, as summarized in Table 2.‡ + +Table 2: Comparisons of computational complexity against baselines. + +
StageComputationLightGCNSGLSimGCLLightGCL
Pre-processingNormalization SVDO(E)O(E)O(E)O(E) O(qE)
TrainingAugmentation Graph Convolution BPRLoss InfoNCE LossO(2ELd) O(2Bd) 1O(2pE) O(2ELd+4pELd) O(2Bd) O(Bd+BMd)O(6ELd) O(2Bd) O(Bd+BMd)O[2ELd+ 2q(I+ J)Ld] O(2Bd) O[(Bd+BMd)L]
+ +• Although our model requires performing the SVD in the pre-processing stage which takes $O ( q E )$ , the computational cost is negligible compared to the training stage since it only needs to be performed once. In fact, by moving the construction of contrastive view to the pre-processing stage, we avoid the repetitive graph augmentation during training, which improves model efficiency. + +• Traditional GCN methods (e.g., LightGCN) only perform convolution on one graph, inducing a complexity of $O ( 2 E L d )$ per batch. For most GCL-based methods, three contrastive views are computed per batch, leading to a complexity of roughly three times of LightGCN. In our model, instead, only two contrastive views are involved. Additionally, due to the low-rank property of SVD-based graph structure learning, our graph encoder takes only $O [ 2 q ( I + J ) L d ]$ time. For most datasets, including the five we use, $\bar { 2 q } ( \bar { I } + J ) < E$ . Therefore, the training complexity of our model is less than half of that of the SOTA efficient model SimGCL. + +# 4.4 RESISTANCE AGAINST DATA SPARSITY AND POPULARITY BIAS (RQ3) + +To evaluate the robustness of our model in alleviating data sparsity, we group the sparse users by their interaction degrees and calculate the Recall $@ 2 0$ of each group on $Y e l p$ and Gowalla datasets. As can be seen from the figures, the performance of HCCF and SimGCL varies across datasets, but our LightGCL consistently outperforms them in all cases. In particular, our model performs notably well on the extremely sparse user group $< 1 5$ interactions), as the Recall $@ 2 0$ of these users is not much lower (and is even higher on Gowalla) than that of the whole dataset. + +![](images/623a815fd9843e8009e8316113509191d0a46ab26971881711679e6dbbf83323.jpg) +Figure 2: Performance on users of different sparsity degrees, in terms of Recall (histograms) and relative Recall w.r.t overall performances (charts). + +![](images/df5b9e7ff55f07f0a4528b1c2879a17d632a723aaf9dab688e698d6d4aa45616.jpg) +Figure 3: LightGCL’s ability to alleviate popularity bias in comparison to SOTA CLbased methods HCCF and SimGCL. + +Additionally, we illustrate our model’s ability to mitigate popularity bias compared to HCCF and SimGCL. Similar to Section 4.4, we group the long-tail items by their degree of interactions. Following Wu et al. (2021), we adopt the decomposed Recall@20 defined as Recall(g) = |(Vurec)(g)∩Vutest||Vu | where $\mathbb { V } _ { t e s t } ^ { u }$ refers to the set of test items for the user $u$ , and $( \mathbb { V } _ { r e c } ^ { u } ) ^ { ( g ) }$ is the set of Top-K recommended items for $u$ that belong to group $g$ . The results are shown in Fig. 3. Similar to the results on sparse users, HCCF and SimGCL’s performance fluctuates a lot with the influence of popularity bias. Our model performs better in most cases, which shows its resistance against popularity bias. Note that since the extremely sparse group ( $< 1 5$ interactions) is significantly larger than the other groups in Gowalla, they contribute to a large fraction of the Recall $@ 2 0$ , resulting in a different trend from that of Yelp in the figure. + +# 4.5 BALANCING BETWEEN OVER-SMOOTHING AND OVER-UNIFORMITY (RQ3) + +In this section, we illustrate the effectiveness of our model in learning a moderately dispersed embedding distribution, by preserving user unique preference pattern and inter-user collaborative dependencies. We randomly sample 2,000 nodes from Yelp and Gowalla and map their embeddings to the 2-D space with t-SNE (Van der Maaten & Hinton, 2008). The visualizations of these embeddings are presented in Fig. 4. We also calculate the Mean Average Distance (MAD) (Chen et al., 2020a) of the embeddings, summarized in Table 3. + +Table 3: Mean Average Distance (MAD) of the embeddings learned by different methods. + +
DatasetMHCNLightGCNLightGCLSGLSimGCL
Yelp0.88060.94690.96570.99620.9956
Gowalla0.92470.95680.97210.98590.9897
+ +![](images/d6c1337ddbfe6f8ae387e691750f79a7851bc9a61889505c08a8bf2562dd6e5b.jpg) +Figure 4: Embedding distributions on Yelp and Gowalla visualized with t-SNE. + +As can be seen from Fig. 4, the embedding distributions of non-CL methods (i.e., LightGCN, MHCN) exhibit indistinguishable clusters in the embedding space, which indicates the limitation of addressing the over-smoothing issue. On the contrary, the existing CL-based methods tend to learn i) over-uniform distributions, e.g., SGL on $Y e l p$ learns a huge cloud of evenly-distanced embeddings with no clear community structure to well capture the collaborative relations between users; ii) highly dispersed small clusters with severe over-smoothing issue inside the clusters, e.g., the embeddings of SimGCL on Gowalla appear to be scattered grained clusters inside which embeddings are highly similar. Compared with them, clear community structures could be identified by our method to capture collaborative effects, while the embeddings inside each community are reasonably dispersed to be reflective of user-specific preference. The MAD of our model’s learned features is also in between of the two types of baselines as shown in Table 3. + +# 4.6 ABLATION STUDY (RQ4) + +To investigate the effectiveness of our SVD-based graph augmentation scheme, we perform the ablation study to answer the question of whether we could provide guidance to the contrastive learning with a different approach of matrix decomposition. To this end, we implement two variants of our model, replacing the approximated SVD algorithm with other matrix decomposition methods: $C L .$ - $M F$ adopts the view generated by a pre-trained MF (Koren et al., 2009); $C L { \cdot } S V D { + } +$ utilizes the $\mathrm { S V D + + }$ (Koren, 2008) which takes implicit user feedback into consideration. As shown in Table 4, with the information distilled from MF or $\mathrm { S V D + + }$ , the model is able to achieve satisfactory results, indicating the effectiveness of using matrix decomposition to empower CL and the flexibility of our proposed framework. However, adopting a pre-trained CL component is not only tedious and timeconsuming but also inferior to utilizing the approximate SVD algorithm in terms of performance. + +Table 4: Ablation study on LightGCL. + +
VariantYelpGowalla
Recall@20NDCG@20Recall@20NDCG@20
CL-MF0.07810.06590.15610.0929
CL-SVD++0.07880.06660.15680.0932
LightGCL0.07930.06680.15780.0935
+ +![](images/c7d33ed9b133947ba2331f5482f7173be513676421a01b0aa11b0d82d20ada44.jpg) +Figure 5: Recall change w.r.t. $q$ + +# 4.7 HYPERPARAMETER ANALYSIS (RQ5) + +In this section, we investigate our model’s sensitivity in relation to several key hyperparameters: the regularization weight for InfoNCE loss $\lambda _ { 1 }$ , the temperature $\tau$ , and the required rank of SVD $q$ . + +• The impact of $\lambda _ { 1 }$ . As illustrated in Fig. 6, for the three datasets Yelp, Gowalla and ML-10M, the model’s performance reaches the peak when $\lambda _ { 1 } = 1 0 ^ { - 7 }$ . It can be noticed that $\lambda _ { 1 }$ with the range of $[ 1 0 ^ { - 6 } , 1 0 ^ { - 8 } ]$ can often lead to performance improvement. + +![](images/7ebdb10fbddab17d11c287f95d8996357b8aceba9a290f32cbc12ce43d03d72c.jpg) +Figure 6: Impact of $\lambda _ { 1 }$ . + +![](images/9fb35ccdc532666d6f84c206ac95d489d159d8d812f187c41a03b6b1c6914a77.jpg) +Figure 7: Impact of $\tau$ + +• The impact of $\tau$ . Fig. 7 indicates that the model’s performance is relatively stable across different selections of $\tau$ from 0.1 to 10, while the best configuration of $\tau$ value varies by datasets. + +• The selection of $q$ . $q$ determines the rank of SVD in our model. Experiments have shown that satisfactory results can be achieved with a small $q$ . Specifically, as in Fig. 5, we observe that $q = 5$ is sufficient to preserve important structures of the user-item interaction graph. + +# 4.8 CASE STUDY (RQ4) + +In this section, we present a case study to intuitively show the effectiveness of our model to identify useful knowledge from noisy user-item interactions and make accurate recommendations accordingly. In Fig. 8, we can see that the venues visited by user $\# 2 6$ in Yelp mainly fall into two communities: Cleveland (where the user probably lives) and Arizona (where the user may have travelled to). In the reconstructed graph, these venues are assigned a new weight according to their potential importance. Note that item $\# 2 5 8 3$ , a car rental agency in Arizona, has been assigned a negative weight, which conforms to our common sense that people generally would not visit multiple car rental agencies in one trip. The SVD-augmented view also provides predictions on invisible links by assigning a large weight§ to potential venues of interest, such as #2647 and #658. Note that when exploiting the graph, the augmented view does not overlook the smaller Arizona community, which enables the model to predict items of minor interests that are usually overshadowed by the majority. + +![](images/0503ae1e241b24805cdeae8bafec2ea5b3c686916a0f5795d1e74dec13771edf.jpg) +Figure 8: Case study on user $\# 2 6$ in Yelp dataset. + +# 5 CONCLUSION + +In this paper, we propose a simple and effective augmentation method to the graph contrastive learning framework for recommendation. Specifically, we explore the key idea of making the singular value decomposition powerful enough to augment user-item interaction graph structures. Our key findings indicate that our graph augmentation scheme exhibits strong ability in resisting data sparsity and popularity bias. Extensive experiments show that our model achieves new state-of-the-art results on several public evaluation datasets. In future work, we plan to explore the potential of incorporating casual analysis into our lightweight graph contrastive learning model to enhance the recommender system with mitigating confounding effects for data augmentation. + +# REFERENCES + +Deli Chen, Yankai Lin, Wei Li, Peng Li, Jie Zhou, and Xu Sun. Measuring and relieving the oversmoothing problem for graph neural networks from the topological view. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 3438–3445, 2020a. + +Lei Chen, Le Wu, Richang Hong, Kun Zhang, and Meng Wang. 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International Joint Conference on Artificial Intelligence (IJCAI), 2022. + +Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive representation learning. arXiv preprint arXiv:2006.04131, 2020. + +Yanqiao Zhu, Yichen Xu, Qiang Liu, and Shu Wu. An empirical study of graph contrastive learning. arXiv preprint arXiv:2109.01116, 2021a. + +Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Graph contrastive learning with adaptive augmentation. In The Web Conference (WWW), pp. 2069–2080, 2021b. + +# A DETAILS OF THE BASELINES + +MLP-enhanced Collaborative Filtering: + +• NCF (He et al., 2017) is a collaborative filtering model that leverages neural network to exploit non-linearity. Two hidden layers are used in our evaluation. + +GNN-based Collaborative Filtering: + +• GCCF (Chen et al., 2020c) strengthens the GNN-based collaborative filtering by implementing a residual network and reducing the non-linear transformation. +• LightGCN (He et al., 2020) adopts a simplified GCN structure without embedding weight matrices and non-linear projection. + +Disentangled Graph Collaborative Filtering: + +• DGCF (Wang et al., 2020b) learns a more sophisticated representation by segmenting the embedding vectors to represent multiple latent intentions. + +Hypergraph-based Collaborative Filtering: + +• HyRec (Wang et al., 2020a) makes use of hypergraph to encode multi-order information between users and items. + +Self-Supervised Learning Recommender Systems: + +• GraphCL (You et al., 2020) utilizes random node dropping and edge masking to generate two contrastive views, which were aligned by optimizing the SSL loss function. + +• GRACE (Zhu et al., 2020) proposes to corrupt the graph structure by both random edge dropout and random node feature dropping, and uses the corrupted graphs as the contrastive views. + +• GCA (Zhu et al., 2021b) adaptively dropout the nodes and edges by their importance calculated with node centrality. + +• MHCN (Yu et al., 2021) creates self-supervised signals for the graph representation learning by graph infomax network. + +• SAIL (Yu et al., 2022b) maximizes the neighborhood predicting probability between GNNgenerated high-level features and input node features. + +• AutoGCL (Yin et al., 2022) uses GNN to learn to mask nodes and edges in the augmented graph. It minimizes the similarity between the augmented and the original graph, while maximizing the similarity of the embeddings generated through them, so as to uncover the most important information in the graph. + +• SimGRACE (Xia et al., 2022a) creates augmented view by randomly perturbing the parameters of the GNN network. + +• SGL (Wu et al., 2021) adopts random walk sampling and probabilistic edge/node dropout to create augmented views for contrastive learning. In our experiments, we adopt the SGL-ED variant, which implements random edge dropout and exhibits the strongest performance according to the original paper. + +• HCCF (Xia et al., 2022b) encodes global graph information with hypergraph and contrasts it against the local information encoded with GCN. In our experiments, the number of hyper-edges are set as 128 following the original paper. + +• SHT (Xia et al., 2022c) adopts a hypergraph transformer framework to exploit global collaborative relationships and distills the global information to generate the cross-view self-supervised signals. In our experiments, the number of hyper-edges are set as 128 following the original paper. + +• SimGCL (Yu et al., 2022a) propose to simplify the graph augmentation process of contrastive learning by directly injecting random noises into the feature representation. + +# B PERFORMANCE COMPARISON WITH BASELINES (CONTINUED) + +In this appendix, we show the performance of NCF, GCCF, GraphCL, SAIL, GRACE, and AutoGCL, which are not shown in Table 1 due to space limit. The results are summarized in Table 5. As can be seen from the table, our model outperforms these baselines consistently. + +Table 5: Performance comparison with baselines on five datasets (continued). + +
DataMetricNCFGCCFGraphCLSAILGRACEAutoGCLLightGCL
YelpR@20N@200.02520.02020.04620.03980.04620.04010.04710.04050.05500.04700.05930.04940.07930.0668
R@40N@400.04870.02890.07600.05080.07640.05110.07730.05160.09170.06050.10090.06500.12920.0852
GowallaR@20N@200.01710.01060.09510.05350.09970.06030.09990.06020.07440.04520.08320.04840.15780.0935
R@40N@400.02160.01180.13920.06840.14730.07270.14720.07250.10710.05390.12910.06050.22450.1108
ML-10MR@20N@200.10970.12970.17420.21090.16590.20380.17280.21180.21070.24760.23250.27550.26130.3106
R@40N@400.16340.14270.26060.23310.25600.22500.26390.23320.30750.27110.34150.30230.37990.3387
AmazonR@20N@200.01420.00850.03170.02430.03600.02660.03570.02640.03600.02710.03250.02410.05850.0436
R@40N@400.02230.01330.04830.02850.05850.03400.05810.03380.05830.03450.05530.03180.09330.0551
TmallR@20N@200.00820.00590.02090.01410.02510.01750.02540.01770.03030.02100.03120.02040.05280.0361
R@40N@400.01400.00790.03560.01960.04160.02330.04240.02360.05050.02810.05240.02780.08520.0473
+ +# C THEORETICAL ANALYSIS + +We conduct theoretical analyses to show that our local-global CL (Eq. 7) is augmented to maximize the similarity between embeddings of potentially related nodes, based on the SVD-based global relation learning. Specifically, for a node $v _ { j } ~ \in { \mathcal { U } }$ , where $\mathcal { U } = \{ u _ { i ^ { \prime } } | \mathcal { A } _ { i , i ^ { \prime } } = 0 , \hat { \mathcal { A } } _ { i , i ^ { \prime } } \neq 0 \}$ , the embeddings are not updated by $s ( z _ { i , l } , g _ { i , l } )$ in the vanilla InfoNCE loss, as $v _ { j }$ is not adjacent to $u _ { i }$ . Instead, our local-global contrastive assigns the following gradients to the embeddings of $v _ { j }$ : + +$$ +\begin{array} { l } { \displaystyle \partial s ( z _ { i , l } , g _ { i , l } ) / \partial g _ { i , l - 1 } = \partial s \left( z _ { i , l } , \sigma ( \displaystyle \sum _ { j \in \mathcal { U } } \alpha _ { i , j } g _ { j , l - 1 } + \displaystyle \sum _ { A _ { i , j ^ { \prime } } \neq 0 } \alpha _ { i , j ^ { \prime } } g _ { j ^ { \prime } , l - 1 } ) \right) / \partial g _ { j , l - 1 } } \\ { = \frac { z _ { i , l } } { \| z _ { i , l } \| \| g _ { i , l } \| } \cdot \boldsymbol { \sigma } ^ { \prime } ( \cdot ) \cdot \boldsymbol { \alpha } _ { i , j } } \end{array} +$$ + +where $\alpha _ { i , j }$ denotes the normalization weight for node $u _ { i }$ and $v _ { j }$ . In this way, the embeddings of nodes in $\mathcal { U }$ are also pulled close to $s _ { i , l }$ , which injects relatedness information learned by the SVD into the local-global CL optimization. + +# D CALCULATION OF COMPLEXITY + +# D.1 ADJACENCY MATRIX NORMALIZATION + +For a sparse user-item matrix stored in the Coordinate Format (COO), it requires visiting every nonzero elements in the matrix to perform normalization. Thus, the computational complexity is in the order of the number of edges ${ \bf \bar { \boldsymbol { O } } } ( E )$ . Note that for the baseline SGL, it requires normalizing the two augmented graph structures during the training phase, each of which contains $\rho E$ edges, so it induces a complexity of $O ( 2 \rho E )$ per batch. + +# D.2 APPROXIMATE SVD ALGORITHM + +We refer the readers to Halko et al. (2011) in which the complexity of the approximate SVD algorithm is explained in detail. + +# D.3 GRAPH CONVOLUTION + +Given a sparse COO matrix $\mathcal { A }$ with $E$ edges and a dense matrix $\pmb { \cal E }$ with dimensions $I ( J ) \times d$ , it takes $O ( E d )$ time to calculate $\mathcal { A } E$ . To perform graph convolution on a graph, we need to multiply the sparse adjacency matrix with $\pmb { { E } } _ { l - 1 } ^ { ( v ) } \in \mathbb { R } ^ { J \times d }$ and its transpose with $E _ { l - 1 } ^ { ( u ) } \in \mathbb { R } ^ { I \times d }$ , which takes $O ( E d )$ each, and $O ( 2 E d )$ in total. For $L$ layers, $O ( 2 E L d )$ is required. For traditional CL-based methods such as SGL and $\mathrm { S i m C G L }$ , a three-view structure is adopted, resulting in a complexity of $O ( 1 2 E L d )$ (for SGL it again varies a bit depending on $\rho$ ). + +For the SVD-view of our model, $\hat { V } _ { q } ^ { \top } E _ { l - 1 } ^ { ( v ) }$ takes $O ( q J d )$ , and multiplying the result with the precalculated $( \hat { U } _ { q } \hat { S } _ { q } )$ takes $O ( q I d )$ ; $\hat { U } _ { q } ^ { \top } E _ { l - 1 } ^ { ( v ) }$ takes $O ( q I d )$ , and multiplying the result with the precalculated $( \hat { V } _ { q } \hat { S } _ { q } )$ takes $O ( q J d )$ . So in total it takes $O ( 2 q ( I + J ) d )$ . + +# D.4 BPR LOSS + +In each batch with $B$ users, calculating the scores for positive and negative items both take $O ( B d )$ , so in total it takes $O ( 2 B d )$ . + +# D.5 CL LOSS + +In each batch with $B$ users, calculating the numerator of InfoNCE loss takes $O ( B d )$ , and calculating the denominator takes $O ( B M d )$ where $M$ denotes the total number of nodes in the batch. Since our model adopts a per layer InfoNCE loss, a factor of $L$ is appended. \ No newline at end of file diff --git a/parse/dev/MAMOi89bOL/MAMOi89bOL.md b/parse/dev/MAMOi89bOL/MAMOi89bOL.md new file mode 100644 index 0000000000000000000000000000000000000000..1957e21a0fd4570ffbe8a1d8d8c82c2dc77d477c --- /dev/null +++ b/parse/dev/MAMOi89bOL/MAMOi89bOL.md @@ -0,0 +1,235 @@ +# Masked Autoencoders that Listen + +Po-Yao Huang1 Hu Xu1 Juncheng Li2 Alexei Baevski1 Michael Auli1 Wojciech Galuba1 Florian Metze1 Christoph Feichtenhofer1 + +1Meta AI 2Carnegie Mellon University + +# Abstract + +This paper studies a simple extension of image-based Masked Autoencoders (MAE) [1] to self-supervised representation learning from audio spectrograms. Following the Transformer encoder-decoder design in MAE, our Audio-MAE first encodes audio spectrogram patches with a high masking ratio, feeding only the non-masked tokens through encoder layers. The decoder then re-orders and decodes the encoded context padded with mask tokens, in order to reconstruct the input spectrogram. We find it beneficial to incorporate local window attention in the decoder, as audio spectrograms are highly correlated in local time and frequency bands. We then fine-tune the encoder with a lower masking ratio on target datasets. Empirically, Audio-MAE sets new state-of-the-art performance on six audio and speech classification tasks, outperforming other recent models that use external supervised pre-training. Our code and models is available at https://github.com/facebookresearch/AudioMAE. + +# 1 Introduction + +Transformers [2] and self-supervised learning [3, 4, 5, 6, 7, 1] are dominating computer vision (CV) and natural language processing (NLP) research. The revolution firstly started in NLP with the invention of the Transformer architecture and self-attention [8]. Masked autoencoding with BERT [3] set a new state-of-the-art on various NLP tasks by self-supervised pre-training on large-scale language corpus. Similarly in the CV community, Vision Transformers (ViT) [9] have become popular for CV tasks, and, for self-supervised image representation learning, Masked Autoencoders (MAE) [1] have brought the CV community closer to the success of BERT in NLP. In addition to the existing masked autoencoders that can read (BERT) or see (MAE), in this work we study those that can listen. + +Transformer-based models have recently refreshed leaderboards for audio understanding tasks. For example, AST [10] and MBT [11] improved the audio classification performance on the AudioSet [12], Event Sound Classification [13], etc. The key technique behind this is initialization of audio model weights with ImageNet pre-trained supervised models (e.g., DeiT [14]) by deflating patch embeddings and interpolating positional embeddings for encoding audio spectrograms. However, exploiting ImageNet pre-trained models could be sub-optimal. Unlike initializing video models with weights from image models (e.g., the initial weights of I3D [15] or 3D-ResNets [16] are inflated from ImageNet pre-trained image models), there are clear and notable discrepancies between spectrograms representing audio content and natural images. It remains unclear why such heterogeneous image-toaudio transfer is useful beyond arguably similar low-level semantics such as shapes of spectrograms and shapes of visual objects. Further, any label bias would inevitably be transferred to audio models. + +Addressing these concerns, self-supervised audio representation learning has recently attracted much research attention. Based on BEiT [17] that learns to reconstruct image patches or learnt patch tokens, SS-AST [18] extends to the audio domain and exploits spectrograms (akin to 1-channel 2D images) and use both contrastive and reconstruction objective as self-supervision. Without using any labels, the key enabler to effective self-supervised representation learning is large-scale pre-training data. In this work we use AudioSet [12] for pre-training, a common dataset containing ${ \sim } 2$ million audio recordings. Performing large-scale training with Transformer architectures is challenging as self-attention in Transformers has quadratic complexity w.r.t. the length of input sequence. + +![](images/7b578eaf9796ad13f1638cd79725ff0dce34d674e35e6e70f80938d322ab078e.jpg) +Figure 1: Audio-MAE for audio self-supervised learning. An audio recording is first transformed into a spectrogram and split into patches. We embed patches and mask out a large subset $( 8 0 \% )$ . An encoder then operates on the visible $( 2 0 \% )$ patch embeddings. Finally, a decoder processes the order-restored embeddings and mask tokens to reconstruct the input. Audio-MAE is minimizing the mean square error (MSE) on the masked portion of the reconstruction and the input spectrogram. + +This computational burden has been addressed in different ways. A popular approach is to reduce the sequence length in self-attention. Various ViT-based architectures have been developed to alleviate such issues for image and video understanding. For example, Swin-Transformer [19] only performs local attention within windows that shift across layers. MViT [20] employs pooling attention to construct a hierarchy of Transformers where sequence lengths are downsampled. For self-supervised learning, MAE [1] efficiently encodes only a small portion $( 2 5 \% )$ of visual patches while the majority of patches is discarded. The simplicity and scalability in MAE make it a promising framework for large-scale self-supervised learning. + +In this work, we study MAE for sound recognition and the unique challenges of the audio domain. We present Audio-MAE (Fig. 1) as unified and scalable framework for learning self-supervised audio representations. Similar to MAE, it is composed of a pair of a Transformer encoder and decoder. Sound is first transformed and embedded into spectrogram patches. Before feeding them into the Transformer encoder, we mask and discard the majority and only feed a small number of non-masked embeddings into the encoder for efficient encoding. After padding encoded patches with learnable embeddings to represent masked patches, it then restores the order of these patches in frequency and time and propagates them through a Transformer decoder to reconstruct the audio spectrogram. + +Different from image patches, spectrogram patches are comparably local-correlated. For example, formants, the vocal tract resonances, are typically grouped and continuous locally in the spectrogram. The location in frequency and time embeds essential information that determines the semantics of a spectrogram patch and how it sounds like. To this end, we further investigate using localized attention and a hybrid architecture in the Transformer decoder to properly decode for reconstruction. This simple-yet-effective upgrade leads to improved performance for Audio-MAE. + +Similar to MAE for images, we minimize the patch-normalized mean square error. At the fine-tuning stage, we discard the decoder and fine-tune the encoder with patch-masking. Empirically, AudioMAE sets a new state-of-the-art performance on six audio and speech classification tasks. It is the first audio-only self-supervised model that achieves state-of-the-art mAP on AudioSet-2M, outperforming other recent models with external supervision. We further provide the visualization and audible examples to qualitatively demonstrate the effectiveness of the Audio-MAE decoder. + +# 2 Related Work + +Visual masked pre-training. Masked/Denoising autoencoders [21, 22, 3] are a general representation learning methodology by reconstructing source from masked or corrupted inputs. In CV, visual masked pre-training has made recent progress [23, 24, 1, 20]. Based on ViT [9] that applies Transformers to image patches, BEiT [17] and MAE [1] present masked image modeling frameworks. BEiT [17] learns to predict discrete visual tokens generated by VAE [25] in masked patches. MAE [1] reduces sequence length by masking a large portion of image patches randomly and encoding only non-masked ones for reconstruction of pixel color information. MaskFeat [20] studies features for masked pre-training and finds that Histograms of Oriented Gradients (HoG) [26], which are in turn related to spectrogram features, perform strongly for image and video classification models. Our work extends the MAE framework for representation learning with audio spectrograms. + +Out-of-domain pre-training for audio. Transferring ImageNet supervised pre-trained ViT [9] or ResNet [27] has become a popular practice for audio models [10, 28, 11, 29, 30, 31]. After pre-training, these models operate over audio spectrograms by deflating from 3-channels (RGB) into 1-channel (spectrogram) in the pre-trained patch embedding in ViT and employing the rest of the transformer blocks on top. For example, HTS-AT [29] encodes spectrograms with hierarchical Transformer initialized from the Swin Transformer [19]. MBT [11] uses ImageNet-21K pre-trained ViT; AST [10] and PaSST [28] employ DeiT [14] as the Transformer backbone. Without using out-of-domain (non-audio) data, the proposed Audio-MAE focuses on audio-only self-supervised pre-training from scratch. + +In-domain pre-training for audio. Existing in-domain (i.e., audio-only) self-supervised methods can be broadly categorized by the input signal type (e.g., raw waveform [32, 33, 34], frame-level features [35, 36, 37], or spectrogram patches [18, 38]); and the objective used for self-supervision (e.g., contrastive [39, 33, 40, 41, 35] or prediction/reconstruction [18, 34, 37, 36]). For example, wav2vec 2.0 [33] takes raw waveform as inputs and exploits contrastive learning to discriminate contextualized representations in different time segments. Mockingjay [42] proposed a masked acoustic model pretext task to reconstruct frame-level Mel-features of masked time frames. SSAST [18] is the closest work to Audio-MAE and is our main benchmark. Inspired by the success of BERT [3], SS-AST proposed a self-supervised learning method which operates over spectrogram patches and employs joint contrastive and reconstructive objectives on masked patches. These previous methods generate audio representations by encoding full-view of both masked and nonmasked time or spectrogram segments for self-supervised pre-training. In contrast, Audio-MAE encodes only the non-masked spectrogram patches. + +Our work is done independently and concurrently with [38, 43, 44] related methods. We also compare our model to these concurrent works in the experiments and showcase the superiority of Audio-MAE. + +# 3 Audio Masked Autoencoders (Audio-MAE) + +Audio-MAE is a conceptually simple extension of MAE to learn self-supervised representations from audio spectrograms. Fig. 1 depicts an overview. The details of each component are as follows. + +Spectrogram Patch Embeddings. Following [10, 18], we transform audio recordings into Melspectrograms and divide them into non-overlapped regular grid patches. These patches are then flattened and embedded by a linear projection. Similar to MAE [1], we add fixed sinusoidal positional embeddings to the embedded patches. + +![](images/31a937966647bc9c70a9fc197b7e10a2aa38720c7ea2a23229a1c5f045f9bb94.jpg) +Figure 2: Audio-MAE’s masking strategies on Mel-spectrograms. + +Masking Strategies. Audio-MAE masks out a large subset of spectrogram patches. As a spectrogram can be viewed as a 2D representation of time and frequency components of a sound, it is reasonable to explore treating time and frequency differently during masking. In this work, we investigate both the unstructured (i.e., random masking without any prior) and structured (i.e., randomly masking a portion of time, frequency, or time $^ +$ frequency of a spectrogram) in the pre-training and fine-tuning phase. Illustrative examples are shown in Fig. 2. We show masked regions with dark overlay. + +The masking mechanism, as introduced in MAE [1], is the key ingredient for efficient self-supervised learning. For a input patch sequence, this can be regarded as a Bernoulli process where each patch is masked/dropped with probability $p$ (masking ratio). Masking reduces input patch sequence length and encourages learning global, contextualized representations from limited “visible” patches. We observe that akin to images, a large masking rate ( $80 \%$ in our experiments for spectrogram patches, which is similar to $7 5 \%$ in MAE for images) is feasible for learning self-supervised audio representations. Unlike BERT [3] that uses $15 \%$ masking rate for self-supervised learning in NLP, most of the tokens/patches can be discarded for spectrograms as well as images due to high redundancy in these modalities. Beyond self-supervised pre-training, we further explore the effectiveness of masking in the supervised fine-tuning stage. Empirically, we found unstructured (random) masking at a higher ratio for pre-training and structured (time+frequency masking) at a lower ratio for fine-tuning provide best accuracy (ablations are in $\ S \_ 4 )$ ). + +Encoder. Audio-MAE uses a stack of standard Transformers [2] as its encoder. The encoder only processes $( 2 0 \% )$ non-masked patches to reduce computation overhead which is quadratic to the input sequence length. We use the 12-layer ViT-Base (ViT-B) [9] Transformer as our default. + +Decoder with Local Attention. The decoder is also composed of standard Transformer blocks. The encoded patches from the encoder are padded with trainable masked tokens. After restoring the original time-frequency order in the audio spectrogram, we add the decoder’s (fixed sinusoidal) positional embeddings and feed the restored sequence into the decoder. At the top of the decoder stack, we add a linear head to predict and reconstruct the input spectrogram. + +To address the unique characteristics of audio spectrograms, our work investigates an enhancement to the vanilla MAE decoder. Image-based MAE uses global self-attention in the Transformer decoder which is appropriate for visual context, because visual objects are typically invariant under translation or scaling, and their exact position may not affect the semantics of an image. In contrast, the position, scale, and translation of spectrogram features however directly affects the sound or semantics of an audio recording. Consequently, global self-attention is sub-optimal for spectrograms if the timefrequency components is predominantly local. For instance, we would have better success to use the harmonics (e.g., Fig. 2a) in lower bands of a vowel to predict the spectrogram patch vertically in a higher frequency band rather than horizontally in the time domain. Similarly, a frictional sound of a consonant likely only correlates to other part of the consonant, and is without dependency to other silence segments in the audio recording. Compared to images, the spectrogram patches are more similar to speech or text tokens where its order and position is more relevant. + +To address the nature of audio spectrograms, in addition to using Transformers with global self-attention as in vanilla MAE, we incorporate the local attention mechanism which groups and separates the spectrogram patches in to local windows in self-attention for decoding. We investigate two types of local attention: (1) Shifted window location: Inspired by the shifted-window in Swin Transformers [19], we shift window attention by $50 \%$ between consecutive Transformer decoder layers. For padding the margin when shifting, we cyclically shift the spectrogram to the top-left direction. Fig. 3 illustrates the localized decoder attention by shifted windows. (2) Hybrid window attention (global+local attention): Inspired by [45], to add better cross-window connections, we design a simple hybrid (global+local) attention that computes local attention within a window in all but the last few top layers. In this way, the input feature maps for the final reconstruction layer also contain global information. For simplicity, we use $_ { n o }$ pooling or hierarchical structure. Decoders with different attention types are compared in $\ S \ O = 4$ . + +![](images/eac4e377de5817b96d0854d87b3e709eb9a487a2db0842240cefe54d8a061b6c.jpg) +Figure 3: Decoder’s local attention and shifted window (right). + +Objective. The Audio-MAE decoder learns to reconstruct the input spectrogram by predicting the values in the spectrogram patches or their per-patch normalized ones. The objective is the mean squared error (MSE) between the prediction and the input spectrogram, averaged over unknown patches. Empirically we found employing the reconstruction loss alone is sufficient while including additional contrastive objectives (e.g., InfoNCE loss [46]) does not improve Audio-MAE. + +Fine-tuning for Downstream Tasks. In the fine-tuning stage, we only keep and fine-tune the AudioMAE encoder and discard the decoder. Different from the original MAE, and inspired by [47, 28], we also explore to employ masking in the fine-tuning stage to remove a portion of patches to further regularize learning from a limited view of spectrogram inputs, which, as a side effect, also reduces computation during fine-tuning. Compared to SpecAug [48] which takes full-length input with the masked portion set to zero as data augmentation, Audio-MAE sees only a subset of real-valued input patches without the nullified ones. Audio-MAE then encodes these non-masked patches and applies an average pooling layer followed by a linear layer on top for fine-tuning in classification tasks. + +# 4 Experiments + +We perform an extensive evaluation on six tasks, including audio classification on AudioSet (AS-2M, AS-20K) and Environmental Sound Classification (ESC-50), and speech classification on Speech Commands (SPC-1 and SPC-2) and VoxCeleb (SID). We use AudioSet for ablation studies. + +# 4.1 Datasets and Tasks + +AudioSet [12] (AS-2M, AS-20K) contains ${ \sim } 2$ million 10-second YouTube clips for audio classification. 527 types of audio events are weakly annotated [49, 50, 51] for each clip. There could be multiple events in a clip. The full training set has 2 subsets: A class-wise balanced (22,176 clips) and an unbalanced (2,042,985 clips) set. The eval set has 20,383 clips. We downloaded and processed around 1.96M unbalanced training, 21K balanced training, and 19K evaluation clips. + +For the AS-2M experiments, we use the union of unbalanced and balanced training audio for pretraining and fine-tuning. For the AS-20K experiments, we use AS-2M for pre-training and the 20K balanced set for fine-tuning. We report the testing mAP on the 19K eval set used by AST [10]. + +Environmental Sound Classification (ESC-50) [13] is an audio classification dataset consists of 2,000 5-second environmental sound recordings. There are 50 classes in ESC. We report accuracy under 5-fold cross-validation with the same split used by [10]. + +Speech Commands (SPC-2, SPC-1) [52] are two keyword spotting tasks. In SPC-2, there are 35 speech commands. The training/validation/testing set has 84,843/9,981/11,005 1-second recordings, respectively. In SPC-1, there are 10 classes of keywords, 1 silence class, and 1 unknown class that includes all the other 20 common speech commands. We use the data and split provided in the SUPERB [53] benchmark to report the testing accuracy. + +VoxCeleb (SID) [54] contains 150K utterances from 1,251 speakers. The speaker identification task (SID) is to classify the utterances to identify its original speaker. We use the V1 standard train (138,361), validation (6,904), testing (8,251) sets and report the testing accuracy. + +# 4.2 Implementation Details + +We use a vanilla 12-layer ViT-B by default as the Transformer encoder. For the decoder, we use a 16-layer Transformer with shifted local attention. We investigate the vanilla (global attention) and hybrid (global+local attention) decoder variants (see Table. 1c). + +Following [10, 11], we transform raw waveform (pre-processed as mono channel under 16,000 sampling rate) into 128 Kaldi [55]-compatible Mel-frequency bands with a $2 5 \mathrm { m s }$ Hanning window that shifts every $1 0 ~ \mathrm { m s }$ . For a 10-second recording in AudioSet, the resulting spectrogram is of $1 \times 1 0 2 4 \times 1 2 8$ dimension. + +For patch embedding, we use convolutional kernels with (16, 16) size and stride in time and frequency (thus, patches are non-overlapping) to avoid short-cuts via overlap in self-supervision (though, at high masking ratios such short-cuts are less severe). By default, we use a masking ratio of 0.8 with (unstructured) random masking for pre-training. During fine-tuning, we employ a lower masking ratio (0.3 in time and 0.3 in frequency). Ablations on these design choices are given in $\ S \ O = 4$ . + +# 4.3 Pre-training and Fine-tuning + +We use AudioSet-2M for pre-training and randomly iterate over all audio recordings. We train for 32 epochs with a batch size of 512 and a 0.0002 learning rate. We distribute the training load over 64 V100 GPUs and the total training time is ${ \sim } 3 6$ hours. For each audio, we randomly sample the starting time, cyclically extract 10-second audio, and randomly jitter its magnitude by up to $\pm 6 \mathrm { d B }$ . We use only natural audio spectrograms and apply no augmentations (e.g., [48, 56, 57]) as we do not find these strong augmentations helpful in the pre-training phase. + +In the fine-tuning phase, we remove the decoder and only fine-tune the encoder. For the supervised fine-tuning on AudioSet-2M, since the size of training samples are uneven across classes (unbalanced), we follow the common practice of using a weighted sampling to balance the classes during training. In each epoch, we sample 200K instances ( $\mathord { \sim } 1 0 \%$ of AudioSet-2M) without replacement. We fine-tune for 100 epochs, which aggregate to ${ \sim } 1 0$ full epochs of AudioSet-2M. The probability of sampling an instance is inversely proportional to the dataset-wise occurrences of its classes. Fine-tuning on 64 GPUs takes ${ \sim } 1 2$ hours. For the smaller balanced AudioSet-20K, we fine-tune on 4 GPUs for 60 epochs without weighted sampling. Please see Supplementary for the details on other datasets. + +![](images/77981a7649efa2be98251d63c87ce0bb29948d0455d85ac60829270b7d834094.jpg) +Figure 4: Masking strategy. For pre-training, a higher ratio and unstructured masking (random) is preferred. For fine-tuning, a lower ratio and structured masking (time $^ +$ frequency) is better. The y-axes are mAP on AS-2M and the $\mathbf { X }$ -axes are masking ratio. This ablation format follows [1]. + +# 4.4 Ablations and Model Properties + +Masking Strategies in Pre-training and Fine-tuning. In Fig. 4, we compare different pre-training and fine-tuning masking strategies for Audio-MAE. First, in Fig. 4a we explore the pre-training masking ratio. We observe, similar as in MAE for images [1], that a high pre-training masking ratio $80 \%$ in our case) is optimal for audio spectrograms. This is due to the fact that both audio spectrograms and images are continuous signals with significant redundancy. Further, we find the unstructured random masking works the best for self-supervised pre-training over more structured masking (e.g., time+frequency). + +Unlike MAE for images, there are clear performance differences among masking strategies when pre-training with audio spectrograms. Comparing Audio-MAE reconstructions between Fig. 6a to 6e and 6d to 6h, under the same masking ratio, we observe the unstructured random masking is comparably easier than structured masking (i.e., time and/or frequency) as the model can guess the missing component by extrapolating nearby context (e.g., formants in vowels and frictional sounds in consonants around). We also observe that for higher masking ratios, the structured masking alternatives drop in performance, presumably because the task becomes too difficult while random masking improves steadily up to $80 \%$ . This result show that designing a pretext task with proper hardness is important for effective self-supervised learning of audio representations. We therefore use random masking with ratio of $80 \%$ as our default for pre-training. + +Fig. 4b studies the effect of masking during the fine-tuning phase. We see that in this case, it is more beneficial to use structured masking: time+frequency performs better than time- or frequency-based masking, and these perform better than unstructured masking. Overall, we see that the optimal masking ratios are lower than for pre-training and we use 0.3 as our default in the fine-tuning phase. + +In general, we observe that for task-agnostic pre-training, unstructured masking with a higher ratio is preferred. While in task-specific fine-tuning, structured masking with lower ratios performs better. + +Impact of Patch Size and Stride. We compare the performance of Audio-MAE trained with different patch sizes and strides in Table 1a. A non-zero overlap (i.e., stride $<$ patch size) between patches will increase the number of patches and quadratically increase computation in floating point operations (FLOPs), as reported in the table. Most prior works follow AST [10] to use overlapped patches (patch $= 1 6$ and stride $= 1 0$ ) to boost end task performance. As shown in Table 1a, we do not observe a performance improvement using overlapped patches for Audio-MAE (both $4 7 . 3 \mathrm { m A P }$ ), presumably because due to overlap, the patch embedding can leak information into the masked patches. The non-overlapped $1 6 \times 1 6$ patches achieve a good balance between computation and performance. By default, we use this setup in our experiments. + +Encoder. We investigate the design choices of encoder and decoder architectures in Audio-MAE. Table 1b shows the trade-off between encoder model size and performance. As expected, larger models achieve better performance, at a cost of computation and memory. The accuracy gain of ViT-L over ViT-B/S is more significant on the smaller and balanced AS-20K. For ViT-S, the performance + +
(16,16), (16,16)64×848.647.3
(16,16), (10,10)101×12130.547.3
(32,16), (16,16)63×847.846.6
(16,32), (16,16)64×742.146.8
+ +
ViT-S22M32.145.0
ViT-B86M37.147.3
ViT-L304M37.647.4
+ +scenario IN-SSL IN-SL AS-SSL AS-20K AS-2M + +
Attention typeAS-20K AS-2M ESC-50 SID
Global(8) (vanilla)36.6 46.8
93.6 94.1 47.3 94.1Local(16) (shifted) 37.1
Hwin (local(8)+ global(4) 36.894.8 47.3 93.8 95.0
+ +![](images/c8970b78f179014b4562bee38cb0d12fbbb43623ba5e47ab3c3b6f7e874f2872.jpg) +(h) External ImageNet (IN) pre-training. SSL: w/ selfsupervised MAE. SL: w/ supervised (fine-tuned) MAE. + +Table 1: Ablation studies on AS-2M. The gray entries are the default Audio-MAE setup (ViT-B encoder, decoder with shifted local attention, pre-trained for 32 epochs). Table format follows [1]. + +gap to ViT-B can be significantly closed $\mathrm { 5 . 0 \to 2 . 3 \ m A P }$ ) when fine-tuning with more in-domain data $( \mathrm { A S } - 2 0 \mathrm { K } \mathrm { A S } - 2 \mathrm { M } )$ ). + +Decoder. Table 1c compares decoder attention types in Audio-MAE. Note that decoders are discarded after pretraining and only the equal-sized ViT-B encoders are fine-tuned for the end task. Our results show that local attention with shifted window achieves the best performance. Combining local and global attention (i.e., hybrid attention, Hwin) also improves vanilla global self-attention. Fig. 5 shows the qualitative reconstruction comparison. In the spectrogram of vowels, the decoder with local attention reconstructs better harmonics and recovers more context in the spectrogram. Similar phenomena are observed in the frictional sound in the middle consonant. + +![](images/2eac13304af84b06c5760b4abdf221924cb45af9f00f2bcb95dca8f6af1aa8b1.jpg) +Figure 5: Decoder reconstruction comparison. + +Table 1d ablates the impact of decoder depth on mAP. A deeper 16-layer decoder achieves better performance against its shallower variants. Note that our decoder uses local window attention by default where only a fraction of tokens $4 { \times } 4$ local windows vs. $6 4 \times 8$ with global attention) are attended. For global attention we find 8-layer decoders to perform better than 16-layer. Table 1e compares decoder width (embedding dimension). A 512-dimension decoder achieves a good trade-off between computation and performance as a wider one is not better. + +Pre-training Data and Setup. Table 1f summarizes the impact of pre-training dataset size. Overall the model performance is monotonically increasing when using more data for pre-training. Comparing the performance of using $1 \%$ well-annotated AS-20K balanced data to using randomly sampled 20K unbalanced data for pre-training, the similar mAPs (39.4 vs 39.6) suggest that the distribution of data classes (balanced vs. unbalanced) is less important for pre-training. Meanwhile, as shown in Table $1 \mathrm { g }$ , training for longer is beneficial yet the performance saturates after the 24-th epoch. + +Out-of-domain Pre-training on ImageNet. Initializing audio models from ImageNet pre-trained weights has become popular for audio classification. However, as there are significant discrepancies between image and audio modalities, it is questionable if out-of-domain pre-training benefits audio representation learning. In Table 1h we design 3 scenarios to investigate this for Audio-MAE: (1) + +Table 2: Comparison with other state-of-the-art models on audio and speech classification tasks. Metrics are mAP for AS and accuracy $( \% )$ for ESC/SPC/SID. For pre-training (PT) dataset, AS:AudioSet, LS:LibriSpeech, and IN:ImageNet. †: Fine-tuning results with additional supervised training on AS-2M. We gray-out models pre-trained with external non-audio datasets (e.g., ImageNet). Best single models in AS-2M are compared (no ensembles). \*: linear evaluation results from [53]. + +
ModerBackboneP1-DataAS-20KAS-ZMIESC-30SPC-2SPC-1SID
No pre-training
ERANN [58]CNN45.089.2
PANN [59]CNN27.843.183.361.8
In-domain self-supervised pre-training
wav2vec 2.0 [33]TransformerLS96.2*75.2*
HuBERT[35]TransformerLS96.3*81.4*
Conformer [37]ConformerAS=41.188.0=--
SS-AST[18]ViT-BAS+LS31.0188.898.096.064.3
Concurrent MAE-based works
MaskSpec [43]ViT-BAS32.347.189.697.7=
MAE-AST[38]ViT-BAS+LS30.6-90.097.995.863.3
Audio-MAE (global)ViT-BAS36.6±.1146.8±.0693.6±.1198.3±.0697.6±.0694.1±.06
Audio-MAE (local)ViT-BAS37.0±.1147.3±.1194.1±.1098.3±.0696.9±.0094.8± .11
Out-of-domain supervised pre-training
PSLA [30]EffNet [60]IN31.944.4=96.3=
AST[10]DeiT-BIN34.745.988.798.195.541.1
MBT[11]ViT-BIN-21K31.344.31-=
HTS-AT [29]Swin-BIN=47.197.0t98.0
PaSST[28]DeiT-BIN47.196.8†-
+ +Audio-only pre-training (AS-SSL) from scratch. We consider this the ideal schema for learning audio representations as it is a simple and clean setup that prevents uncontrollable bias transfer from other modalities. (2) Directly using self-supervised ImageNet MAE models (IN-SSL) and its fine-tuned variant (IN-SL). (3) Audio-MAE self-supervised pre-training on top of these ImageNet weights. + +The results show that (1) from-scratch audio-only pre-training is the best. For scenarios (2) and (3), we observe that ImageNet pre-training alone (2) is not sufficient (especially when the downstream data is smaller, AS-20K), and, in self-supervised pre-training on AudioSet, ImageNet initialization (3) does not help but degrades accuracy. Also in (3), supervised ImageNet pre-training (IN-SL) seems harmful. Consequently, the result suggests that out-of-domain pre-training (i.e., ImageNet) is not helpful for Audio-MAE, possibly due to domain shift. + +# 4.5 Comparison with the State-of-the-art + +Table 2 compares Audio-MAE (with 3-run error bars) to prior state-of-the-art. We categorize the comparison into 3 groups. For fair comparison, our main benchmark is the models in the middle group with self-supervised pre-training on in-domain (audio) datasets (AudioSet and LibriSpeech). For reference we also list other models without pre-training (the top group) and other models with supervised pre-training on out-of-domain ImageNet (the bottom group), where the latter contains previous best systems on the datasets. + +Pre-trained on AudioSet, Audio-MAE achieves the best performance across all tasks compared to other models with in-domain self-supervised pre-training. On AudioSet-20K, its $3 7 . 1 \ \mathrm { m A P }$ significantly outperforms all other approaches including concurrent works and other models with outof-domain pre-training. On AudioSet-2M and ESC-50, our method also outperforms Conformer [37] and SS-AST [18]. Notably, unlike SS-AST and concurrent MAE-AST [38], which trained with additional 1,000 hours of speech in Librispeech, we use only AudioSet for pre-training. + +In the bottom group of Table 2, Audio-MAE also outperforms previous state-of-the-art models with ImageNet supervised pre-training. Note that the proposed Audio-MAE does not rely on any out-ofdomain data and labels, nor using knowledge distillation (e.g., DeiT) from additional CNN-based models. Also, compared to HTS-AT [29] and PaSST [28], Audio-MAE is trained with audio under 16K sampling rate. As experimented in [59], there could be up to 0.4 potential mAP improvement for Audio-MAE if audio with 32K sampling rate are available. + +![](images/6244fd1a7a46c5c39262c96d30a4dfd6f022b65c3eb7f73ba45382c9f667a734.jpg) +Figure 6: Spectrogram reconstruction visualizations on the AudioSet eval set. Column-wise type: speech, music, event, others. Masking type: (a-d) unstructured (random); (e-h) structured (time $^ +$ frequency). Masking Ratio: $70 \%$ . In each group, we show the original spectrogram (1, top), masked input (2, middle), and MAE output (3, bottom). The spectrogram size is $1 0 2 4 \times 1 2 8$ ; patch size is $1 6 \times 1 6$ . Each sample has $6 4 \times 8 = 5 1 2$ patches with 154 ( $70 \%$ masked) patches being visible to Audio-MAE. Please click (1 2 3) for audible .wavs. More audible examples are in Supplementary. + +For the speech tasks (SPC-1, SPC-2, and SID), Audio-MAE outperforms other models without pre-training (ERANN [58], PANN [59]), supervised (AST) and self-supervised models (SS-AST, MAE-AST). We further list other works (marked with \*) to include the latest results introduced in the SUPERB [53] benchmark. But note that these results are not strictly comparable since SUPERB employs linear evaluation where the underlying pre-trained models are not end-to-end fine-tuned. + +In summary, with audio-only from-scratch pre-training on AudioSet, our Audio-MAE performs well for both the audio and speech classification tasks. + +# 4.6 Visualization and Audible Examples by Audio-MAE Decoder + +For better visualization, we follow MAE [1] to use MSE over non-normalized spectrograms as the selfsupervised objective. We use ViT-L as the Audio-MAE encoder for visualization. Fig. 6 illustrates the reconstruction results sampled from the AudioSet-2M eval set. We further reconstruct .wavs using the Griffin-Lim [61] algorithm, audible under the anonymous links (accessible in respective 1 2 3). + +As can be seen and heard, for various masking strategies and different sounds, our Audio-MAE generates reasonable reconstruction. It works well for noisy event sounds (e.g., the reconstructed siren in Fig. 6c-3), as well as speech and music (e.g., the reconstructed singing in Fig. 6b-3). Notably, unlike visual contents that are typically scale/translation/position invariant [19], absolute positions and arrangement of spectrogram components are critical for humans to understand sound [62]. For example, shifting a pitch will make an audio sounds completely different. Also, phoneme sequences in time are important cues for speech understanding. Consequently, unstructured masking produces better aligned outputs that are closer to the ground-truth (top row in each subfigure) as the model can make better predictions based on nearby spectrogram patches; while structured masking is harder (less accurate or with words missing), especially when masking is performed over the time axis. A failure example (missing words) is the reconstructed speech in Fig. 6e-3. + +# 5 Conclusion + +We have explored a simple extension of MAE [1] to audio data. Our Audio-MAE learns to reconstruct masked spectrogram patches from audio recordings and achieves state-of-the-art performance on six audio and speech classification tasks. We have drawn four interesting observations: First, a simple MAE approach works surprisingly well for audio spectrograms. Second, we find that it is possible to learn stronger representations with local self-attention in the decoder. Third, we show that masking can be applied to both pre-training and fine-tuning, improving accuracy and reducing training computation. The optimal strategy depends on the nature of the data (audio, image, etc.) and the learning type (self-/supervised). Fourth, the best performance can be achieved by pre-training and fine-tuning under the same modality, without reliance on cross-modality transfer learning. In future work, we aim to explore multimodal self-supervised learning with a joint audio-visual MAE approach as these domains share natural correspondences in video data. + +Acknowledgements. We thank Kaiming He and Luke Zettlemoyer for their feedback and discussions. + +# References + +[1] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” arXiv preprint arXiv:2111.06377, 2021. [2] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. 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Takeshima, “Equal-loudness-level contours for pure tones,” The Journal of the Acoustical Society of America, vol. 116, no. 2, pp. 918–933, 2004. \ No newline at end of file diff --git a/parse/dev/MAMOi89bOL/MAMOi89bOL_content_list.json b/parse/dev/MAMOi89bOL/MAMOi89bOL_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a58ce64dd7c56a066cdaaff6df89c88e7ff1fbad --- /dev/null +++ b/parse/dev/MAMOi89bOL/MAMOi89bOL_content_list.json @@ -0,0 +1,1050 @@ +[ + { + "type": "text", + "text": "Masked Autoencoders that Listen ", + "text_level": 1, + "bbox": [ + 294, + 122, + 704, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Po-Yao Huang1 Hu Xu1 Juncheng Li2 Alexei Baevski1 Michael Auli1 Wojciech Galuba1 Florian Metze1 Christoph Feichtenhofer1 ", + "bbox": [ + 212, + 195, + 784, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Meta AI 2Carnegie Mellon University ", + "bbox": [ + 356, + 238, + 638, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 277, + 535, + 292 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper studies a simple extension of image-based Masked Autoencoders (MAE) [1] to self-supervised representation learning from audio spectrograms. Following the Transformer encoder-decoder design in MAE, our Audio-MAE first encodes audio spectrogram patches with a high masking ratio, feeding only the non-masked tokens through encoder layers. The decoder then re-orders and decodes the encoded context padded with mask tokens, in order to reconstruct the input spectrogram. We find it beneficial to incorporate local window attention in the decoder, as audio spectrograms are highly correlated in local time and frequency bands. We then fine-tune the encoder with a lower masking ratio on target datasets. Empirically, Audio-MAE sets new state-of-the-art performance on six audio and speech classification tasks, outperforming other recent models that use external supervised pre-training. Our code and models is available at https://github.com/facebookresearch/AudioMAE. ", + "bbox": [ + 233, + 297, + 766, + 477 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 491, + 310, + 508 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformers [2] and self-supervised learning [3, 4, 5, 6, 7, 1] are dominating computer vision (CV) and natural language processing (NLP) research. The revolution firstly started in NLP with the invention of the Transformer architecture and self-attention [8]. Masked autoencoding with BERT [3] set a new state-of-the-art on various NLP tasks by self-supervised pre-training on large-scale language corpus. Similarly in the CV community, Vision Transformers (ViT) [9] have become popular for CV tasks, and, for self-supervised image representation learning, Masked Autoencoders (MAE) [1] have brought the CV community closer to the success of BERT in NLP. In addition to the existing masked autoencoders that can read (BERT) or see (MAE), in this work we study those that can listen. ", + "bbox": [ + 174, + 516, + 825, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Transformer-based models have recently refreshed leaderboards for audio understanding tasks. For example, AST [10] and MBT [11] improved the audio classification performance on the AudioSet [12], Event Sound Classification [13], etc. The key technique behind this is initialization of audio model weights with ImageNet pre-trained supervised models (e.g., DeiT [14]) by deflating patch embeddings and interpolating positional embeddings for encoding audio spectrograms. However, exploiting ImageNet pre-trained models could be sub-optimal. Unlike initializing video models with weights from image models (e.g., the initial weights of I3D [15] or 3D-ResNets [16] are inflated from ImageNet pre-trained image models), there are clear and notable discrepancies between spectrograms representing audio content and natural images. It remains unclear why such heterogeneous image-toaudio transfer is useful beyond arguably similar low-level semantics such as shapes of spectrograms and shapes of visual objects. Further, any label bias would inevitably be transferred to audio models. ", + "bbox": [ + 174, + 633, + 825, + 785 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Addressing these concerns, self-supervised audio representation learning has recently attracted much research attention. Based on BEiT [17] that learns to reconstruct image patches or learnt patch tokens, SS-AST [18] extends to the audio domain and exploits spectrograms (akin to 1-channel 2D images) and use both contrastive and reconstruction objective as self-supervision. Without using any labels, the key enabler to effective self-supervised representation learning is large-scale pre-training data. In this work we use AudioSet [12] for pre-training, a common dataset containing ${ \\sim } 2$ million audio recordings. Performing large-scale training with Transformer architectures is challenging as self-attention in Transformers has quadratic complexity w.r.t. the length of input sequence. ", + "bbox": [ + 174, + 791, + 825, + 902 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/7b578eaf9796ad13f1638cd79725ff0dce34d674e35e6e70f80938d322ab078e.jpg", + "image_caption": [ + "Figure 1: Audio-MAE for audio self-supervised learning. An audio recording is first transformed into a spectrogram and split into patches. We embed patches and mask out a large subset $( 8 0 \\% )$ . An encoder then operates on the visible $( 2 0 \\% )$ patch embeddings. Finally, a decoder processes the order-restored embeddings and mask tokens to reconstruct the input. Audio-MAE is minimizing the mean square error (MSE) on the masked portion of the reconstruction and the input spectrogram. " + ], + "image_footnote": [], + "bbox": [ + 171, + 0, + 826, + 222 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This computational burden has been addressed in different ways. A popular approach is to reduce the sequence length in self-attention. Various ViT-based architectures have been developed to alleviate such issues for image and video understanding. For example, Swin-Transformer [19] only performs local attention within windows that shift across layers. MViT [20] employs pooling attention to construct a hierarchy of Transformers where sequence lengths are downsampled. For self-supervised learning, MAE [1] efficiently encodes only a small portion $( 2 5 \\% )$ of visual patches while the majority of patches is discarded. The simplicity and scalability in MAE make it a promising framework for large-scale self-supervised learning. ", + "bbox": [ + 174, + 310, + 825, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we study MAE for sound recognition and the unique challenges of the audio domain. We present Audio-MAE (Fig. 1) as unified and scalable framework for learning self-supervised audio representations. Similar to MAE, it is composed of a pair of a Transformer encoder and decoder. Sound is first transformed and embedded into spectrogram patches. Before feeding them into the Transformer encoder, we mask and discard the majority and only feed a small number of non-masked embeddings into the encoder for efficient encoding. After padding encoded patches with learnable embeddings to represent masked patches, it then restores the order of these patches in frequency and time and propagates them through a Transformer decoder to reconstruct the audio spectrogram. ", + "bbox": [ + 174, + 428, + 825, + 539 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Different from image patches, spectrogram patches are comparably local-correlated. For example, formants, the vocal tract resonances, are typically grouped and continuous locally in the spectrogram. The location in frequency and time embeds essential information that determines the semantics of a spectrogram patch and how it sounds like. To this end, we further investigate using localized attention and a hybrid architecture in the Transformer decoder to properly decode for reconstruction. This simple-yet-effective upgrade leads to improved performance for Audio-MAE. ", + "bbox": [ + 174, + 546, + 825, + 628 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Similar to MAE for images, we minimize the patch-normalized mean square error. At the fine-tuning stage, we discard the decoder and fine-tune the encoder with patch-masking. Empirically, AudioMAE sets a new state-of-the-art performance on six audio and speech classification tasks. It is the first audio-only self-supervised model that achieves state-of-the-art mAP on AudioSet-2M, outperforming other recent models with external supervision. We further provide the visualization and audible examples to qualitatively demonstrate the effectiveness of the Audio-MAE decoder. ", + "bbox": [ + 174, + 635, + 825, + 718 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 737, + 321, + 753 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Visual masked pre-training. Masked/Denoising autoencoders [21, 22, 3] are a general representation learning methodology by reconstructing source from masked or corrupted inputs. In CV, visual masked pre-training has made recent progress [23, 24, 1, 20]. Based on ViT [9] that applies Transformers to image patches, BEiT [17] and MAE [1] present masked image modeling frameworks. BEiT [17] learns to predict discrete visual tokens generated by VAE [25] in masked patches. MAE [1] reduces sequence length by masking a large portion of image patches randomly and encoding only non-masked ones for reconstruction of pixel color information. MaskFeat [20] studies features for masked pre-training and finds that Histograms of Oriented Gradients (HoG) [26], which are in turn related to spectrogram features, perform strongly for image and video classification models. Our work extends the MAE framework for representation learning with audio spectrograms. ", + "bbox": [ + 174, + 772, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Out-of-domain pre-training for audio. Transferring ImageNet supervised pre-trained ViT [9] or ResNet [27] has become a popular practice for audio models [10, 28, 11, 29, 30, 31]. After pre-training, these models operate over audio spectrograms by deflating from 3-channels (RGB) into 1-channel (spectrogram) in the pre-trained patch embedding in ViT and employing the rest of the transformer blocks on top. For example, HTS-AT [29] encodes spectrograms with hierarchical Transformer initialized from the Swin Transformer [19]. MBT [11] uses ImageNet-21K pre-trained ViT; AST [10] and PaSST [28] employ DeiT [14] as the Transformer backbone. Without using out-of-domain (non-audio) data, the proposed Audio-MAE focuses on audio-only self-supervised pre-training from scratch. ", + "bbox": [ + 173, + 90, + 825, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In-domain pre-training for audio. Existing in-domain (i.e., audio-only) self-supervised methods can be broadly categorized by the input signal type (e.g., raw waveform [32, 33, 34], frame-level features [35, 36, 37], or spectrogram patches [18, 38]); and the objective used for self-supervision (e.g., contrastive [39, 33, 40, 41, 35] or prediction/reconstruction [18, 34, 37, 36]). For example, wav2vec 2.0 [33] takes raw waveform as inputs and exploits contrastive learning to discriminate contextualized representations in different time segments. Mockingjay [42] proposed a masked acoustic model pretext task to reconstruct frame-level Mel-features of masked time frames. SSAST [18] is the closest work to Audio-MAE and is our main benchmark. Inspired by the success of BERT [3], SS-AST proposed a self-supervised learning method which operates over spectrogram patches and employs joint contrastive and reconstructive objectives on masked patches. These previous methods generate audio representations by encoding full-view of both masked and nonmasked time or spectrogram segments for self-supervised pre-training. In contrast, Audio-MAE encodes only the non-masked spectrogram patches. ", + "bbox": [ + 173, + 226, + 826, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our work is done independently and concurrently with [38, 43, 44] related methods. We also compare our model to these concurrent works in the experiments and showcase the superiority of Audio-MAE. ", + "bbox": [ + 174, + 412, + 825, + 440 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Audio Masked Autoencoders (Audio-MAE) ", + "text_level": 1, + "bbox": [ + 174, + 462, + 565, + 479 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Audio-MAE is a conceptually simple extension of MAE to learn self-supervised representations from audio spectrograms. Fig. 1 depicts an overview. The details of each component are as follows. ", + "bbox": [ + 174, + 494, + 823, + 523 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Spectrogram Patch Embeddings. Following [10, 18], we transform audio recordings into Melspectrograms and divide them into non-overlapped regular grid patches. These patches are then flattened and embedded by a linear projection. Similar to MAE [1], we add fixed sinusoidal positional embeddings to the embedded patches. ", + "bbox": [ + 174, + 529, + 825, + 585 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/31a937966647bc9c70a9fc197b7e10a2aa38720c7ea2a23229a1c5f045f9bb94.jpg", + "image_caption": [ + "Figure 2: Audio-MAE’s masking strategies on Mel-spectrograms. " + ], + "image_footnote": [], + "bbox": [ + 183, + 601, + 813, + 685 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Masking Strategies. Audio-MAE masks out a large subset of spectrogram patches. As a spectrogram can be viewed as a 2D representation of time and frequency components of a sound, it is reasonable to explore treating time and frequency differently during masking. In this work, we investigate both the unstructured (i.e., random masking without any prior) and structured (i.e., randomly masking a portion of time, frequency, or time $^ +$ frequency of a spectrogram) in the pre-training and fine-tuning phase. Illustrative examples are shown in Fig. 2. We show masked regions with dark overlay. ", + "bbox": [ + 173, + 724, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The masking mechanism, as introduced in MAE [1], is the key ingredient for efficient self-supervised learning. For a input patch sequence, this can be regarded as a Bernoulli process where each patch is masked/dropped with probability $p$ (masking ratio). Masking reduces input patch sequence length and encourages learning global, contextualized representations from limited “visible” patches. We observe that akin to images, a large masking rate ( $80 \\%$ in our experiments for spectrogram patches, which is similar to $7 5 \\%$ in MAE for images) is feasible for learning self-supervised audio representations. Unlike BERT [3] that uses $15 \\%$ masking rate for self-supervised learning in NLP, most of the tokens/patches can be discarded for spectrograms as well as images due to high redundancy in these modalities. Beyond self-supervised pre-training, we further explore the effectiveness of masking in the supervised fine-tuning stage. Empirically, we found unstructured (random) masking at a higher ratio for pre-training and structured (time+frequency masking) at a lower ratio for fine-tuning provide best accuracy (ablations are in $\\ S \\_ 4 )$ ). ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 161 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Encoder. Audio-MAE uses a stack of standard Transformers [2] as its encoder. The encoder only processes $( 2 0 \\% )$ non-masked patches to reduce computation overhead which is quadratic to the input sequence length. We use the 12-layer ViT-Base (ViT-B) [9] Transformer as our default. ", + "bbox": [ + 174, + 166, + 823, + 209 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Decoder with Local Attention. The decoder is also composed of standard Transformer blocks. The encoded patches from the encoder are padded with trainable masked tokens. After restoring the original time-frequency order in the audio spectrogram, we add the decoder’s (fixed sinusoidal) positional embeddings and feed the restored sequence into the decoder. At the top of the decoder stack, we add a linear head to predict and reconstruct the input spectrogram. ", + "bbox": [ + 174, + 215, + 825, + 285 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To address the unique characteristics of audio spectrograms, our work investigates an enhancement to the vanilla MAE decoder. Image-based MAE uses global self-attention in the Transformer decoder which is appropriate for visual context, because visual objects are typically invariant under translation or scaling, and their exact position may not affect the semantics of an image. In contrast, the position, scale, and translation of spectrogram features however directly affects the sound or semantics of an audio recording. Consequently, global self-attention is sub-optimal for spectrograms if the timefrequency components is predominantly local. For instance, we would have better success to use the harmonics (e.g., Fig. 2a) in lower bands of a vowel to predict the spectrogram patch vertically in a higher frequency band rather than horizontally in the time domain. Similarly, a frictional sound of a consonant likely only correlates to other part of the consonant, and is without dependency to other silence segments in the audio recording. Compared to images, the spectrogram patches are more similar to speech or text tokens where its order and position is more relevant. ", + "bbox": [ + 174, + 290, + 825, + 457 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To address the nature of audio spectrograms, in addition to using Transformers with global self-attention as in vanilla MAE, we incorporate the local attention mechanism which groups and separates the spectrogram patches in to local windows in self-attention for decoding. We investigate two types of local attention: (1) Shifted window location: Inspired by the shifted-window in Swin Transformers [19], we shift window attention by $50 \\%$ between consecutive Transformer decoder layers. For padding the margin when shifting, we cyclically shift the spectrogram to the top-left direction. Fig. 3 illustrates the localized decoder attention by shifted windows. (2) Hybrid window attention (global+local attention): Inspired by [45], to add better cross-window connections, we design a simple hybrid (global+local) attention that computes local attention within a window in all but the last few top layers. In this way, the input feature maps for the final reconstruction layer also contain global information. For simplicity, we use $_ { n o }$ pooling or hierarchical structure. Decoders with different attention types are compared in $\\ S \\ O = 4$ . ", + "bbox": [ + 174, + 463, + 576, + 601 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/eac4e377de5817b96d0854d87b3e709eb9a487a2db0842240cefe54d8a061b6c.jpg", + "image_caption": [ + "Figure 3: Decoder’s local attention and shifted window (right). " + ], + "image_footnote": [], + "bbox": [ + 591, + 476, + 821, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 602, + 825, + 684 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Objective. The Audio-MAE decoder learns to reconstruct the input spectrogram by predicting the values in the spectrogram patches or their per-patch normalized ones. The objective is the mean squared error (MSE) between the prediction and the input spectrogram, averaged over unknown patches. Empirically we found employing the reconstruction loss alone is sufficient while including additional contrastive objectives (e.g., InfoNCE loss [46]) does not improve Audio-MAE. ", + "bbox": [ + 174, + 690, + 825, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Fine-tuning for Downstream Tasks. In the fine-tuning stage, we only keep and fine-tune the AudioMAE encoder and discard the decoder. Different from the original MAE, and inspired by [47, 28], we also explore to employ masking in the fine-tuning stage to remove a portion of patches to further regularize learning from a limited view of spectrogram inputs, which, as a side effect, also reduces computation during fine-tuning. Compared to SpecAug [48] which takes full-length input with the masked portion set to zero as data augmentation, Audio-MAE sees only a subset of real-valued input patches without the nullified ones. Audio-MAE then encodes these non-masked patches and applies an average pooling layer followed by a linear layer on top for fine-tuning in classification tasks. ", + "bbox": [ + 174, + 766, + 825, + 877 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 89, + 312, + 107 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We perform an extensive evaluation on six tasks, including audio classification on AudioSet (AS-2M, AS-20K) and Environmental Sound Classification (ESC-50), and speech classification on Speech Commands (SPC-1 and SPC-2) and VoxCeleb (SID). We use AudioSet for ablation studies. ", + "bbox": [ + 174, + 121, + 825, + 162 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 Datasets and Tasks ", + "text_level": 1, + "bbox": [ + 174, + 181, + 346, + 196 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "AudioSet [12] (AS-2M, AS-20K) contains ${ \\sim } 2$ million 10-second YouTube clips for audio classification. 527 types of audio events are weakly annotated [49, 50, 51] for each clip. There could be multiple events in a clip. The full training set has 2 subsets: A class-wise balanced (22,176 clips) and an unbalanced (2,042,985 clips) set. The eval set has 20,383 clips. We downloaded and processed around 1.96M unbalanced training, 21K balanced training, and 19K evaluation clips. ", + "bbox": [ + 174, + 207, + 825, + 277 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For the AS-2M experiments, we use the union of unbalanced and balanced training audio for pretraining and fine-tuning. For the AS-20K experiments, we use AS-2M for pre-training and the 20K balanced set for fine-tuning. We report the testing mAP on the 19K eval set used by AST [10]. ", + "bbox": [ + 174, + 284, + 825, + 325 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Environmental Sound Classification (ESC-50) [13] is an audio classification dataset consists of 2,000 5-second environmental sound recordings. There are 50 classes in ESC. We report accuracy under 5-fold cross-validation with the same split used by [10]. ", + "bbox": [ + 174, + 332, + 825, + 373 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Speech Commands (SPC-2, SPC-1) [52] are two keyword spotting tasks. In SPC-2, there are 35 speech commands. The training/validation/testing set has 84,843/9,981/11,005 1-second recordings, respectively. In SPC-1, there are 10 classes of keywords, 1 silence class, and 1 unknown class that includes all the other 20 common speech commands. We use the data and split provided in the SUPERB [53] benchmark to report the testing accuracy. ", + "bbox": [ + 174, + 380, + 825, + 450 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "VoxCeleb (SID) [54] contains 150K utterances from 1,251 speakers. The speaker identification task (SID) is to classify the utterances to identify its original speaker. We use the V1 standard train (138,361), validation (6,904), testing (8,251) sets and report the testing accuracy. ", + "bbox": [ + 174, + 455, + 825, + 497 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 Implementation Details ", + "text_level": 1, + "bbox": [ + 176, + 515, + 375, + 530 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use a vanilla 12-layer ViT-B by default as the Transformer encoder. For the decoder, we use a 16-layer Transformer with shifted local attention. We investigate the vanilla (global attention) and hybrid (global+local attention) decoder variants (see Table. 1c). ", + "bbox": [ + 176, + 541, + 823, + 583 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Following [10, 11], we transform raw waveform (pre-processed as mono channel under 16,000 sampling rate) into 128 Kaldi [55]-compatible Mel-frequency bands with a $2 5 \\mathrm { m s }$ Hanning window that shifts every $1 0 ~ \\mathrm { m s }$ . For a 10-second recording in AudioSet, the resulting spectrogram is of $1 \\times 1 0 2 4 \\times 1 2 8$ dimension. ", + "bbox": [ + 174, + 590, + 825, + 645 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For patch embedding, we use convolutional kernels with (16, 16) size and stride in time and frequency (thus, patches are non-overlapping) to avoid short-cuts via overlap in self-supervision (though, at high masking ratios such short-cuts are less severe). By default, we use a masking ratio of 0.8 with (unstructured) random masking for pre-training. During fine-tuning, we employ a lower masking ratio (0.3 in time and 0.3 in frequency). Ablations on these design choices are given in $\\ S \\ O = 4$ . ", + "bbox": [ + 174, + 651, + 825, + 722 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 Pre-training and Fine-tuning ", + "text_level": 1, + "bbox": [ + 176, + 739, + 415, + 755 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use AudioSet-2M for pre-training and randomly iterate over all audio recordings. We train for 32 epochs with a batch size of 512 and a 0.0002 learning rate. We distribute the training load over 64 V100 GPUs and the total training time is ${ \\sim } 3 6$ hours. For each audio, we randomly sample the starting time, cyclically extract 10-second audio, and randomly jitter its magnitude by up to $\\pm 6 \\mathrm { d B }$ . We use only natural audio spectrograms and apply no augmentations (e.g., [48, 56, 57]) as we do not find these strong augmentations helpful in the pre-training phase. ", + "bbox": [ + 174, + 766, + 825, + 849 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the fine-tuning phase, we remove the decoder and only fine-tune the encoder. For the supervised fine-tuning on AudioSet-2M, since the size of training samples are uneven across classes (unbalanced), we follow the common practice of using a weighted sampling to balance the classes during training. In each epoch, we sample 200K instances ( $\\mathord { \\sim } 1 0 \\%$ of AudioSet-2M) without replacement. We fine-tune for 100 epochs, which aggregate to ${ \\sim } 1 0$ full epochs of AudioSet-2M. The probability of sampling an instance is inversely proportional to the dataset-wise occurrences of its classes. Fine-tuning on 64 GPUs takes ${ \\sim } 1 2$ hours. For the smaller balanced AudioSet-20K, we fine-tune on 4 GPUs for 60 epochs without weighted sampling. Please see Supplementary for the details on other datasets. ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/77981a7649efa2be98251d63c87ce0bb29948d0455d85ac60829270b7d834094.jpg", + "image_caption": [ + "Figure 4: Masking strategy. For pre-training, a higher ratio and unstructured masking (random) is preferred. For fine-tuning, a lower ratio and structured masking (time $^ +$ frequency) is better. The y-axes are mAP on AS-2M and the $\\mathbf { X }$ -axes are masking ratio. This ablation format follows [1]. " + ], + "image_footnote": [], + "bbox": [ + 181, + 87, + 812, + 198 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 265, + 825, + 320 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.4 Ablations and Model Properties ", + "text_level": 1, + "bbox": [ + 176, + 340, + 434, + 354 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Masking Strategies in Pre-training and Fine-tuning. In Fig. 4, we compare different pre-training and fine-tuning masking strategies for Audio-MAE. First, in Fig. 4a we explore the pre-training masking ratio. We observe, similar as in MAE for images [1], that a high pre-training masking ratio $80 \\%$ in our case) is optimal for audio spectrograms. This is due to the fact that both audio spectrograms and images are continuous signals with significant redundancy. Further, we find the unstructured random masking works the best for self-supervised pre-training over more structured masking (e.g., time+frequency). ", + "bbox": [ + 174, + 371, + 825, + 468 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Unlike MAE for images, there are clear performance differences among masking strategies when pre-training with audio spectrograms. Comparing Audio-MAE reconstructions between Fig. 6a to 6e and 6d to 6h, under the same masking ratio, we observe the unstructured random masking is comparably easier than structured masking (i.e., time and/or frequency) as the model can guess the missing component by extrapolating nearby context (e.g., formants in vowels and frictional sounds in consonants around). We also observe that for higher masking ratios, the structured masking alternatives drop in performance, presumably because the task becomes too difficult while random masking improves steadily up to $80 \\%$ . This result show that designing a pretext task with proper hardness is important for effective self-supervised learning of audio representations. We therefore use random masking with ratio of $80 \\%$ as our default for pre-training. ", + "bbox": [ + 174, + 474, + 825, + 613 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Fig. 4b studies the effect of masking during the fine-tuning phase. We see that in this case, it is more beneficial to use structured masking: time+frequency performs better than time- or frequency-based masking, and these perform better than unstructured masking. Overall, we see that the optimal masking ratios are lower than for pre-training and we use 0.3 as our default in the fine-tuning phase. ", + "bbox": [ + 174, + 619, + 825, + 675 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In general, we observe that for task-agnostic pre-training, unstructured masking with a higher ratio is preferred. While in task-specific fine-tuning, structured masking with lower ratios performs better. ", + "bbox": [ + 174, + 681, + 823, + 709 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Impact of Patch Size and Stride. We compare the performance of Audio-MAE trained with different patch sizes and strides in Table 1a. A non-zero overlap (i.e., stride $<$ patch size) between patches will increase the number of patches and quadratically increase computation in floating point operations (FLOPs), as reported in the table. Most prior works follow AST [10] to use overlapped patches (patch $= 1 6$ and stride $= 1 0$ ) to boost end task performance. As shown in Table 1a, we do not observe a performance improvement using overlapped patches for Audio-MAE (both $4 7 . 3 \\mathrm { m A P }$ ), presumably because due to overlap, the patch embedding can leak information into the masked patches. The non-overlapped $1 6 \\times 1 6$ patches achieve a good balance between computation and performance. By default, we use this setup in our experiments. ", + "bbox": [ + 173, + 719, + 825, + 844 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Encoder. We investigate the design choices of encoder and decoder architectures in Audio-MAE. Table 1b shows the trade-off between encoder model size and performance. As expected, larger models achieve better performance, at a cost of computation and memory. The accuracy gain of ViT-L over ViT-B/S is more significant on the smaller and balanced AS-20K. For ViT-S, the performance ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/6f59a1c28e76f7aa6d92f54dc2336e032fe1fec05ebb7a82d7a9a980d84c69bf.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(16,16), (16,16)64×848.647.3
(16,16), (10,10)101×12130.547.3
(32,16), (16,16)63×847.846.6
(16,32), (16,16)64×742.146.8
", + "bbox": [ + 200, + 95, + 459, + 166 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/cd323607a9ef4d1435f53f5a4694198cfcaa19480ab62f7674c6087d91c726d9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ViT-S22M32.145.0
ViT-B86M37.147.3
ViT-L304M37.647.4
", + "bbox": [ + 539, + 103, + 758, + 147 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/2e64e580b9f04e22b287bd69d5bfdd39254f01c2e6f243ca7c94c7792437d8fd.jpg", + "table_caption": [ + "scenario IN-SSL IN-SL AS-SSL AS-20K AS-2M " + ], + "table_footnote": [], + "table_body": "
Attention typeAS-20K AS-2M ESC-50 SID
Global(8) (vanilla)36.6 46.8
93.6 94.1 47.3 94.1Local(16) (shifted) 37.1
Hwin (local(8)+ global(4) 36.894.8 47.3 93.8 95.0
", + "bbox": [ + 192, + 188, + 539, + 251 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c8970b78f179014b4562bee38cb0d12fbbb43623ba5e47ab3c3b6f7e874f2872.jpg", + "image_caption": [ + "(h) External ImageNet (IN) pre-training. SSL: w/ selfsupervised MAE. SL: w/ supervised (fine-tuned) MAE. " + ], + "image_footnote": [], + "bbox": [ + 503, + 291, + 828, + 363 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 1: Ablation studies on AS-2M. The gray entries are the default Audio-MAE setup (ViT-B encoder, decoder with shifted local attention, pre-trained for 32 epochs). Table format follows [1]. ", + "bbox": [ + 174, + 396, + 825, + 428 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "gap to ViT-B can be significantly closed $\\mathrm { 5 . 0 \\to 2 . 3 \\ m A P }$ ) when fine-tuning with more in-domain data $( \\mathrm { A S } - 2 0 \\mathrm { K } \\mathrm { A S } - 2 \\mathrm { M } )$ ). ", + "bbox": [ + 173, + 443, + 823, + 470 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Decoder. Table 1c compares decoder attention types in Audio-MAE. Note that decoders are discarded after pretraining and only the equal-sized ViT-B encoders are fine-tuned for the end task. Our results show that local attention with shifted window achieves the best performance. Combining local and global attention (i.e., hybrid attention, Hwin) also improves vanilla global self-attention. Fig. 5 shows the qualitative reconstruction comparison. In the spectrogram of vowels, the decoder with local attention reconstructs better harmonics and recovers more context in the spectrogram. Similar phenomena are observed in the frictional sound in the middle consonant. ", + "bbox": [ + 174, + 482, + 531, + 660 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/2eac13304af84b06c5760b4abdf221924cb45af9f00f2bcb95dca8f6af1aa8b1.jpg", + "image_caption": [ + "Figure 5: Decoder reconstruction comparison. " + ], + "image_footnote": [], + "bbox": [ + 544, + 498, + 823, + 607 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 1d ablates the impact of decoder depth on mAP. A deeper 16-layer decoder achieves better performance against its shallower variants. Note that our decoder uses local window attention by default where only a fraction of tokens $4 { \\times } 4$ local windows vs. $6 4 \\times 8$ with global attention) are attended. For global attention we find 8-layer decoders to perform better than 16-layer. Table 1e compares decoder width (embedding dimension). A 512-dimension decoder achieves a good trade-off between computation and performance as a wider one is not better. ", + "bbox": [ + 173, + 667, + 825, + 751 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Pre-training Data and Setup. Table 1f summarizes the impact of pre-training dataset size. Overall the model performance is monotonically increasing when using more data for pre-training. Comparing the performance of using $1 \\%$ well-annotated AS-20K balanced data to using randomly sampled 20K unbalanced data for pre-training, the similar mAPs (39.4 vs 39.6) suggest that the distribution of data classes (balanced vs. unbalanced) is less important for pre-training. Meanwhile, as shown in Table $1 \\mathrm { g }$ , training for longer is beneficial yet the performance saturates after the 24-th epoch. ", + "bbox": [ + 174, + 761, + 825, + 844 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Out-of-domain Pre-training on ImageNet. Initializing audio models from ImageNet pre-trained weights has become popular for audio classification. However, as there are significant discrepancies between image and audio modalities, it is questionable if out-of-domain pre-training benefits audio representation learning. In Table 1h we design 3 scenarios to investigate this for Audio-MAE: (1) ", + "bbox": [ + 174, + 856, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e0146f12659732d22bc5a656b8f385365a1efb9fcc9883c795d053f067db9e94.jpg", + "table_caption": [ + "Table 2: Comparison with other state-of-the-art models on audio and speech classification tasks. Metrics are mAP for AS and accuracy $( \\% )$ for ESC/SPC/SID. For pre-training (PT) dataset, AS:AudioSet, LS:LibriSpeech, and IN:ImageNet. †: Fine-tuning results with additional supervised training on AS-2M. We gray-out models pre-trained with external non-audio datasets (e.g., ImageNet). Best single models in AS-2M are compared (no ensembles). \\*: linear evaluation results from [53]. " + ], + "table_footnote": [], + "table_body": "
ModerBackboneP1-DataAS-20KAS-ZMIESC-30SPC-2SPC-1SID
No pre-training
ERANN [58]CNN45.089.2
PANN [59]CNN27.843.183.361.8
In-domain self-supervised pre-training
wav2vec 2.0 [33]TransformerLS96.2*75.2*
HuBERT[35]TransformerLS96.3*81.4*
Conformer [37]ConformerAS=41.188.0=--
SS-AST[18]ViT-BAS+LS31.0188.898.096.064.3
Concurrent MAE-based works
MaskSpec [43]ViT-BAS32.347.189.697.7=
MAE-AST[38]ViT-BAS+LS30.6-90.097.995.863.3
Audio-MAE (global)ViT-BAS36.6±.1146.8±.0693.6±.1198.3±.0697.6±.0694.1±.06
Audio-MAE (local)ViT-BAS37.0±.1147.3±.1194.1±.1098.3±.0696.9±.0094.8± .11
Out-of-domain supervised pre-training
PSLA [30]EffNet [60]IN31.944.4=96.3=
AST[10]DeiT-BIN34.745.988.798.195.541.1
MBT[11]ViT-BIN-21K31.344.31-=
HTS-AT [29]Swin-BIN=47.197.0t98.0
PaSST[28]DeiT-BIN47.196.8†-
", + "bbox": [ + 176, + 82, + 820, + 345 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Audio-only pre-training (AS-SSL) from scratch. We consider this the ideal schema for learning audio representations as it is a simple and clean setup that prevents uncontrollable bias transfer from other modalities. (2) Directly using self-supervised ImageNet MAE models (IN-SSL) and its fine-tuned variant (IN-SL). (3) Audio-MAE self-supervised pre-training on top of these ImageNet weights. ", + "bbox": [ + 174, + 462, + 825, + 517 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The results show that (1) from-scratch audio-only pre-training is the best. For scenarios (2) and (3), we observe that ImageNet pre-training alone (2) is not sufficient (especially when the downstream data is smaller, AS-20K), and, in self-supervised pre-training on AudioSet, ImageNet initialization (3) does not help but degrades accuracy. Also in (3), supervised ImageNet pre-training (IN-SL) seems harmful. Consequently, the result suggests that out-of-domain pre-training (i.e., ImageNet) is not helpful for Audio-MAE, possibly due to domain shift. ", + "bbox": [ + 173, + 523, + 825, + 607 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.5 Comparison with the State-of-the-art ", + "text_level": 1, + "bbox": [ + 174, + 623, + 472, + 638 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 2 compares Audio-MAE (with 3-run error bars) to prior state-of-the-art. We categorize the comparison into 3 groups. For fair comparison, our main benchmark is the models in the middle group with self-supervised pre-training on in-domain (audio) datasets (AudioSet and LibriSpeech). For reference we also list other models without pre-training (the top group) and other models with supervised pre-training on out-of-domain ImageNet (the bottom group), where the latter contains previous best systems on the datasets. ", + "bbox": [ + 174, + 648, + 825, + 732 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Pre-trained on AudioSet, Audio-MAE achieves the best performance across all tasks compared to other models with in-domain self-supervised pre-training. On AudioSet-20K, its $3 7 . 1 \\ \\mathrm { m A P }$ significantly outperforms all other approaches including concurrent works and other models with outof-domain pre-training. On AudioSet-2M and ESC-50, our method also outperforms Conformer [37] and SS-AST [18]. Notably, unlike SS-AST and concurrent MAE-AST [38], which trained with additional 1,000 hours of speech in Librispeech, we use only AudioSet for pre-training. ", + "bbox": [ + 174, + 738, + 825, + 821 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In the bottom group of Table 2, Audio-MAE also outperforms previous state-of-the-art models with ImageNet supervised pre-training. Note that the proposed Audio-MAE does not rely on any out-ofdomain data and labels, nor using knowledge distillation (e.g., DeiT) from additional CNN-based models. Also, compared to HTS-AT [29] and PaSST [28], Audio-MAE is trained with audio under 16K sampling rate. As experimented in [59], there could be up to 0.4 potential mAP improvement for Audio-MAE if audio with 32K sampling rate are available. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6244fd1a7a46c5c39262c96d30a4dfd6f022b65c3eb7f73ba45382c9f667a734.jpg", + "image_caption": [ + "Figure 6: Spectrogram reconstruction visualizations on the AudioSet eval set. Column-wise type: speech, music, event, others. Masking type: (a-d) unstructured (random); (e-h) structured (time $^ +$ frequency). Masking Ratio: $70 \\%$ . In each group, we show the original spectrogram (1, top), masked input (2, middle), and MAE output (3, bottom). The spectrogram size is $1 0 2 4 \\times 1 2 8$ ; patch size is $1 6 \\times 1 6$ . Each sample has $6 4 \\times 8 = 5 1 2$ patches with 154 ( $70 \\%$ masked) patches being visible to Audio-MAE. Please click (1 2 3) for audible .wavs. More audible examples are in Supplementary. " + ], + "image_footnote": [], + "bbox": [ + 174, + 89, + 820, + 428 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For the speech tasks (SPC-1, SPC-2, and SID), Audio-MAE outperforms other models without pre-training (ERANN [58], PANN [59]), supervised (AST) and self-supervised models (SS-AST, MAE-AST). We further list other works (marked with \\*) to include the latest results introduced in the SUPERB [53] benchmark. But note that these results are not strictly comparable since SUPERB employs linear evaluation where the underlying pre-trained models are not end-to-end fine-tuned. ", + "bbox": [ + 174, + 547, + 825, + 617 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In summary, with audio-only from-scratch pre-training on AudioSet, our Audio-MAE performs well for both the audio and speech classification tasks. ", + "bbox": [ + 174, + 623, + 825, + 652 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.6 Visualization and Audible Examples by Audio-MAE Decoder ", + "text_level": 1, + "bbox": [ + 174, + 670, + 637, + 685 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For better visualization, we follow MAE [1] to use MSE over non-normalized spectrograms as the selfsupervised objective. We use ViT-L as the Audio-MAE encoder for visualization. Fig. 6 illustrates the reconstruction results sampled from the AudioSet-2M eval set. We further reconstruct .wavs using the Griffin-Lim [61] algorithm, audible under the anonymous links (accessible in respective 1 2 3). ", + "bbox": [ + 174, + 696, + 825, + 752 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "As can be seen and heard, for various masking strategies and different sounds, our Audio-MAE generates reasonable reconstruction. It works well for noisy event sounds (e.g., the reconstructed siren in Fig. 6c-3), as well as speech and music (e.g., the reconstructed singing in Fig. 6b-3). Notably, unlike visual contents that are typically scale/translation/position invariant [19], absolute positions and arrangement of spectrogram components are critical for humans to understand sound [62]. For example, shifting a pitch will make an audio sounds completely different. Also, phoneme sequences in time are important cues for speech understanding. Consequently, unstructured masking produces better aligned outputs that are closer to the ground-truth (top row in each subfigure) as the model can make better predictions based on nearby spectrogram patches; while structured masking is harder (less accurate or with words missing), especially when masking is performed over the time axis. A failure example (missing words) is the reconstructed speech in Fig. 6e-3. ", + "bbox": [ + 174, + 758, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 89, + 299, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have explored a simple extension of MAE [1] to audio data. Our Audio-MAE learns to reconstruct masked spectrogram patches from audio recordings and achieves state-of-the-art performance on six audio and speech classification tasks. We have drawn four interesting observations: First, a simple MAE approach works surprisingly well for audio spectrograms. Second, we find that it is possible to learn stronger representations with local self-attention in the decoder. Third, we show that masking can be applied to both pre-training and fine-tuning, improving accuracy and reducing training computation. The optimal strategy depends on the nature of the data (audio, image, etc.) and the learning type (self-/supervised). Fourth, the best performance can be achieved by pre-training and fine-tuning under the same modality, without reliance on cross-modality transfer learning. In future work, we aim to explore multimodal self-supervised learning with a joint audio-visual MAE approach as these domains share natural correspondences in video data. ", + "bbox": [ + 174, + 119, + 825, + 272 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgements. We thank Kaiming He and Luke Zettlemoyer for their feedback and discussions. ", + "bbox": [ + 173, + 282, + 823, + 297 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 316, + 266, + 333 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] K. He, X. Chen, S. Xie, Y. Li, P. Dollár, and R. Girshick, “Masked autoencoders are scalable vision learners,” arXiv preprint arXiv:2111.06377, 2021. [2] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” in Proceedings of the 31st International Conference on Neural Information Processing Systems, ser. NIPS’17. USA: Curran Associates Inc., 2017, pp. 6000–6010. \n[3] J. Devlin, M. Chang, K. Lee, and K. 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The decoder then re-orders and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 470, + 303 + ], + "score": 1.0, + "content": "decodes the encoded context padded with mask tokens, in order to reconstruct", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 302, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 142, + 302, + 470, + 313 + ], + "score": 1.0, + "content": "the input spectrogram. We find it beneficial to incorporate local window atten-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 470, + 325 + ], + "score": 1.0, + "content": "tion in the decoder, as audio spectrograms are highly correlated in local time and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 324, + 469, + 335 + ], + "score": 1.0, + "content": "frequency bands. We then fine-tune the encoder with a lower masking ratio on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 335, + 469, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 469, + 347 + ], + "score": 1.0, + "content": "target datasets. Empirically, Audio-MAE sets new state-of-the-art performance", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 346, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 346, + 469, + 357 + ], + "score": 1.0, + "content": "on six audio and speech classification tasks, outperforming other recent models", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 356, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 142, + 356, + 469, + 367 + ], + "score": 1.0, + "content": "that use external supervised pre-training. Our code and models is available at", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 367, + 377, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 377, + 378 + ], + "score": 1.0, + "content": "https://github.com/facebookresearch/AudioMAE.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 236, + 471, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 389, + 190, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 192, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 192, + 405 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 409, + 505, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 422 + ], + "score": 1.0, + "content": "Transformers [2] and self-supervised learning [3, 4, 5, 6, 7, 1] are dominating computer vision (CV)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "and natural language processing (NLP) research. 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Similarly in the CV community, Vision Transformers (ViT) [9] have become popular for CV", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "tasks, and, for self-supervised image representation learning, Masked Autoencoders (MAE) [1] have", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "brought the CV community closer to the success of BERT in NLP. 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An audio recording is first transformed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 477, + 204 + ], + "score": 1.0, + "content": "into a spectrogram and split into patches. 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We further provide the visualization and audible", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 558, + 443, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 443, + 570 + ], + "score": 1.0, + "content": "examples to qualitatively demonstrate the effectiveness of the Audio-MAE decoder.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 501, + 506, + 570 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 584, + 197, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 198, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 198, + 599 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Visual masked pre-training. Masked/Denoising autoencoders [21, 22, 3] are a general represen-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "tation learning methodology by reconstructing source from masked or corrupted inputs. In CV,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "visual masked pre-training has made recent progress [23, 24, 1, 20]. Based on ViT [9] that applies", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "Transformers to image patches, BEiT [17] and MAE [1] present masked image modeling frameworks.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 669 + ], + "score": 1.0, + "content": "BEiT [17] learns to predict discrete visual tokens generated by VAE [25] in masked patches. MAE [1]", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "reduces sequence length by masking a large portion of image patches randomly and encoding only", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "non-masked ones for reconstruction of pixel color information. MaskFeat [20] studies features for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "masked pre-training and finds that Histograms of Oriented Gradients (HoG) [26], which are in turn", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "related to spectrogram features, perform strongly for image and video classification models. Our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 458, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 458, + 723 + ], + "score": 1.0, + "content": "work extends the MAE framework for representation learning with audio spectrograms.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 612, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "Out-of-domain pre-training for audio. Transferring ImageNet supervised pre-trained ViT [9]", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "or ResNet [27] has become a popular practice for audio models [10, 28, 11, 29, 30, 31]. After", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "pre-training, these models operate over audio spectrograms by deflating from 3-channels (RGB)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "into 1-channel (spectrogram) in the pre-trained patch embedding in ViT and employing the rest of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "the transformer blocks on top. For example, HTS-AT [29] encodes spectrograms with hierarchical", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "Transformer initialized from the Swin Transformer [19]. MBT [11] uses ImageNet-21K pre-trained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 136, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 152 + ], + "score": 1.0, + "content": "ViT; AST [10] and PaSST [28] employ DeiT [14] as the Transformer backbone. Without using", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "out-of-domain (non-audio) data, the proposed Audio-MAE focuses on audio-only self-supervised", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 161, + 211, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 211, + 172 + ], + "score": 1.0, + "content": "pre-training from scratch.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 506, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 505, + 192 + ], + "score": 1.0, + "content": "In-domain pre-training for audio. Existing in-domain (i.e., audio-only) self-supervised methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "score": 1.0, + "content": "can be broadly categorized by the input signal type (e.g., raw waveform [32, 33, 34], frame-level", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 200, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "features [35, 36, 37], or spectrogram patches [18, 38]); and the objective used for self-supervision", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "(e.g., contrastive [39, 33, 40, 41, 35] or prediction/reconstruction [18, 34, 37, 36]). For example,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "wav2vec 2.0 [33] takes raw waveform as inputs and exploits contrastive learning to discriminate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "contextualized representations in different time segments. Mockingjay [42] proposed a masked", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "score": 1.0, + "content": "acoustic model pretext task to reconstruct frame-level Mel-features of masked time frames. SS-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 257, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 506, + 268 + ], + "score": 1.0, + "content": "AST [18] is the closest work to Audio-MAE and is our main benchmark. Inspired by the success of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "BERT [3], SS-AST proposed a self-supervised learning method which operates over spectrogram", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "patches and employs joint contrastive and reconstructive objectives on masked patches. These", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "previous methods generate audio representations by encoding full-view of both masked and non-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "masked time or spectrogram segments for self-supervised pre-training. In contrast, Audio-MAE", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 313, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 313, + 323 + ], + "score": 1.0, + "content": "encodes only the non-masked spectrogram patches.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "Our work is done independently and concurrently with [38, 43, 44] related methods. We also compare", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "score": 1.0, + "content": "our model to these concurrent works in the experiments and showcase the superiority of Audio-MAE.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 346, + 380 + ], + "lines": [ + { + "bbox": [ + 104, + 365, + 347, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 347, + 383 + ], + "score": 1.0, + "content": "3 Audio Masked Autoencoders (Audio-MAE)", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 504, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "Audio-MAE is a conceptually simple extension of MAE to learn self-supervised representations from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 484, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 484, + 416 + ], + "score": 1.0, + "content": "audio spectrograms. Fig. 1 depicts an overview. The details of each component are as follows.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "score": 1.0, + "content": "Spectrogram Patch Embeddings. Following [10, 18], we transform audio recordings into Mel-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "spectrograms and divide them into non-overlapped regular grid patches. These patches are then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "flattened and embedded by a linear projection. 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Audio-MAE masks out a large subset of spectrogram patches. As a spectrogram", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "can be viewed as a 2D representation of time and frequency components of a sound, it is reasonable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "to explore treating time and frequency differently during masking. 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Illustrative examples are shown in Fig. 2. We show masked regions with dark overlay.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "The masking mechanism, as introduced in MAE [1], is the key ingredient for efficient self-supervised", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "learning. For a input patch sequence, this can be regarded as a Bernoulli process where each patch is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 237, + 680 + ], + "score": 1.0, + "content": "masked/dropped with probability", + "type": "text" + }, + { + "bbox": [ + 238, + 669, + 244, + 679 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "(masking ratio). Masking reduces input patch sequence length and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 690 + ], + "score": 1.0, + "content": "encourages learning global, contextualized representations from limited “visible” patches. We observe", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 276, + 701 + ], + "score": 1.0, + "content": "that akin to images, a large masking rate (", + "type": "text" + }, + { + "bbox": [ + 276, + 689, + 296, + 699 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "in our experiments for spectrogram patches, which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 156, + 713 + ], + "score": 1.0, + "content": "is similar to", + "type": "text" + }, + { + "bbox": [ + 156, + 700, + 176, + 711 + ], + "score": 0.88, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "in MAE for images) is feasible for learning self-supervised audio representations.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 222, + 724 + ], + "score": 1.0, + "content": "Unlike BERT [3] that uses", + "type": "text" + }, + { + "bbox": [ + 222, + 711, + 242, + 721 + ], + "score": 0.85, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "masking rate for self-supervised learning in NLP, most of the", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "Out-of-domain pre-training for audio. Transferring ImageNet supervised pre-trained ViT [9]", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "or ResNet [27] has become a popular practice for audio models [10, 28, 11, 29, 30, 31]. After", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "pre-training, these models operate over audio spectrograms by deflating from 3-channels (RGB)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "into 1-channel (spectrogram) in the pre-trained patch embedding in ViT and employing the rest of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 130 + ], + "score": 1.0, + "content": "the transformer blocks on top. For example, HTS-AT [29] encodes spectrograms with hierarchical", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "Transformer initialized from the Swin Transformer [19]. MBT [11] uses ImageNet-21K pre-trained", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 136, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 152 + ], + "score": 1.0, + "content": "ViT; AST [10] and PaSST [28] employ DeiT [14] as the Transformer backbone. Without using", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "out-of-domain (non-audio) data, the proposed Audio-MAE focuses on audio-only self-supervised", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 161, + 211, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 211, + 172 + ], + "score": 1.0, + "content": "pre-training from scratch.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 73, + 506, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 506, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 505, + 192 + ], + "score": 1.0, + "content": "In-domain pre-training for audio. Existing in-domain (i.e., audio-only) self-supervised methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 506, + 204 + ], + "score": 1.0, + "content": "can be broadly categorized by the input signal type (e.g., raw waveform [32, 33, 34], frame-level", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 200, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "features [35, 36, 37], or spectrogram patches [18, 38]); and the objective used for self-supervision", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 212, + 506, + 226 + ], + "score": 1.0, + "content": "(e.g., contrastive [39, 33, 40, 41, 35] or prediction/reconstruction [18, 34, 37, 36]). For example,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "wav2vec 2.0 [33] takes raw waveform as inputs and exploits contrastive learning to discriminate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "contextualized representations in different time segments. Mockingjay [42] proposed a masked", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "score": 1.0, + "content": "acoustic model pretext task to reconstruct frame-level Mel-features of masked time frames. SS-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 257, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 506, + 268 + ], + "score": 1.0, + "content": "AST [18] is the closest work to Audio-MAE and is our main benchmark. Inspired by the success of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "BERT [3], SS-AST proposed a self-supervised learning method which operates over spectrogram", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "patches and employs joint contrastive and reconstructive objectives on masked patches. These", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "previous methods generate audio representations by encoding full-view of both masked and non-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "masked time or spectrogram segments for self-supervised pre-training. In contrast, Audio-MAE", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 313, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 313, + 323 + ], + "score": 1.0, + "content": "encodes only the non-masked spectrogram patches.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 179, + 506, + 323 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "Our work is done independently and concurrently with [38, 43, 44] related methods. We also compare", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 350 + ], + "score": 1.0, + "content": "our model to these concurrent works in the experiments and showcase the superiority of Audio-MAE.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 326, + 506, + 350 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 366, + 346, + 380 + ], + "lines": [ + { + "bbox": [ + 104, + 365, + 347, + 383 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 347, + 383 + ], + "score": 1.0, + "content": "3 Audio Masked Autoencoders (Audio-MAE)", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 504, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "Audio-MAE is a conceptually simple extension of MAE to learn self-supervised representations from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 484, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 484, + 416 + ], + "score": 1.0, + "content": "audio spectrograms. Fig. 1 depicts an overview. The details of each component are as follows.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 392, + 505, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 507, + 432 + ], + "score": 1.0, + "content": "Spectrogram Patch Embeddings. Following [10, 18], we transform audio recordings into Mel-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "spectrograms and divide them into non-overlapped regular grid patches. These patches are then", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "flattened and embedded by a linear projection. 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Audio-MAE masks out a large subset of spectrogram patches. As a spectrogram", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "can be viewed as a 2D representation of time and frequency components of a sound, it is reasonable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "to explore treating time and frequency differently during masking. In this work, we investigate both", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "the unstructured (i.e., random masking without any prior) and structured (i.e., randomly masking a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 245, + 632 + ], + "score": 1.0, + "content": "portion of time, frequency, or time", + "type": "text" + }, + { + "bbox": [ + 245, + 619, + 253, + 628 + ], + "score": 0.33, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 617, + 506, + 632 + ], + "score": 1.0, + "content": "frequency of a spectrogram) in the pre-training and fine-tuning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 480, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 480, + 642 + ], + "score": 1.0, + "content": "phase. Illustrative examples are shown in Fig. 2. We show masked regions with dark overlay.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 573, + 506, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "The masking mechanism, as introduced in MAE [1], is the key ingredient for efficient self-supervised", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "learning. For a input patch sequence, this can be regarded as a Bernoulli process where each patch is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 237, + 680 + ], + "score": 1.0, + "content": "masked/dropped with probability", + "type": "text" + }, + { + "bbox": [ + 238, + 669, + 244, + 679 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "(masking ratio). Masking reduces input patch sequence length and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 690 + ], + "score": 1.0, + "content": "encourages learning global, contextualized representations from limited “visible” patches. We observe", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 276, + 701 + ], + "score": 1.0, + "content": "that akin to images, a large masking rate (", + "type": "text" + }, + { + "bbox": [ + 276, + 689, + 296, + 699 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "in our experiments for spectrogram patches, which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 156, + 713 + ], + "score": 1.0, + "content": "is similar to", + "type": "text" + }, + { + "bbox": [ + 156, + 700, + 176, + 711 + ], + "score": 0.88, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "in MAE for images) is feasible for learning self-supervised audio representations.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 222, + 724 + ], + "score": 1.0, + "content": "Unlike BERT [3] that uses", + "type": "text" + }, + { + "bbox": [ + 222, + 711, + 242, + 721 + ], + "score": 0.85, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "masking rate for self-supervised learning in NLP, most of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "tokens/patches can be discarded for spectrograms as well as images due to high redundancy in these", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "modalities. Beyond self-supervised pre-training, we further explore the effectiveness of masking in", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "the supervised fine-tuning stage. 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Beyond self-supervised pre-training, we further explore the effectiveness of masking in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "the supervised fine-tuning stage. 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Audio-MAE uses a stack of standard Transformers [2] as its encoder. The encoder only", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 147, + 156 + ], + "score": 1.0, + "content": "processes", + "type": "text" + }, + { + "bbox": [ + 147, + 144, + 172, + 155 + ], + "score": 0.87, + "content": "( 2 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "non-masked patches to reduce computation overhead which is quadratic to the input", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 456, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 456, + 167 + ], + "score": 1.0, + "content": "sequence length. We use the 12-layer ViT-Base (ViT-B) [9] Transformer as our default.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "Decoder with Local Attention. The decoder is also composed of standard Transformer blocks.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "The encoded patches from the encoder are padded with trainable masked tokens. After restoring", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "the original time-frequency order in the audio spectrogram, we add the decoder’s (fixed sinusoidal)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "positional embeddings and feed the restored sequence into the decoder. At the top of the decoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 412, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 412, + 228 + ], + "score": 1.0, + "content": "stack, we add a linear head to predict and reconstruct the input spectrogram.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "To address the unique characteristics of audio spectrograms, our work investigates an enhancement to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "the vanilla MAE decoder. Image-based MAE uses global self-attention in the Transformer decoder", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "which is appropriate for visual context, because visual objects are typically invariant under translation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "or scaling, and their exact position may not affect the semantics of an image. In contrast, the position,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "scale, and translation of spectrogram features however directly affects the sound or semantics of an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "audio recording. Consequently, global self-attention is sub-optimal for spectrograms if the time-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "frequency components is predominantly local. For instance, we would have better success to use the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "harmonics (e.g., Fig. 2a) in lower bands of a vowel to predict the spectrogram patch vertically in a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "higher frequency band rather than horizontally in the time domain. Similarly, a frictional sound of a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "consonant likely only correlates to other part of the consonant, and is without dependency to other", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "silence segments in the audio recording. Compared to images, the spectrogram patches are more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 415, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 415, + 363 + ], + "score": 1.0, + "content": "similar to speech or text tokens where its order and position is more relevant.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 353, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 355, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 355, + 379 + ], + "score": 1.0, + "content": "To address the nature of audio spectrograms, in addition to us-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 378, + 354, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 354, + 390 + ], + "score": 1.0, + "content": "ing Transformers with global self-attention as in vanilla MAE,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 354, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 354, + 402 + ], + "score": 1.0, + "content": "we incorporate the local attention mechanism which groups", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 354, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 354, + 412 + ], + "score": 1.0, + "content": "and separates the spectrogram patches in to local windows", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 410, + 354, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 354, + 423 + ], + "score": 1.0, + "content": "in self-attention for decoding. We investigate two types of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 353, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 353, + 434 + ], + "score": 1.0, + "content": "local attention: (1) Shifted window location: Inspired by the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 433, + 354, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 354, + 444 + ], + "score": 1.0, + "content": "shifted-window in Swin Transformers [19], we shift window", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 443, + 354, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 157, + 455 + ], + "score": 1.0, + "content": "attention by", + "type": "text" + }, + { + "bbox": [ + 158, + 444, + 178, + 455 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 443, + 354, + 455 + ], + "score": 1.0, + "content": "between consecutive Transformer decoder", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 455, + 353, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 353, + 467 + ], + "score": 1.0, + "content": "layers. For padding the margin when shifting, we cyclically", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 466, + 354, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 354, + 477 + ], + "score": 1.0, + "content": "shift the spectrogram to the top-left direction. Fig. 3 illustrates", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 362, + 377, + 503, + 437 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 362, + 377, + 503, + 437 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 362, + 377, + 503, + 437 + ], + "spans": [ + { + "bbox": [ + 362, + 377, + 503, + 437 + ], + "score": 0.847, + "type": "image", + "image_path": "eac4e377de5817b96d0854d87b3e709eb9a487a2db0842240cefe54d8a061b6c.jpg" + } + ] + } + ], + "index": 30.0, + "virtual_lines": [ + { + "bbox": [ + 362, + 377, + 503, + 407.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 362, + 407.0, + 503, + 437.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 361, + 442, + 504, + 465 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 361, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 361, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "Figure 3: Decoder’s local attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 361, + 453, + 471, + 464 + ], + "spans": [ + { + "bbox": [ + 361, + 453, + 471, + 464 + ], + "score": 1.0, + "content": "and shifted window (right).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.0 + } + ], + "index": 33.0 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "the localized decoder attention by shifted windows. (2) Hybrid window attention (global+local", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "attention): Inspired by [45], to add better cross-window connections, we design a simple hybrid", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "score": 1.0, + "content": "(global+local) attention that computes local attention within a window in all but the last few top layers.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 510, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 506, + 521 + ], + "score": 1.0, + "content": "In this way, the input feature maps for the final reconstruction layer also contain global information.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 197, + 533 + ], + "score": 1.0, + "content": "For simplicity, we use", + "type": "text" + }, + { + "bbox": [ + 198, + 522, + 209, + 530 + ], + "score": 0.27, + "content": "_ { n o }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "pooling or hierarchical structure. Decoders with different attention types", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 531, + 195, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 173, + 543 + ], + "score": 1.0, + "content": "are compared in", + "type": "text" + }, + { + "bbox": [ + 173, + 531, + 192, + 542 + ], + "score": 0.83, + "content": "\\ S \\ O = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 531, + 195, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "Objective. The Audio-MAE decoder learns to reconstruct the input spectrogram by predicting the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "values in the spectrogram patches or their per-patch normalized ones. The objective is the mean", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "score": 1.0, + "content": "squared error (MSE) between the prediction and the input spectrogram, averaged over unknown", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "patches. Empirically we found employing the reconstruction loss alone is sufficient while including", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 591, + 466, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 466, + 604 + ], + "score": 1.0, + "content": "additional contrastive objectives (e.g., InfoNCE loss [46]) does not improve Audio-MAE.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "Fine-tuning for Downstream Tasks. In the fine-tuning stage, we only keep and fine-tune the Audio-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "MAE encoder and discard the decoder. Different from the original MAE, and inspired by [47, 28],", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "score": 1.0, + "content": "we also explore to employ masking in the fine-tuning stage to remove a portion of patches to further", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "regularize learning from a limited view of spectrogram inputs, which, as a side effect, also reduces", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "computation during fine-tuning. Compared to SpecAug [48] which takes full-length input with the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "masked portion set to zero as data augmentation, Audio-MAE sees only a subset of real-valued input", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "score": 1.0, + "content": "patches without the nullified ones. Audio-MAE then encodes these non-masked patches and applies", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 684, + 490, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 490, + 696 + ], + "score": 1.0, + "content": "an average pooling layer followed by a linear layer on top for fine-tuning in classification tasks.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 53.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 128 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 73, + 505, + 131 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 504, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "Encoder. Audio-MAE uses a stack of standard Transformers [2] as its encoder. The encoder only", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 147, + 156 + ], + "score": 1.0, + "content": "processes", + "type": "text" + }, + { + "bbox": [ + 147, + 144, + 172, + 155 + ], + "score": 0.87, + "content": "( 2 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "non-masked patches to reduce computation overhead which is quadratic to the input", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 456, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 456, + 167 + ], + "score": 1.0, + "content": "sequence length. We use the 12-layer ViT-Base (ViT-B) [9] Transformer as our default.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 133, + 506, + 167 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 171, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "Decoder with Local Attention. The decoder is also composed of standard Transformer blocks.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 195 + ], + "score": 1.0, + "content": "The encoded patches from the encoder are padded with trainable masked tokens. After restoring", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "the original time-frequency order in the audio spectrogram, we add the decoder’s (fixed sinusoidal)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "positional embeddings and feed the restored sequence into the decoder. At the top of the decoder", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 412, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 412, + 228 + ], + "score": 1.0, + "content": "stack, we add a linear head to predict and reconstruct the input spectrogram.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 171, + 506, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 230, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "To address the unique characteristics of audio spectrograms, our work investigates an enhancement to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "the vanilla MAE decoder. Image-based MAE uses global self-attention in the Transformer decoder", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "which is appropriate for visual context, because visual objects are typically invariant under translation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "or scaling, and their exact position may not affect the semantics of an image. In contrast, the position,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "scale, and translation of spectrogram features however directly affects the sound or semantics of an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "audio recording. Consequently, global self-attention is sub-optimal for spectrograms if the time-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "frequency components is predominantly local. For instance, we would have better success to use the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "harmonics (e.g., Fig. 2a) in lower bands of a vowel to predict the spectrogram patch vertically in a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 330 + ], + "score": 1.0, + "content": "higher frequency band rather than horizontally in the time domain. Similarly, a frictional sound of a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 104, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "consonant likely only correlates to other part of the consonant, and is without dependency to other", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "silence segments in the audio recording. Compared to images, the spectrogram patches are more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 415, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 415, + 363 + ], + "score": 1.0, + "content": "similar to speech or text tokens where its order and position is more relevant.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 230, + 506, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 353, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 355, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 355, + 379 + ], + "score": 1.0, + "content": "To address the nature of audio spectrograms, in addition to us-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 378, + 354, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 354, + 390 + ], + "score": 1.0, + "content": "ing Transformers with global self-attention as in vanilla MAE,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 388, + 354, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 354, + 402 + ], + "score": 1.0, + "content": "we incorporate the local attention mechanism which groups", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 400, + 354, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 354, + 412 + ], + "score": 1.0, + "content": "and separates the spectrogram patches in to local windows", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 410, + 354, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 354, + 423 + ], + "score": 1.0, + "content": "in self-attention for decoding. We investigate two types of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 353, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 353, + 434 + ], + "score": 1.0, + "content": "local attention: (1) Shifted window location: Inspired by the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 433, + 354, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 354, + 444 + ], + "score": 1.0, + "content": "shifted-window in Swin Transformers [19], we shift window", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 443, + 354, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 157, + 455 + ], + "score": 1.0, + "content": "attention by", + "type": "text" + }, + { + "bbox": [ + 158, + 444, + 178, + 455 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 443, + 354, + 455 + ], + "score": 1.0, + "content": "between consecutive Transformer decoder", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 455, + 353, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 353, + 467 + ], + "score": 1.0, + "content": "layers. For padding the margin when shifting, we cyclically", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 466, + 354, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 354, + 477 + ], + "score": 1.0, + "content": "shift the spectrogram to the top-left direction. Fig. 3 illustrates", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "the localized decoder attention by shifted windows. (2) Hybrid window attention (global+local", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "attention): Inspired by [45], to add better cross-window connections, we design a simple hybrid", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 507, + 511 + ], + "score": 1.0, + "content": "(global+local) attention that computes local attention within a window in all but the last few top layers.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 510, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 506, + 521 + ], + "score": 1.0, + "content": "In this way, the input feature maps for the final reconstruction layer also contain global information.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 197, + 533 + ], + "score": 1.0, + "content": "For simplicity, we use", + "type": "text" + }, + { + "bbox": [ + 198, + 522, + 209, + 530 + ], + "score": 0.27, + "content": "_ { n o }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "pooling or hierarchical structure. Decoders with different attention types", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 531, + 195, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 173, + 543 + ], + "score": 1.0, + "content": "are compared in", + "type": "text" + }, + { + "bbox": [ + 173, + 531, + 192, + 542 + ], + "score": 0.83, + "content": "\\ S \\ O = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 531, + 195, + 543 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 366, + 355, + 477 + ] + }, + { + "type": "image", + "bbox": [ + 362, + 377, + 503, + 437 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 362, + 377, + 503, + 437 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 362, + 377, + 503, + 437 + ], + "spans": [ + { + "bbox": [ + 362, + 377, + 503, + 437 + ], + "score": 0.847, + "type": "image", + "image_path": "eac4e377de5817b96d0854d87b3e709eb9a487a2db0842240cefe54d8a061b6c.jpg" + } + ] + } + ], + "index": 30.0, + "virtual_lines": [ + { + "bbox": [ + 362, + 377, + 503, + 407.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 362, + 407.0, + 503, + 437.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 361, + 442, + 504, + 465 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 361, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 361, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "Figure 3: Decoder’s local attention", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 361, + 453, + 471, + 464 + ], + "spans": [ + { + "bbox": [ + 361, + 453, + 471, + 464 + ], + "score": 1.0, + "content": "and shifted window (right).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.0 + } + ], + "index": 33.0 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 542 + ], + "lines": [], + "index": 41.5, + "bbox_fs": [ + 105, + 476, + 507, + 543 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "Objective. The Audio-MAE decoder learns to reconstruct the input spectrogram by predicting the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "values in the spectrogram patches or their per-patch normalized ones. The objective is the mean", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "score": 1.0, + "content": "squared error (MSE) between the prediction and the input spectrogram, averaged over unknown", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "patches. Empirically we found employing the reconstruction loss alone is sufficient while including", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 591, + 466, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 466, + 604 + ], + "score": 1.0, + "content": "additional contrastive objectives (e.g., InfoNCE loss [46]) does not improve Audio-MAE.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 546, + 506, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 620 + ], + "score": 1.0, + "content": "Fine-tuning for Downstream Tasks. In the fine-tuning stage, we only keep and fine-tune the Audio-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "MAE encoder and discard the decoder. Different from the original MAE, and inspired by [47, 28],", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 506, + 642 + ], + "score": 1.0, + "content": "we also explore to employ masking in the fine-tuning stage to remove a portion of patches to further", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "regularize learning from a limited view of spectrogram inputs, which, as a side effect, also reduces", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 663 + ], + "score": 1.0, + "content": "computation during fine-tuning. Compared to SpecAug [48] which takes full-length input with the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 505, + 674 + ], + "score": 1.0, + "content": "masked portion set to zero as data augmentation, Audio-MAE sees only a subset of real-valued input", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 506, + 686 + ], + "score": 1.0, + "content": "patches without the nullified ones. Audio-MAE then encodes these non-masked patches and applies", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 684, + 490, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 490, + 696 + ], + "score": 1.0, + "content": "an average pooling layer followed by a linear layer on top for fine-tuning in classification tasks.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 607, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 191, + 85 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 193, + 88 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 193, + 88 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 96, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 506, + 109 + ], + "score": 1.0, + "content": "We perform an extensive evaluation on six tasks, including audio classification on AudioSet (AS-2M,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 121 + ], + "score": 1.0, + "content": "AS-20K) and Environmental Sound Classification (ESC-50), and speech classification on Speech", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 119, + 474, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 474, + 131 + ], + "score": 1.0, + "content": "Commands (SPC-1 and SPC-2) and VoxCeleb (SID). We use AudioSet for ablation studies.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 144, + 212, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 212, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 212, + 156 + ], + "score": 1.0, + "content": "4.1 Datasets and Tasks", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 282, + 177 + ], + "score": 1.0, + "content": "AudioSet [12] (AS-2M, AS-20K) contains", + "type": "text" + }, + { + "bbox": [ + 282, + 165, + 295, + 175 + ], + "score": 0.78, + "content": "{ \\sim } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "million 10-second YouTube clips for audio classifi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "cation. 527 types of audio events are weakly annotated [49, 50, 51] for each clip. There could be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "multiple events in a clip. The full training set has 2 subsets: A class-wise balanced (22,176 clips) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "an unbalanced (2,042,985 clips) set. The eval set has 20,383 clips. We downloaded and processed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 446, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 446, + 222 + ], + "score": 1.0, + "content": "around 1.96M unbalanced training, 21K balanced training, and 19K evaluation clips.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "For the AS-2M experiments, we use the union of unbalanced and balanced training audio for pre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "training and fine-tuning. For the AS-20K experiments, we use AS-2M for pre-training and the 20K", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 246, + 485, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 485, + 259 + ], + "score": 1.0, + "content": "balanced set for fine-tuning. We report the testing mAP on the 19K eval set used by AST [10].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "Environmental Sound Classification (ESC-50) [13] is an audio classification dataset consists of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "2,000 5-second environmental sound recordings. There are 50 classes in ESC. We report accuracy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 357, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 357, + 298 + ], + "score": 1.0, + "content": "under 5-fold cross-validation with the same split used by [10].", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Speech Commands (SPC-2, SPC-1) [52] are two keyword spotting tasks. In SPC-2, there are 35", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "speech commands. The training/validation/testing set has 84,843/9,981/11,005 1-second recordings,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "respectively. In SPC-1, there are 10 classes of keywords, 1 silence class, and 1 unknown class that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "includes all the other 20 common speech commands. 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For the decoder, we use a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "16-layer Transformer with shifted local attention. 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By default, we use a masking ratio of 0.8 with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "(unstructured) random masking for pre-training. During fine-tuning, we employ a lower masking", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 476, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 453, + 573 + ], + "score": 1.0, + "content": "ratio (0.3 in time and 0.3 in frequency). Ablations on these design choices are given in", + "type": "text" + }, + { + "bbox": [ + 454, + 561, + 473, + 572 + ], + "score": 0.83, + "content": "\\ S \\ O = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 560, + 476, + 573 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 108, + 586, + 254, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 255, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 255, + 601 + ], + "score": 1.0, + "content": "4.3 Pre-training and Fine-tuning", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "We use AudioSet-2M for pre-training and randomly iterate over all audio recordings. We train for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "32 epochs with a batch size of 512 and a 0.0002 learning rate. We distribute the training load over", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 289, + 641 + ], + "score": 1.0, + "content": "64 V100 GPUs and the total training time is", + "type": "text" + }, + { + "bbox": [ + 289, + 629, + 307, + 640 + ], + "score": 0.85, + "content": "{ \\sim } 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "hours. 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For the supervised", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "fine-tuning on AudioSet-2M, since the size of training samples are uneven across classes (unbalanced),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "we follow the common practice of using a weighted sampling to balance the classes during training. In", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 267, + 723 + ], + "score": 1.0, + "content": "each epoch, we sample 200K instances (", + "type": "text" + }, + { + "bbox": [ + 267, + 711, + 293, + 721 + ], + "score": 0.88, + "content": "\\mathord { \\sim } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "of AudioSet-2M) without replacement. 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We use AudioSet for ablation studies.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 96, + 506, + 131 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 144, + 212, + 156 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 212, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 212, + 156 + ], + "score": 1.0, + "content": "4.1 Datasets and Tasks", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 164, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 282, + 177 + ], + "score": 1.0, + "content": "AudioSet [12] (AS-2M, AS-20K) contains", + "type": "text" + }, + { + "bbox": [ + 282, + 165, + 295, + 175 + ], + "score": 0.78, + "content": "{ \\sim } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "million 10-second YouTube clips for audio classifi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "cation. 527 types of audio events are weakly annotated [49, 50, 51] for each clip. There could be", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "multiple events in a clip. The full training set has 2 subsets: A class-wise balanced (22,176 clips) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "an unbalanced (2,042,985 clips) set. The eval set has 20,383 clips. We downloaded and processed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 446, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 446, + 222 + ], + "score": 1.0, + "content": "around 1.96M unbalanced training, 21K balanced training, and 19K evaluation clips.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 165, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 225, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "For the AS-2M experiments, we use the union of unbalanced and balanced training audio for pre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "training and fine-tuning. For the AS-20K experiments, we use AS-2M for pre-training and the 20K", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 246, + 485, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 485, + 259 + ], + "score": 1.0, + "content": "balanced set for fine-tuning. We report the testing mAP on the 19K eval set used by AST [10].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 224, + 506, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "Environmental Sound Classification (ESC-50) [13] is an audio classification dataset consists of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 288 + ], + "score": 1.0, + "content": "2,000 5-second environmental sound recordings. There are 50 classes in ESC. We report accuracy", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 357, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 357, + 298 + ], + "score": 1.0, + "content": "under 5-fold cross-validation with the same split used by [10].", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 263, + 505, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "Speech Commands (SPC-2, SPC-1) [52] are two keyword spotting tasks. In SPC-2, there are 35", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "speech commands. The training/validation/testing set has 84,843/9,981/11,005 1-second recordings,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "respectively. In SPC-1, there are 10 classes of keywords, 1 silence class, and 1 unknown class that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "includes all the other 20 common speech commands. We use the data and split provided in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 332, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 332, + 359 + ], + "score": 1.0, + "content": "SUPERB [53] benchmark to report the testing accuracy.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 301, + 506, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "VoxCeleb (SID) [54] contains 150K utterances from 1,251 speakers. The speaker identification", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "task (SID) is to classify the utterances to identify its original speaker. We use the V1 standard train", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 431, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 431, + 397 + ], + "score": 1.0, + "content": "(138,361), validation (6,904), testing (8,251) sets and report the testing accuracy.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 360, + 505, + 397 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 230, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 231, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 231, + 422 + ], + "score": 1.0, + "content": "4.2 Implementation Details", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 429, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "We use a vanilla 12-layer ViT-B by default as the Transformer encoder. For the decoder, we use a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 453 + ], + "score": 1.0, + "content": "16-layer Transformer with shifted local attention. We investigate the vanilla (global attention) and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 452, + 365, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 365, + 464 + ], + "score": 1.0, + "content": "hybrid (global+local attention) decoder variants (see Table. 1c).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 428, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "Following [10, 11], we transform raw waveform (pre-processed as mono channel under 16,000", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 409, + 491 + ], + "score": 1.0, + "content": "sampling rate) into 128 Kaldi [55]-compatible Mel-frequency bands with a", + "type": "text" + }, + { + "bbox": [ + 409, + 479, + 433, + 489 + ], + "score": 0.44, + "content": "2 5 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Hanning window", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 176, + 502 + ], + "score": 1.0, + "content": "that shifts every", + "type": "text" + }, + { + "bbox": [ + 176, + 490, + 202, + 500 + ], + "score": 0.54, + "content": "1 0 ~ \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 489, + 506, + 502 + ], + "score": 1.0, + "content": ". For a 10-second recording in AudioSet, the resulting spectrogram is of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 499, + 210, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 163, + 511 + ], + "score": 0.89, + "content": "1 \\times 1 0 2 4 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 499, + 210, + 513 + ], + "score": 1.0, + "content": "dimension.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 467, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "For patch embedding, we use convolutional kernels with (16, 16) size and stride in time and frequency", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "(thus, patches are non-overlapping) to avoid short-cuts via overlap in self-supervision (though, at", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "high masking ratios such short-cuts are less severe). By default, we use a masking ratio of 0.8 with", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "(unstructured) random masking for pre-training. During fine-tuning, we employ a lower masking", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 560, + 476, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 453, + 573 + ], + "score": 1.0, + "content": "ratio (0.3 in time and 0.3 in frequency). Ablations on these design choices are given in", + "type": "text" + }, + { + "bbox": [ + 454, + 561, + 473, + 572 + ], + "score": 0.83, + "content": "\\ S \\ O = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 560, + 476, + 573 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 516, + 506, + 573 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 586, + 254, + 598 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 255, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 255, + 601 + ], + "score": 1.0, + "content": "4.3 Pre-training and Fine-tuning", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "We use AudioSet-2M for pre-training and randomly iterate over all audio recordings. We train for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "32 epochs with a batch size of 512 and a 0.0002 learning rate. We distribute the training load over", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 289, + 641 + ], + "score": 1.0, + "content": "64 V100 GPUs and the total training time is", + "type": "text" + }, + { + "bbox": [ + 289, + 629, + 307, + 640 + ], + "score": 0.85, + "content": "{ \\sim } 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "hours. For each audio, we randomly sample the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 640, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 474, + 651 + ], + "score": 1.0, + "content": "starting time, cyclically extract 10-second audio, and randomly jitter its magnitude by up to", + "type": "text" + }, + { + "bbox": [ + 474, + 640, + 503, + 651 + ], + "score": 0.66, + "content": "\\pm 6 \\mathrm { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 640, + 506, + 651 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 664 + ], + "score": 1.0, + "content": "We use only natural audio spectrograms and apply no augmentations (e.g., [48, 56, 57]) as we do not", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 661, + 368, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 368, + 675 + ], + "score": 1.0, + "content": "find these strong augmentations helpful in the pre-training phase.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 607, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "In the fine-tuning phase, we remove the decoder and only fine-tune the encoder. For the supervised", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "fine-tuning on AudioSet-2M, since the size of training samples are uneven across classes (unbalanced),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "we follow the common practice of using a weighted sampling to balance the classes during training. In", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 267, + 723 + ], + "score": 1.0, + "content": "each epoch, we sample 200K instances (", + "type": "text" + }, + { + "bbox": [ + 267, + 711, + 293, + 721 + ], + "score": 0.88, + "content": "\\mathord { \\sim } 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "of AudioSet-2M) without replacement. We fine-tune", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 209, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 251, + 224 + ], + "score": 1.0, + "content": "for 100 epochs, which aggregate to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 251, + 210, + 269, + 221 + ], + "score": 0.85, + "content": "{ \\sim } 1 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 270, + 209, + 505, + 224 + ], + "score": 1.0, + "content": "full epochs of AudioSet-2M. The probability of sampling", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "an instance is inversely proportional to the dataset-wise occurrences of its classes. Fine-tuning on", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 168, + 244 + ], + "score": 1.0, + "content": "64 GPUs takes", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 168, + 232, + 186, + 243 + ], + "score": 0.83, + "content": "{ \\sim } 1 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 186, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "hours. For the smaller balanced AudioSet-20K, we fine-tune on 4 GPUs for 60", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 244, + 486, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 486, + 255 + ], + "score": 1.0, + "content": "epochs without weighted sampling. Please see Supplementary for the details on other datasets.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 69, + 497, + 157 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 69, + 497, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 69, + 497, + 157 + ], + "spans": [ + { + "bbox": [ + 111, + 69, + 497, + 157 + ], + "score": 0.891, + "type": "image", + "image_path": "77981a7649efa2be98251d63c87ce0bb29948d0455d85ac60829270b7d834094.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 69, + 497, + 98.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 98.33333333333333, + 497, + 127.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 127.66666666666666, + 497, + 157.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 163, + 505, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "score": 1.0, + "content": "Figure 4: Masking strategy. For pre-training, a higher ratio and unstructured masking (random)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 397, + 186 + ], + "score": 1.0, + "content": "is preferred. For fine-tuning, a lower ratio and structured masking (time", + "type": "text" + }, + { + "bbox": [ + 397, + 175, + 404, + 185 + ], + "score": 0.39, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "frequency) is better. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 185, + 484, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 248, + 198 + ], + "score": 1.0, + "content": "y-axes are mAP on AS-2M and the", + "type": "text" + }, + { + "bbox": [ + 248, + 187, + 255, + 195 + ], + "score": 0.31, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 185, + 484, + 198 + ], + "score": 1.0, + "content": "-axes are masking ratio. This ablation format follows [1].", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 251, + 224 + ], + "score": 1.0, + "content": "for 100 epochs, which aggregate to", + "type": "text" + }, + { + "bbox": [ + 251, + 210, + 269, + 221 + ], + "score": 0.85, + "content": "{ \\sim } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 209, + 505, + 224 + ], + "score": 1.0, + "content": "full epochs of AudioSet-2M. The probability of sampling", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "an instance is inversely proportional to the dataset-wise occurrences of its classes. Fine-tuning on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 168, + 244 + ], + "score": 1.0, + "content": "64 GPUs takes", + "type": "text" + }, + { + "bbox": [ + 168, + 232, + 186, + 243 + ], + "score": 0.83, + "content": "{ \\sim } 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "hours. For the smaller balanced AudioSet-20K, we fine-tune on 4 GPUs for 60", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 244, + 486, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 486, + 255 + ], + "score": 1.0, + "content": "epochs without weighted sampling. Please see Supplementary for the details on other datasets.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 270, + 266, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 267, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 267, + 283 + ], + "score": 1.0, + "content": "4.4 Ablations and Model Properties", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 308 + ], + "score": 1.0, + "content": "Masking Strategies in Pre-training and Fine-tuning. In Fig. 4, we compare different pre-training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 319 + ], + "score": 1.0, + "content": "and fine-tuning masking strategies for Audio-MAE. First, in Fig. 4a we explore the pre-training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "masking ratio. We observe, similar as in MAE for images [1], that a high pre-training masking", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 131, + 339 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 132, + 327, + 151, + 338 + ], + "score": 0.85, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "in our case) is optimal for audio spectrograms. This is due to the fact that both audio", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 339, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 350 + ], + "score": 1.0, + "content": "spectrograms and images are continuous signals with significant redundancy. Further, we find the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "unstructured random masking works the best for self-supervised pre-training over more structured", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 360, + 239, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 239, + 373 + ], + "score": 1.0, + "content": "masking (e.g., time+frequency).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "Unlike MAE for images, there are clear performance differences among masking strategies when", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 401 + ], + "score": 1.0, + "content": "pre-training with audio spectrograms. Comparing Audio-MAE reconstructions between Fig. 6a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "to 6e and 6d to 6h, under the same masking ratio, we observe the unstructured random masking", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "is comparably easier than structured masking (i.e., time and/or frequency) as the model can guess", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "the missing component by extrapolating nearby context (e.g., formants in vowels and frictional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "sounds in consonants around). We also observe that for higher masking ratios, the structured masking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "alternatives drop in performance, presumably because the task becomes too difficult while random", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 241, + 466 + ], + "score": 1.0, + "content": "masking improves steadily up to", + "type": "text" + }, + { + "bbox": [ + 242, + 453, + 262, + 463 + ], + "score": 0.88, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 452, + 505, + 466 + ], + "score": 1.0, + "content": ". This result show that designing a pretext task with proper", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "hardness is important for effective self-supervised learning of audio representations. We therefore", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 387, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 243, + 489 + ], + "score": 1.0, + "content": "use random masking with ratio of", + "type": "text" + }, + { + "bbox": [ + 243, + 475, + 263, + 485 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 474, + 387, + 489 + ], + "score": 1.0, + "content": "as our default for pre-training.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Fig. 4b studies the effect of masking during the fine-tuning phase. We see that in this case, it is more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "beneficial to use structured masking: time+frequency performs better than time- or frequency-based", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "masking, and these perform better than unstructured masking. Overall, we see that the optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "masking ratios are lower than for pre-training and we use 0.3 as our default in the fine-tuning phase.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "In general, we observe that for task-agnostic pre-training, unstructured masking with a higher ratio is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 551, + 501, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 501, + 564 + ], + "score": 1.0, + "content": "preferred. While in task-specific fine-tuning, structured masking with lower ratios performs better.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 570, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "Impact of Patch Size and Stride. We compare the performance of Audio-MAE trained with different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 366, + 595 + ], + "score": 1.0, + "content": "patch sizes and strides in Table 1a. A non-zero overlap (i.e., stride", + "type": "text" + }, + { + "bbox": [ + 366, + 583, + 376, + 592 + ], + "score": 0.75, + "content": "<", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "patch size) between patches will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "increase the number of patches and quadratically increase computation in floating point operations", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 616 + ], + "score": 1.0, + "content": "(FLOPs), as reported in the table. Most prior works follow AST [10] to use overlapped patches", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 132, + 628 + ], + "score": 1.0, + "content": "(patch", + "type": "text" + }, + { + "bbox": [ + 132, + 615, + 155, + 625 + ], + "score": 0.74, + "content": "= 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 614, + 195, + 628 + ], + "score": 1.0, + "content": "and stride", + "type": "text" + }, + { + "bbox": [ + 195, + 615, + 217, + 625 + ], + "score": 0.84, + "content": "= 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 614, + 506, + 628 + ], + "score": 1.0, + "content": ") to boost end task performance. As shown in Table 1a, we do not observe", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 409, + 638 + ], + "score": 1.0, + "content": "a performance improvement using overlapped patches for Audio-MAE (both", + "type": "text" + }, + { + "bbox": [ + 409, + 626, + 451, + 636 + ], + "score": 0.45, + "content": "4 7 . 3 \\mathrm { m A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "), presumably", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "because due to overlap, the patch embedding can leak information into the masked patches. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 172, + 660 + ], + "score": 1.0, + "content": "non-overlapped", + "type": "text" + }, + { + "bbox": [ + 172, + 647, + 201, + 658 + ], + "score": 0.89, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "patches achieve a good balance between computation and performance. By", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 657, + 288, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 288, + 672 + ], + "score": 1.0, + "content": "default, we use this setup in our experiments.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "Encoder. We investigate the design choices of encoder and decoder architectures in Audio-MAE.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "Table 1b shows the trade-off between encoder model size and performance. As expected, larger", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 700, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 507, + 713 + ], + "score": 1.0, + "content": "models achieve better performance, at a cost of computation and memory. The accuracy gain of ViT-L", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "over ViT-B/S is more significant on the smaller and balanced AS-20K. For ViT-S, the performance", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 69, + 497, + 157 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 69, + 497, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 69, + 497, + 157 + ], + "spans": [ + { + "bbox": [ + 111, + 69, + 497, + 157 + ], + "score": 0.891, + "type": "image", + "image_path": "77981a7649efa2be98251d63c87ce0bb29948d0455d85ac60829270b7d834094.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 69, + 497, + 98.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 98.33333333333333, + 497, + 127.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 127.66666666666666, + 497, + 157.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 163, + 505, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 506, + 176 + ], + "score": 1.0, + "content": "Figure 4: Masking strategy. For pre-training, a higher ratio and unstructured masking (random)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 397, + 186 + ], + "score": 1.0, + "content": "is preferred. For fine-tuning, a lower ratio and structured masking (time", + "type": "text" + }, + { + "bbox": [ + 397, + 175, + 404, + 185 + ], + "score": 0.39, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "frequency) is better. The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 185, + 484, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 248, + 198 + ], + "score": 1.0, + "content": "y-axes are mAP on AS-2M and the", + "type": "text" + }, + { + "bbox": [ + 248, + 187, + 255, + 195 + ], + "score": 0.31, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 185, + 484, + 198 + ], + "score": 1.0, + "content": "-axes are masking ratio. This ablation format follows [1].", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 254 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 209, + 505, + 255 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 270, + 266, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 267, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 267, + 283 + ], + "score": 1.0, + "content": "4.4 Ablations and Model Properties", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 308 + ], + "score": 1.0, + "content": "Masking Strategies in Pre-training and Fine-tuning. In Fig. 4, we compare different pre-training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 304, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 319 + ], + "score": 1.0, + "content": "and fine-tuning masking strategies for Audio-MAE. First, in Fig. 4a we explore the pre-training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "masking ratio. We observe, similar as in MAE for images [1], that a high pre-training masking", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 327, + 504, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 131, + 339 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 132, + 327, + 151, + 338 + ], + "score": 0.85, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 327, + 504, + 339 + ], + "score": 1.0, + "content": "in our case) is optimal for audio spectrograms. This is due to the fact that both audio", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 339, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 350 + ], + "score": 1.0, + "content": "spectrograms and images are continuous signals with significant redundancy. Further, we find the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "unstructured random masking works the best for self-supervised pre-training over more structured", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 360, + 239, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 239, + 373 + ], + "score": 1.0, + "content": "masking (e.g., time+frequency).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 293, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "Unlike MAE for images, there are clear performance differences among masking strategies when", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 506, + 401 + ], + "score": 1.0, + "content": "pre-training with audio spectrograms. Comparing Audio-MAE reconstructions between Fig. 6a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "to 6e and 6d to 6h, under the same masking ratio, we observe the unstructured random masking", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "is comparably easier than structured masking (i.e., time and/or frequency) as the model can guess", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "the missing component by extrapolating nearby context (e.g., formants in vowels and frictional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "sounds in consonants around). We also observe that for higher masking ratios, the structured masking", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "alternatives drop in performance, presumably because the task becomes too difficult while random", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 241, + 466 + ], + "score": 1.0, + "content": "masking improves steadily up to", + "type": "text" + }, + { + "bbox": [ + 242, + 453, + 262, + 463 + ], + "score": 0.88, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 452, + 505, + 466 + ], + "score": 1.0, + "content": ". This result show that designing a pretext task with proper", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "hardness is important for effective self-supervised learning of audio representations. We therefore", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 474, + 387, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 243, + 489 + ], + "score": 1.0, + "content": "use random masking with ratio of", + "type": "text" + }, + { + "bbox": [ + 243, + 475, + 263, + 485 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 474, + 387, + 489 + ], + "score": 1.0, + "content": "as our default for pre-training.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 375, + 506, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 535 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Fig. 4b studies the effect of masking during the fine-tuning phase. We see that in this case, it is more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "beneficial to use structured masking: time+frequency performs better than time- or frequency-based", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "masking, and these perform better than unstructured masking. Overall, we see that the optimal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "masking ratios are lower than for pre-training and we use 0.3 as our default in the fine-tuning phase.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 491, + 506, + 536 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 504, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "In general, we observe that for task-agnostic pre-training, unstructured masking with a higher ratio is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 551, + 501, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 501, + 564 + ], + "score": 1.0, + "content": "preferred. 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Attention typeAS-20K AS-2M ESC-50 SID
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A deeper 16-layer decoder achieves better", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "performance against its shallower variants. Note that our decoder uses local window attention by", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 273, + 564 + ], + "score": 1.0, + "content": "default where only a fraction of tokens", + "type": "text" + }, + { + "bbox": [ + 273, + 551, + 293, + 561 + ], + "score": 0.86, + "content": "4 { \\times } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 550, + 372, + 564 + ], + "score": 1.0, + "content": "local windows vs.", + "type": "text" + }, + { + "bbox": [ + 372, + 551, + 397, + 561 + ], + "score": 0.89, + "content": "6 4 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "with global attention) are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "attended. For global attention we find 8-layer decoders to perform better than 16-layer. Table 1e", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "compares decoder width (embedding dimension). A 512-dimension decoder achieves a good trade-off", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 584, + 375, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 375, + 595 + ], + "score": 1.0, + "content": "between computation and performance as a wider one is not better.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 528, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "Pre-training Data and Setup. Table 1f summarizes the impact of pre-training dataset size. Overall", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "the model performance is monotonically increasing when using more data for pre-training. Comparing", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 625, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 208, + 637 + ], + "score": 1.0, + "content": "the performance of using", + "type": "text" + }, + { + "bbox": [ + 208, + 626, + 222, + 636 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 625, + 505, + 637 + ], + "score": 1.0, + "content": "well-annotated AS-20K balanced data to using randomly sampled 20K", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 636, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 648 + ], + "score": 1.0, + "content": "unbalanced data for pre-training, the similar mAPs (39.4 vs 39.6) suggest that the distribution of", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "data classes (balanced vs. unbalanced) is less important for pre-training. Meanwhile, as shown in", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 658, + 478, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 131, + 671 + ], + "score": 1.0, + "content": "Table", + "type": "text" + }, + { + "bbox": [ + 131, + 659, + 142, + 670 + ], + "score": 0.27, + "content": "1 \\mathrm { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 658, + 478, + 671 + ], + "score": 1.0, + "content": ", training for longer is beneficial yet the performance saturates after the 24-th epoch.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 604, + 506, + 671 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "Out-of-domain Pre-training on ImageNet. Initializing audio models from ImageNet pre-trained", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "weights has become popular for audio classification. However, as there are significant discrepancies", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "between image and audio modalities, it is questionable if out-of-domain pre-training benefits audio", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "representation learning. In Table 1h we design 3 scenarios to investigate this for Audio-MAE: (1)", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 55.5, + "bbox_fs": [ + 105, + 678, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 65, + 502, + 274 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 65, + 502, + 274 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 65, + 502, + 274 + ], + "spans": [ + { + "bbox": [ + 108, + 65, + 502, + 274 + ], + "score": 0.972, + "html": "
ModerBackboneP1-DataAS-20KAS-ZMIESC-30SPC-2SPC-1SID
No pre-training
ERANN [58]CNN45.089.2
PANN [59]CNN27.843.183.361.8
In-domain self-supervised pre-training
wav2vec 2.0 [33]TransformerLS96.2*75.2*
HuBERT[35]TransformerLS96.3*81.4*
Conformer [37]ConformerAS=41.188.0=--
SS-AST[18]ViT-BAS+LS31.0188.898.096.064.3
Concurrent MAE-based works
MaskSpec [43]ViT-BAS32.347.189.697.7=
MAE-AST[38]ViT-BAS+LS30.6-90.097.995.863.3
Audio-MAE (global)ViT-BAS36.6±.1146.8±.0693.6±.1198.3±.0697.6±.0694.1±.06
Audio-MAE (local)ViT-BAS37.0±.1147.3±.1194.1±.1098.3±.0696.9±.0094.8± .11
Out-of-domain supervised pre-training
PSLA [30]EffNet [60]IN31.944.4=96.3=
AST[10]DeiT-BIN34.745.988.798.195.541.1
MBT[11]ViT-BIN-21K31.344.31-=
HTS-AT [29]Swin-BIN=47.197.0t98.0
PaSST[28]DeiT-BIN47.196.8†-
", + "type": "table", + "image_path": "e0146f12659732d22bc5a656b8f385365a1efb9fcc9883c795d053f067db9e94.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 65, + 502, + 134.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666669, + 502, + 204.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 204.33333333333337, + 502, + 274.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 278, + 506, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "Table 2: Comparison with other state-of-the-art models on audio and speech classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 289, + 302 + ], + "score": 1.0, + "content": "tasks. Metrics are mAP for AS and accuracy", + "type": "text" + }, + { + "bbox": [ + 290, + 289, + 306, + 300 + ], + "score": 0.7, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "for ESC/SPC/SID. For pre-training (PT) dataset,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "AS:AudioSet, LS:LibriSpeech, and IN:ImageNet. †: Fine-tuning results with additional supervised", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "training on AS-2M. We gray-out models pre-trained with external non-audio datasets (e.g., ImageNet).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 322, + 500, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 500, + 334 + ], + "score": 1.0, + "content": "Best single models in AS-2M are compared (no ensembles). *: linear evaluation results from [53].", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "Audio-only pre-training (AS-SSL) from scratch. We consider this the ideal schema for learning audio", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "representations as it is a simple and clean setup that prevents uncontrollable bias transfer from other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "modalities. (2) Directly using self-supervised ImageNet MAE models (IN-SSL) and its fine-tuned", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 399, + 491, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 491, + 411 + ], + "score": 1.0, + "content": "variant (IN-SL). (3) Audio-MAE self-supervised pre-training on top of these ImageNet weights.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "The results show that (1) from-scratch audio-only pre-training is the best. For scenarios (2) and (3),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "we observe that ImageNet pre-training alone (2) is not sufficient (especially when the downstream", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 436, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 451 + ], + "score": 1.0, + "content": "data is smaller, AS-20K), and, in self-supervised pre-training on AudioSet, ImageNet initialization", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "(3) does not help but degrades accuracy. Also in (3), supervised ImageNet pre-training (IN-SL) seems", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "harmful. Consequently, the result suggests that out-of-domain pre-training (i.e., ImageNet) is not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 470, + 325, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 325, + 482 + ], + "score": 1.0, + "content": "helpful for Audio-MAE, possibly due to domain shift.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 494, + 289, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 290, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 290, + 507 + ], + "score": 1.0, + "content": "4.5 Comparison with the State-of-the-art", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "Table 2 compares Audio-MAE (with 3-run error bars) to prior state-of-the-art. We categorize the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 526, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 537 + ], + "score": 1.0, + "content": "comparison into 3 groups. For fair comparison, our main benchmark is the models in the middle", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "group with self-supervised pre-training on in-domain (audio) datasets (AudioSet and LibriSpeech).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "For reference we also list other models without pre-training (the top group) and other models with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "supervised pre-training on out-of-domain ImageNet (the bottom group), where the latter contains", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 569, + 259, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 259, + 581 + ], + "score": 1.0, + "content": "previous best systems on the datasets.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "Pre-trained on AudioSet, Audio-MAE achieves the best performance across all tasks compared", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 460, + 609 + ], + "score": 1.0, + "content": "to other models with in-domain self-supervised pre-training. On AudioSet-20K, its", + "type": "text" + }, + { + "bbox": [ + 460, + 596, + 505, + 607 + ], + "score": 0.51, + "content": "3 7 . 1 \\ \\mathrm { m A P }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 606, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 507, + 621 + ], + "score": 1.0, + "content": "significantly outperforms all other approaches including concurrent works and other models with out-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "of-domain pre-training. On AudioSet-2M and ESC-50, our method also outperforms Conformer [37]", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "and SS-AST [18]. Notably, unlike SS-AST and concurrent MAE-AST [38], which trained with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 457, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 457, + 654 + ], + "score": 1.0, + "content": "additional 1,000 hours of speech in Librispeech, we use only AudioSet for pre-training.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "In the bottom group of Table 2, Audio-MAE also outperforms previous state-of-the-art models with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "ImageNet supervised pre-training. Note that the proposed Audio-MAE does not rely on any out-of-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "domain data and labels, nor using knowledge distillation (e.g., DeiT) from additional CNN-based", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "models. Also, compared to HTS-AT [29] and PaSST [28], Audio-MAE is trained with audio under", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "16K sampling rate. As experimented in [59], there could be up to 0.4 potential mAP improvement for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 343, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 343, + 723 + ], + "score": 1.0, + "content": "Audio-MAE if audio with 32K sampling rate are available.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 194, + 60, + 481, + 71 + ], + "lines": [ + { + "bbox": [ + 191, + 56, + 483, + 74 + ], + "spans": [ + { + "bbox": [ + 191, + 56, + 483, + 74 + ], + "score": 1.0, + "content": "Backbone PT-Data AS-20K AS-2M ESC-50 SPC-2 SPC-1 SID", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 111, + 61, + 136, + 70 + ], + "lines": [ + { + "bbox": [ + 109, + 60, + 138, + 72 + ], + "spans": [ + { + "bbox": [ + 109, + 60, + 138, + 72 + ], + "score": 1.0, + "content": "Model", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 65, + 502, + 274 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 65, + 502, + 274 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 65, + 502, + 274 + ], + "spans": [ + { + "bbox": [ + 108, + 65, + 502, + 274 + ], + "score": 0.972, + "html": "
ModerBackboneP1-DataAS-20KAS-ZMIESC-30SPC-2SPC-1SID
No pre-training
ERANN [58]CNN45.089.2
PANN [59]CNN27.843.183.361.8
In-domain self-supervised pre-training
wav2vec 2.0 [33]TransformerLS96.2*75.2*
HuBERT[35]TransformerLS96.3*81.4*
Conformer [37]ConformerAS=41.188.0=--
SS-AST[18]ViT-BAS+LS31.0188.898.096.064.3
Concurrent MAE-based works
MaskSpec [43]ViT-BAS32.347.189.697.7=
MAE-AST[38]ViT-BAS+LS30.6-90.097.995.863.3
Audio-MAE (global)ViT-BAS36.6±.1146.8±.0693.6±.1198.3±.0697.6±.0694.1±.06
Audio-MAE (local)ViT-BAS37.0±.1147.3±.1194.1±.1098.3±.0696.9±.0094.8± .11
Out-of-domain supervised pre-training
PSLA [30]EffNet [60]IN31.944.4=96.3=
AST[10]DeiT-BIN34.745.988.798.195.541.1
MBT[11]ViT-BIN-21K31.344.31-=
HTS-AT [29]Swin-BIN=47.197.0t98.0
PaSST[28]DeiT-BIN47.196.8†-
", + "type": "table", + "image_path": "e0146f12659732d22bc5a656b8f385365a1efb9fcc9883c795d053f067db9e94.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 65, + 502, + 134.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666669, + 502, + 204.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 204.33333333333337, + 502, + 274.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 278, + 506, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "Table 2: Comparison with other state-of-the-art models on audio and speech classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 288, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 289, + 302 + ], + "score": 1.0, + "content": "tasks. Metrics are mAP for AS and accuracy", + "type": "text" + }, + { + "bbox": [ + 290, + 289, + 306, + 300 + ], + "score": 0.7, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 288, + 506, + 302 + ], + "score": 1.0, + "content": "for ESC/SPC/SID. For pre-training (PT) dataset,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "AS:AudioSet, LS:LibriSpeech, and IN:ImageNet. †: Fine-tuning results with additional supervised", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 323 + ], + "score": 1.0, + "content": "training on AS-2M. We gray-out models pre-trained with external non-audio datasets (e.g., ImageNet).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 322, + 500, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 500, + 334 + ], + "score": 1.0, + "content": "Best single models in AS-2M are compared (no ensembles). *: linear evaluation results from [53].", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "Audio-only pre-training (AS-SSL) from scratch. We consider this the ideal schema for learning audio", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "representations as it is a simple and clean setup that prevents uncontrollable bias transfer from other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "modalities. (2) Directly using self-supervised ImageNet MAE models (IN-SSL) and its fine-tuned", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 399, + 491, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 491, + 411 + ], + "score": 1.0, + "content": "variant (IN-SL). (3) Audio-MAE self-supervised pre-training on top of these ImageNet weights.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 366, + 505, + 411 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "The results show that (1) from-scratch audio-only pre-training is the best. For scenarios (2) and (3),", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "we observe that ImageNet pre-training alone (2) is not sufficient (especially when the downstream", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 436, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 506, + 451 + ], + "score": 1.0, + "content": "data is smaller, AS-20K), and, in self-supervised pre-training on AudioSet, ImageNet initialization", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "(3) does not help but degrades accuracy. Also in (3), supervised ImageNet pre-training (IN-SL) seems", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "harmful. Consequently, the result suggests that out-of-domain pre-training (i.e., ImageNet) is not", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 470, + 325, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 325, + 482 + ], + "score": 1.0, + "content": "helpful for Audio-MAE, possibly due to domain shift.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 415, + 506, + 482 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 494, + 289, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 290, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 290, + 507 + ], + "score": 1.0, + "content": "4.5 Comparison with the State-of-the-art", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "Table 2 compares Audio-MAE (with 3-run error bars) to prior state-of-the-art. We categorize the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 526, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 537 + ], + "score": 1.0, + "content": "comparison into 3 groups. For fair comparison, our main benchmark is the models in the middle", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "group with self-supervised pre-training on in-domain (audio) datasets (AudioSet and LibriSpeech).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "For reference we also list other models without pre-training (the top group) and other models with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "supervised pre-training on out-of-domain ImageNet (the bottom group), where the latter contains", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 569, + 259, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 259, + 581 + ], + "score": 1.0, + "content": "previous best systems on the datasets.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 514, + 506, + 581 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 585, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "Pre-trained on AudioSet, Audio-MAE achieves the best performance across all tasks compared", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 460, + 609 + ], + "score": 1.0, + "content": "to other models with in-domain self-supervised pre-training. On AudioSet-20K, its", + "type": "text" + }, + { + "bbox": [ + 460, + 596, + 505, + 607 + ], + "score": 0.51, + "content": "3 7 . 1 \\ \\mathrm { m A P }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 606, + 507, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 507, + 621 + ], + "score": 1.0, + "content": "significantly outperforms all other approaches including concurrent works and other models with out-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "of-domain pre-training. On AudioSet-2M and ESC-50, our method also outperforms Conformer [37]", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "and SS-AST [18]. Notably, unlike SS-AST and concurrent MAE-AST [38], which trained with", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 638, + 457, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 457, + 654 + ], + "score": 1.0, + "content": "additional 1,000 hours of speech in Librispeech, we use only AudioSet for pre-training.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 585, + 507, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "In the bottom group of Table 2, Audio-MAE also outperforms previous state-of-the-art models with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "ImageNet supervised pre-training. Note that the proposed Audio-MAE does not rely on any out-of-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "domain data and labels, nor using knowledge distillation (e.g., DeiT) from additional CNN-based", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "models. Also, compared to HTS-AT [29] and PaSST [28], Audio-MAE is trained with audio under", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "16K sampling rate. As experimented in [59], there could be up to 0.4 potential mAP improvement for", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 343, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 343, + 723 + ], + "score": 1.0, + "content": "Audio-MAE if audio with 32K sampling rate are available.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 655, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 71, + 502, + 339 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 71, + 502, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 71, + 502, + 339 + ], + "spans": [ + { + "bbox": [ + 107, + 71, + 502, + 339 + ], + "score": 0.976, + "type": "image", + "image_path": "6244fd1a7a46c5c39262c96d30a4dfd6f022b65c3eb7f73ba45382c9f667a734.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 71, + 502, + 160.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 160.33333333333331, + 502, + 249.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 249.66666666666663, + 502, + 338.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 345, + 506, + 411 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "Figure 6: Spectrogram reconstruction visualizations on the AudioSet eval set. Column-wise", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "type: speech, music, event, others. Masking type: (a-d) unstructured (random); (e-h) structured", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 128, + 380 + ], + "score": 1.0, + "content": "(time", + "type": "text" + }, + { + "bbox": [ + 128, + 368, + 136, + 377 + ], + "score": 0.28, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 367, + 249, + 380 + ], + "score": 1.0, + "content": "frequency). Masking Ratio:", + "type": "text" + }, + { + "bbox": [ + 249, + 367, + 269, + 378 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ". In each group, we show the original spectrogram (1, top),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 432, + 390 + ], + "score": 1.0, + "content": "masked input (2, middle), and MAE output (3, bottom). The spectrogram size is", + "type": "text" + }, + { + "bbox": [ + 433, + 378, + 477, + 388 + ], + "score": 0.9, + "content": "1 0 2 4 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "; patch", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 389, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 133, + 400 + ], + "score": 1.0, + "content": "size is", + "type": "text" + }, + { + "bbox": [ + 134, + 389, + 162, + 399 + ], + "score": 0.88, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 389, + 235, + 400 + ], + "score": 1.0, + "content": ". Each sample has", + "type": "text" + }, + { + "bbox": [ + 235, + 389, + 280, + 399 + ], + "score": 0.89, + "content": "6 4 \\times 8 = 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 389, + 353, + 400 + ], + "score": 1.0, + "content": "patches with 154 (", + "type": "text" + }, + { + "bbox": [ + 353, + 389, + 372, + 399 + ], + "score": 0.83, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 389, + 505, + 400 + ], + "score": 1.0, + "content": "masked) patches being visible to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 398, + 500, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 500, + 413 + ], + "score": 1.0, + "content": "Audio-MAE. Please click (1 2 3) for audible .wavs. More audible examples are in Supplementary.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "For the speech tasks (SPC-1, SPC-2, and SID), Audio-MAE outperforms other models without", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "pre-training (ERANN [58], PANN [59]), supervised (AST) and self-supervised models (SS-AST,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "MAE-AST). We further list other works (marked with *) to include the latest results introduced in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "the SUPERB [53] benchmark. But note that these results are not strictly comparable since SUPERB", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 478, + 497, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 497, + 491 + ], + "score": 1.0, + "content": "employs linear evaluation where the underlying pre-trained models are not end-to-end fine-tuned.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "In summary, with audio-only from-scratch pre-training on AudioSet, our Audio-MAE performs well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 506, + 305, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 305, + 518 + ], + "score": 1.0, + "content": "for both the audio and speech classification tasks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 390, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 391, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 391, + 545 + ], + "score": 1.0, + "content": "4.6 Visualization and Audible Examples by Audio-MAE Decoder", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "For better visualization, we follow MAE [1] to use MSE over non-normalized spectrograms as the self-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "supervised objective. We use ViT-L as the Audio-MAE encoder for visualization. Fig. 6 illustrates", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "the reconstruction results sampled from the AudioSet-2M eval set. We further reconstruct .wavs using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 584, + 503, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 503, + 598 + ], + "score": 1.0, + "content": "the Griffin-Lim [61] algorithm, audible under the anonymous links (accessible in respective 1 2 3).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "As can be seen and heard, for various masking strategies and different sounds, our Audio-MAE", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "generates reasonable reconstruction. It works well for noisy event sounds (e.g., the reconstructed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "siren in Fig. 6c-3), as well as speech and music (e.g., the reconstructed singing in Fig. 6b-3). Notably,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "unlike visual contents that are typically scale/translation/position invariant [19], absolute positions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "and arrangement of spectrogram components are critical for humans to understand sound [62]. For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "example, shifting a pitch will make an audio sounds completely different. Also, phoneme sequences", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "in time are important cues for speech understanding. Consequently, unstructured masking produces", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "better aligned outputs that are closer to the ground-truth (top row in each subfigure) as the model can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "make better predictions based on nearby spectrogram patches; while structured masking is harder", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "(less accurate or with words missing), especially when masking is performed over the time axis. 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Column-wise", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "type: speech, music, event, others. Masking type: (a-d) unstructured (random); (e-h) structured", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 128, + 380 + ], + "score": 1.0, + "content": "(time", + "type": "text" + }, + { + "bbox": [ + 128, + 368, + 136, + 377 + ], + "score": 0.28, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 367, + 249, + 380 + ], + "score": 1.0, + "content": "frequency). Masking Ratio:", + "type": "text" + }, + { + "bbox": [ + 249, + 367, + 269, + 378 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ". In each group, we show the original spectrogram (1, top),", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 378, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 432, + 390 + ], + "score": 1.0, + "content": "masked input (2, middle), and MAE output (3, bottom). The spectrogram size is", + "type": "text" + }, + { + "bbox": [ + 433, + 378, + 477, + 388 + ], + "score": 0.9, + "content": "1 0 2 4 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 378, + 506, + 390 + ], + "score": 1.0, + "content": "; patch", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 389, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 133, + 400 + ], + "score": 1.0, + "content": "size is", + "type": "text" + }, + { + "bbox": [ + 134, + 389, + 162, + 399 + ], + "score": 0.88, + "content": "1 6 \\times 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 389, + 235, + 400 + ], + "score": 1.0, + "content": ". Each sample has", + "type": "text" + }, + { + "bbox": [ + 235, + 389, + 280, + 399 + ], + "score": 0.89, + "content": "6 4 \\times 8 = 5 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 389, + 353, + 400 + ], + "score": 1.0, + "content": "patches with 154 (", + "type": "text" + }, + { + "bbox": [ + 353, + 389, + 372, + 399 + ], + "score": 0.83, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 389, + 505, + 400 + ], + "score": 1.0, + "content": "masked) patches being visible to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 398, + 500, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 500, + 413 + ], + "score": 1.0, + "content": "Audio-MAE. Please click (1 2 3) for audible .wavs. More audible examples are in Supplementary.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "For the speech tasks (SPC-1, SPC-2, and SID), Audio-MAE outperforms other models without", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "pre-training (ERANN [58], PANN [59]), supervised (AST) and self-supervised models (SS-AST,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "MAE-AST). We further list other works (marked with *) to include the latest results introduced in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 504, + 479 + ], + "score": 1.0, + "content": "the SUPERB [53] benchmark. But note that these results are not strictly comparable since SUPERB", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 478, + 497, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 497, + 491 + ], + "score": 1.0, + "content": "employs linear evaluation where the underlying pre-trained models are not end-to-end fine-tuned.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 434, + 506, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "score": 1.0, + "content": "In summary, with audio-only from-scratch pre-training on AudioSet, our Audio-MAE performs well", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 506, + 305, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 305, + 518 + ], + "score": 1.0, + "content": "for both the audio and speech classification tasks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 494, + 505, + 518 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 390, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 391, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 391, + 545 + ], + "score": 1.0, + "content": "4.6 Visualization and Audible Examples by Audio-MAE Decoder", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "For better visualization, we follow MAE [1] to use MSE over non-normalized spectrograms as the self-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "supervised objective. We use ViT-L as the Audio-MAE encoder for visualization. Fig. 6 illustrates", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "the reconstruction results sampled from the AudioSet-2M eval set. We further reconstruct .wavs using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 584, + 503, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 503, + 598 + ], + "score": 1.0, + "content": "the Griffin-Lim [61] algorithm, audible under the anonymous links (accessible in respective 1 2 3).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 552, + 506, + 598 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "As can be seen and heard, for various masking strategies and different sounds, our Audio-MAE", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "generates reasonable reconstruction. It works well for noisy event sounds (e.g., the reconstructed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "score": 1.0, + "content": "siren in Fig. 6c-3), as well as speech and music (e.g., the reconstructed singing in Fig. 6b-3). Notably,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "unlike visual contents that are typically scale/translation/position invariant [19], absolute positions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "and arrangement of spectrogram components are critical for humans to understand sound [62]. For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "example, shifting a pitch will make an audio sounds completely different. Also, phoneme sequences", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "in time are important cues for speech understanding. Consequently, unstructured masking produces", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "better aligned outputs that are closer to the ground-truth (top row in each subfigure) as the model can", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "make better predictions based on nearby spectrogram patches; while structured masking is harder", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "(less accurate or with words missing), especially when masking is performed over the time axis. A", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 711, + 399, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 399, + 723 + ], + "score": 1.0, + "content": "failure example (missing words) is the reconstructed speech in Fig. 6e-3.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 601, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 71, + 183, + 84 + ], + "lines": [ + { + "bbox": [ + 104, + 69, + 185, + 87 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 185, + 87 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 95, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 506, + 108 + ], + "score": 1.0, + "content": "We have explored a simple extension of MAE [1] to audio data. Our Audio-MAE learns to reconstruct", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "masked spectrogram patches from audio recordings and achieves state-of-the-art performance on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "six audio and speech classification tasks. We have drawn four interesting observations: First, a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "simple MAE approach works surprisingly well for audio spectrograms. Second, we find that it is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "possible to learn stronger representations with local self-attention in the decoder. Third, we show", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "score": 1.0, + "content": "that masking can be applied to both pre-training and fine-tuning, improving accuracy and reducing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "training computation. The optimal strategy depends on the nature of the data (audio, image, etc.) and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 186 + ], + "score": 1.0, + "content": "the learning type (self-/supervised). Fourth, the best performance can be achieved by pre-training", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "and fine-tuning under the same modality, without reliance on cross-modality transfer learning. In", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "future work, we aim to explore multimodal self-supervised learning with a joint audio-visual MAE", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 394, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 394, + 217 + ], + "score": 1.0, + "content": "approach as these domains share natural correspondences in video data.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 224, + 504, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 237 + ], + "score": 1.0, + "content": "Acknowledgements. 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ModerBackboneP1-DataAS-20KAS-ZMIESC-30SPC-2SPC-1SID
No pre-training
ERANN [58]CNN45.089.2
PANN [59]CNN27.843.183.361.8
In-domain self-supervised pre-training
wav2vec 2.0 [33]TransformerLS96.2*75.2*
HuBERT[35]TransformerLS96.3*81.4*
Conformer [37]ConformerAS=41.188.0=--
SS-AST[18]ViT-BAS+LS31.0188.898.096.064.3
Concurrent MAE-based works
MaskSpec [43]ViT-BAS32.347.189.697.7=
MAE-AST[38]ViT-BAS+LS30.6-90.097.995.863.3
Audio-MAE (global)ViT-BAS36.6±.1146.8±.0693.6±.1198.3±.0697.6±.0694.1±.06
Audio-MAE (local)ViT-BAS37.0±.1147.3±.1194.1±.1098.3±.0696.9±.0094.8± .11
Out-of-domain supervised pre-training
PSLA [30]EffNet [60]IN31.944.4=96.3=
AST[10]DeiT-BIN34.745.988.798.195.541.1
MBT[11]ViT-BIN-21K31.344.31-=
HTS-AT [29]Swin-BIN=47.197.0t98.0
PaSST[28]DeiT-BIN47.196.8†-
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a/parse/dev/_XNtisL32jv/_XNtisL32jv_content_list.json b/parse/dev/_XNtisL32jv/_XNtisL32jv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..aac8ecb4dc4e1e2c4513da1316ef408b187dec02 --- /dev/null +++ b/parse/dev/_XNtisL32jv/_XNtisL32jv_content_list.json @@ -0,0 +1,1803 @@ +[ + { + "type": "text", + "text": "TEMPORAL EFFICIENT TRAINING OF SPIKINGNEURAL NETWORK VIA GRADIENT RE-WEIGHTING", + "text_level": 1, + "bbox": [ + 176, + 99, + 797, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Shikuang Deng1,2, Yuhang $\\mathbf { L i ^ { 3 } }$ , Shanghang Zhang4 & Shi $\\mathbf { G u } ^ { 1 , 2 , 5 \\boxtimes }$ ", + "bbox": [ + 186, + 169, + 650, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1University of Electronic Science and Technology of China, \n2Shenzhen Institute for Advanced Study, UESTC \n3Yale University, 4Peking University ,5Peng Cheng Laboratory \ndengsk119@std.uestc.edu.cn, yuhang.li@yale.edu, gus@uestc.edu.cn ", + "bbox": [ + 184, + 185, + 810, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 279, + 544, + 294 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, brain-inspired spiking neuron networks (SNNs) have attracted widespread research interest because of their event-driven and energy-efficient characteristics. Still, it is difficult to efficiently train deep SNNs due to the nondifferentiability of its activation function, which disables the typically used gradient descent approaches for traditional artificial neural networks (ANNs). Although the adoption of surrogate gradient (SG) formally allows for the back-propagation of losses, the discrete spiking mechanism actually differentiates the loss landscape of SNNs from that of ANNs, failing the surrogate gradient methods to achieve comparable accuracy as for ANNs. In this paper, we first analyze why the current direct training approach with surrogate gradient results in SNNs with poor generalizability. Then we introduce the temporal efficient training (TET) approach to compensate for the loss of momentum in the gradient descent with SG so that the training process can converge into flatter minima with better generalizability. Meanwhile, we demonstrate that TET improves the temporal scalability of SNN and induces a temporal inheritable training for acceleration. Our method consistently outperforms the SOTA on all reported mainstream datasets, including CIFAR-10/100 and ImageNet. Remarkably on DVS-CIFAR10, we obtained $8 3 \\%$ top-1 accuracy, over $\\bar { 1 0 \\% }$ improvement compared to existing state of the art. Codes are available at https://github.com/Gus-Lab/temporal_ efficient_training. ", + "bbox": [ + 233, + 309, + 764, + 587 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 613, + 336, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The advantages of Spiking neuron networks (SNNs) lie in their energy-saving and fast-inference computation when embedded on neuromorphic hardware such as TrueNorth (DeBole et al., 2019) and Loihi (Davies et al., 2018). Such advantages originate from the biology-inspired binary spike transmitted mechanism, by which the networks avoid multiplication during inference. On the other hand, this mechanism also leads to difficulty in training very deep SNNs from scratch because the non-differentiable spike transmission hinders the powerful back-propagation approaches like gradient descents. Recently, many studies on converting artificial neuron networks (ANNs) to SNNs have demonstrated SNNs’ comparable power in feature representation as ANNs (Han & Roy, 2020; Deng & Gu, 2020; Li et al., 2021a). Nevertheless, it is commonly agreed that the direct training method for high-performance SNN is still crucial since it distinguishes SNNs from converted ANNs, especially on neuromorphic datasets. ", + "bbox": [ + 174, + 645, + 825, + 796 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The output layer’s spike frequency or the average membrane potential increment is commonly used as inference indicators in SNNs (Shrestha & Orchard, 2018; Kim et al., 2019). The current standard direct training (SDT) methods regard the SNN as RNN and optimize inference indicators’ distribution (Wu et al., 2018). They adopt surrogate gradients (SG) to relieve the non-differentiability (Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021). However, the gradient descent with SG does not match with the loss landscape in SNN and is easy to get trapped in a local minimum with low generalizability. Although using suitable optimizers and weight decay help ease this problem, the performance of deep SNNs trained from scratch still suffers a big deficit compared to that of ANNs Deng et al. (2020). Another training issue is the memory and time consumption, which increases linearly with the simulation time. Rathi & Roy (2020) initializes the target network by a converted SNN to shorten the training epochs, indicating the possibility of high-performance SNN with limited activation time. The training problem due to the non-differentiable activation function has become the main obstruction of spiking neural network development. ", + "bbox": [ + 174, + 804, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/1c0df3e5e42eb5f12b364ae3e1b86c171367176d7d8e201b7cccf8c9ba08687d.jpg", + "image_caption": [ + "Figure 1: Workflow of temporal efficient training (TET). To obtain a more generalized SNN, we modify the optimization target to adjust each moment’s output distribution. " + ], + "image_footnote": [], + "bbox": [ + 179, + 111, + 823, + 330 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 398, + 825, + 483 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we examine the limitation of the traditional direct training approach with SG and propose the temporal efficient training (TET) algorithm. Instead of directly optimizing the integrated potential, TET optimizes every moment’s pre-synaptic inputs. As a result, it avoids the trap into local minima with low prediction error but a high second-order moment. Furthermore, since the TET applies optimization on each time point, the network naturally has more robust time scalability. Based on this characteristic, we propose the time inheritance training (TIT), which reduces the training time by initializing the SNN with a smaller simulation length. With the help of TET, the performance of SNNs has improved on both static datasets and neuromorphic datasets. Figure 1 depicts the workflow of our approach. ", + "bbox": [ + 174, + 489, + 825, + 614 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The following summarizes our main contributions: ", + "bbox": [ + 174, + 621, + 506, + 636 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We analyze the problem of training SNN with SG and propose the TET method, a new loss and gradient descent regime that succeeds in obtaining more generalizable SNNs. • We analyze the feasibility of TET and picture the loss landscape under both the SDT and TET setups to demonstrate TET’s advantage in better generalization. • Our sufficient experiments on both static datasets and neuromorphic datasets prove the effectiveness of the TET method. Especially on DVS-CIFAR10, we report $8 3 . 1 \\bar { 7 } \\%$ top-1 accuracy for the first time, which is over $1 0 \\%$ better than the current state-of-the-art result. ", + "bbox": [ + 217, + 647, + 825, + 753 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 773, + 344, + 790 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In recent years, SNNs have developed rapidly and received more and more attention from the research community. However, lots of challenging problems remain to be unsolved. In general, most works on SNN training have been carried out in two strategies: ANN-to-SNN conversion and direct training from scratch. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "ANN-to-SNN Conversion. Conversion approaches avoid the training problem by trading high accuracy through high latency. They convert a high-performing ANN to SNN and adjust the SNN parameters w.r.t the ANN activation value layer-by-layer (Diehl et al., 2015; 2016). Some special techniques have been proposed to reduce the inference latency, such as the subtraction mechanism (Rueckauer et al., 2016; Han et al., 2020), robust normalization Rueckauer et al. (2016), spike-norm (Sengupta et al., 2018), and channel-wise normalization (Kim et al., 2019). Recently, Deng & Gu (2020) decompose the conversion error to each layer and reduce it by bias shift. Li et al. (2021a) suggest using adaptive threshold and layer-wise calibration to obtain high-performance SNNs that require a simulation length of less than 50. However, converted methods significantly extend the inference latency, and they are not suitable for neuromorphic data (Deng et al., 2020). ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Direct training. In this area, SNNs are regarded as special RNNs and training with BPTT (Neftci et al., 2019). On the backpropagation process, The non-differentiable activation term is replaced with a surrogate gradient (Lee et al., 2016). Compared with ANN-to-SNN conversion, direct training achieves high accuracy with few time steps but suffers more training costs (Deng et al., 2020). Several studies suggest that surrogate gradient (SG) is helpful to obtain high-performance SNNs on both static datasets and neuromorphic datasets (Wu et al., 2019; Shrestha & Orchard, 2018; Li et al., 2021b). On the backpropagation process, SG replaces the Dirac function with various shapes of curves. Exceptionally, Wu et al. (2018) first propose the STBP method and train SNNs on the ANN programming platform, which significantly promotes direct training development. Zheng et al. (2021) further proposes the tdBN algorithm to smooth the loss function and first realize training a large-scale SNN on ImageNet. Zhang & Li (2020) proposes TSSL-BP to break down error backpropagation across two types of inter-neuron and intra-neuron dependencies and achieve low-latency and high accuracy SNNs. Recently, Yang et al. (2021) designed a neighborhood aggregation (NA) method to use the multiple perturbed membrane potential waveforms in the neighborhood to compute the finite difference gradients and guide the weight updates. They significantly decrease the required training iterations and improve the SNN performance. ", + "bbox": [ + 174, + 194, + 825, + 416 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PRELIMINARY ", + "text_level": 1, + "bbox": [ + 176, + 436, + 325, + 453 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 ITERATIVE LIF MODEL ", + "text_level": 1, + "bbox": [ + 176, + 467, + 379, + 482 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We adopt the Leaky Integrate-and-Fire (LIF) model and translate it to an iterative expression with the Euler method (Wu et al., 2019). Mathematically, the membrane potential is updated as ", + "bbox": [ + 173, + 493, + 823, + 522 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/79d0ddccc8ccc58411e76edbec4fc81d48d5beb97e2427fcc4feed59b3981d00.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { u } ( t + 1 ) = \\tau \\pmb { u } ( t ) + \\pmb { I } ( t ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 527, + 584, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\tau$ is the constant leaky factor, ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\mathbf \\Psi \\Psi \\mathbf { \\mathbf } \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ is the membrane potential at time $t$ , and $\\mathbf { } I ( t )$ denotes the pre-synaptic inputs, which is the product of synaptic weight $\\mathbf { W }$ and spiking input ${ \\mathbf { } } x ( t )$ . Given a specific threshold $V _ { t h }$ , the neuron fires a spike and ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf { \\mathbf } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ reset to 0 when the ${ \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } { \\mathbf { } } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf { \\Psi \\mathbf } \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf $ exceeds the threshold. So the firing function and hard reset mechanism can be described as ", + "bbox": [ + 173, + 549, + 825, + 606 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/20e0ed1a52553143a3bef8068e7733454312aa974301f895a48de2c6e6b19bc9.jpg", + "text": "$$\n\\pmb { a } ( t + 1 ) = \\pmb { \\Theta } ( \\pmb { u } ( t + 1 ) - V _ { t h } )\n$$", + "text_format": "latex", + "bbox": [ + 393, + 611, + 604, + 628 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2aa71469768ad37328e20be11ce6b924f417ac2ecbe65d9957fbda1dc4dbcc1c.jpg", + "text": "$$\n\\pmb { u } ( t + 1 ) = \\pmb { u } ( t + 1 ) \\cdot ( 1 - \\pmb { a } ( t + 1 ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 632, + 625, + 648 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\Theta$ denotes the Heaviside step function. The output spike $\\mathbf { \\delta } \\mathbf { \\ } \\mathbf { \\em a } ( t + 1 )$ will become the post synaptic spike and propagate to the next layer. In this study, we set the starting membrane $\\pmb { u } ( 0 )$ to 0, the threshold $V _ { t h }$ to 1, and the leaky factor $\\tau$ to 0.5 for all experiments. ", + "bbox": [ + 176, + 651, + 823, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The last layer’s spike frequency is typically used as the final classification index. However, adopting the LIF model on the last layer will lose information on the membrane potential and damage the performance, especially on complex tasks (Kim et al., 2019). Instead, we integrate the pre-synaptic inputs $\\mathbf { } I ( t )$ with no decay or firing (Rathi & Roy, 2020; Fang et al., 2021). Finally, we set the average membrane potential as the classification index and calculate the cross-entropy loss for training. ", + "bbox": [ + 173, + 699, + 825, + 770 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 SURROGATE GRADIENT ", + "text_level": 1, + "bbox": [ + 176, + 786, + 377, + 801 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Following the concept of direct training, we regard the SNN as RNN and calculate the gradients through spatial-temporal backpropagation (STBP) (Wu et al., 2018): ", + "bbox": [ + 174, + 813, + 821, + 842 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cd21c6adb4f30e317bf469128694fc231f872f68c84c7a80998a1e60e25c08bb.jpg", + "text": "$$\n\\frac { \\partial L } { \\partial \\mathbf { W } } = \\sum _ { t } \\frac { \\partial L } { \\partial \\pmb { a } ( t ) } \\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { a } ( t ) } \\frac { \\partial \\pmb { u } ( t ) } { \\partial \\pmb { I } ( t ) } \\frac { \\partial \\pmb { I } ( t ) } { \\partial \\mathbf { W } } ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 845, + 627, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the term $\\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { u } ( t ) }$ is the gradient of the non-differentiability step function involving the derivative of Dirac’s $\\delta$ -function that is typically replaced by surrogate gradients with a derivable curve. So far, ", + "bbox": [ + 174, + 891, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "there are various shapes of surrogate gradients, such as rectangular (Wu et al., 2018; 2019), triangle (Esser et al., 2016; Rathi & Roy, 2020), and exponential (Shrestha & Orchard, 2018) curve. In this work, we choose the surrogate gradients shaped like triangles. Mathematically, it can describe as ", + "bbox": [ + 173, + 103, + 826, + 147 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0d996846155207901ccfa55f7bd0020fc5d421b0bad5b204f7ff5dc37acc1d79.jpg", + "text": "$$\n\\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { u } ( t ) } = \\frac { 1 } { \\gamma ^ { 2 } } \\mathrm { m a x } ( 0 , \\gamma - | \\pmb { u } ( t ) - V _ { t h } | ) ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 156, + 629, + 190 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the $\\gamma$ denotes the constraint factor that determines the sample range to activate the gradient. ", + "bbox": [ + 171, + 199, + 815, + 214 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 BATCH NORMALIZATION FOR SNN ", + "text_level": 1, + "bbox": [ + 176, + 234, + 460, + 250 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Batch Normalization (BN) (Ioffe & Szegedy, 2015) is beneficial to accelerate training and increase performance since it can smooth the loss landscape during training (Santurkar et al., 2018). Zheng et al. (2021) modified the forward time loop form and proposed threshold-dependent Batch Normalization (tdBN) to normalize the pre-synaptic inputs $\\pmb { I }$ in both spatial and temporal paradigms so that the BN can support spatial-temporal input. We adopt this setup with the extension of the time dimension to batch dimension 1. In the inference process, the BN layer will be merged into the pre-convolutional layer, thus the inference rule of SNN remain the same but with modified weight: ", + "bbox": [ + 173, + 262, + 825, + 361 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/73c9e2e71e72668daac6ce033bd93a8a69776856e2f35751fa47b8f51dec6235.jpg", + "text": "$$\n\\hat { \\mathbf { W } } \\gets \\mathbf { W } \\frac { \\gamma } { \\alpha } , \\hat { \\pmb { b } } \\gets \\beta + ( \\pmb { b } - \\mu ) \\frac { \\gamma } { \\alpha } ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 369, + 609, + 397 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mu , \\alpha$ are the running mean and standard deviation on both spatial and temporal paradigm, $\\gamma , \\beta$ are the affine transformation parameters, and $\\mathbf { W } , b$ are the parameters of the pre-convolutional layer. ", + "bbox": [ + 176, + 406, + 823, + 435 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 174, + 459, + 341, + 476 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 FORMULA OF TRAINING SNN WITH SURROGATE GRADIENTS", + "text_level": 1, + "bbox": [ + 174, + 492, + 637, + 508 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Standard Direct Training. We use $O ( t )$ to represent pre-synaptic input $\\mathbf { } I ( t )$ of the output layer and calculate the cross-entropy loss. The loss function of standard direct training ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ is: ", + "bbox": [ + 173, + 520, + 828, + 550 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8f4533824d324c069c3603f39de4b247d351ea6cf666d20b7c5b6f05cc4ffc71.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { S D T } } = \\mathcal { L } _ { \\mathrm { C E } } \\big ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } O ( t ) , { \\pmb y } \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 559, + 598, + 603 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $T$ is the total simulation time, $\\mathcal { L } _ { \\mathrm { C E } }$ denotes the cross-entropy loss, and $\\textbf { { y } }$ represents the target label. Following the chain rule, we obtain the gradient of $\\mathbf { W }$ with softmax $S ( \\cdot )$ inference function : ", + "bbox": [ + 173, + 612, + 826, + 641 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2291a0db108c1baa17edc7f234f088fa3a88711355c3d68e0d15b610f7cddd0d.jpg", + "text": "$$\n\\frac { \\partial \\mathcal { L } _ { \\mathrm { S D T } } } { \\partial { \\bf W } } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } [ S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb { y } } ] \\frac { \\partial O ( t ) } { \\partial { \\bf W } } ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 651, + 632, + 695 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $O _ { \\mathrm { m e a n } }$ denotes the average of the output $O ( t )$ over time, and $\\hat { y }$ is the one-hot coding of $\\textbf { { y } }$ . ", + "bbox": [ + 173, + 704, + 803, + 720 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Temporal Efficient Training. In this section, we come up with a new kind of loss function ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ to realize temporal efficient training (TET). It constrains the output (pre-synaptic inputs) at each moment to be close to the target distribution. It is described as: ", + "bbox": [ + 174, + 727, + 820, + 768 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5b2fde9662b1a8301b77818a0131d21844eaca42707e3ef63dd52e680943a97b.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { T E T } } = \\frac { 1 } { T } \\cdot \\sum _ { t = 1 } ^ { T } \\mathcal { L } _ { \\mathrm { C E } } [ O ( t ) , { \\pmb y } ] .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 779, + 601, + 823 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Recalculate the gradient of weights under the loss function ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ , and we have: ", + "bbox": [ + 173, + 832, + 687, + 847 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c30cf63716931d277601336b8854b8a69f7d6fe85744cd096f5dd6904c14578f.jpg", + "text": "$$\n\\frac { \\partial \\mathcal { L } _ { \\mathrm { T E T } } } { \\partial { \\bf W } } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } [ S ( \\pmb { O } ( t ) ) - \\pmb { \\hat { y } } ] \\cdot \\frac { \\partial \\pmb { O } ( t ) } { \\partial { \\bf W } } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 856, + 635, + 900 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 CONVERGENCE OF GRADIENT DESCENT FOR SDT V.S. TET ", + "text_level": 1, + "bbox": [ + 173, + 103, + 632, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the case of SDT, the gradient consists of two parts, the error term $( S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb y } )$ and the partial derivative of output $\\partial { \\cal O } \\bar { ( } t ) / \\partial { \\bf W }$ . When the training process reaches near a local minimum, the term $( S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb y } )$ approximates 0 for all $t = 1 , . . . , T$ , ignorant of the term $\\partial O ( t ) / \\partial \\mathbf { W }$ . For traditional ANNs, the accumulated momentum may help get out of the local minima (e.g. saddle point) that typically implies bad generalizability (Kingma & Ba, 2014; Kidambi et al., 2018). However, when the SNN is trained with surrogate gradients, the accumulated momentum could be extremely small, considering the mismatch of gradients and losses. The fact that the activation function is a step one while the SG is bounded with integral constraints. This mismatch dissipates the momentum around a local minimum and stops the SDT from searching for a flatter minimum that may suggest better generalizability. ", + "bbox": [ + 173, + 131, + 825, + 270 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the case of TET, this issue of mismatch is relieved by reweighting the contribution of $\\partial { \\cal O } ( t ) / \\partial { \\bf W }$ . Indeed, considering the fact that the first term $( S ( O ( t ) ) - \\hat { { \\mathbf { y } } } )$ is impossible to be 0 at every moment of SNN since the early output accuracy on the training set is not $1 0 0 \\%$ . So TET needs the second term $\\partial { \\cal O } ( t ) / \\partial { \\bf W }$ close to 0 to make the ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ convergence. This mechanism increases the norm of gradients around sharp local minima and drives the TET to search for a flat local minimum where the disturbance of weight does not cause a huge change in $O ( t )$ . ", + "bbox": [ + 174, + 276, + 825, + 362 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Further, to ensure that the convergence with TET implies the convergence of SDT, we prove the following lemma: ", + "bbox": [ + 174, + 367, + 823, + 396 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 4.1. $\\mathcal { L } _ { S D T }$ is upper bounded by $\\mathcal { L } _ { T E T }$ ", + "bbox": [ + 174, + 401, + 472, + 417 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. Suppose $O _ { i } ( t )$ and $\\hat { y } _ { i }$ denote the i-th component of $O ( t )$ and $\\hat { y }$ , respectively. Expand Eqn.9, we have: ", + "bbox": [ + 173, + 443, + 821, + 473 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/af76f778462d27eb467b31d9a012ce480c4641edb15c8934625506b0e9b4fb94.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { T E T } } = - \\displaystyle \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log S ( \\boldsymbol { O } _ { i } ( t ) ) = - \\frac { 1 } { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad = - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) ^ { \\frac { 1 } { T } } \\geq - \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad \\geq - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( S ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } O _ { i } ( t ) ) ) = \\mathcal { L } _ { \\mathrm { S D T } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 479, + 733, + 613 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the first inequality is given by the Arithmetic Mean-Geometric Mean Inequality, and the second one is given by Jensen Inequality since the softmax function is convex. As a corollary, once the ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ gets closed to zero, the original loss function ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ also approaches zero. □ ", + "bbox": [ + 176, + 621, + 823, + 664 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Furthermore, the network output $O ( t )$ at a particular time point may be a particular outlier that dramatically affects the total output since the output of the SNN has the same weight at every moment under the rule of integration. Thus it is necessary to add a regularization term like $\\mathcal { L } _ { \\mathrm { M S E } }$ loss to confine each moment’s output to reduce the risk of outliers: ", + "bbox": [ + 174, + 690, + 825, + 747 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c03a2ccbd27a1cfad5ad85cefa96a16c4c0e3ccba38d4f927a7bb4ab70ca5181.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { M S E } } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathrm { M S E } ( \\mathbf { O } ( t ) , \\phi ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 757, + 599, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\phi$ is a constant used to regularize the membrane potential distribution. And we set $\\phi = V _ { t h }$ in our experiments. In practice, we use a hyperparameter $\\lambda$ to adjust the proportion of the regular term, we have: ", + "bbox": [ + 173, + 810, + 825, + 853 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2d4ca1af8e92f1b5e37f29d8225e52d8f739328ae7f8fbd21a9e43ec0522cfeb.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { T O T A L } } = ( 1 - \\lambda ) \\mathcal { L } _ { \\mathrm { T E T } } + \\lambda \\mathcal { L } _ { \\mathrm { M S E } } .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 856, + 612, + 873 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "It is worth noting that we only changed the loss function in the training process and did not change SNN’s inference rules in the testing phase for a fair comparison. This algorithm is detailed in Algo.1. ", + "bbox": [ + 173, + 881, + 823, + 910 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/65e0c9778d5b8d9266b65a187986cadf1c733a981e2dc311d68d5eed5c5bed81.jpg", + "table_caption": [ + "" + ], + "table_footnote": [], + "table_body": "
Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset;
total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival
for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi;
Compute the SNN output Oi(t) of eatch time step;
Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE =
(1-λ):¹∑t=1LcE(O²(t),Yi)+>·¹∑t=1 MSE(Oi(t),𝜙);
Backpropagation and update model parameters;
end for all i= 1,2,..Ival iteration do
Get mini-batch validation data,and class label: Yi;
T
Compute the SNN average output Omean = ∑T=1 O(t) over al time step;
Compare the clasification factor Omean and Yi for classification; end
", + "bbox": [ + 171, + 111, + 807, + 342 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 TIME INHERITANCE TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 371, + 428, + 386 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "SNN demands simulation length long enough to obtain a satisfying performance, but the training time consumption will increase linearly as the simulation length grows. So how to shorten the training time is also an essential problem in the direct training field. Traditional loss function ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ only optimizes the whole network output under a specific $T$ , so its temporal scalability is poor. Unlike the standard training, TET algorithm optimizes each moment’s output, enabling us to extend the simulation time naturally. We introduce Time Inheritance Training (TIT) to alleviate the training time problem. We first use long epochs to train an SNN with a short simulation time T, e.g., 2. Then, we increase the simulation time to the target value and retrain with short epochs. We discover that TIT performs better than training from scratch on accuracy and significantly saves the training time. Assuming that training an SNN with simulation length $T = 1$ cost $t s$ time per epoch, the SNN needs 300 epochs to train from scratch, and the TIT needs 50 epochs for finetuning. So we need $1 8 0 0 t s$ time to train an SNN with $T = 6$ from scratch, but following the TIT pipeline with the initial $T = 2$ only requires $9 0 0 t s$ . As a result, the TIT can reduce the training time cost by half. ", + "bbox": [ + 173, + 397, + 825, + 578 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 598, + 326, + 614 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We validate our proposed TET algorithm and compare it with existing works on both static and neuromorphic datasets. The network architectures in this paper include ResNet-19 (Zheng et al., 2021), Spiking-ResNet34 (Zheng et al., 2021), SEW-ResNet34 (Fang et al., 2021), SNN-5, and VGGSNN. SNN-5 (16C3-64C5-AP2-128C5-AP2-256C5-AP2-512C3-AP2-FC) is a simple convolutional SNN suitable for multiple runs to discover statistical rules (Figure A. 7). The architecture of VGGSNN (64C3-128C3-AP2-256C3-256C3-AP2-512C3-512C3-AP2-512C3-512C3-AP2- FC) is based on VGG11 with two fully connected layers removed as we found that additional fully connected layers were unnecessary for neuromorphic datasets. ", + "bbox": [ + 173, + 630, + 825, + 741 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 MODEL VALIDATION AND ABLATION STUDY", + "text_level": 1, + "bbox": [ + 174, + 758, + 524, + 772 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Effectiveness of TET over SDT with SG. We first examine whether the mismatch between SG and loss causes the convergence problem. For this purpose, we set the simulation length to 4 and change the spike function $\\Theta$ in Eqn.2 to Sigmoid $\\sigma ( \\bar { k } \\cdot \\mathrm { { i n p u t } } )$ . We find that the TET and SDT achieved similar accuracy (Table 2) when $k = 1 , 1 0 , 2 0$ . This indicates that both TET and SDT work when the gradient and loss function match each other. Next, we compare the results training with ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ and ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ on SNNs (ResNet-19 on CIFAR100) training with surrogate gradient for three runs. As shown in Table 1, our proposed new TET training strategy dramatically increases the accuracy by $3 . 2 5 \\%$ when the simulation time is 4 and $3 . 5 3 \\%$ when the simulation time is 6. These results quantitatively support the effectiveness of TET in solving the mismatch between gradient and loss in training SNNs with SG. ", + "bbox": [ + 173, + 784, + 825, + 922 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/aff831f07e0e329cffab78a14e6c6b9cb6a33fb6860a562c194fcda2f2064b92.jpg", + "image_caption": [ + "Figure 2: Loss landscape of VGGSNN. The 2D landscape of ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ and ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ from two different training methods. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 821, + 212 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/a12356c0a887b45ae0ed74a50360f2129b99cacc177540d45e375f3846d166c8.jpg", + "table_caption": [ + "Table 1: Comparison between SDT and TET. We adopt the SNN architecture ResNet-19 with SG on CIFAR100 and record the results with three different simulation lengths 2, 4, and 6. " + ], + "table_footnote": [], + "table_body": "
MethodT=2T=4T=6
Direct training69.41±0.0870.86±0.2271.12±0.57
TET72.37±0.2174.11±0.1874.65±0.12
", + "bbox": [ + 176, + 354, + 488, + 390 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/2b28a47a94520362d7566ceb1821586f201923734a5ba4241ac94a4a29bed241.jpg", + "table_caption": [ + "Table 2: Comparison of SDT and TET with sigmoid function $\\sigma ( k { \\cdot } \\mathrm { i n p u t } )$ . We fix the simulation length to 4 and record the results of CNN-5 under three different $k$ on CIFAR10. " + ], + "table_footnote": [], + "table_body": "
Methodk=1k=10k=20
Direct training88.00±0.1588.83±0.3288.50±0.32
TET87.63±0.3889.31±0.1588.64±0.28
", + "bbox": [ + 513, + 354, + 823, + 390 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Loss Landscape around Local Minima. We further inspect the 2D landscapes (Li et al., 2018) of $\\mathcal { L } _ { \\mathrm { S D T } }$ and ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ around their local minima (see Figure. 2) to demonstrate why TET generalizes better than SDT and how TET helps the training process jump out of the sharp local minima typically found by SDT. First, comparing Figure. $2 \\textrm { A }$ and C, we can see that although the values of local minima achieved by SDT and TET are similar in $\\mathcal { L } _ { \\mathrm { S D T } }$ , the local minima of TET (Figure. $2 \\textrm { C }$ ) is flatter than that of SDT (Figure. $2 \\mathrm { \\ A }$ ). This indicates that the TET is effective in finding flatter minima that are typically more generalizable even w.r.t the original loss in TET. Next, we examine the two local minima under ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ to see how it helps jump out the local minima found by SDT. When comparing Figure. 2 B and D, we observe that the local minima found by SDT (Figure. 2 B) is not only sharper than that found by TET (Figure. $2 \\mathbf { D }$ ) under ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ but also maintains a higher loss value. This supports our claim that TET loss cannot be easily minimized around sharp local minima (Figure. $2 \\mathrm { \\ B }$ ), thus preferable to converge into flatter local minima (Figure. $2 \\mathrm { D }$ ). Put together, the results here provide evidence for our reasoning in Section 4.2. ", + "bbox": [ + 173, + 420, + 825, + 599 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training from SDT to TET. In this part, we further validate the ability of TET to escape from the local minimum found by SDT. We adopt the VGGSNN with 300 epochs training on DVS-CIFAR10. First, we optimize ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ for 200 epochs and then change the loss function to ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ after epoch 200. Figure 3 demonstrates the accuracy and loss change on the test set. After 200 epochs training, SDT gets trapped into a local minimum, and the ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ no longer decreases. The ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ is much higher than $\\mathcal { L } _ { \\mathrm { S D T } }$ since SDT does not optimize it. Nevertheless, after we change the loss function to ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ , the ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ and $\\mathcal { L } _ { \\mathrm { S D T } }$ on the test set both have a rapid decline. This phenomenon illustrates the TET ability to help the SNN efficiently jump out of the local minimum with poor generalization and find another flatter local minimum. ", + "bbox": [ + 174, + 607, + 825, + 662 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/3337c633c01451316aa0cc30732fe1eaeb582425bcdf30fbe135431d8bf36214.jpg", + "image_caption": [ + "Figure 3: TET helps to jump out the local minimum point. We provide the test accuracy (A) and loss $( B )$ change after changing the SDT to TET at epoch 200. TET efficiently improves the test performance and reduces the two kinds of loss. " + ], + "image_footnote": [], + "bbox": [ + 232, + 688, + 754, + 859 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/086a43f62fe160235967e86bf1fd8d81f8153af1874a2e9a74e352d77d8e5e20.jpg", + "image_caption": [ + "Figure 4: Time scalability robustness and network efficiency of ResNet-19 on CIFAR100. (A) The comparison of training from scratch (dots) and inheriting from a small simulation length (lines). $( B )$ SNN network performance changes with energy consumption. " + ], + "image_footnote": [], + "bbox": [ + 240, + 107, + 759, + 284 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 377, + 825, + 448 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Time Scalability Robustness. Here, we study the time scalability robustness of SNNs trained with TET $( \\mathcal { L } _ { \\mathrm { T E T } } )$ . First, we use 300 epochs to train a small simulation length ResNet-19 on CIFAR100 as the initial SNN. Then, we directly change the simulation length from 2 to 8 without finetuning and report the network accuracy on the test set. Figure. 4. A displays the results after changing the simulation length. We use 2, 3, and 4, respectively, as the simulation length of the initial network. When we increase the simulation length, the accuracy of all networks gradually increases. After the simulation time reaches a certain value, the network performance will slightly decrease. Interestingly, SNNs trained from scratch ( $\\mathrm { T } { = } 4$ and ${ \\mathrm { T } } { = } 6$ ) are not as good as those trained following the TIT procedure. ", + "bbox": [ + 174, + 455, + 825, + 580 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Network Efficiency. In this section, we measure the relationship between energy consumption and network performance. SNN avoids multiplication on the inference since its binary activation and event-based operation. The addition operation in SNN costs $0 . 9 p J$ energy while multiplication operation consumes $4 . 6 p J$ measured in $4 5 \\mathrm { n m }$ CMOS technology (Rathi & Roy, 2020). In our SNN model, the first layer has multiplication operations, while the other layers only have addition operations. Figure 4. B summarizes the results of different simulation times. In all cases, the SNN obtained by TET has higher efficiency. ", + "bbox": [ + 174, + 587, + 825, + 684 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 COMPARISON TO EXITING WORKS ", + "text_level": 1, + "bbox": [ + 178, + 708, + 449, + 720 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we compare our experimental results with previous works. We validate the full TIT algorithm $( \\mathcal { L } _ { \\mathrm { T O T A L } } )$ both on the static dataset and neuromorphic dataset. All of the experiment results are summarized in Table 5.2. We specify all the training details in the appendix A.1. ", + "bbox": [ + 174, + 736, + 823, + 777 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CIFAR. We apply TET and TIT algorithm on CIFAR (Krizhevsky et al., 2009), and report the mean and standard deviation of 3 runs under different random seeds. The $\\lambda$ is set to 0.05. On CIFAR10, our TET method achieves the highest accuracy above all existing approaches. Even when $T = 2$ , there is a $1 . 8 2 \\%$ increment compare to STBP-tdBN with simulation length $T = 6$ . It is worth noting that our method is only $0 . 4 7 \\%$ lower than the ANN performance. TET algorithm demonstrates a more excellent ability on CIFAR100. It has an accuracy increase greater than $3 \\%$ on all report simulation lengths. In addition, when $T = 6$ , the reported accuracy is only $0 . 6 3 \\%$ lower than that of ANN. We can see that the proposed TET’s improvement is even higher on complex data like CIFAR100, where the generalizability of the model distinguishes a lot among minima with different flatness. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/dadbd052f80ebc40da5335c537abae5ee726446e1d387a7f4fc73961ba3ebea6.jpg", + "table_caption": [ + "Table 3: Compare with existing works. Our method improves network performance across all tasks. \\* denotes self-implementation results. † denotes data augmentation (Li et al., 2022). " + ], + "table_footnote": [], + "table_body": "
DatasetModelMethodsArchitectureSimulationLengthAccuracy
CIFAR10Rathi et al. (2019)Hybrid training Diet-SNNResNet-2025092.22
Rathi & Roy (2020)ResNet-201092.54
Wu et al. (2018)STBPCIFARNet1289.83
Wu et al. (2019)STBP NeuNormCIFARNet1290.53
Zhang & Li (2020)TSSL-BPCIFARNet591.41
Zheng et al. (2021)STBP-tdBNResNet-196 493.16 92.92
our modelTET292.34
694.50±0.07
494.44±0.08
ResNet-19294.16±0.03
CIFAR100ANN*ANNResNet-19194.97
Rathi et al. (2019) Rathi & Roy (2020)Hybrid trainingVGG-1112567.87
Diet-SNNResNet-20564.07 71.12±0.57
Zheng et al. (2021)*STBP-tdBNResNet-19670.86±0.22
4 269.41±0.08
674.72±0.28
our modelTETResNet-19474.47±0.15
272.87±0.10
ANN* Rathi etal. (2019)ANNResNet-19175.35
Hybrid training SPIKE-NORMResNet-3425061.48
Sengupta et al. (2018) Zheng et al. (2021)STBP-tdBNResNet-34250069.96
ImageNetFang et al. (2021)SEWResNetSpiking-ResNet-34663.72
TETSEW-ResNet-34 Spiking-ResNet-34467.04 64.79
our modelTETSEW-ResNet-346 468.00
Zheng et al. (2021)STBP-tdBNResNet-191067.8
Kugele et al. (2020)Streaming RolloutDenseNet1066.8
DVS-CIFAR10Wu et al. (2021)Conv3DLIAF-Net71.70
Wu et al. (2021)LIAFLIAF-Net10 1070.40
our modelTETVGGSNN1077.33±0.21
TETtVGGSNN83.17±0.15
10
", + "bbox": [ + 176, + 147, + 826, + 523 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ImageNet. The training set of ImageNet (Krizhevsky et al., 2012) provides $1 . 2 8 \\mathrm { k }$ training samples for each label. We choose the two most representative ResNet-34 to verify our algorithm on ImageNet with $\\lambda = 0 . 0 0 1$ . SEW-ResNet34 is not a typical SNN since it adopts the IF model and modifies the Residual structure. Although we only train our model for 120 epochs, the TET algorithm achieves a $1 . 0 7 \\%$ increment on Spiking-ResNet-34 and a $0 . 9 6 \\%$ increment on SEW-ResNet34. ", + "bbox": [ + 174, + 556, + 823, + 626 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "DVS-CIFAR10. The neuromorphic datasets suffer much more noise than static datasets. Thus the well-trained SNN is easier to overfit on these datasets than static datasets. DVS-CIFAR10 (Li et al., 2017), which provides each label with $0 . 9 \\mathrm { k }$ training samples, is the most challenging mainstream neuromorphic dataset. Recent works prefer to deal with this dataset by complex architectures, which are more susceptible to overfitting and do not result in very high accuracy. Here, we adopt VGGSNN on the DVS-CIFAR10 dataset, set $\\lambda = 0 . 0 0 1$ , and report the mean and standard deviation of 3 runs under different random seeds. Along with data augmentation methods, VGGSNN can achieve an accuracy of $7 7 . 4 \\%$ . Then we apply the TET method to obtain a more generalizable optima. The accuracy rises to $8 3 . 1 7 \\%$ . Our TET method outperforms existing state-of-the-art by $1 1 . 4 7 \\%$ accuracy. Without data augmentation methods, VGGSNN obtains $7 \\bar { 3 } . 3 \\%$ accuracy by SDT and $7 7 . 3 \\%$ accuracy by TET. ", + "bbox": [ + 174, + 633, + 825, + 786 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 808, + 318, + 824 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This paper focuses on the SNN generalization problem, which is described as the direct training SNN performs well on the training set but poor on the test set. We find this phenomenon is due to the incorrect SG that makes the SNN easily trapped into a local minimum with poor generalization. To solve this problem, we propose the temporal efficient training algorithm (TET). Extensive experiments verify that our proposed method consistently achieves better performance than the SDT process. Furthermore, TET significantly improves the time scalability robustness of SNN, which enables us to propose the time inheritance training (TIT) to significantly reduce the training time consumption by almost a half. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 825, + 132 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "7 ACKNOWLEDGMENT ", + "text_level": 1, + "bbox": [ + 176, + 151, + 379, + 167 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This project is supported by NSFC 61876032 and JCYJ20210324140807019. Y. Li completed this work during his prior research assistantship in UESTC. ", + "bbox": [ + 174, + 183, + 821, + 212 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 233, + 287, + 247 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, et al. Loihi: A neuromorphic manycore processor with on-chip learning. Ieee Micro, 38(1):82–99, 2018. 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Temporal spike sequence learning via backpropagation for deep spiking neural networks. arXiv preprint arXiv:2002.10085, 2020. ", + "bbox": [ + 171, + 474, + 823, + 503 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guoqi Li. Going deeper with directly-trained larger spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 11062–11070, 2021. ", + "bbox": [ + 174, + 511, + 825, + 555 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 DATASET AND TRAINING DETAIL ", + "text_level": 1, + "bbox": [ + 178, + 133, + 446, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "CIFAR. The CIFAR dataset (Krizhevsky et al., 2009) consists of 50k training images and 10k testing images with the size of $3 2 \\times 3 2$ . We use ResNet-19 for both CIFAR10 and CIFAR100. Moreover, random horizontal flip and crop are applied to the training images the augmentation. First, we use 300 epoch to train the SNN with the simulation length $T = 2$ . We use an Adam optimizer with a learning rate of 0.01 and cosine decay to 0. Next, following the TIT algorithm, we increase the simulation time (to 4 and 6) and continue training the SNN for only 50 epochs, with the learning rate changing to $1 e - 4$ . ", + "bbox": [ + 173, + 160, + 825, + 257 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "ImageNet. ImageNet (Deng et al., 2009) contains more than $1 2 5 0 \\mathrm { k }$ training images and $5 0 \\mathrm { k }$ validation images. We crop the images to $2 2 4 \\times 2 2 4$ and using the standard augmentation for the training data. We use an SGD optimizer with 0.9 momentum and weight decay $4 e - 5$ . The learning rate is set to 0.1 and cosine decay to 0. We train the SEW-ResNet34 (Fang et al., 2021) with $T = 4$ for 120 epochs. As for the Spiking-ResNet34 (Zheng et al., 2021), we use TIT algorithm to train 90 epochs with $T = 4$ first, then change the simulation time to 6 and finetune the network for 30 epochs. We adopt an Adam optimizer on the finetune phase and change the learning rate to $1 e - 4$ . TIT algorithm significantly reduces the training time consumption since training the Spiking-ResNet34 is extremely slow. ", + "bbox": [ + 173, + 263, + 825, + 390 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "DVS-CIFAR10. DVS-CIFAR10 (Li et al., 2017), the most challenging mainstream neuromorphic data set, is converted from CIFAR10. It has 10k images with the size $1 2 8 \\times 1 2 8$ . Following Samadzadeh et al. (2020), we divide the data stream into 10 blocks by time and accumulate the spikes in each block. Then, we split the dataset into $9 \\mathrm { k }$ training images and $1 \\mathrm { k }$ test images and reduce the spatial resolution to $4 8 \\times 4 8$ . Random horizontal flip and random roll within 5 pixels are taken as augmentation (Li et al., 2022). We adopt VGGSNN architecture with 300 epochs training on this classification task. And we use an Adam optimizer with the learning rate $1 e - 3$ and cosine decay to 0. As for the case that does not apply any augmentation, we add a weight decay of 5e-4 to the optimizer. ", + "bbox": [ + 173, + 396, + 825, + 522 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 LSDT LOSS LANDSCAPE OF RESNET-19 ", + "text_level": 1, + "bbox": [ + 176, + 540, + 486, + 554 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Here we compare the classification loss $( \\mathcal { L } _ { \\mathrm { S D T } } )$ landscapes of ResNet-19 on CIFAR100. The position around the local minimal value found by the SDT $( \\mathcal { L } _ { \\mathrm { { S D T } } } )$ is very sharp. However, the area around the local minimum found by TET $( \\mathcal { L } _ { \\mathrm { T E T } } )$ is much smoother (Figure 5), which indicates that TET effectively improves the network generalization. Such improvements could be further utilized to other techniques like privacy-preserving data generalization (Kim et al., 2021) and neural architecture search (Kim et al., 2022). ", + "bbox": [ + 173, + 565, + 825, + 648 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/ec3a5c919a88e624c8ee236df840321cb1292d402cae6ed2e4a7f134e697de42.jpg", + "image_caption": [ + "Figure 5: STD loss landscape of ResNet-19 on CIFAR100 from different training approaches. " + ], + "image_footnote": [], + "bbox": [ + 274, + 665, + 723, + 810 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.3 EFFECT OF $\\mathcal { L } _ { \\mathrm { M S E } }$ ", + "text_level": 1, + "bbox": [ + 176, + 868, + 334, + 883 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this part, we examine the effect of the regular term $\\mathcal { L } _ { \\mathrm { M S E } }$ with 5 different levels of $\\lambda$ . Figure 6 Summarizes the final results. The regular term $\\mathcal { L } _ { \\mathrm { M S E } }$ effectively increases the performance of both ResNet-19 on CIFAR100 and VGGSNN on DVS-CIFAR10. The static dataset CIFAR100 is more suitable for larger $\\lambda$ , while smaller $\\lambda$ is suitable for DVS-CIFAR10. Theoretically, it is hard to obtain satisfying performance at the early simulation moment due to the sparseness of neuromorphic datasets. So too large regular term $\\mathcal { L } _ { \\mathrm { M S E } }$ is not suitable for the neuromorphic dataset. Furthermore, we find that a high $\\lambda$ may harm the early training phase on ImageNet, especially if zero-initialize (Goyal et al., 2017) is not performed. As a result, we set $\\lambda$ to $5 e - 2$ for CIFAR10 and CIFAR100, $1 e - 3$ for ImageNet and DVS-CIFAR10. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 826, + 200 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/1dd2e09ae0d1ba0fa6f9db557415a2e61436ffe4627041352f5dae8ffd64e9b0.jpg", + "image_caption": [ + "Figure 6: The accuracy under different levels of $\\lambda$ . " + ], + "image_footnote": [], + "bbox": [ + 282, + 218, + 710, + 357 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.4 STATISTICAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 417, + 372, + 431 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we provide statistical results (Figure 7) to prove that the total SNN accuracy is positively associated with every average of moment’s output test accuracy. We train CNN-5 on CIFAR10 for a total of 20 runs with SDT and 5 runs with TET. ", + "bbox": [ + 174, + 443, + 825, + 484 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/295735f1db530423a1bc9f14bdb244ee4d3157db65b06f1b197a62a8ce989d2a.jpg", + "image_caption": [ + "Figure 7: Statistical results. The overall performance of SNN is highly positively associated with the average accuracy of each moment. The standard training obtains the green dots, while the red dots are trained by the TET method. " + ], + "image_footnote": [], + "bbox": [ + 351, + 507, + 627, + 674 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.5 TIME SCALABILITY ROBUSTNESS OF SDT AND TET. ", + "text_level": 1, + "bbox": [ + 174, + 765, + 581, + 779 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Here we first show the test accuracy (ResNet19 on CIFAR100) of the membrane potential increment at each moment instead of the integrated membrane potential. We set the initial simulation length of the SNNs to 3 or 4 and trained them for a full 300 epochs. Then we expand their simulation length to 8. As shown in table 4, TET $( \\mathcal { L } _ { \\mathrm { T E T } } )$ makes the membrane potential increment at each moment have a higher classification ability than SDT $( \\mathcal { L } _ { \\mathrm { { S D T } } } )$ . And TET (1.41 and 0.08) also acquires a low accuracy variance than SDT (3.81 and 4.04). ", + "bbox": [ + 173, + 790, + 825, + 875 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Then we compare the time scalability robustness between SDT $( \\mathcal { L } _ { \\mathrm { { S D T } } } )$ and TET $( \\mathcal { L } _ { \\mathrm { T E T } } )$ . We set the initial simulation length of ResNet19 SNNs to 2, 3, 4 and train with SDT or TET. Then we gradually increase SNN simulation length to 64 and record test accuracy of the integrated membrane potential. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/8be4907531399e22bedf40aaa0bdeff1ad93013af87033e52d12b9d582926cdd.jpg", + "image_caption": [ + "Figure 8: The accuracy after increasing the simulation length. We first train the SNN with TET (only use $\\mathcal { L } _ { \\mathrm { T E T } } ,$ ) and SDT ${ ( \\mathcal { L } _ { \\mathrm { { S D T } } } ) }$ with simulation length (T) is 2, 3, or 4. Then, we increase the simulation to 64 without finetuning and record the test the classification accuracy (A) and the accuracy relative growth rate (B) of the total SNN output (integrate membrane potential) at each simulation time. " + ], + "image_footnote": [], + "bbox": [ + 184, + 102, + 789, + 300 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "As we increase the simulation length, all the SNNs’ accuracy will first increase and then be stable in a certain area. Meanwhile, TET (1.80) has a small accuracy variance than the SDT (11.13) after increasing the simulation length. This phenomenon indicates that the initialization steps of TIT only need a small simulation length SNN for TET but a sufficiently large simulation (or enough epochs for finetuning step) for SDT. ", + "bbox": [ + 173, + 401, + 825, + 470 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/52e3fc01f3ea8e2391bfd6f898f93f9ad9dfdc42e86e4f748550727f3448b45b.jpg", + "table_caption": [ + "Table 4: Accuracy of each moment’s membrane potential increment. We use ${ \\mathcal { L } } _ { \\mathrm { S D T } }$ or ${ \\mathcal { L } } _ { \\mathrm { T E T } }$ to train the networks with simulation length 3 or 4. Then directly increase their simulation length to 8 and record each moment’s potential increment test accuracy. " + ], + "table_footnote": [], + "table_body": "
MethodT=1T=2T=3T=4T=5T=6T=7T=8
SDT (T=3)55.6157.9556.8755.0957.5653.5457.7254.04
SDT (T=4)37.9661.7855.0356.6457.4754.2458.7455.48
TET (T=3)65.9772.2271.7870.5571.9069.5772.1569.78
TET (T=4)62.1771.5771.0572.0871.7771.2371.8171.36
", + "bbox": [ + 223, + 536, + 769, + 609 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/dev/_XNtisL32jv/_XNtisL32jv_middle.json b/parse/dev/_XNtisL32jv/_XNtisL32jv_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..87ba3496c6dc63febc05185ebc28230ae8095dcb --- /dev/null +++ b/parse/dev/_XNtisL32jv/_XNtisL32jv_middle.json @@ -0,0 +1,38084 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 488, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 443, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 443, + 97 + ], + "score": 1.0, + "content": "TEMPORAL EFFICIENT TRAINING OF SPIKING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 489, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 489, + 118 + ], + "score": 1.0, + "content": "NEURAL NETWORK VIA GRADIENT RE-WEIGHTING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 134, + 398, + 147 + ], + "lines": [ + { + "bbox": [ + 111, + 133, + 398, + 149 + ], + "spans": [ + { + "bbox": [ + 111, + 133, + 228, + 149 + ], + "score": 1.0, + "content": "Shikuang Deng1,2, Yuhang", + "type": "text" + }, + { + "bbox": [ + 228, + 135, + 243, + 146 + ], + "score": 0.8, + "content": "\\mathbf { L i ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 133, + 359, + 149 + ], + "score": 1.0, + "content": ", Shanghang Zhang4 & Shi", + "type": "text" + }, + { + "bbox": [ + 359, + 134, + 398, + 146 + ], + "score": 0.36, + "content": "\\mathbf { G u } ^ { 1 , 2 , 5 \\boxtimes }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 113, + 147, + 496, + 192 + ], + "lines": [ + { + "bbox": [ + 111, + 145, + 355, + 159 + ], + "spans": [ + { + "bbox": [ + 111, + 145, + 355, + 159 + ], + "score": 1.0, + "content": "1University of Electronic Science and Technology of China,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 156, + 312, + 171 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 312, + 171 + ], + "score": 1.0, + "content": "2Shenzhen Institute for Advanced Study, UESTC", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 111, + 168, + 366, + 183 + ], + "spans": [ + { + "bbox": [ + 111, + 168, + 366, + 183 + ], + "score": 1.0, + "content": "3Yale University, 4Peking University ,5Peng Cheng Laboratory", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 112, + 181, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 112, + 181, + 498, + 194 + ], + "score": 1.0, + "content": "dengsk119@std.uestc.edu.cn, yuhang.li@yale.edu, gus@uestc.edu.cn", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 278, + 221, + 333, + 233 + ], + "lines": [ + { + "bbox": [ + 276, + 221, + 335, + 234 + ], + "spans": [ + { + "bbox": [ + 276, + 221, + 335, + 234 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 143, + 245, + 468, + 465 + ], + "lines": [ + { + "bbox": [ + 142, + 246, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 142, + 246, + 469, + 258 + ], + "score": 1.0, + "content": "Recently, brain-inspired spiking neuron networks (SNNs) have attracted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "widespread research interest because of their event-driven and energy-efficient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 469, + 280 + ], + "score": 1.0, + "content": "characteristics. Still, it is difficult to efficiently train deep SNNs due to the non-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "score": 1.0, + "content": "differentiability of its activation function, which disables the typically used gradi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "ent descent approaches for traditional artificial neural networks (ANNs). Although", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "score": 1.0, + "content": "the adoption of surrogate gradient (SG) formally allows for the back-propagation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 325 + ], + "score": 1.0, + "content": "of losses, the discrete spiking mechanism actually differentiates the loss landscape", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "of SNNs from that of ANNs, failing the surrogate gradient methods to achieve", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 334, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 469, + 345 + ], + "score": 1.0, + "content": "comparable accuracy as for ANNs. In this paper, we first analyze why the current", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 357 + ], + "score": 1.0, + "content": "direct training approach with surrogate gradient results in SNNs with poor gen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "eralizability. Then we introduce the temporal efficient training (TET) approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 378 + ], + "score": 1.0, + "content": "to compensate for the loss of momentum in the gradient descent with SG so that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "score": 1.0, + "content": "the training process can converge into flatter minima with better generalizabil-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "score": 1.0, + "content": "ity. Meanwhile, we demonstrate that TET improves the temporal scalability of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 470, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 470, + 411 + ], + "score": 1.0, + "content": "SNN and induces a temporal inheritable training for acceleration. Our method", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "score": 1.0, + "content": "consistently outperforms the SOTA on all reported mainstream datasets, includ-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "score": 1.0, + "content": "ing CIFAR-10/100 and ImageNet. Remarkably on DVS-CIFAR10, we obtained", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 430, + 470, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 162, + 443 + ], + "score": 0.9, + "content": "8 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 430, + 252, + 444 + ], + "score": 1.0, + "content": "top-1 accuracy, over", + "type": "text" + }, + { + "bbox": [ + 252, + 432, + 272, + 442 + ], + "score": 0.87, + "content": "\\bar { 1 0 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 430, + 470, + 444 + ], + "score": 1.0, + "content": "improvement compared to existing state of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 442, + 469, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 469, + 455 + ], + "score": 1.0, + "content": "art. Codes are available at https://github.com/Gus-Lab/temporal_", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 453, + 255, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 255, + 466 + ], + "score": 1.0, + "content": "efficient_training.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 206, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 208, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 208, + 501 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 523 + ], + "score": 1.0, + "content": "The advantages of Spiking neuron networks (SNNs) lie in their energy-saving and fast-inference", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 504, + 534 + ], + "score": 1.0, + "content": "computation when embedded on neuromorphic hardware such as TrueNorth (DeBole et al., 2019)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "and Loihi (Davies et al., 2018). Such advantages originate from the biology-inspired binary spike", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "transmitted mechanism, by which the networks avoid multiplication during inference. On the other", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "hand, this mechanism also leads to difficulty in training very deep SNNs from scratch because", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "the non-differentiable spike transmission hinders the powerful back-propagation approaches like", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "gradient descents. Recently, many studies on converting artificial neuron networks (ANNs) to SNNs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "have demonstrated SNNs’ comparable power in feature representation as ANNs (Han & Roy, 2020;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "Deng & Gu, 2020; Li et al., 2021a). Nevertheless, it is commonly agreed that the direct training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "method for high-performance SNN is still crucial since it distinguishes SNNs from converted ANNs,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 257, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 257, + 632 + ], + "score": 1.0, + "content": "especially on neuromorphic datasets.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "The output layer’s spike frequency or the average membrane potential increment is commonly used", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "as inference indicators in SNNs (Shrestha & Orchard, 2018; Kim et al., 2019). The current standard", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "direct training (SDT) methods regard the SNN as RNN and optimize inference indicators’ distri-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "bution (Wu et al., 2018). They adopt surrogate gradients (SG) to relieve the non-differentiability", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "(Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021). However, the gradient descent with SG does", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "not match with the loss landscape in SNN and is easy to get trapped in a local minimum with low", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 506, + 716 + ], + "score": 1.0, + "content": "generalizability. Although using suitable optimizers and weight decay help ease this problem, the", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 123, + 722, + 212, + 731 + ], + "lines": [ + { + "bbox": [ + 122, + 720, + 214, + 734 + ], + "spans": [ + { + "bbox": [ + 122, + 720, + 214, + 734 + ], + "score": 1.0, + "content": "B Corresponding author", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 488, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 443, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 443, + 97 + ], + "score": 1.0, + "content": "TEMPORAL EFFICIENT TRAINING OF SPIKING", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 489, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 489, + 118 + ], + "score": 1.0, + "content": "NEURAL NETWORK VIA GRADIENT RE-WEIGHTING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 114, + 134, + 398, + 147 + ], + "lines": [ + { + "bbox": [ + 111, + 133, + 398, + 149 + ], + "spans": [ + { + "bbox": [ + 111, + 133, + 228, + 149 + ], + "score": 1.0, + "content": "Shikuang Deng1,2, Yuhang", + "type": "text" + }, + { + "bbox": [ + 228, + 135, + 243, + 146 + ], + "score": 0.8, + "content": "\\mathbf { L i ^ { 3 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 133, + 359, + 149 + ], + "score": 1.0, + "content": ", Shanghang Zhang4 & Shi", + "type": "text" + }, + { + "bbox": [ + 359, + 134, + 398, + 146 + ], + "score": 0.36, + "content": "\\mathbf { G u } ^ { 1 , 2 , 5 \\boxtimes }", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 111, + 133, + 398, + 149 + ] + }, + { + "type": "list", + "bbox": [ + 113, + 147, + 496, + 192 + ], + "lines": [ + { + "bbox": [ + 111, + 145, + 355, + 159 + ], + "spans": [ + { + "bbox": [ + 111, + 145, + 355, + 159 + ], + "score": 1.0, + "content": "1University of Electronic Science and Technology of China,", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 156, + 312, + 171 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 312, + 171 + ], + "score": 1.0, + "content": "2Shenzhen Institute for Advanced Study, UESTC", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 168, + 366, + 183 + ], + "spans": [ + { + "bbox": [ + 111, + 168, + 366, + 183 + ], + "score": 1.0, + "content": "3Yale University, 4Peking University ,5Peng Cheng Laboratory", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 181, + 498, + 194 + ], + "spans": [ + { + "bbox": [ + 112, + 181, + 498, + 194 + ], + "score": 1.0, + "content": "dengsk119@std.uestc.edu.cn, yuhang.li@yale.edu, gus@uestc.edu.cn", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + } + ], + "index": 4.5, + "bbox_fs": [ + 111, + 145, + 498, + 194 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 221, + 333, + 233 + ], + "lines": [ + { + "bbox": [ + 276, + 221, + 335, + 234 + ], + "spans": [ + { + "bbox": [ + 276, + 221, + 335, + 234 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 143, + 245, + 468, + 465 + ], + "lines": [ + { + "bbox": [ + 142, + 246, + 469, + 258 + ], + "spans": [ + { + "bbox": [ + 142, + 246, + 469, + 258 + ], + "score": 1.0, + "content": "Recently, brain-inspired spiking neuron networks (SNNs) have attracted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 142, + 257, + 469, + 268 + ], + "score": 1.0, + "content": "widespread research interest because of their event-driven and energy-efficient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 469, + 280 + ], + "score": 1.0, + "content": "characteristics. Still, it is difficult to efficiently train deep SNNs due to the non-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 470, + 291 + ], + "score": 1.0, + "content": "differentiability of its activation function, which disables the typically used gradi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 469, + 302 + ], + "score": 1.0, + "content": "ent descent approaches for traditional artificial neural networks (ANNs). Although", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 470, + 313 + ], + "score": 1.0, + "content": "the adoption of surrogate gradient (SG) formally allows for the back-propagation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 325 + ], + "score": 1.0, + "content": "of losses, the discrete spiking mechanism actually differentiates the loss landscape", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "of SNNs from that of ANNs, failing the surrogate gradient methods to achieve", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 142, + 334, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 142, + 334, + 469, + 345 + ], + "score": 1.0, + "content": "comparable accuracy as for ANNs. In this paper, we first analyze why the current", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 469, + 357 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 469, + 357 + ], + "score": 1.0, + "content": "direct training approach with surrogate gradient results in SNNs with poor gen-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "eralizability. Then we introduce the temporal efficient training (TET) approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 378 + ], + "score": 1.0, + "content": "to compensate for the loss of momentum in the gradient descent with SG so that", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 470, + 389 + ], + "score": 1.0, + "content": "the training process can converge into flatter minima with better generalizabil-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 470, + 401 + ], + "score": 1.0, + "content": "ity. Meanwhile, we demonstrate that TET improves the temporal scalability of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 399, + 470, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 470, + 411 + ], + "score": 1.0, + "content": "SNN and induces a temporal inheritable training for acceleration. Our method", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 421 + ], + "score": 1.0, + "content": "consistently outperforms the SOTA on all reported mainstream datasets, includ-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "spans": [ + { + "bbox": [ + 141, + 421, + 470, + 434 + ], + "score": 1.0, + "content": "ing CIFAR-10/100 and ImageNet. Remarkably on DVS-CIFAR10, we obtained", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 142, + 430, + 470, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 432, + 162, + 443 + ], + "score": 0.9, + "content": "8 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 430, + 252, + 444 + ], + "score": 1.0, + "content": "top-1 accuracy, over", + "type": "text" + }, + { + "bbox": [ + 252, + 432, + 272, + 442 + ], + "score": 0.87, + "content": "\\bar { 1 0 \\% }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 430, + 470, + 444 + ], + "score": 1.0, + "content": "improvement compared to existing state of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 442, + 469, + 455 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 469, + 455 + ], + "score": 1.0, + "content": "art. Codes are available at https://github.com/Gus-Lab/temporal_", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 453, + 255, + 466 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 255, + 466 + ], + "score": 1.0, + "content": "efficient_training.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 17.5, + "bbox_fs": [ + 141, + 246, + 470, + 466 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 206, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 208, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 208, + 501 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 523 + ], + "score": 1.0, + "content": "The advantages of Spiking neuron networks (SNNs) lie in their energy-saving and fast-inference", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 504, + 534 + ], + "score": 1.0, + "content": "computation when embedded on neuromorphic hardware such as TrueNorth (DeBole et al., 2019)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "and Loihi (Davies et al., 2018). Such advantages originate from the biology-inspired binary spike", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "transmitted mechanism, by which the networks avoid multiplication during inference. On the other", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "hand, this mechanism also leads to difficulty in training very deep SNNs from scratch because", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "the non-differentiable spike transmission hinders the powerful back-propagation approaches like", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "gradient descents. Recently, many studies on converting artificial neuron networks (ANNs) to SNNs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 599 + ], + "score": 1.0, + "content": "have demonstrated SNNs’ comparable power in feature representation as ANNs (Han & Roy, 2020;", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 611 + ], + "score": 1.0, + "content": "Deng & Gu, 2020; Li et al., 2021a). Nevertheless, it is commonly agreed that the direct training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "method for high-performance SNN is still crucial since it distinguishes SNNs from converted ANNs,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 257, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 257, + 632 + ], + "score": 1.0, + "content": "especially on neuromorphic datasets.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 510, + 505, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "score": 1.0, + "content": "The output layer’s spike frequency or the average membrane potential increment is commonly used", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 660 + ], + "score": 1.0, + "content": "as inference indicators in SNNs (Shrestha & Orchard, 2018; Kim et al., 2019). The current standard", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "direct training (SDT) methods regard the SNN as RNN and optimize inference indicators’ distri-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "bution (Wu et al., 2018). They adopt surrogate gradients (SG) to relieve the non-differentiability", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "(Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021). However, the gradient descent with SG does", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "not match with the loss landscape in SNN and is easy to get trapped in a local minimum with low", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 506, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 506, + 716 + ], + "score": 1.0, + "content": "generalizability. Although using suitable optimizers and weight decay help ease this problem, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "performance of deep SNNs trained from scratch still suffers a big deficit compared to that of ANNs", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Deng et al. (2020). Another training issue is the memory and time consumption, which increases", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "linearly with the simulation time. Rathi & Roy (2020) initializes the target network by a converted", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "SNN to shorten the training epochs, indicating the possibility of high-performance SNN with limited", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "activation time. The training problem due to the non-differentiable activation function has become", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 371, + 351, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 351, + 383 + ], + "score": 1.0, + "content": "the main obstruction of spiking neural network development.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 637, + 506, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 88, + 504, + 262 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 88, + 504, + 262 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 88, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 110, + 88, + 504, + 262 + ], + "score": 0.972, + "type": "image", + "image_path": "1c0df3e5e42eb5f12b364ae3e1b86c171367176d7d8e201b7cccf8c9ba08687d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 88, + 504, + 146.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 146.0, + 504, + 204.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 204.0, + 504, + 262.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 274, + 506, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "Figure 1: Workflow of temporal efficient training (TET). To obtain a more generalized SNN, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 286, + 408, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 408, + 298 + ], + "score": 1.0, + "content": "modify the optimization target to adjust each moment’s output distribution.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "performance of deep SNNs trained from scratch still suffers a big deficit compared to that of ANNs", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "Deng et al. (2020). Another training issue is the memory and time consumption, which increases", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "linearly with the simulation time. Rathi & Roy (2020) initializes the target network by a converted", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "SNN to shorten the training epochs, indicating the possibility of high-performance SNN with limited", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "activation time. The training problem due to the non-differentiable activation function has become", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 371, + 351, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 351, + 383 + ], + "score": 1.0, + "content": "the main obstruction of spiking neural network development.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "In this work, we examine the limitation of the traditional direct training approach with SG and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "propose the temporal efficient training (TET) algorithm. Instead of directly optimizing the integrated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "potential, TET optimizes every moment’s pre-synaptic inputs. As a result, it avoids the trap into", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "local minima with low prediction error but a high second-order moment. Furthermore, since the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "TET applies optimization on each time point, the network naturally has more robust time scalability.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Based on this characteristic, we propose the time inheritance training (TIT), which reduces the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "training time by initializing the SNN with a smaller simulation length. With the help of TET, the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 465, + 504, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 504, + 478 + ], + "score": 1.0, + "content": "performance of SNNs has improved on both static datasets and neuromorphic datasets. Figure 1", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 260, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 260, + 488 + ], + "score": 1.0, + "content": "depicts the workflow of our approach.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 310, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 312, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 312, + 506 + ], + "score": 1.0, + "content": "The following summarizes our main contributions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 133, + 513, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 133, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 133, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "• We analyze the problem of training SNN with SG and propose the TET method, a new loss", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 524, + 469, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 524, + 469, + 536 + ], + "score": 1.0, + "content": "and gradient descent regime that succeeds in obtaining more generalizable SNNs.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 132, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 132, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "• We analyze the feasibility of TET and picture the loss landscape under both the SDT and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 549, + 419, + 562 + ], + "spans": [ + { + "bbox": [ + 141, + 549, + 419, + 562 + ], + "score": 1.0, + "content": "TET setups to demonstrate TET’s advantage in better generalization.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 132, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 132, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "• Our sufficient experiments on both static datasets and neuromorphic datasets prove the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 575, + 447, + 587 + ], + "score": 1.0, + "content": "effectiveness of the TET method. Especially on DVS-CIFAR10, we report", + "type": "text" + }, + { + "bbox": [ + 447, + 575, + 480, + 586 + ], + "score": 0.88, + "content": "8 3 . 1 \\bar { 7 } \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "top-1", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 586, + 504, + 597 + ], + "spans": [ + { + "bbox": [ + 141, + 586, + 305, + 597 + ], + "score": 1.0, + "content": "accuracy for the first time, which is over", + "type": "text" + }, + { + "bbox": [ + 306, + 586, + 325, + 596 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 586, + 504, + 597 + ], + "score": 1.0, + "content": "better than the current state-of-the-art result.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 613, + 211, + 626 + ], + "lines": [ + { + "bbox": [ + 104, + 612, + 213, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 612, + 213, + 628 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "In recent years, SNNs have developed rapidly and received more and more attention from the re-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "search community. 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To obtain a more generalized SNN, we", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 286, + 408, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 408, + 298 + ], + "score": 1.0, + "content": "modify the optimization target to adjust each moment’s output distribution.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 383 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 316, + 506, + 383 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "In this work, we examine the limitation of the traditional direct training approach with SG and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "propose the temporal efficient training (TET) algorithm. Instead of directly optimizing the integrated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "potential, TET optimizes every moment’s pre-synaptic inputs. As a result, it avoids the trap into", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "local minima with low prediction error but a high second-order moment. Furthermore, since the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "TET applies optimization on each time point, the network naturally has more robust time scalability.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "Based on this characteristic, we propose the time inheritance training (TIT), which reduces the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "training time by initializing the SNN with a smaller simulation length. 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Figure 1", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 260, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 260, + 488 + ], + "score": 1.0, + "content": "depicts the workflow of our approach.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 389, + 506, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 310, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 312, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 312, + 506 + ], + "score": 1.0, + "content": "The following summarizes our main contributions:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 491, + 312, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 133, + 513, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 133, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 133, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "• We analyze the problem of training SNN with SG and propose the TET method, a new loss", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 524, + 469, + 536 + ], + "spans": [ + { + "bbox": [ + 142, + 524, + 469, + 536 + ], + "score": 1.0, + "content": "and gradient descent regime that succeeds in obtaining more generalizable SNNs.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 132, + 537, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 132, + 537, + 506, + 552 + ], + "score": 1.0, + "content": "• We analyze the feasibility of TET and picture the loss landscape under both the SDT and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 549, + 419, + 562 + ], + "spans": [ + { + "bbox": [ + 141, + 549, + 419, + 562 + ], + "score": 1.0, + "content": "TET setups to demonstrate TET’s advantage in better generalization.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 132, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 132, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "• Our sufficient experiments on both static datasets and neuromorphic datasets prove the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 141, + 575, + 447, + 587 + ], + "score": 1.0, + "content": "effectiveness of the TET method. 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Some special", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "techniques have been proposed to reduce the inference latency, such as the subtraction mechanism", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "(Rueckauer et al., 2016; Han et al., 2020), robust normalization Rueckauer et al. (2016), spike-norm", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "(Sengupta et al., 2018), and channel-wise normalization (Kim et al., 2019). Recently, Deng & Gu", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 116 + ], + "score": 1.0, + "content": "(2020) decompose the conversion error to each layer and reduce it by bias shift. Li et al. (2021a)", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "suggest using adaptive threshold and layer-wise calibration to obtain high-performance SNNs that", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "require a simulation length of less than 50. 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(2021a)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "suggest using adaptive threshold and layer-wise calibration to obtain high-performance SNNs that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "require a simulation length of less than 50. However, converted methods significantly extend the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 450, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 450, + 149 + ], + "score": 1.0, + "content": "inference latency, and they are not suitable for neuromorphic data (Deng et al., 2020).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "Direct training. In this area, SNNs are regarded as special RNNs and training with BPTT (Neftci", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "et al., 2019). On the backpropagation process, The non-differentiable activation term is replaced", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "with a surrogate gradient (Lee et al., 2016). Compared with ANN-to-SNN conversion, direct train-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "score": 1.0, + "content": "ing achieves high accuracy with few time steps but suffers more training costs (Deng et al., 2020).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "Several studies suggest that surrogate gradient (SG) is helpful to obtain high-performance SNNs", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "on both static datasets and neuromorphic datasets (Wu et al., 2019; Shrestha & Orchard, 2018; Li", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "et al., 2021b). On the backpropagation process, SG replaces the Dirac function with various shapes", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "of curves. Exceptionally, Wu et al. (2018) first propose the STBP method and train SNNs on the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "ANN programming platform, which significantly promotes direct training development. Zheng et al.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "(2021) further proposes the tdBN algorithm to smooth the loss function and first realize training a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "large-scale SNN on ImageNet. Zhang & Li (2020) proposes TSSL-BP to break down error back-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 288 + ], + "score": 1.0, + "content": "propagation across two types of inter-neuron and intra-neuron dependencies and achieve low-latency", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 504, + 298 + ], + "score": 1.0, + "content": "and high accuracy SNNs. Recently, Yang et al. (2021) designed a neighborhood aggregation (NA)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "method to use the multiple perturbed membrane potential waveforms in the neighborhood to com-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "pute the finite difference gradients and guide the weight updates. They significantly decrease the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 361, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 361, + 331 + ], + "score": 1.0, + "content": "required training iterations and improve the SNN performance.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 346, + 199, + 359 + ], + "lines": [ + { + "bbox": [ + 104, + 344, + 200, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 344, + 200, + 362 + ], + "score": 1.0, + "content": "3 PRELIMINARY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 232, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 233, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 233, + 383 + ], + "score": 1.0, + "content": "3.1 ITERATIVE LIF MODEL", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 391, + 504, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "We adopt the Leaky Integrate-and-Fire (LIF) model and translate it to an iterative expression with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 402, + 468, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 468, + 415 + ], + "score": 1.0, + "content": "the Euler method (Wu et al., 2019). 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\\mathbf \\Psi \\mathbf { } \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf \\Psi \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\Psi \\mathbf { } \\mathbf \\mathbf \\mathbf \\Psi \\Psi \\mathbf { \\mathbf } \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf 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The output spike", + "type": "text" + }, + { + "bbox": [ + 376, + 516, + 413, + 528 + ], + "score": 0.93, + "content": "\\mathbf { \\delta } \\mathbf { \\ } \\mathbf { \\em a } ( t + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "will become the post", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 472, + 540 + ], + "score": 1.0, + "content": "synaptic spike and propagate to the next layer. 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\\mathbf \\Psi \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf \\mathbf ", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "exceeds the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 468, + 423, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 423, + 480 + ], + "score": 1.0, + "content": "threshold. 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The output spike", + "type": "text" + }, + { + "bbox": [ + 376, + 516, + 413, + 528 + ], + "score": 0.93, + "content": "\\mathbf { \\delta } \\mathbf { \\ } \\mathbf { \\em a } ( t + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "will become the post", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 472, + 540 + ], + "score": 1.0, + "content": "synaptic spike and propagate to the next layer. In this study, we set the starting membrane", + "type": "text" + }, + { + "bbox": [ + 473, + 527, + 493, + 539 + ], + "score": 0.91, + "content": "\\pmb { u } ( 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 538, + 402, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 170, + 550 + ], + "score": 1.0, + "content": "0, the threshold", + "type": "text" + }, + { + "bbox": [ + 171, + 538, + 186, + 549 + ], + "score": 0.9, + "content": "V _ { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 538, + 288, + 550 + ], + "score": 1.0, + "content": "to 1, and the leaky factor", + "type": "text" + }, + { + "bbox": [ + 289, + 540, + 295, + 548 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 538, + 402, + 550 + ], + "score": 1.0, + "content": "to 0.5 for all experiments.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 515, + 506, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "The last layer’s spike frequency is typically used as the final classification index. However, adopting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "the LIF model on the last layer will lose information on the membrane potential and damage the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "performance, especially on complex tasks (Kim et al., 2019). Instead, we integrate the pre-synaptic", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 586, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 133, + 602 + ], + "score": 1.0, + "content": "inputs", + "type": "text" + }, + { + "bbox": [ + 133, + 588, + 151, + 600 + ], + "score": 0.91, + "content": "\\mathbf { } I ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 586, + 506, + 602 + ], + "score": 1.0, + "content": "with no decay or firing (Rathi & Roy, 2020; Fang et al., 2021). Finally, we set the average", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 597, + 487, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 487, + 613 + ], + "score": 1.0, + "content": "membrane potential as the classification index and calculate the cross-entropy loss for training.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 554, + 506, + 613 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 231, + 635 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 233, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 233, + 636 + ], + "score": 1.0, + "content": "3.2 SURROGATE GRADIENT", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 503, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 657 + ], + "score": 1.0, + "content": "Following the concept of direct training, we regard the SNN as RNN and calculate the gradients", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 655, + 382, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 382, + 668 + ], + "score": 1.0, + "content": "through spatial-temporal backpropagation (STBP) (Wu et al., 2018):", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 642, + 505, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 670, + 384, + 700 + ], + "lines": [ + { + "bbox": [ + 226, + 670, + 384, + 700 + ], + "spans": [ + { + "bbox": [ + 226, + 670, + 384, + 700 + ], + "score": 0.95, + "content": "\\frac { \\partial L } { \\partial \\mathbf { W } } = \\sum _ { t } \\frac { \\partial L } { \\partial \\pmb { a } ( t ) } \\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { a } ( t ) } \\frac { \\partial \\pmb { u } ( t ) } { \\partial \\pmb { I } ( t ) } \\frac { \\partial \\pmb { I } ( t ) } { \\partial \\mathbf { W } } ,", + "type": "interline_equation", + "image_path": "cd21c6adb4f30e317bf469128694fc231f872f68c84c7a80998a1e60e25c08bb.jpg" + } + ] + } + ], + "index": 44.5, + "virtual_lines": [ + { + "bbox": [ + 226, + 670, + 384, + 685.0 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 226, + 685.0, + 384, + 700.0 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 706, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 703, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 168, + 724 + ], + "score": 1.0, + "content": "where the term", + "type": "text" + }, + { + "bbox": [ + 169, + 706, + 190, + 722 + ], + "score": 0.94, + "content": "\\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { u } ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 703, + 506, + 724 + ], + "score": 1.0, + "content": "is the gradient of the non-differentiability step function involving the derivative", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 149, + 732 + ], + "score": 1.0, + "content": "of Dirac’s", + "type": "text" + }, + { + "bbox": [ + 149, + 722, + 154, + 730 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "-function that is typically replaced by surrogate gradients with a derivable curve. So far,", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 703, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "there are various shapes of surrogate gradients, such as rectangular (Wu et al., 2018; 2019), triangle", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "(Esser et al., 2016; Rathi & Roy, 2020), and exponential (Shrestha & Orchard, 2018) curve. In this", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 495, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 495, + 116 + ], + "score": 1.0, + "content": "work, we choose the surrogate gradients shaped like triangles. Mathematically, it can describe as", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 124, + 385, + 151 + ], + "lines": [ + { + "bbox": [ + 226, + 124, + 385, + 151 + ], + "spans": [ + { + "bbox": [ + 226, + 124, + 385, + 151 + ], + "score": 0.95, + "content": "\\frac { \\partial \\pmb { a } ( t ) } { \\partial \\pmb { u } ( t ) } = \\frac { 1 } { \\gamma ^ { 2 } } \\mathrm { m a x } ( 0 , \\gamma - | \\pmb { u } ( t ) - V _ { t h } | ) ,", + "type": "interline_equation", + "image_path": "0d996846155207901ccfa55f7bd0020fc5d421b0bad5b204f7ff5dc37acc1d79.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 226, + 124, + 385, + 151 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 158, + 499, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 157, + 500, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 147, + 172 + ], + "score": 1.0, + "content": "where the", + "type": "text" + }, + { + "bbox": [ + 147, + 160, + 155, + 170 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 157, + 500, + 172 + ], + "score": 1.0, + "content": "denotes the constraint factor that determines the sample range to activate the gradient.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 282, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 282, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 282, + 199 + ], + "score": 1.0, + "content": "3.3 BATCH NORMALIZATION FOR SNN", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "Batch Normalization (BN) (Ioffe & Szegedy, 2015) is beneficial to accelerate training and increase", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 218, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 506, + 232 + ], + "score": 1.0, + "content": "performance since it can smooth the loss landscape during training (Santurkar et al., 2018). Zheng", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 504, + 242 + ], + "score": 1.0, + "content": "et al. (2021) modified the forward time loop form and proposed threshold-dependent Batch Nor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 242, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 334, + 253 + ], + "score": 1.0, + "content": "malization (tdBN) to normalize the pre-synaptic inputs", + "type": "text" + }, + { + "bbox": [ + 334, + 242, + 341, + 251 + ], + "score": 0.71, + "content": "\\pmb { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 242, + 505, + 253 + ], + "score": 1.0, + "content": "in both spatial and temporal paradigms", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 252, + 504, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 504, + 263 + ], + "score": 1.0, + "content": "so that the BN can support spatial-temporal input. We adopt this setup with the extension of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "time dimension to batch dimension 1. 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We use", + "type": "text" + }, + { + "bbox": [ + 257, + 413, + 278, + 425 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 412, + 406, + 427 + ], + "score": 1.0, + "content": "to represent pre-synaptic input", + "type": "text" + }, + { + "bbox": [ + 407, + 413, + 425, + 425 + ], + "score": 0.91, + "content": "\\mathbf { } I ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "of the output layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 423, + 466, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 431, + 438 + ], + "score": 1.0, + "content": "and calculate the cross-entropy loss. The loss function of standard direct training", + "type": "text" + }, + { + "bbox": [ + 431, + 425, + 453, + 435 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 423, + 466, + 438 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 412, + 506, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 443, + 366, + 478 + ], + "lines": [ + { + "bbox": [ + 244, + 443, + 366, + 478 + ], + "spans": [ + { + "bbox": [ + 244, + 443, + 366, + 478 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { \\mathrm { S D T } } = \\mathcal { L } _ { \\mathrm { C E } } \\big ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } O ( t ) , { \\pmb y } \\big ) ,", + "type": "interline_equation", + "image_path": "8f4533824d324c069c3603f39de4b247d351ea6cf666d20b7c5b6f05cc4ffc71.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 244, + 443, + 366, + 460.5 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 244, + 460.5, + 366, + 478.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 506, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 133, + 498 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 486, + 142, + 496 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 485, + 254, + 498 + ], + "score": 1.0, + "content": "is the total simulation time,", + "type": "text" + }, + { + "bbox": [ + 254, + 486, + 272, + 497 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 485, + 414, + 498 + ], + "score": 1.0, + "content": "denotes the cross-entropy loss, and", + "type": "text" + }, + { + "bbox": [ + 414, + 488, + 421, + 497 + ], + "score": 0.82, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "represents the target", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 496, + 504, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 336, + 509 + ], + "score": 1.0, + "content": "label. 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TET", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 104, + 103, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 380, + 119 + ], + "score": 1.0, + "content": "In the case of SDT, the gradient consists of two parts, the error term", + "type": "text" + }, + { + "bbox": [ + 380, + 104, + 444, + 117 + ], + "score": 0.9, + "content": "( S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 103, + 506, + 119 + ], + "score": 1.0, + "content": "and the partial", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 186, + 128 + ], + "score": 1.0, + "content": "derivative of output", + "type": "text" + }, + { + "bbox": [ + 187, + 115, + 235, + 127 + ], + "score": 0.92, + "content": "\\partial { \\cal O } \\bar { ( } t ) / \\partial { \\bf W }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ". 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For traditional", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "ANNs, the accumulated momentum may help get out of the local minima (e.g. saddle point) that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "typically implies bad generalizability (Kingma & Ba, 2014; Kidambi et al., 2018). 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This mismatch dissipates the momentum around", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "a local minimum and stops the SDT from searching for a flatter minimum that may suggest better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 173, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 173, + 216 + ], + "score": 1.0, + "content": "generalizability.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 453, + 232 + ], + "score": 1.0, + "content": "In the case of TET, this issue of mismatch is relieved by reweighting the contribution of", + "type": "text" + }, + { + "bbox": [ + 453, + 219, + 501, + 232 + ], + "score": 0.91, + "content": "\\partial { \\cal O } ( t ) / \\partial { \\bf W }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 219, + 505, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 290, + 244 + ], + "score": 1.0, + "content": "Indeed, considering the fact that the first term", + "type": "text" + }, + { + "bbox": [ + 290, + 231, + 350, + 243 + ], + "score": 0.92, + "content": "( S ( O ( t ) ) - \\hat { { \\mathbf { y } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "is impossible to be 0 at every moment", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 370, + 255 + ], + "score": 1.0, + "content": "of SNN since the early output accuracy on the training set is not", + "type": "text" + }, + { + "bbox": [ + 370, + 242, + 394, + 253 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 241, + 506, + 255 + ], + "score": 1.0, + "content": ". 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This mechanism increases the norm of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "gradients around sharp local minima and drives the TET to search for a flat local minimum where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 366, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 340, + 288 + ], + "score": 1.0, + "content": "the disturbance of weight does not cause a huge change in", + "type": "text" + }, + { + "bbox": [ + 340, + 275, + 361, + 287 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 274, + 366, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "Further, to ensure that the convergence with TET implies the convergence of SDT, we prove the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 181, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 181, + 316 + ], + "score": 1.0, + "content": "following lemma:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 289, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 288, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 161, + 334 + ], + "score": 1.0, + "content": "Lemma 4.1.", + "type": "text" + }, + { + "bbox": [ + 162, + 319, + 183, + 330 + ], + "score": 0.7, + "content": "\\mathcal { L } _ { S D T }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 316, + 267, + 334 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 267, + 319, + 288, + 330 + ], + "score": 0.54, + "content": "\\mathcal { L } _ { T E T }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 503, + 375 + ], + "lines": [ + { + "bbox": [ + 104, + 349, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 172, + 366 + ], + "score": 1.0, + "content": "Proof. Suppose", + "type": "text" + }, + { + "bbox": [ + 172, + 352, + 196, + 364 + ], + "score": 0.93, + "content": "O _ { i } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 349, + 214, + 366 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 353, + 224, + 364 + ], + "score": 0.88, + "content": "\\hat { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 349, + 342, + 366 + ], + "score": 1.0, + "content": "denote the i-th component of", + "type": "text" + }, + { + "bbox": [ + 342, + 352, + 362, + 364 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 349, + 380, + 366 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 353, + 388, + 364 + ], + "score": 0.84, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 349, + 506, + 366 + ], + "score": 1.0, + "content": ", respectively. Expand Eqn.9,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 145, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 145, + 376 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 380, + 449, + 486 + ], + "lines": [ + { + "bbox": [ + 160, + 380, + 449, + 486 + ], + "spans": [ + { + "bbox": [ + 160, + 380, + 449, + 486 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { T E T } } = - \\displaystyle \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log S ( \\boldsymbol { O } _ { i } ( t ) ) = - \\frac { 1 } { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad = - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) ^ { \\frac { 1 } { T } } \\geq - \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad \\geq - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( S ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } O _ { i } ( t ) ) ) = \\mathcal { L } _ { \\mathrm { S D T } } , } \\end{array}", + "type": "interline_equation", + "image_path": "af76f778462d27eb467b31d9a012ce480c4641edb15c8934625506b0e9b4fb94.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 160, + 380, + 449, + 415.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 160, + 415.3333333333333, + 449, + 450.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 160, + 450.66666666666663, + 449, + 485.99999999999994 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 492, + 504, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "where the first inequality is given by the Arithmetic Mean-Geometric Mean Inequality, and the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 504, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 516 + ], + "score": 1.0, + "content": "second one is given by Jensen Inequality since the softmax function is convex. 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In practice, we use a hyperparameter", + "type": "text" + }, + { + "bbox": [ + 342, + 654, + 349, + 663 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "to adjust the proportion of the regular", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 665, + 167, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 167, + 676 + ], + "score": 1.0, + "content": "term, we have:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 678, + 375, + 692 + ], + "lines": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "spans": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { T O T A L } } = ( 1 - \\lambda ) \\mathcal { L } _ { \\mathrm { T E T } } + \\lambda \\mathcal { L } _ { \\mathrm { M S E } } .", + "type": "interline_equation", + "image_path": "2d4ca1af8e92f1b5e37f29d8225e52d8f739328ae7f8fbd21a9e43ec0522cfeb.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 504, + 721 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "It is worth noting that we only changed the loss function in the training process and did not change", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "SNN’s inference rules in the testing phase for a fair comparison. This algorithm is detailed in Algo.1.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 387, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 388, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 388, + 95 + ], + "score": 1.0, + "content": "4.2 CONVERGENCE OF GRADIENT DESCENT FOR SDT V.S. TET", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 104, + 103, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 103, + 380, + 119 + ], + "score": 1.0, + "content": "In the case of SDT, the gradient consists of two parts, the error term", + "type": "text" + }, + { + "bbox": [ + 380, + 104, + 444, + 117 + ], + "score": 0.9, + "content": "( S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 103, + 506, + 119 + ], + "score": 1.0, + "content": "and the partial", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 186, + 128 + ], + "score": 1.0, + "content": "derivative of output", + "type": "text" + }, + { + "bbox": [ + 187, + 115, + 235, + 127 + ], + "score": 0.92, + "content": "\\partial { \\cal O } \\bar { ( } t ) / \\partial { \\bf W }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ". When the training process reaches near a local minimum, the term", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 107, + 126, + 170, + 138 + ], + "score": 0.91, + "content": "( S ( O _ { \\mathrm { m e a n } } ) - \\hat { \\pmb y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 126, + 261, + 138 + ], + "score": 1.0, + "content": "approximates 0 for all", + "type": "text" + }, + { + "bbox": [ + 262, + 127, + 309, + 138 + ], + "score": 0.91, + "content": "t = 1 , . . . , T", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 126, + 393, + 138 + ], + "score": 1.0, + "content": ", ignorant of the term", + "type": "text" + }, + { + "bbox": [ + 393, + 126, + 442, + 138 + ], + "score": 0.92, + "content": "\\partial O ( t ) / \\partial \\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 126, + 505, + 138 + ], + "score": 1.0, + "content": ". For traditional", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "ANNs, the accumulated momentum may help get out of the local minima (e.g. saddle point) that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 160 + ], + "score": 1.0, + "content": "typically implies bad generalizability (Kingma & Ba, 2014; Kidambi et al., 2018). However, when", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "the SNN is trained with surrogate gradients, the accumulated momentum could be extremely small,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "considering the mismatch of gradients and losses. The fact that the activation function is a step one", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "while the SG is bounded with integral constraints. This mismatch dissipates the momentum around", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "a local minimum and stops the SDT from searching for a flatter minimum that may suggest better", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 203, + 173, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 173, + 216 + ], + "score": 1.0, + "content": "generalizability.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 103, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 453, + 232 + ], + "score": 1.0, + "content": "In the case of TET, this issue of mismatch is relieved by reweighting the contribution of", + "type": "text" + }, + { + "bbox": [ + 453, + 219, + 501, + 232 + ], + "score": 0.91, + "content": "\\partial { \\cal O } ( t ) / \\partial { \\bf W }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 219, + 505, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 290, + 244 + ], + "score": 1.0, + "content": "Indeed, considering the fact that the first term", + "type": "text" + }, + { + "bbox": [ + 290, + 231, + 350, + 243 + ], + "score": 0.92, + "content": "( S ( O ( t ) ) - \\hat { { \\mathbf { y } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "is impossible to be 0 at every moment", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 370, + 255 + ], + "score": 1.0, + "content": "of SNN since the early output accuracy on the training set is not", + "type": "text" + }, + { + "bbox": [ + 370, + 242, + 394, + 253 + ], + "score": 0.88, + "content": "1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 241, + 506, + 255 + ], + "score": 1.0, + "content": ". So TET needs the second", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 127, + 266 + ], + "score": 1.0, + "content": "term", + "type": "text" + }, + { + "bbox": [ + 128, + 253, + 176, + 265 + ], + "score": 0.92, + "content": "\\partial { \\cal O } ( t ) / \\partial { \\bf W }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 253, + 268, + 266 + ], + "score": 1.0, + "content": "close to 0 to make the", + "type": "text" + }, + { + "bbox": [ + 268, + 253, + 290, + 264 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "convergence. This mechanism increases the norm of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "gradients around sharp local minima and drives the TET to search for a flat local minimum where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 366, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 340, + 288 + ], + "score": 1.0, + "content": "the disturbance of weight does not cause a huge change in", + "type": "text" + }, + { + "bbox": [ + 340, + 275, + 361, + 287 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 274, + 366, + 288 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 219, + 506, + 288 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "Further, to ensure that the convergence with TET implies the convergence of SDT, we prove the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 301, + 181, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 181, + 316 + ], + "score": 1.0, + "content": "following lemma:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 290, + 505, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 289, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 288, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 161, + 334 + ], + "score": 1.0, + "content": "Lemma 4.1.", + "type": "text" + }, + { + "bbox": [ + 162, + 319, + 183, + 330 + ], + "score": 0.7, + "content": "\\mathcal { L } _ { S D T }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 316, + 267, + 334 + ], + "score": 1.0, + "content": "is upper bounded by", + "type": "text" + }, + { + "bbox": [ + 267, + 319, + 288, + 330 + ], + "score": 0.54, + "content": "\\mathcal { L } _ { T E T }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 316, + 288, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 351, + 503, + 375 + ], + "lines": [ + { + "bbox": [ + 104, + 349, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 172, + 366 + ], + "score": 1.0, + "content": "Proof. Suppose", + "type": "text" + }, + { + "bbox": [ + 172, + 352, + 196, + 364 + ], + "score": 0.93, + "content": "O _ { i } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 349, + 214, + 366 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 353, + 224, + 364 + ], + "score": 0.88, + "content": "\\hat { y } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 349, + 342, + 366 + ], + "score": 1.0, + "content": "denote the i-th component of", + "type": "text" + }, + { + "bbox": [ + 342, + 352, + 362, + 364 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 349, + 380, + 366 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 353, + 388, + 364 + ], + "score": 0.84, + "content": "\\hat { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 349, + 506, + 366 + ], + "score": 1.0, + "content": ", respectively. Expand Eqn.9,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 361, + 145, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 145, + 376 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 349, + 506, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 380, + 449, + 486 + ], + "lines": [ + { + "bbox": [ + 160, + 380, + 449, + 486 + ], + "spans": [ + { + "bbox": [ + 160, + 380, + 449, + 486 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { T E T } } = - \\displaystyle \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log S ( \\boldsymbol { O } _ { i } ( t ) ) = - \\frac { 1 } { T } \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad = - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\prod _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) ^ { \\frac { 1 } { T } } \\geq - \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } S ( \\boldsymbol { O } _ { i } ( t ) ) ) } \\\\ & { \\quad \\quad \\quad \\geq - \\displaystyle \\sum _ { i = 1 } ^ { n } \\hat { y } _ { i } \\log ( S ( \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } O _ { i } ( t ) ) ) = \\mathcal { L } _ { \\mathrm { S D T } } , } \\end{array}", + "type": "interline_equation", + "image_path": "af76f778462d27eb467b31d9a012ce480c4641edb15c8934625506b0e9b4fb94.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 160, + 380, + 449, + 415.3333333333333 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 160, + 415.3333333333333, + 449, + 450.66666666666663 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 160, + 450.66666666666663, + 449, + 485.99999999999994 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 492, + 504, + 526 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "where the first inequality is given by the Arithmetic Mean-Geometric Mean Inequality, and the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 504, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 506, + 516 + ], + "score": 1.0, + "content": "second one is given by Jensen Inequality since the softmax function is convex. As a corollary, once", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 121, + 528 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 515, + 143, + 526 + ], + "score": 0.87, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 514, + 325, + 528 + ], + "score": 1.0, + "content": "gets closed to zero, the original loss function", + "type": "text" + }, + { + "bbox": [ + 325, + 515, + 347, + 526 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 514, + 435, + 528 + ], + "score": 1.0, + "content": "also approaches zero.", + "type": "text" + }, + { + "bbox": [ + 494, + 515, + 506, + 525 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 493, + 506, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 239, + 560 + ], + "score": 1.0, + "content": "Furthermore, the network output", + "type": "text" + }, + { + "bbox": [ + 239, + 547, + 260, + 560 + ], + "score": 0.92, + "content": "O ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "at a particular time point may be a particular outlier that dra-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 571 + ], + "score": 1.0, + "content": "matically affects the total output since the output of the SNN has the same weight at every moment", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 450, + 582 + ], + "score": 1.0, + "content": "under the rule of integration. 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In practice, we use a hyperparameter", + "type": "text" + }, + { + "bbox": [ + 342, + 654, + 349, + 663 + ], + "score": 0.82, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "to adjust the proportion of the regular", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 665, + 167, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 167, + 676 + ], + "score": 1.0, + "content": "term, we have:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 641, + 505, + 676 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 678, + 375, + 692 + ], + "lines": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "spans": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { T O T A L } } = ( 1 - \\lambda ) \\mathcal { L } _ { \\mathrm { T E T } } + \\lambda \\mathcal { L } _ { \\mathrm { M S E } } .", + "type": "interline_equation", + "image_path": "2d4ca1af8e92f1b5e37f29d8225e52d8f739328ae7f8fbd21a9e43ec0522cfeb.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 236, + 678, + 375, + 692 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 698, + 504, + 721 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "It is worth noting that we only changed the loss function in the training process and did not change", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "SNN’s inference rules in the testing phase for a fair comparison. This algorithm is detailed in Algo.1.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 698, + 505, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 88, + 494, + 271 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 85, + 330, + 97 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 85, + 330, + 97 + ], + "spans": [], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 88, + 494, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 494, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 494, + 271 + ], + "score": 0.896, + "html": "
Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset;
total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival
for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi;
Compute the SNN output Oi(t) of eatch time step;
Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE =
(1-λ):¹∑t=1LcE(O²(t),Yi)+>·¹∑t=1 MSE(Oi(t),𝜙);
Backpropagation and update model parameters;
end for all i= 1,2,..Ival iteration do
Get mini-batch validation data,and class label: Yi;
T
Compute the SNN average output Omean = ∑T=1 O(t) over al time step;
Compare the clasification factor Omean and Yi for classification; end
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So how to shorten the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 336, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 482, + 352 + ], + "score": 1.0, + "content": "training time is also an essential problem in the direct training field. Traditional loss function", + "type": "text" + }, + { + "bbox": [ + 482, + 338, + 504, + 349 + ], + "score": 0.82, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 348, + 361 + ], + "score": 1.0, + "content": "only optimizes the whole network output under a specific", + "type": "text" + }, + { + "bbox": [ + 348, + 349, + 357, + 358 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 348, + 505, + 361 + ], + "score": 1.0, + "content": ", so its temporal scalability is poor.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "Unlike the standard training, TET algorithm optimizes each moment’s output, enabling us to extend", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "the simulation time naturally. We introduce Time Inheritance Training (TIT) to alleviate the training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 380, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 395 + ], + "score": 1.0, + "content": "time problem. We first use long epochs to train an SNN with a short simulation time T, e.g., 2. Then,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "we increase the simulation time to the target value and retrain with short epochs. We discover that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "TIT performs better than training from scratch on accuracy and significantly saves the training time.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 324, + 426 + ], + "score": 1.0, + "content": "Assuming that training an SNN with simulation length", + "type": "text" + }, + { + "bbox": [ + 324, + 415, + 351, + 424 + ], + "score": 0.9, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 415, + 370, + 426 + ], + "score": 1.0, + "content": "cost", + "type": "text" + }, + { + "bbox": [ + 370, + 415, + 380, + 424 + ], + "score": 0.68, + "content": "t s", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "time per epoch, the SNN needs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 425, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 475, + 438 + ], + "score": 1.0, + "content": "300 epochs to train from scratch, and the TIT needs 50 epochs for finetuning. So we need", + "type": "text" + }, + { + "bbox": [ + 475, + 425, + 504, + 435 + ], + "score": 0.78, + "content": "1 8 0 0 t s", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 436, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 211, + 449 + ], + "score": 1.0, + "content": "time to train an SNN with", + "type": "text" + }, + { + "bbox": [ + 211, + 436, + 239, + 446 + ], + "score": 0.9, + "content": "T = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 436, + 477, + 449 + ], + "score": 1.0, + "content": "from scratch, but following the TIT pipeline with the initial", + "type": "text" + }, + { + "bbox": [ + 477, + 436, + 504, + 446 + ], + "score": 0.87, + "content": "T = 2", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 447, + 435, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 161, + 459 + ], + "score": 1.0, + "content": "only requires", + "type": "text" + }, + { + "bbox": [ + 161, + 447, + 186, + 457 + ], + "score": 0.76, + "content": "9 0 0 t s", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 447, + 435, + 459 + ], + "score": 1.0, + "content": ". As a result, the TIT can reduce the training time cost by half.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 474, + 200, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 201, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 201, + 489 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "We validate our proposed TET algorithm and compare it with existing works on both static and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 510, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 504, + 522 + ], + "score": 1.0, + "content": "neuromorphic datasets. The network architectures in this paper include ResNet-19 (Zheng et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "2021), Spiking-ResNet34 (Zheng et al., 2021), SEW-ResNet34 (Fang et al., 2021), SNN-5, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "VGGSNN. SNN-5 (16C3-64C5-AP2-128C5-AP2-256C5-AP2-512C3-AP2-FC) is a simple convo-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 543, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 504, + 555 + ], + "score": 1.0, + "content": "lutional SNN suitable for multiple runs to discover statistical rules (Figure A. 7). The architec-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "ture of VGGSNN (64C3-128C3-AP2-256C3-256C3-AP2-512C3-512C3-AP2-512C3-512C3-AP2-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "FC) is based on VGG11 with two fully connected layers removed as we found that additional fully", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 576, + 358, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 358, + 590 + ], + "score": 1.0, + "content": "connected layers were unnecessary for neuromorphic datasets.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 601, + 321, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 323, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 323, + 614 + ], + "score": 1.0, + "content": "5.1 MODEL VALIDATION AND ABLATION STUDY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Effectiveness of TET over SDT with SG. We first examine whether the mismatch between SG and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "loss causes the convergence problem. For this purpose, we set the simulation length to 4 and change", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 182, + 655 + ], + "score": 1.0, + "content": "the spike function", + "type": "text" + }, + { + "bbox": [ + 183, + 644, + 193, + 654 + ], + "score": 0.78, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 644, + 281, + 655 + ], + "score": 1.0, + "content": "in Eqn.2 to Sigmoid", + "type": "text" + }, + { + "bbox": [ + 281, + 644, + 330, + 655 + ], + "score": 0.76, + "content": "\\sigma ( \\bar { k } \\cdot \\mathrm { { i n p u t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 644, + 505, + 655 + ], + "score": 1.0, + "content": ". We find that the TET and SDT achieved", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 239, + 667 + ], + "score": 1.0, + "content": "similar accuracy (Table 2) when", + "type": "text" + }, + { + "bbox": [ + 239, + 655, + 294, + 666 + ], + "score": 0.85, + "content": "k = 1 , 1 0 , 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ". This indicates that both TET and SDT work when", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 465, + 678 + ], + "score": 1.0, + "content": "the gradient and loss function match each other. Next, we compare the results training with", + "type": "text" + }, + { + "bbox": [ + 466, + 666, + 487, + 677 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 128, + 688 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "on SNNs (ResNet-19 on CIFAR100) training with surrogate gradient for three runs. As shown", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 476, + 700 + ], + "score": 1.0, + "content": "in Table 1, our proposed new TET training strategy dramatically increases the accuracy by", + "type": "text" + }, + { + "bbox": [ + 477, + 687, + 504, + 699 + ], + "score": 0.88, + "content": "3 . 2 5 \\%", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 243, + 712 + ], + "score": 1.0, + "content": "when the simulation time is 4 and", + "type": "text" + }, + { + "bbox": [ + 243, + 699, + 270, + 709 + ], + "score": 0.89, + "content": "3 . 5 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "when the simulation time is 6. These results quantitatively", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "support the effectiveness of TET in solving the mismatch between gradient and loss in training SNNs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 145, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 145, + 732 + ], + "score": 1.0, + "content": "with SG.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 88, + 494, + 271 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 85, + 330, + 97 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 85, + 330, + 97 + ], + "spans": [], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 88, + 494, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 494, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 494, + 271 + ], + "score": 0.896, + "html": "
Algorithm1:Temporalefficienttrainingforoneepoch Input: SNN model; Simulation length: T; Threshold: Vth; Training dataset; Validation dataset;
total training iteration in one epoch: Itrain; total validation iteration in one epoch: Ival
for all i= 1,2,...Itrain iteration do Get mini-batch training data,and class label: Yi;
Compute the SNN output Oi(t) of eatch time step;
Calculate loss function: LTOTAL = (1-λ)LTET + 入LMSE =
(1-λ):¹∑t=1LcE(O²(t),Yi)+>·¹∑t=1 MSE(Oi(t),𝜙);
Backpropagation and update model parameters;
end for all i= 1,2,..Ival iteration do
Get mini-batch validation data,and class label: Yi;
T
Compute the SNN average output Omean = ∑T=1 O(t) over al time step;
Compare the clasification factor Omean and Yi for classification; end
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So how to shorten the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 336, + 504, + 352 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 482, + 352 + ], + "score": 1.0, + "content": "training time is also an essential problem in the direct training field. Traditional loss function", + "type": "text" + }, + { + "bbox": [ + 482, + 338, + 504, + 349 + ], + "score": 0.82, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 348, + 361 + ], + "score": 1.0, + "content": "only optimizes the whole network output under a specific", + "type": "text" + }, + { + "bbox": [ + 348, + 349, + 357, + 358 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 348, + 505, + 361 + ], + "score": 1.0, + "content": ", so its temporal scalability is poor.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "Unlike the standard training, TET algorithm optimizes each moment’s output, enabling us to extend", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "score": 1.0, + "content": "the simulation time naturally. We introduce Time Inheritance Training (TIT) to alleviate the training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 380, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 395 + ], + "score": 1.0, + "content": "time problem. We first use long epochs to train an SNN with a short simulation time T, e.g., 2. Then,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 404 + ], + "score": 1.0, + "content": "we increase the simulation time to the target value and retrain with short epochs. We discover that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 416 + ], + "score": 1.0, + "content": "TIT performs better than training from scratch on accuracy and significantly saves the training time.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 324, + 426 + ], + "score": 1.0, + "content": "Assuming that training an SNN with simulation length", + "type": "text" + }, + { + "bbox": [ + 324, + 415, + 351, + 424 + ], + "score": 0.9, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 415, + 370, + 426 + ], + "score": 1.0, + "content": "cost", + "type": "text" + }, + { + "bbox": [ + 370, + 415, + 380, + 424 + ], + "score": 0.68, + "content": "t s", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "time per epoch, the SNN needs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 425, + 504, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 475, + 438 + ], + "score": 1.0, + "content": "300 epochs to train from scratch, and the TIT needs 50 epochs for finetuning. So we need", + "type": "text" + }, + { + "bbox": [ + 475, + 425, + 504, + 435 + ], + "score": 0.78, + "content": "1 8 0 0 t s", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 436, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 211, + 449 + ], + "score": 1.0, + "content": "time to train an SNN with", + "type": "text" + }, + { + "bbox": [ + 211, + 436, + 239, + 446 + ], + "score": 0.9, + "content": "T = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 436, + 477, + 449 + ], + "score": 1.0, + "content": "from scratch, but following the TIT pipeline with the initial", + "type": "text" + }, + { + "bbox": [ + 477, + 436, + 504, + 446 + ], + "score": 0.87, + "content": "T = 2", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 447, + 435, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 161, + 459 + ], + "score": 1.0, + "content": "only requires", + "type": "text" + }, + { + "bbox": [ + 161, + 447, + 186, + 457 + ], + "score": 0.76, + "content": "9 0 0 t s", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 447, + 435, + 459 + ], + "score": 1.0, + "content": ". As a result, the TIT can reduce the training time cost by half.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 313, + 506, + 459 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 474, + 200, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 201, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 201, + 489 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "We validate our proposed TET algorithm and compare it with existing works on both static and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 510, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 504, + 522 + ], + "score": 1.0, + "content": "neuromorphic datasets. The network architectures in this paper include ResNet-19 (Zheng et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "2021), Spiking-ResNet34 (Zheng et al., 2021), SEW-ResNet34 (Fang et al., 2021), SNN-5, and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "VGGSNN. SNN-5 (16C3-64C5-AP2-128C5-AP2-256C5-AP2-512C3-AP2-FC) is a simple convo-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 543, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 504, + 555 + ], + "score": 1.0, + "content": "lutional SNN suitable for multiple runs to discover statistical rules (Figure A. 7). The architec-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "ture of VGGSNN (64C3-128C3-AP2-256C3-256C3-AP2-512C3-512C3-AP2-512C3-512C3-AP2-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "FC) is based on VGG11 with two fully connected layers removed as we found that additional fully", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 576, + 358, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 358, + 590 + ], + "score": 1.0, + "content": "connected layers were unnecessary for neuromorphic datasets.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 500, + 505, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 601, + 321, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 323, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 323, + 614 + ], + "score": 1.0, + "content": "5.1 MODEL VALIDATION AND ABLATION STUDY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Effectiveness of TET over SDT with SG. We first examine whether the mismatch between SG and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "loss causes the convergence problem. For this purpose, we set the simulation length to 4 and change", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 182, + 655 + ], + "score": 1.0, + "content": "the spike function", + "type": "text" + }, + { + "bbox": [ + 183, + 644, + 193, + 654 + ], + "score": 0.78, + "content": "\\Theta", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 644, + 281, + 655 + ], + "score": 1.0, + "content": "in Eqn.2 to Sigmoid", + "type": "text" + }, + { + "bbox": [ + 281, + 644, + 330, + 655 + ], + "score": 0.76, + "content": "\\sigma ( \\bar { k } \\cdot \\mathrm { { i n p u t } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 644, + 505, + 655 + ], + "score": 1.0, + "content": ". We find that the TET and SDT achieved", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 239, + 667 + ], + "score": 1.0, + "content": "similar accuracy (Table 2) when", + "type": "text" + }, + { + "bbox": [ + 239, + 655, + 294, + 666 + ], + "score": 0.85, + "content": "k = 1 , 1 0 , 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ". This indicates that both TET and SDT work when", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 465, + 678 + ], + "score": 1.0, + "content": "the gradient and loss function match each other. Next, we compare the results training with", + "type": "text" + }, + { + "bbox": [ + 466, + 666, + 487, + 677 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 128, + 688 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "on SNNs (ResNet-19 on CIFAR100) training with surrogate gradient for three runs. As shown", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 476, + 700 + ], + "score": 1.0, + "content": "in Table 1, our proposed new TET training strategy dramatically increases the accuracy by", + "type": "text" + }, + { + "bbox": [ + 477, + 687, + 504, + 699 + ], + "score": 0.88, + "content": "3 . 2 5 \\%", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 243, + 712 + ], + "score": 1.0, + "content": "when the simulation time is 4 and", + "type": "text" + }, + { + "bbox": [ + 243, + 699, + 270, + 709 + ], + "score": 0.89, + "content": "3 . 5 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "when the simulation time is 6. These results quantitatively", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "support the effectiveness of TET in solving the mismatch between gradient and loss in training SNNs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 145, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 145, + 732 + ], + "score": 1.0, + "content": "with SG.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 622, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 79, + 503, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 79, + 503, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 79, + 503, + 168 + ], + "spans": [ + { + "bbox": [ + 107, + 79, + 503, + 168 + ], + "score": 0.971, + "type": "image", + "image_path": "aff831f07e0e329cffab78a14e6c6b9cb6a33fb6860a562c194fcda2f2064b92.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 79, + 503, + 108.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 108.66666666666667, + 503, + 138.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 138.33333333333334, + 503, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 184, + 504, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 183, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 362, + 199 + ], + "score": 1.0, + "content": "Figure 2: Loss landscape of VGGSNN. The 2D landscape of", + "type": "text" + }, + { + "bbox": [ + 362, + 185, + 384, + 196 + ], + "score": 0.92, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 183, + 403, + 199 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 185, + 426, + 196 + ], + "score": 0.9, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 183, + 506, + 199 + ], + "score": 1.0, + "content": "from two different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 196, + 179, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 179, + 208 + ], + "score": 1.0, + "content": "training methods.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 108, + 281, + 299, + 309 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 229, + 299, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 228, + 299, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 299, + 241 + ], + "score": 1.0, + "content": "Table 1: Comparison between SDT and TET.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 240, + 299, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 299, + 252 + ], + "score": 1.0, + "content": "We adopt the SNN architecture ResNet-19 with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 251, + 299, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 299, + 263 + ], + "score": 1.0, + "content": "SG on CIFAR100 and record the results with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 262, + 288, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 288, + 274 + ], + "score": 1.0, + "content": "three different simulation lengths 2, 4, and 6.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 281, + 299, + 309 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 281, + 299, + 309 + ], + "spans": [ + { + "bbox": [ + 108, + 281, + 299, + 309 + ], + "score": 0.957, + "html": "
MethodT=2T=4T=6
Direct training69.41±0.0870.86±0.2271.12±0.57
TET72.37±0.2174.11±0.1874.65±0.12
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Methodk=1k=10k=20
Direct training88.00±0.1588.83±0.3288.50±0.32
TET87.63±0.3889.31±0.1588.64±0.28
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We further inspect the 2D landscapes (Li et al., 2018) of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 128, + 355 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 343, + 146, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 344, + 167, + 355 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "around their local minima (see Figure. 2) to demonstrate why TET generalizes better", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "than SDT and how TET helps the training process jump out of the sharp local minima typically", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 272, + 378 + ], + "score": 1.0, + "content": "found by SDT. First, comparing Figure.", + "type": "text" + }, + { + "bbox": [ + 273, + 366, + 290, + 376 + ], + "score": 0.41, + "content": "2 \\textrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "and C, we can see that although the values of local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 314, + 389 + ], + "score": 1.0, + "content": "minima achieved by SDT and TET are similar in", + "type": "text" + }, + { + "bbox": [ + 314, + 377, + 336, + 388 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 376, + 483, + 389 + ], + "score": 1.0, + "content": ", the local minima of TET (Figure.", + "type": "text" + }, + { + "bbox": [ + 483, + 377, + 501, + 387 + ], + "score": 0.26, + "content": "2 \\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 376, + 505, + 389 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 250, + 400 + ], + "score": 1.0, + "content": "is flatter than that of SDT (Figure.", + "type": "text" + }, + { + "bbox": [ + 250, + 388, + 267, + 398 + ], + "score": 0.32, + "content": "2 \\mathrm { \\ A }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "). This indicates that the TET is effective in finding flatter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "minima that are typically more generalizable even w.r.t the original loss in TET. Next, we examine", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 223, + 422 + ], + "score": 1.0, + "content": "the two local minima under", + "type": "text" + }, + { + "bbox": [ + 224, + 410, + 245, + 420 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "to see how it helps jump out the local minima found by SDT.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "When comparing Figure. 2 B and D, we observe that the local minima found by SDT (Figure. 2 B)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 310, + 444 + ], + "score": 1.0, + "content": "is not only sharper than that found by TET (Figure.", + "type": "text" + }, + { + "bbox": [ + 311, + 432, + 327, + 442 + ], + "score": 0.3, + "content": "2 \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 431, + 356, + 444 + ], + "score": 1.0, + "content": ") under", + "type": "text" + }, + { + "bbox": [ + 356, + 432, + 378, + 443 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "but also maintains a higher loss", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "value. This supports our claim that TET loss cannot be easily minimized around sharp local minima", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 140, + 467 + ], + "score": 1.0, + "content": "(Figure.", + "type": "text" + }, + { + "bbox": [ + 141, + 453, + 158, + 464 + ], + "score": 0.33, + "content": "2 \\mathrm { \\ B }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 452, + 412, + 467 + ], + "score": 1.0, + "content": "), thus preferable to converge into flatter local minima (Figure.", + "type": "text" + }, + { + "bbox": [ + 412, + 453, + 430, + 464 + ], + "score": 0.33, + "content": "2 \\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "). Put together, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 356, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 356, + 476 + ], + "score": 1.0, + "content": "results here provide evidence for our reasoning in Section 4.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "Training from SDT to TET. In this part, we further validate the ability of TET to escape from the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "local minimum found by SDT. We adopt the VGGSNN with 300 epochs training on DVS-CIFAR10.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 181, + 516 + ], + "score": 1.0, + "content": "First, we optimize", + "type": "text" + }, + { + "bbox": [ + 182, + 504, + 204, + 514 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 502, + 415, + 516 + ], + "score": 1.0, + "content": "for 200 epochs and then change the loss function to", + "type": "text" + }, + { + "bbox": [ + 415, + 503, + 437, + 514 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "after epoch 200.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "Figure 3 demonstrates the accuracy and loss change on the test set. 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We provide the test accuracy (A) and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 125, + 720 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 126, + 708, + 139, + 719 + ], + "score": 0.58, + "content": "( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "change after changing the SDT to TET at epoch 200. 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MethodT=2T=4T=6
Direct training69.41±0.0870.86±0.2271.12±0.57
TET72.37±0.2174.11±0.1874.65±0.12
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Methodk=1k=10k=20
Direct training88.00±0.1588.83±0.3288.50±0.32
TET87.63±0.3889.31±0.1588.64±0.28
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We further inspect the 2D landscapes (Li et al., 2018) of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 128, + 355 + ], + "score": 0.84, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 343, + 146, + 356 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 344, + 167, + 355 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "around their local minima (see Figure. 2) to demonstrate why TET generalizes better", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 368 + ], + "score": 1.0, + "content": "than SDT and how TET helps the training process jump out of the sharp local minima typically", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 272, + 378 + ], + "score": 1.0, + "content": "found by SDT. First, comparing Figure.", + "type": "text" + }, + { + "bbox": [ + 273, + 366, + 290, + 376 + ], + "score": 0.41, + "content": "2 \\textrm { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "and C, we can see that although the values of local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 314, + 389 + ], + "score": 1.0, + "content": "minima achieved by SDT and TET are similar in", + "type": "text" + }, + { + "bbox": [ + 314, + 377, + 336, + 388 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 376, + 483, + 389 + ], + "score": 1.0, + "content": ", the local minima of TET (Figure.", + "type": "text" + }, + { + "bbox": [ + 483, + 377, + 501, + 387 + ], + "score": 0.26, + "content": "2 \\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 376, + 505, + 389 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 250, + 400 + ], + "score": 1.0, + "content": "is flatter than that of SDT (Figure.", + "type": "text" + }, + { + "bbox": [ + 250, + 388, + 267, + 398 + ], + "score": 0.32, + "content": "2 \\mathrm { \\ A }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "). This indicates that the TET is effective in finding flatter", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "minima that are typically more generalizable even w.r.t the original loss in TET. Next, we examine", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 223, + 422 + ], + "score": 1.0, + "content": "the two local minima under", + "type": "text" + }, + { + "bbox": [ + 224, + 410, + 245, + 420 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 408, + 505, + 422 + ], + "score": 1.0, + "content": "to see how it helps jump out the local minima found by SDT.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "When comparing Figure. 2 B and D, we observe that the local minima found by SDT (Figure. 2 B)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 310, + 444 + ], + "score": 1.0, + "content": "is not only sharper than that found by TET (Figure.", + "type": "text" + }, + { + "bbox": [ + 311, + 432, + 327, + 442 + ], + "score": 0.3, + "content": "2 \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 431, + 356, + 444 + ], + "score": 1.0, + "content": ") under", + "type": "text" + }, + { + "bbox": [ + 356, + 432, + 378, + 443 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "but also maintains a higher loss", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "value. This supports our claim that TET loss cannot be easily minimized around sharp local minima", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 140, + 467 + ], + "score": 1.0, + "content": "(Figure.", + "type": "text" + }, + { + "bbox": [ + 141, + 453, + 158, + 464 + ], + "score": 0.33, + "content": "2 \\mathrm { \\ B }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 452, + 412, + 467 + ], + "score": 1.0, + "content": "), thus preferable to converge into flatter local minima (Figure.", + "type": "text" + }, + { + "bbox": [ + 412, + 453, + 430, + 464 + ], + "score": 0.33, + "content": "2 \\mathrm { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "). Put together, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 356, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 356, + 476 + ], + "score": 1.0, + "content": "results here provide evidence for our reasoning in Section 4.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 332, + 506, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "Training from SDT to TET. In this part, we further validate the ability of TET to escape from the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "local minimum found by SDT. We adopt the VGGSNN with 300 epochs training on DVS-CIFAR10.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 502, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 181, + 516 + ], + "score": 1.0, + "content": "First, we optimize", + "type": "text" + }, + { + "bbox": [ + 182, + 504, + 204, + 514 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 502, + 415, + 516 + ], + "score": 1.0, + "content": "for 200 epochs and then change the loss function to", + "type": "text" + }, + { + "bbox": [ + 415, + 503, + 437, + 514 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 502, + 506, + 516 + ], + "score": 1.0, + "content": "after epoch 200.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "Figure 3 demonstrates the accuracy and loss change on the test set. After 200 epochs training, SDT", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 286, + 312 + ], + "score": 1.0, + "content": "gets trapped into a local minimum, and the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 287, + 300, + 309, + 311 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 309, + 299, + 417, + 312 + ], + "score": 1.0, + "content": "no longer decreases. The", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 418, + 300, + 440, + 311 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 440, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "is much higher", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 126, + 325 + ], + "score": 1.0, + "content": "than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 126, + 311, + 148, + 322 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 149, + 309, + 480, + 325 + ], + "score": 1.0, + "content": "since SDT does not optimize it. Nevertheless, after we change the loss function to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 480, + 311, + 501, + 322 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 501, + 309, + 506, + 325 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 121, + 335 + ], + "score": 1.0, + "content": "the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 122, + 322, + 144, + 333 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 144, + 321, + 162, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 162, + 322, + 184, + 333 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 185, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "on the test set both have a rapid decline. This phenomenon illustrates the TET", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "ability to help the SNN efficiently jump out of the local minimum with poor generalization and find", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 230, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 230, + 356 + ], + "score": 1.0, + "content": "another flatter local minimum.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 480, + 506, + 527 + ] + }, + { + "type": "image", + "bbox": [ + 142, + 545, + 462, + 681 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 142, + 545, + 462, + 681 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 142, + 545, + 462, + 681 + ], + "spans": [ + { + "bbox": [ + 142, + 545, + 462, + 681 + ], + "score": 0.976, + "type": "image", + "image_path": "3337c633c01451316aa0cc30732fe1eaeb582425bcdf30fbe135431d8bf36214.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 142, + 545, + 462, + 590.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 142, + 590.3333333333334, + 462, + 635.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 142, + 635.6666666666667, + 462, + 681.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 696, + 505, + 730 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 696, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 505, + 708 + ], + "score": 1.0, + "content": "Figure 3: TET helps to jump out the local minimum point. We provide the test accuracy (A) and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 125, + 720 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 126, + 708, + 139, + 719 + ], + "score": 0.58, + "content": "( B )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "change after changing the SDT to TET at epoch 200. TET efficiently improves the test", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 718, + 297, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 297, + 730 + ], + "score": 1.0, + "content": "performance and reduces the two kinds of loss.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 36.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 147, + 85, + 465, + 225 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 85, + 465, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 85, + 465, + 225 + ], + "spans": [ + { + "bbox": [ + 147, + 85, + 465, + 225 + ], + "score": 0.974, + "type": "image", + "image_path": "086a43f62fe160235967e86bf1fd8d81f8153af1874a2e9a74e352d77d8e5e20.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 85, + 465, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 131.66666666666666, + 465, + 178.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 178.33333333333331, + 465, + 224.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 504, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "Figure 4: Time scalability robustness and network efficiency of ResNet-19 on CIFAR100. (A) The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 489, + 265 + ], + "score": 1.0, + "content": "comparison of training from scratch (dots) and inheriting from a small simulation length (lines).", + "type": "text" + }, + { + "bbox": [ + 490, + 253, + 504, + 264 + ], + "score": 0.51, + "content": "( B )", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 357, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 357, + 276 + ], + "score": 1.0, + "content": "SNN network performance changes with energy consumption.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 355 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 286, + 312 + ], + "score": 1.0, + "content": "gets trapped into a local minimum, and the", + "type": "text" + }, + { + "bbox": [ + 287, + 300, + 309, + 311 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 299, + 417, + 312 + ], + "score": 1.0, + "content": "no longer decreases. The", + "type": "text" + }, + { + "bbox": [ + 418, + 300, + 440, + 311 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "is much higher", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 126, + 325 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 126, + 311, + 148, + 322 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 309, + 480, + 325 + ], + "score": 1.0, + "content": "since SDT does not optimize it. Nevertheless, after we change the loss function to", + "type": "text" + }, + { + "bbox": [ + 480, + 311, + 501, + 322 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 309, + 506, + 325 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 121, + 335 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 322, + 144, + 333 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\mathrm { T E T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 321, + 162, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 322, + 184, + 333 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { S D T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "on the test set both have a rapid decline. This phenomenon illustrates the TET", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 345 + ], + "score": 1.0, + "content": "ability to help the SNN efficiently jump out of the local minimum with poor generalization and find", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 343, + 230, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 230, + 356 + ], + "score": 1.0, + "content": "another flatter local minimum.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "Time Scalability Robustness. Here, we study the time scalability robustness of SNNs trained with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 128, + 384 + ], + "score": 1.0, + "content": "TET", + "type": "text" + }, + { + "bbox": [ + 128, + 372, + 155, + 383 + ], + "score": 0.8, + "content": "( \\mathcal { L } _ { \\mathrm { T E T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". First, we use 300 epochs to train a small simulation length ResNet-19 on CIFAR100", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 381, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 397 + ], + "score": 1.0, + "content": "as the initial SNN. Then, we directly change the simulation length from 2 to 8 without finetuning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and report the network accuracy on the test set. Figure. 4. A displays the results after changing the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "simulation length. We use 2, 3, and 4, respectively, as the simulation length of the initial network.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "When we increase the simulation length, the accuracy of all networks gradually increases. After the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "simulation time reaches a certain value, the network performance will slightly decrease. Interest-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 244, + 450 + ], + "score": 1.0, + "content": "ingly, SNNs trained from scratch (", + "type": "text" + }, + { + "bbox": [ + 244, + 438, + 263, + 448 + ], + "score": 0.74, + "content": "\\mathrm { T } { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 437, + 280, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 438, + 299, + 448 + ], + "score": 0.75, + "content": "{ \\mathrm { T } } { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ") are not as good as those trained following the TIT", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 448, + 151, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 151, + 461 + ], + "score": 1.0, + "content": "procedure.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "Network Efficiency. In this section, we measure the relationship between energy consumption", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "and network performance. SNN avoids multiplication on the inference since its binary activation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 365, + 500 + ], + "score": 1.0, + "content": "and event-based operation. The addition operation in SNN costs", + "type": "text" + }, + { + "bbox": [ + 366, + 487, + 391, + 498 + ], + "score": 0.89, + "content": "0 . 9 p J", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "energy while multiplication", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 191, + 511 + ], + "score": 1.0, + "content": "operation consumes", + "type": "text" + }, + { + "bbox": [ + 191, + 498, + 217, + 510 + ], + "score": 0.9, + "content": "4 . 6 p J", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 498, + 272, + 511 + ], + "score": 1.0, + "content": "measured in", + "type": "text" + }, + { + "bbox": [ + 273, + 498, + 298, + 509 + ], + "score": 0.55, + "content": "4 5 \\mathrm { n m }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "CMOS technology (Rathi & Roy, 2020). In our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "SNN model, the first layer has multiplication operations, while the other layers only have addition", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "operations. Figure 4. B summarizes the results of different simulation times. In all cases, the SNN", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 530, + 263, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 263, + 545 + ], + "score": 1.0, + "content": "obtained by TET has higher efficiency.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 109, + 561, + 275, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 277, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 277, + 573 + ], + "score": 1.0, + "content": "5.2 COMPARISON TO EXITING WORKS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "In this section, we compare our experimental results with previous works. We validate the full TIT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 149, + 607 + ], + "score": 1.0, + "content": "algorithm", + "type": "text" + }, + { + "bbox": [ + 150, + 594, + 183, + 605 + ], + "score": 0.69, + "content": "( \\mathcal { L } _ { \\mathrm { T O T A L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "both on the static dataset and neuromorphic dataset. All of the experiment results", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 444, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 444, + 617 + ], + "score": 1.0, + "content": "are summarized in Table 5.2. We specify all the training details in the appendix A.1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "CIFAR. We apply TET and TIT algorithm on CIFAR (Krizhevsky et al., 2009), and report the mean", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 633, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 380, + 644 + ], + "score": 1.0, + "content": "and standard deviation of 3 runs under different random seeds. The", + "type": "text" + }, + { + "bbox": [ + 380, + 633, + 388, + 643 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 634, + 505, + 644 + ], + "score": 1.0, + "content": "is set to 0.05. On CIFAR10,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 472, + 657 + ], + "score": 1.0, + "content": "our TET method achieves the highest accuracy above all existing approaches. 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It is worth noting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 207, + 678 + ], + "score": 1.0, + "content": "that our method is only", + "type": "text" + }, + { + "bbox": [ + 208, + 666, + 235, + 676 + ], + "score": 0.89, + "content": "0 . 4 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "lower than the ANN performance. TET algorithm demonstrates", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 435, + 690 + ], + "score": 1.0, + "content": "a more excellent ability on CIFAR100. It has an accuracy increase greater than", + "type": "text" + }, + { + "bbox": [ + 435, + 677, + 450, + 687 + ], + "score": 0.86, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "on all report", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 261, + 699 + ], + "score": 1.0, + "content": "simulation lengths. 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We can see that the proposed TET’s improvement is even higher on complex data like", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "CIFAR100, where the generalizability of the model distinguishes a lot among minima with different", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 142, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 142, + 732 + ], + "score": 1.0, + "content": "flatness.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 147, + 85, + 465, + 225 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 85, + 465, + 225 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 85, + 465, + 225 + ], + "spans": [ + { + "bbox": [ + 147, + 85, + 465, + 225 + ], + "score": 0.974, + "type": "image", + "image_path": "086a43f62fe160235967e86bf1fd8d81f8153af1874a2e9a74e352d77d8e5e20.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 85, + 465, + 131.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 131.66666666666666, + 465, + 178.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 178.33333333333331, + 465, + 224.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 241, + 504, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "Figure 4: Time scalability robustness and network efficiency of ResNet-19 on CIFAR100. (A) The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 489, + 265 + ], + "score": 1.0, + "content": "comparison of training from scratch (dots) and inheriting from a small simulation length (lines).", + "type": "text" + }, + { + "bbox": [ + 490, + 253, + 504, + 264 + ], + "score": 0.51, + "content": "( B )", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 264, + 357, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 357, + 276 + ], + "score": 1.0, + "content": "SNN network performance changes with energy consumption.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 505, + 355 + ], + "lines": [], + "index": 8, + "bbox_fs": [ + 105, + 299, + 506, + 356 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 374 + ], + "score": 1.0, + "content": "Time Scalability Robustness. Here, we study the time scalability robustness of SNNs trained with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 128, + 384 + ], + "score": 1.0, + "content": "TET", + "type": "text" + }, + { + "bbox": [ + 128, + 372, + 155, + 383 + ], + "score": 0.8, + "content": "( \\mathcal { L } _ { \\mathrm { T E T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 371, + 506, + 384 + ], + "score": 1.0, + "content": ". First, we use 300 epochs to train a small simulation length ResNet-19 on CIFAR100", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 381, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 397 + ], + "score": 1.0, + "content": "as the initial SNN. Then, we directly change the simulation length from 2 to 8 without finetuning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and report the network accuracy on the test set. Figure. 4. A displays the results after changing the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 419 + ], + "score": 1.0, + "content": "simulation length. We use 2, 3, and 4, respectively, as the simulation length of the initial network.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "When we increase the simulation length, the accuracy of all networks gradually increases. After the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "simulation time reaches a certain value, the network performance will slightly decrease. Interest-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 244, + 450 + ], + "score": 1.0, + "content": "ingly, SNNs trained from scratch (", + "type": "text" + }, + { + "bbox": [ + 244, + 438, + 263, + 448 + ], + "score": 0.74, + "content": "\\mathrm { T } { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 437, + 280, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 280, + 438, + 299, + 448 + ], + "score": 0.75, + "content": "{ \\mathrm { T } } { = } 6", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 437, + 506, + 450 + ], + "score": 1.0, + "content": ") are not as good as those trained following the TIT", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 448, + 151, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 151, + 461 + ], + "score": 1.0, + "content": "procedure.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 360, + 506, + 461 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "Network Efficiency. In this section, we measure the relationship between energy consumption", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "and network performance. SNN avoids multiplication on the inference since its binary activation", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 365, + 500 + ], + "score": 1.0, + "content": "and event-based operation. The addition operation in SNN costs", + "type": "text" + }, + { + "bbox": [ + 366, + 487, + 391, + 498 + ], + "score": 0.89, + "content": "0 . 9 p J", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "energy while multiplication", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 191, + 511 + ], + "score": 1.0, + "content": "operation consumes", + "type": "text" + }, + { + "bbox": [ + 191, + 498, + 217, + 510 + ], + "score": 0.9, + "content": "4 . 6 p J", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 498, + 272, + 511 + ], + "score": 1.0, + "content": "measured in", + "type": "text" + }, + { + "bbox": [ + 273, + 498, + 298, + 509 + ], + "score": 0.55, + "content": "4 5 \\mathrm { n m }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "CMOS technology (Rathi & Roy, 2020). In our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "SNN model, the first layer has multiplication operations, while the other layers only have addition", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "operations. Figure 4. B summarizes the results of different simulation times. In all cases, the SNN", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 530, + 263, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 263, + 545 + ], + "score": 1.0, + "content": "obtained by TET has higher efficiency.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 464, + 505, + 545 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 561, + 275, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 277, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 277, + 573 + ], + "score": 1.0, + "content": "5.2 COMPARISON TO EXITING WORKS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 504, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "In this section, we compare our experimental results with previous works. We validate the full TIT", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 149, + 607 + ], + "score": 1.0, + "content": "algorithm", + "type": "text" + }, + { + "bbox": [ + 150, + 594, + 183, + 605 + ], + "score": 0.69, + "content": "( \\mathcal { L } _ { \\mathrm { T O T A L } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "both on the static dataset and neuromorphic dataset. All of the experiment results", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 605, + 444, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 444, + 617 + ], + "score": 1.0, + "content": "are summarized in Table 5.2. We specify all the training details in the appendix A.1.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 582, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "CIFAR. We apply TET and TIT algorithm on CIFAR (Krizhevsky et al., 2009), and report the mean", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 633, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 380, + 644 + ], + "score": 1.0, + "content": "and standard deviation of 3 runs under different random seeds. The", + "type": "text" + }, + { + "bbox": [ + 380, + 633, + 388, + 643 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 634, + 505, + 644 + ], + "score": 1.0, + "content": "is set to 0.05. 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Even when", + "type": "text" + }, + { + "bbox": [ + 473, + 644, + 501, + 654 + ], + "score": 0.89, + "content": "T = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 643, + 506, + 657 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 144, + 668 + ], + "score": 1.0, + "content": "there is a", + "type": "text" + }, + { + "bbox": [ + 145, + 655, + 172, + 666 + ], + "score": 0.88, + "content": "1 . 8 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 653, + 403, + 668 + ], + "score": 1.0, + "content": "increment compare to STBP-tdBN with simulation length", + "type": "text" + }, + { + "bbox": [ + 403, + 655, + 430, + 665 + ], + "score": 0.9, + "content": "T = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 653, + 506, + 668 + ], + "score": 1.0, + "content": ". It is worth noting", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 207, + 678 + ], + "score": 1.0, + "content": "that our method is only", + "type": "text" + }, + { + "bbox": [ + 208, + 666, + 235, + 676 + ], + "score": 0.89, + "content": "0 . 4 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "lower than the ANN performance. TET algorithm demonstrates", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 435, + 690 + ], + "score": 1.0, + "content": "a more excellent ability on CIFAR100. It has an accuracy increase greater than", + "type": "text" + }, + { + "bbox": [ + 435, + 677, + 450, + 687 + ], + "score": 0.86, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "on all report", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 261, + 699 + ], + "score": 1.0, + "content": "simulation lengths. In addition, when", + "type": "text" + }, + { + "bbox": [ + 261, + 688, + 289, + 698 + ], + "score": 0.9, + "content": "T = 6", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 688, + 413, + 699 + ], + "score": 1.0, + "content": ", the reported accuracy is only", + "type": "text" + }, + { + "bbox": [ + 413, + 687, + 441, + 699 + ], + "score": 0.89, + "content": "0 . 6 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "lower than that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "of ANN. We can see that the proposed TET’s improvement is even higher on complex data like", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "CIFAR100, where the generalizability of the model distinguishes a lot among minima with different", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 142, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 142, + 732 + ], + "score": 1.0, + "content": "flatness.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 621, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 117, + 506, + 415 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "score": 1.0, + "content": "Table 3: Compare with existing works. Our method improves network performance across all tasks.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 443, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 443, + 112 + ], + "score": 1.0, + "content": "* denotes self-implementation results. † denotes data augmentation (Li et al., 2022).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 117, + 506, + 415 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 117, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 108, + 117, + 506, + 415 + ], + "score": 0.984, + "html": "
DatasetModelMethodsArchitectureSimulationLengthAccuracy
CIFAR10Rathi et al. (2019)Hybrid training Diet-SNNResNet-2025092.22
Rathi & Roy (2020)ResNet-201092.54
Wu et al. (2018)STBPCIFARNet1289.83
Wu et al. (2019)STBP NeuNormCIFARNet1290.53
Zhang & Li (2020)TSSL-BPCIFARNet591.41
Zheng et al. (2021)STBP-tdBNResNet-196 493.16 92.92
our modelTET292.34
694.50±0.07
494.44±0.08
ResNet-19294.16±0.03
CIFAR100ANN*ANNResNet-19194.97
Rathi et al. (2019) Rathi & Roy (2020)Hybrid trainingVGG-1112567.87
Diet-SNNResNet-20564.07 71.12±0.57
Zheng et al. (2021)*STBP-tdBNResNet-19670.86±0.22
4 269.41±0.08
674.72±0.28
our modelTETResNet-19474.47±0.15
272.87±0.10
ANN* Rathi etal. (2019)ANNResNet-19175.35
Hybrid training SPIKE-NORMResNet-3425061.48
Sengupta et al. (2018) Zheng et al. (2021)STBP-tdBNResNet-34250069.96
ImageNetFang et al. (2021)SEWResNetSpiking-ResNet-34663.72
TETSEW-ResNet-34 Spiking-ResNet-34467.04 64.79
our modelTETSEW-ResNet-346 468.00
Zheng et al. (2021)STBP-tdBNResNet-191067.8
Kugele et al. (2020)Streaming RolloutDenseNet1066.8
DVS-CIFAR10Wu et al. (2021)Conv3DLIAF-Net71.70
Wu et al. (2021)LIAFLIAF-Net10 1070.40
our modelTETVGGSNN1077.33±0.21
TETtVGGSNN83.17±0.15
10
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The training set of ImageNet (Krizhevsky et al., 2012) provides", + "type": "text" + }, + { + "bbox": [ + 412, + 441, + 436, + 452 + ], + "score": 0.74, + "content": "1 . 2 8 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "training samples", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "for each label. We choose the two most representative ResNet-34 to verify our algorithm on Ima-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 153, + 476 + ], + "score": 1.0, + "content": "geNet with", + "type": "text" + }, + { + "bbox": [ + 153, + 463, + 196, + 474 + ], + "score": 0.9, + "content": "\\lambda = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 462, + 506, + 476 + ], + "score": 1.0, + "content": ". SEW-ResNet34 is not a typical SNN since it adopts the IF model and mod-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "ifies the Residual structure. 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The neuromorphic datasets suffer much more noise than static datasets. Thus the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "well-trained SNN is easier to overfit on these datasets than static datasets. DVS-CIFAR10 (Li et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 265, + 537 + ], + "score": 1.0, + "content": "2017), which provides each label with", + "type": "text" + }, + { + "bbox": [ + 265, + 524, + 285, + 534 + ], + "score": 0.72, + "content": "0 . 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "training samples, is the most challenging mainstream", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "neuromorphic dataset. Recent works prefer to deal with this dataset by complex architectures, which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "are more susceptible to overfitting and do not result in very high accuracy. Here, we adopt VGGSNN", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 249, + 569 + ], + "score": 1.0, + "content": "on the DVS-CIFAR10 dataset, set", + "type": "text" + }, + { + "bbox": [ + 250, + 557, + 297, + 568 + ], + "score": 0.9, + "content": "\\lambda = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 556, + 506, + 569 + ], + "score": 1.0, + "content": ", and report the mean and standard deviation of 3", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "runs under different random seeds. Along with data augmentation methods, VGGSNN can achieve", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 170, + 591 + ], + "score": 1.0, + "content": "an accuracy of", + "type": "text" + }, + { + "bbox": [ + 171, + 578, + 198, + 589 + ], + "score": 0.9, + "content": "7 7 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 578, + 505, + 591 + ], + "score": 1.0, + "content": ". Then we apply the TET method to obtain a more generalizable optima.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 589, + 504, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 196, + 602 + ], + "score": 1.0, + "content": "The accuracy rises to", + "type": "text" + }, + { + "bbox": [ + 197, + 590, + 228, + 600 + ], + "score": 0.89, + "content": "8 3 . 1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 589, + 472, + 602 + ], + "score": 1.0, + "content": ". Our TET method outperforms existing state-of-the-art by", + "type": "text" + }, + { + "bbox": [ + 472, + 590, + 504, + 600 + ], + "score": 0.9, + "content": "1 1 . 4 7 \\%", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 383, + 613 + ], + "score": 1.0, + "content": "accuracy. Without data augmentation methods, VGGSNN obtains", + "type": "text" + }, + { + "bbox": [ + 383, + 600, + 411, + 612 + ], + "score": 0.87, + "content": "7 \\bar { 3 } . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "accuracy by SDT and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 611, + 207, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 134, + 623 + ], + "score": 0.89, + "content": "7 7 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 611, + 207, + 624 + ], + "score": 1.0, + "content": "accuracy by TET.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 195, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 197, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 197, + 656 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "score": 1.0, + "content": "This paper focuses on the SNN generalization problem, which is described as the direct training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "SNN performs well on the training set but poor on the test set. We find this phenomenon is due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the incorrect SG that makes the SNN easily trapped into a local minimum with poor generalization.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "To solve this problem, we propose the temporal efficient training algorithm (TET). Extensive ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "periments verify that our proposed method consistently achieves better performance than the SDT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "process. Furthermore, TET significantly improves the time scalability robustness of SNN, which", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 117, + 506, + 415 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "score": 1.0, + "content": "Table 3: Compare with existing works. Our method improves network performance across all tasks.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 443, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 443, + 112 + ], + "score": 1.0, + "content": "* denotes self-implementation results. † denotes data augmentation (Li et al., 2022).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 117, + 506, + 415 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 117, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 108, + 117, + 506, + 415 + ], + "score": 0.984, + "html": "
DatasetModelMethodsArchitectureSimulationLengthAccuracy
CIFAR10Rathi et al. (2019)Hybrid training Diet-SNNResNet-2025092.22
Rathi & Roy (2020)ResNet-201092.54
Wu et al. (2018)STBPCIFARNet1289.83
Wu et al. (2019)STBP NeuNormCIFARNet1290.53
Zhang & Li (2020)TSSL-BPCIFARNet591.41
Zheng et al. (2021)STBP-tdBNResNet-196 493.16 92.92
our modelTET292.34
694.50±0.07
494.44±0.08
ResNet-19294.16±0.03
CIFAR100ANN*ANNResNet-19194.97
Rathi et al. (2019) Rathi & Roy (2020)Hybrid trainingVGG-1112567.87
Diet-SNNResNet-20564.07 71.12±0.57
Zheng et al. (2021)*STBP-tdBNResNet-19670.86±0.22
4 269.41±0.08
674.72±0.28
our modelTETResNet-19474.47±0.15
272.87±0.10
ANN* Rathi etal. (2019)ANNResNet-19175.35
Hybrid training SPIKE-NORMResNet-3425061.48
Sengupta et al. (2018) Zheng et al. (2021)STBP-tdBNResNet-34250069.96
ImageNetFang et al. (2021)SEWResNetSpiking-ResNet-34663.72
TETSEW-ResNet-34 Spiking-ResNet-34467.04 64.79
our modelTETSEW-ResNet-346 468.00
Zheng et al. (2021)STBP-tdBNResNet-191067.8
Kugele et al. (2020)Streaming RolloutDenseNet1066.8
DVS-CIFAR10Wu et al. (2021)Conv3DLIAF-Net71.70
Wu et al. (2021)LIAFLIAF-Net10 1070.40
our modelTETVGGSNN1077.33±0.21
TETtVGGSNN83.17±0.15
10
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The training set of ImageNet (Krizhevsky et al., 2012) provides", + "type": "text" + }, + { + "bbox": [ + 412, + 441, + 436, + 452 + ], + "score": 0.74, + "content": "1 . 2 8 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "training samples", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "for each label. We choose the two most representative ResNet-34 to verify our algorithm on Ima-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 462, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 153, + 476 + ], + "score": 1.0, + "content": "geNet with", + "type": "text" + }, + { + "bbox": [ + 153, + 463, + 196, + 474 + ], + "score": 0.9, + "content": "\\lambda = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 462, + 506, + 476 + ], + "score": 1.0, + "content": ". SEW-ResNet34 is not a typical SNN since it adopts the IF model and mod-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "ifies the Residual structure. Although we only train our model for 120 epochs, the TET algorithm", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 484, + 487, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 150, + 498 + ], + "score": 1.0, + "content": "achieves a", + "type": "text" + }, + { + "bbox": [ + 150, + 485, + 177, + 496 + ], + "score": 0.91, + "content": "1 . 0 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 484, + 335, + 498 + ], + "score": 1.0, + "content": "increment on Spiking-ResNet-34 and a", + "type": "text" + }, + { + "bbox": [ + 336, + 485, + 363, + 496 + ], + "score": 0.89, + "content": "0 . 9 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 484, + 487, + 498 + ], + "score": 1.0, + "content": "increment on SEW-ResNet34.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 440, + 506, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 513 + ], + "score": 1.0, + "content": "DVS-CIFAR10. The neuromorphic datasets suffer much more noise than static datasets. Thus the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "well-trained SNN is easier to overfit on these datasets than static datasets. DVS-CIFAR10 (Li et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 265, + 537 + ], + "score": 1.0, + "content": "2017), which provides each label with", + "type": "text" + }, + { + "bbox": [ + 265, + 524, + 285, + 534 + ], + "score": 0.72, + "content": "0 . 9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "training samples, is the most challenging mainstream", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "neuromorphic dataset. Recent works prefer to deal with this dataset by complex architectures, which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "are more susceptible to overfitting and do not result in very high accuracy. Here, we adopt VGGSNN", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 249, + 569 + ], + "score": 1.0, + "content": "on the DVS-CIFAR10 dataset, set", + "type": "text" + }, + { + "bbox": [ + 250, + 557, + 297, + 568 + ], + "score": 0.9, + "content": "\\lambda = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 556, + 506, + 569 + ], + "score": 1.0, + "content": ", and report the mean and standard deviation of 3", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "runs under different random seeds. Along with data augmentation methods, VGGSNN can achieve", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 170, + 591 + ], + "score": 1.0, + "content": "an accuracy of", + "type": "text" + }, + { + "bbox": [ + 171, + 578, + 198, + 589 + ], + "score": 0.9, + "content": "7 7 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 578, + 505, + 591 + ], + "score": 1.0, + "content": ". Then we apply the TET method to obtain a more generalizable optima.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 589, + 504, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 196, + 602 + ], + "score": 1.0, + "content": "The accuracy rises to", + "type": "text" + }, + { + "bbox": [ + 197, + 590, + 228, + 600 + ], + "score": 0.89, + "content": "8 3 . 1 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 589, + 472, + 602 + ], + "score": 1.0, + "content": ". Our TET method outperforms existing state-of-the-art by", + "type": "text" + }, + { + "bbox": [ + 472, + 590, + 504, + 600 + ], + "score": 0.9, + "content": "1 1 . 4 7 \\%", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 383, + 613 + ], + "score": 1.0, + "content": "accuracy. Without data augmentation methods, VGGSNN obtains", + "type": "text" + }, + { + "bbox": [ + 383, + 600, + 411, + 612 + ], + "score": 0.87, + "content": "7 \\bar { 3 } . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "accuracy by SDT and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 611, + 207, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 134, + 623 + ], + "score": 0.89, + "content": "7 7 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 611, + 207, + 624 + ], + "score": 1.0, + "content": "accuracy by TET.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 502, + 506, + 624 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 640, + 195, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 197, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 197, + 656 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "score": 1.0, + "content": "This paper focuses on the SNN generalization problem, which is described as the direct training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "SNN performs well on the training set but poor on the test set. We find this phenomenon is due to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the incorrect SG that makes the SNN easily trapped into a local minimum with poor generalization.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "To solve this problem, we propose the temporal efficient training algorithm (TET). Extensive ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "periments verify that our proposed method consistently achieves better performance than the SDT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "process. Furthermore, TET significantly improves the time scalability robustness of SNN, which", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "enables us to propose the time inheritance training (TIT) to significantly reduce the training time", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 230, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 230, + 105 + ], + "score": 1.0, + "content": "consumption by almost a half.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 664, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "enables us to propose the time inheritance training (TIT) to significantly reduce the training time", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 230, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 230, + 105 + ], + "score": 1.0, + "content": "consumption by almost a half.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "title", + "bbox": [ + 108, + 120, + 232, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 232, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 232, + 135 + ], + "score": 1.0, + "content": "7 ACKNOWLEDGMENT", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 145, + 503, + 168 + ], + "lines": [ + { + "bbox": [ + 106, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "This project is supported by NSFC 61876032 and JCYJ20210324140807019. Y. 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The learning rate is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 446, + 255 + ], + "score": 1.0, + "content": "set to 0.1 and cosine decay to 0. We train the SEW-ResNet34 (Fang et al., 2021) with", + "type": "text" + }, + { + "bbox": [ + 446, + 243, + 473, + 253 + ], + "score": 0.89, + "content": "T = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "for 120", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "epochs. As for the Spiking-ResNet34 (Zheng et al., 2021), we use TIT algorithm to train 90 epochs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 265, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 127, + 276 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 265, + 159, + 275 + ], + "score": 0.91, + "content": "T = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 265, + 504, + 276 + ], + "score": 1.0, + "content": "first, then change the simulation time to 6 and finetune the network for 30 epochs.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 451, + 288 + ], + "score": 1.0, + "content": "We adopt an Adam optimizer on the finetune phase and change the learning rate to", + "type": "text" + }, + { + "bbox": [ + 451, + 276, + 480, + 286 + ], + "score": 0.79, + "content": "1 e - 4", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 275, + 505, + 288 + ], + "score": 1.0, + "content": ". 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DVS-CIFAR10 (Li et al., 2017), the most challenging mainstream neuromor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 416, + 339 + ], + "score": 1.0, + "content": "phic data set, is converted from CIFAR10. It has 10k images with the size", + "type": "text" + }, + { + "bbox": [ + 417, + 326, + 455, + 336 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 324, + 506, + 339 + ], + "score": 1.0, + "content": ". Following", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Samadzadeh et al. (2020), we divide the data stream into 10 blocks by time and accumulate the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 327, + 360 + ], + "score": 1.0, + "content": "spikes in each block. Then, we split the dataset into", + "type": "text" + }, + { + "bbox": [ + 327, + 348, + 339, + 358 + ], + "score": 0.31, + "content": "9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 347, + 425, + 360 + ], + "score": 1.0, + "content": "training images and", + "type": "text" + }, + { + "bbox": [ + 426, + 348, + 437, + 358 + ], + "score": 0.34, + "content": "1 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "test images and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 232, + 370 + ], + "score": 1.0, + "content": "reduce the spatial resolution to", + "type": "text" + }, + { + "bbox": [ + 232, + 358, + 261, + 369 + ], + "score": 0.9, + "content": "4 8 \\times 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 359, + 505, + 370 + ], + "score": 1.0, + "content": ". Random horizontal flip and random roll within 5 pixels are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 367, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 384 + ], + "score": 1.0, + "content": "taken as augmentation (Li et al., 2022). We adopt VGGSNN architecture with 300 epochs training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 430, + 393 + ], + "score": 1.0, + "content": "on this classification task. And we use an Adam optimizer with the learning rate", + "type": "text" + }, + { + "bbox": [ + 430, + 381, + 459, + 391 + ], + "score": 0.84, + "content": "1 e - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "and cosine", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "decay to 0. As for the case that does not apply any augmentation, we add a weight decay of 5e-4 to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 164, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 164, + 415 + ], + "score": 1.0, + "content": "the optimizer.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 298, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 300, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 300, + 441 + ], + "score": 1.0, + "content": "A.2 LSDT LOSS LANDSCAPE OF RESNET-19", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 272, + 462 + ], + "score": 1.0, + "content": "Here we compare the classification loss", + "type": "text" + }, + { + "bbox": [ + 272, + 449, + 299, + 460 + ], + "score": 0.81, + "content": "( \\mathcal { L } _ { \\mathrm { S D T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "landscapes of ResNet-19 on CIFAR100. The po-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 340, + 473 + ], + "score": 1.0, + "content": "sition around the local minimal value found by the SDT", + "type": "text" + }, + { + "bbox": [ + 340, + 460, + 367, + 471 + ], + "score": 0.82, + "content": "( \\mathcal { L } _ { \\mathrm { { S D T } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "is very sharp. However, the area", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 275, + 483 + ], + "score": 1.0, + "content": "around the local minimum found by TET", + "type": "text" + }, + { + "bbox": [ + 275, + 471, + 302, + 482 + ], + "score": 0.83, + "content": "( \\mathcal { L } _ { \\mathrm { T E T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "is much smoother (Figure 5), which indicates that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "TET effectively improves the network generalization. Such improvements could be further utilized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "to other techniques like privacy-preserving data generalization (Kim et al., 2021) and neural archi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 240, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 240, + 516 + ], + "score": 1.0, + "content": "tecture search (Kim et al., 2022).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "image", + "bbox": [ + 168, + 527, + 443, + 642 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 168, + 527, + 443, + 642 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 527, + 443, + 642 + ], + "spans": [ + { + "bbox": [ + 168, + 527, + 443, + 642 + ], + "score": 0.971, + "type": "image", + "image_path": "ec3a5c919a88e624c8ee236df840321cb1292d402cae6ed2e4a7f134e697de42.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 168, + 527, + 443, + 565.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 168, + 565.3333333333334, + 443, + 603.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 168, + 603.6666666666667, + 443, + 642.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 113, + 656, + 493, + 668 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 653, + 494, + 671 + ], + "spans": [ + { + "bbox": [ + 116, + 653, + 494, + 671 + ], + "score": 1.0, + "content": "Figure 5: STD loss landscape of ResNet-19 on CIFAR100 from different training approaches.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + } + ], + "index": 36.0 + }, + { + "type": "title", + "bbox": [ + 108, + 688, + 205, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 205, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 181, + 703 + ], + "score": 1.0, + "content": "A.3 EFFECT OF", + "type": "text" + }, + { + "bbox": [ + 181, + 689, + 205, + 700 + ], + "score": 0.51, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 332, + 722 + ], + "score": 1.0, + "content": "In this part, we examine the effect of the regular term", + "type": "text" + }, + { + "bbox": [ + 332, + 710, + 355, + 721 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 709, + 462, + 722 + ], + "score": 1.0, + "content": "with 5 different levels of", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 469, + 720 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ". 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The regular term", + "type": "text" + }, + { + "bbox": [ + 313, + 721, + 336, + 732 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "effectively increases the performance of", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 184, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 184, + 96 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 109, + 106, + 273, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 276, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 276, + 119 + ], + "score": 1.0, + "content": "A.1 DATASET AND TRAINING DETAIL", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 127, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "CIFAR. The CIFAR dataset (Krizhevsky et al., 2009) consists of 50k training images and 10k testing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 202, + 151 + ], + "score": 1.0, + "content": "images with the size of", + "type": "text" + }, + { + "bbox": [ + 202, + 138, + 236, + 149 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 138, + 506, + 151 + ], + "score": 1.0, + "content": ". We use ResNet-19 for both CIFAR10 and CIFAR100. Moreover,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "random horizontal flip and crop are applied to the training images the augmentation. First, we use", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 504, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 333, + 172 + ], + "score": 1.0, + "content": "300 epoch to train the SNN with the simulation length", + "type": "text" + }, + { + "bbox": [ + 333, + 160, + 363, + 171 + ], + "score": 0.9, + "content": "T = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 161, + 504, + 172 + ], + "score": 1.0, + "content": ". We use an Adam optimizer with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "a learning rate of 0.01 and cosine decay to 0. Next, following the TIT algorithm, we increase the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "simulation time (to 4 and 6) and continue training the SNN for only 50 epochs, with the learning", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 193, + 206, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 173, + 205 + ], + "score": 1.0, + "content": "rate changing to", + "type": "text" + }, + { + "bbox": [ + 173, + 194, + 201, + 204 + ], + "score": 0.8, + "content": "1 e - 4", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 193, + 206, + 205 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 126, + 506, + 205 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 209, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 350, + 222 + ], + "score": 1.0, + "content": "ImageNet. ImageNet (Deng et al., 2009) contains more than", + "type": "text" + }, + { + "bbox": [ + 351, + 210, + 376, + 221 + ], + "score": 0.54, + "content": "1 2 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 210, + 457, + 222 + ], + "score": 1.0, + "content": "training images and", + "type": "text" + }, + { + "bbox": [ + 458, + 210, + 474, + 221 + ], + "score": 0.58, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 210, + 504, + 222 + ], + "score": 1.0, + "content": "valida-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 221, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 252, + 234 + ], + "score": 1.0, + "content": "tion images. We crop the images to", + "type": "text" + }, + { + "bbox": [ + 252, + 221, + 291, + 232 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 221, + 505, + 234 + ], + "score": 1.0, + "content": "and using the standard augmentation for the training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 392, + 244 + ], + "score": 1.0, + "content": "data. We use an SGD optimizer with 0.9 momentum and weight decay", + "type": "text" + }, + { + "bbox": [ + 392, + 232, + 420, + 243 + ], + "score": 0.85, + "content": "4 e - 5", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 231, + 505, + 244 + ], + "score": 1.0, + "content": ". The learning rate is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 446, + 255 + ], + "score": 1.0, + "content": "set to 0.1 and cosine decay to 0. We train the SEW-ResNet34 (Fang et al., 2021) with", + "type": "text" + }, + { + "bbox": [ + 446, + 243, + 473, + 253 + ], + "score": 0.89, + "content": "T = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "for 120", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "epochs. As for the Spiking-ResNet34 (Zheng et al., 2021), we use TIT algorithm to train 90 epochs", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 265, + 504, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 127, + 276 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 265, + 159, + 275 + ], + "score": 0.91, + "content": "T = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 265, + 504, + 276 + ], + "score": 1.0, + "content": "first, then change the simulation time to 6 and finetune the network for 30 epochs.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 451, + 288 + ], + "score": 1.0, + "content": "We adopt an Adam optimizer on the finetune phase and change the learning rate to", + "type": "text" + }, + { + "bbox": [ + 451, + 276, + 480, + 286 + ], + "score": 0.79, + "content": "1 e - 4", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 275, + 505, + 288 + ], + "score": 1.0, + "content": ". TIT", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "algorithm significantly reduces the training time consumption since training the Spiking-ResNet34", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 180, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 180, + 310 + ], + "score": 1.0, + "content": "is extremely slow.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 210, + 505, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 327 + ], + "score": 1.0, + "content": "DVS-CIFAR10. DVS-CIFAR10 (Li et al., 2017), the most challenging mainstream neuromor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 416, + 339 + ], + "score": 1.0, + "content": "phic data set, is converted from CIFAR10. It has 10k images with the size", + "type": "text" + }, + { + "bbox": [ + 417, + 326, + 455, + 336 + ], + "score": 0.89, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 324, + 506, + 339 + ], + "score": 1.0, + "content": ". Following", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Samadzadeh et al. (2020), we divide the data stream into 10 blocks by time and accumulate the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 327, + 360 + ], + "score": 1.0, + "content": "spikes in each block. Then, we split the dataset into", + "type": "text" + }, + { + "bbox": [ + 327, + 348, + 339, + 358 + ], + "score": 0.31, + "content": "9 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 347, + 425, + 360 + ], + "score": 1.0, + "content": "training images and", + "type": "text" + }, + { + "bbox": [ + 426, + 348, + 437, + 358 + ], + "score": 0.34, + "content": "1 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "test images and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 232, + 370 + ], + "score": 1.0, + "content": "reduce the spatial resolution to", + "type": "text" + }, + { + "bbox": [ + 232, + 358, + 261, + 369 + ], + "score": 0.9, + "content": "4 8 \\times 4 8", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 359, + 505, + 370 + ], + "score": 1.0, + "content": ". Random horizontal flip and random roll within 5 pixels are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 367, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 384 + ], + "score": 1.0, + "content": "taken as augmentation (Li et al., 2022). We adopt VGGSNN architecture with 300 epochs training", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 430, + 393 + ], + "score": 1.0, + "content": "on this classification task. And we use an Adam optimizer with the learning rate", + "type": "text" + }, + { + "bbox": [ + 430, + 381, + 459, + 391 + ], + "score": 0.84, + "content": "1 e - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "and cosine", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "decay to 0. As for the case that does not apply any augmentation, we add a weight decay of 5e-4 to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 164, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 164, + 415 + ], + "score": 1.0, + "content": "the optimizer.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 314, + 506, + 415 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 298, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 300, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 300, + 441 + ], + "score": 1.0, + "content": "A.2 LSDT LOSS LANDSCAPE OF RESNET-19", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 272, + 462 + ], + "score": 1.0, + "content": "Here we compare the classification loss", + "type": "text" + }, + { + "bbox": [ + 272, + 449, + 299, + 460 + ], + "score": 0.81, + "content": "( \\mathcal { L } _ { \\mathrm { S D T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "landscapes of ResNet-19 on CIFAR100. The po-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 340, + 473 + ], + "score": 1.0, + "content": "sition around the local minimal value found by the SDT", + "type": "text" + }, + { + "bbox": [ + 340, + 460, + 367, + 471 + ], + "score": 0.82, + "content": "( \\mathcal { L } _ { \\mathrm { { S D T } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "is very sharp. However, the area", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 275, + 483 + ], + "score": 1.0, + "content": "around the local minimum found by TET", + "type": "text" + }, + { + "bbox": [ + 275, + 471, + 302, + 482 + ], + "score": 0.83, + "content": "( \\mathcal { L } _ { \\mathrm { T E T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "is much smoother (Figure 5), which indicates that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 493 + ], + "score": 1.0, + "content": "TET effectively improves the network generalization. Such improvements could be further utilized", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "to other techniques like privacy-preserving data generalization (Kim et al., 2021) and neural archi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 240, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 240, + 516 + ], + "score": 1.0, + "content": "tecture search (Kim et al., 2022).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 447, + 506, + 516 + ] + }, + { + "type": "image", + "bbox": [ + 168, + 527, + 443, + 642 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 168, + 527, + 443, + 642 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 168, + 527, + 443, + 642 + ], + "spans": [ + { + "bbox": [ + 168, + 527, + 443, + 642 + ], + "score": 0.971, + "type": "image", + "image_path": "ec3a5c919a88e624c8ee236df840321cb1292d402cae6ed2e4a7f134e697de42.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 168, + 527, + 443, + 565.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 168, + 565.3333333333334, + 443, + 603.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 168, + 603.6666666666667, + 443, + 642.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 113, + 656, + 493, + 668 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 653, + 494, + 671 + ], + "spans": [ + { + "bbox": [ + 116, + 653, + 494, + 671 + ], + "score": 1.0, + "content": "Figure 5: STD loss landscape of ResNet-19 on CIFAR100 from different training approaches.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + } + ], + "index": 36.0 + }, + { + "type": "title", + "bbox": [ + 108, + 688, + 205, + 700 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 205, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 181, + 703 + ], + "score": 1.0, + "content": "A.3 EFFECT OF", + "type": "text" + }, + { + "bbox": [ + 181, + 689, + 205, + 700 + ], + "score": 0.51, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 332, + 722 + ], + "score": 1.0, + "content": "In this part, we examine the effect of the regular term", + "type": "text" + }, + { + "bbox": [ + 332, + 710, + 355, + 721 + ], + "score": 0.86, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 709, + 462, + 722 + ], + "score": 1.0, + "content": "with 5 different levels of", + "type": "text" + }, + { + "bbox": [ + 462, + 710, + 469, + 720 + ], + "score": 0.76, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ". Figure", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 313, + 733 + ], + "score": 1.0, + "content": "6 Summarizes the final results. The regular term", + "type": "text" + }, + { + "bbox": [ + 313, + 721, + 336, + 732 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "effectively increases the performance of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "both ResNet-19 on CIFAR100 and VGGSNN on DVS-CIFAR10. The static dataset CIFAR100 is", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 202, + 106 + ], + "score": 1.0, + "content": "more suitable for larger", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 202, + 94, + 209, + 104 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 209, + 93, + 268, + 106 + ], + "score": 1.0, + "content": ", while smaller", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 269, + 94, + 276, + 104 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 276, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "is suitable for DVS-CIFAR10. Theoretically, it is hard to", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "obtain satisfying performance at the early simulation moment due to the sparseness of neuromorphic", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 246, + 128 + ], + "score": 1.0, + "content": "datasets. So too large regular term", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 116, + 270, + 127 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 270, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "is not suitable for the neuromorphic dataset. Furthermore,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 186, + 140 + ], + "score": 1.0, + "content": "we find that a high", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 186, + 127, + 194, + 137 + ], + "score": 0.64, + "content": "\\lambda", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 194, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "may harm the early training phase on ImageNet, especially if zero-initialize", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 335, + 149 + ], + "score": 1.0, + "content": "(Goyal et al., 2017) is not performed. As a result, we set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 335, + 138, + 343, + 147 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 343, + 137, + 354, + 149 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 137, + 383, + 148 + ], + "score": 0.75, + "content": "5 e - 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 383, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "for CIFAR10 and CIFAR100,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 274, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 135, + 159 + ], + "score": 0.84, + "content": "1 e - 3", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 135, + 149, + 274, + 160 + ], + "score": 1.0, + "content": "for ImageNet and DVS-CIFAR10.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "both ResNet-19 on CIFAR100 and VGGSNN on DVS-CIFAR10. The static dataset CIFAR100 is", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 202, + 106 + ], + "score": 1.0, + "content": "more suitable for larger", + "type": "text" + }, + { + "bbox": [ + 202, + 94, + 209, + 104 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 93, + 268, + 106 + ], + "score": 1.0, + "content": ", while smaller", + "type": "text" + }, + { + "bbox": [ + 269, + 94, + 276, + 104 + ], + "score": 0.77, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "is suitable for DVS-CIFAR10. Theoretically, it is hard to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "obtain satisfying performance at the early simulation moment due to the sparseness of neuromorphic", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 246, + 128 + ], + "score": 1.0, + "content": "datasets. So too large regular term", + "type": "text" + }, + { + "bbox": [ + 246, + 116, + 270, + 127 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { \\mathrm { M S E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "is not suitable for the neuromorphic dataset. Furthermore,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 186, + 140 + ], + "score": 1.0, + "content": "we find that a high", + "type": "text" + }, + { + "bbox": [ + 186, + 127, + 194, + 137 + ], + "score": 0.64, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "may harm the early training phase on ImageNet, especially if zero-initialize", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 335, + 149 + ], + "score": 1.0, + "content": "(Goyal et al., 2017) is not performed. As a result, we set", + "type": "text" + }, + { + "bbox": [ + 335, + 138, + 343, + 147 + ], + "score": 0.72, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 137, + 354, + 149 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 354, + 137, + 383, + 148 + ], + "score": 0.75, + "content": "5 e - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "for CIFAR10 and CIFAR100,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 274, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 135, + 159 + ], + "score": 0.84, + "content": "1 e - 3", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 149, + 274, + 160 + ], + "score": 1.0, + "content": "for ImageNet and DVS-CIFAR10.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 173, + 173, + 435, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 173, + 173, + 435, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 173, + 435, + 283 + ], + "spans": [ + { + "bbox": [ + 173, + 173, + 435, + 283 + ], + "score": 0.967, + "type": "image", + "image_path": "1dd2e09ae0d1ba0fa6f9db557415a2e61436ffe4627041352f5dae8ffd64e9b0.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 173, + 435, + 209.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 209.66666666666666, + 435, + 246.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 173, + 246.33333333333331, + 435, + 283.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 203, + 300, + 406, + 312 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 203, + 299, + 407, + 313 + ], + "spans": [ + { + "bbox": [ + 203, + 299, + 397, + 313 + ], + "score": 1.0, + "content": "Figure 6: The accuracy under different levels of", + "type": "text" + }, + { + "bbox": [ + 397, + 300, + 404, + 310 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 299, + 407, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 228, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 229, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 229, + 343 + ], + "score": 1.0, + "content": "A.4 STATISTICAL RESULTS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "Here we provide statistical results (Figure 7) to prove that the total SNN accuracy is positively", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "associated with every average of moment’s output test accuracy. We train CNN-5 on CIFAR10 for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 374, + 304, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 304, + 385 + ], + "score": 1.0, + "content": "a total of 20 runs with SDT and 5 runs with TET.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "image", + "bbox": [ + 215, + 402, + 384, + 534 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 215, + 402, + 384, + 534 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 215, + 402, + 384, + 534 + ], + "spans": [ + { + "bbox": [ + 215, + 402, + 384, + 534 + ], + "score": 0.962, + "type": "image", + "image_path": "295735f1db530423a1bc9f14bdb244ee4d3157db65b06f1b197a62a8ce989d2a.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 402, + 384, + 415.2 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 215, + 415.2, + 384, + 428.4 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 215, + 428.4, + 384, + 441.59999999999997 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 215, + 441.59999999999997, + 384, + 454.79999999999995 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 215, + 454.79999999999995, + 384, + 467.99999999999994 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 215, + 467.99999999999994, + 384, + 481.19999999999993 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 215, + 481.19999999999993, + 384, + 494.3999999999999 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 215, + 494.3999999999999, + 384, + 507.5999999999999 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 215, + 507.5999999999999, + 384, + 520.8 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 215, + 520.8, + 384, + 534.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 551, + 505, + 585 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "Figure 7: Statistical results. The overall performance of SNN is highly positively associated with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "the average accuracy of each moment. The standard training obtains the green dots, while the red", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 574, + 253, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 253, + 586 + ], + "score": 1.0, + "content": "dots are trained by the TET method.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + } + ], + "index": 22.75 + }, + { + "type": "title", + "bbox": [ + 107, + 606, + 356, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 357, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 357, + 619 + ], + "score": 1.0, + "content": "A.5 TIME SCALABILITY ROBUSTNESS OF SDT AND TET.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "Here we first show the test accuracy (ResNet19 on CIFAR100) of the membrane potential increment", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "at each moment instead of the integrated membrane potential. We set the initial simulation length of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "the SNNs to 3 or 4 and trained them for a full 300 epochs. Then we expand their simulation length", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 239, + 672 + ], + "score": 1.0, + "content": "to 8. As shown in table 4, TET", + "type": "text" + }, + { + "bbox": [ + 240, + 660, + 266, + 671 + ], + "score": 0.79, + "content": "( \\mathcal { L } _ { \\mathrm { T E T } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "makes the membrane potential increment at each moment", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 289, + 684 + ], + "score": 1.0, + "content": "have a higher classification ability than SDT", + "type": "text" + }, + { + "bbox": [ + 290, + 671, + 315, + 682 + ], + "score": 0.81, + "content": "( \\mathcal { L } _ { \\mathrm { { S D T } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 671, + 505, + 684 + ], + "score": 1.0, + "content": ". 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We set the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "initial simulation length of ResNet19 SNNs to 2, 3, 4 and train with SDT or TET. Then we gradually", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "increase SNN simulation length to 64 and record test accuracy of the integrated membrane potential.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 159 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 83, + 506, + 160 + ], + "lines_deleted": true + }, + { + "type": "image", + "bbox": [ + 173, + 173, + 435, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 173, + 173, + 435, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 173, + 173, + 435, + 283 + ], + "spans": [ + { + "bbox": [ + 173, + 173, + 435, + 283 + ], + "score": 0.967, + "type": "image", + "image_path": "1dd2e09ae0d1ba0fa6f9db557415a2e61436ffe4627041352f5dae8ffd64e9b0.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 173, + 173, + 435, + 209.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 209.66666666666666, + 435, + 246.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 173, + 246.33333333333331, + 435, + 283.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 203, + 300, + 406, + 312 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 203, + 299, + 407, + 313 + ], + "spans": [ + { + "bbox": [ + 203, + 299, + 397, + 313 + ], + "score": 1.0, + "content": "Figure 6: The accuracy under different levels of", + "type": "text" + }, + { + "bbox": [ + 397, + 300, + 404, + 310 + ], + "score": 0.69, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 299, + 407, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 228, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 229, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 229, + 343 + ], + "score": 1.0, + "content": "A.4 STATISTICAL RESULTS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "Here we provide statistical results (Figure 7) to prove that the total SNN accuracy is positively", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "associated with every average of moment’s output test accuracy. 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The overall performance of SNN is highly positively associated with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 505, + 575 + ], + "score": 1.0, + "content": "the average accuracy of each moment. 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Since better memory would typically prompt better generation (we define this as primal problem). The traditional approach for memory retrieval involves selecting memory that exhibits the highest similarity to the input. However, this method is constrained by the quality of the fixed corpus from which memory is retrieved. In this paper, by exploring the duality of the primal problem: better generation also prompts better memory, we propose a novel framework, Selfmem, which addresses this limitation by iteratively employing a retrieval-augmented generator to create an unbounded memory pool and using a memory selector to choose one output as memory for the subsequent generation round. This enables the model to leverage its own output, referred to as self-memory, for improved generation. We evaluate the effectiveness of Selfmem on three distinct text generation tasks: neural machine translation, abstractive text summarization, and dialogue generation, under two generation paradigms: fine-tuned small model and few-shot LLM. Our approach achieves state-of-the-art results in four directions in JRC-Acquis translation dataset, 50.3 ROUGE-1 in XSum, and 62.9 ROUGE-1 in BigPatent, demonstrating the potential of self-memory in enhancing retrieval-augmented generation models. Furthermore, we conduct thorough analyses of each component in the Selfmem framework to identify current system bottlenecks and provide insights for future research1. + +# 1 Introduction + +In recent years, retrieval-augmented text generation has attracted growing interest across various fields, including neural machine translation[28, 17, 2], dialogue response generation[81, 6, 46], and language modeling[36, 77, 19]. This innovative generation paradigm initially equips a fine-tuned small model or a large language model (LLM) with access to an external database (typically the training corpus) using information retrieval techniques. Subsequently, the generation process is conducted based on both the input text and the retrieved memory. + +In this paradigm, the guiding principle for memory retrieval is to find the memory that exhibits the highest similarity to the current input [36, 96, 49]. This aligns with the human intuition that a more similar demonstration sample typically offers more hints. As demonstrated in Figure 1, for a retrieval-augmented translation model, the memory similarity alone exhibits a strong correlation with the final translation quality, regardless of other factors that may influence translation quality (e.g., polysemy, morphology, and coreference). We define this as the primal problem: better memory prompts better generation. Consequently, numerous studies have focused on how to retrieve better memory, ranging from sparse retrieval to dense retrieval [10, 63], from a fixed retriever to a learnable retriever [41, 8], and from sentence-level memory to more fine-grained token-level memory [36, 35]. + +However, a fundamental limitation exists in all previous works: the memory is retrieved from a fixed corpus and is constrained by the corpus’s quality. Due to the finite retrieval space, bounded memory significantly restricts the potential of memory-augmented generation models [97]. In this paper, we explore the duality of the primal problem, which posits that better generation also prompts better memory. We propose a novel framework called Selfmem, which iteratively employs a retrieval-augmented generator to create an unbounded memory pool and uses a memory selector to choose one output as memory for the subsequent generation round. By combining the primal and dual problem, a retrievalaugmented generation model can elevate itself using its own output, referred to as self-memory. The key insight behind Selfmem is that the text more closely resembling the data distribution during inference is not the training data [87], but the model’s own output. + +![](images/d932460f60720219128e0be0fd5297215e94face0e4c47c4f2c984e573d265a1.jpg) +Figure 1: Relation between memory and hypothesis on JRC-Acquis E $_ { 1 \mathrm { D e } }$ dataset. The hypothesis is generated by a retrievalaugmented translator whose memory is retrieved from the training set. The $\mathbf { X }$ -axis represents the similarity between memory and the reference. + +Selfmem consists of two complementary components: + +a retrieval-augmented generator and a memory selector. The generator operates under two distinct paradigms: fine-tuning a small model or few-shot prompting an LLM. For the former, we train the generator with labeled data and retrieved memory, while for the latter, we employ a fixed black-box LLM exclusively for inference alongside retrieved in-context learning samples. We then use the generator’s output to train a memory selector based on a specific performance metric. By simply replacing the retrieved memory with unbounded generated memory, we achieve higher-quality generation output (primal problem), which subsequently serves as memory for the next round after being refined by the memory selector (dual problem). + +To evaluate the efficacy of the Selfmem, we carry out comprehensive experiments in three distinct text generation tasks: neural machine translation, abstractive text summarization, and dialogue generation. We witness substantial enhancements over robust baselines, attaining state-of-the-art outcomes in JRC-Acquis (four directions), XSum (50.3 ROUGE-1), and BigPatent (62.9 ROUGE-1). To gain deeper insights into the Selfmem, we meticulously investigate each crucial component and pinpoint the existing system bottleneck to guide future research endeavors. + +# 2 Related Work + +# 2.1 Retrieval-augmented Text Generation + +Since the world is not a snapshot once the training corpus is collected, we can never expect an ever-large model to capture everything in its parameters, even for LLMs like GPT-4 [62]. Therefore, it is crucial to equip these models with an external memory bank to store additional knowledge or useful demonstration examples for solving various NLP tasks[41, 78, 95]. + +In the translation domain, retrieval techniques have long been employed by the localization industry to enhance human translators’ productivity and consistency even before the advent of machine translation [94]. Early works on machine translation primarily focused on utilizing memory for statistical machine translation (SMT) systems [80, 50]. For neural machine translation (NMT), [28] were the first to use search engines to retrieve memory from the training set and incorporate it with an external memory network. Subsequent research explored various aspects of retrievalaugmented NMT, such as memory encoding methods [92, 93, 31], joint training of retrievers and generators with monolingual data [8], memory granularity [35], and memory diversity [17]. For few-shot LLM generation, strategies for in-context example selection have been proposed to improve translation quality [2]. Furthermore, in-context machine translation has been shown to be effective for on-the-fly adaptation [79]. For dialogue response generation tasks, employing exemplar/template retrieval as an intermediate step has proven advantageous for generating informative responses [89, 91, 6, 7]. In-context learning example retrieval also aids in controllable dialogue [46]. Other applications include abstractive summarization [64, 14, 18, 15], code generation [30], paraphrase generation [34, 83], language modeling [36, 105], counterfactual data generation [24], open domain question answering [12, 33] and semantic parsing [99]. + +# 2.2 Neural Text Reranking + +By alleviating the discrepancy between training and inference (i.e., exposure bias) and directly optimizing desired metrics, two-stage reranking methods have facilitated significant progress in various text generation tasks. In machine translation, pioneering works by [75] and [61] introduced and popularized discriminative reranking for SMT. In the context of NMT, research has focused on two primary reranking approaches: generative reranking [56, 32, 88] and discriminative reranking [39, 71, 23]. For syntactic parsing, [21] were the first to employ a two-stage reranking method to select outputs from a base parser, while [11] introduced a maximum entropy reranker. In text summarization, RefSum [53] proposed a second-stage summarization framework to address train-test distribution mismatches. SimCLS [54] used pairwise Learning To Rank (LTR) to select candidates with the highest matching scores. SummaReranker [68] adopted a multi-task mixture-of-experts framework to leverage different metrics capturing various aspects of generated candidates. BRIO [55] reused the base model for a second round of fine-tuning with both cross-entropy loss and a candidate-level ranking loss. JGR [76] employed an alternate training paradigm to train the generator and reranker. + +A key limitation of these reranking methods is that they only represent a one-way process, wherein the selected candidates become the system’s final output. In contrast, our framework innovatively utilizes the chosen candidates as memory for the subsequent generation round of a retrieval-augmented generator, which can produce better candidates with enhanced memory. + +# 3 Methods + +In this section, we begin with a motivating experiment on generation as memory $( \ S 3 . 1 )$ . Then, we introduce Selfmem, a framework comprising a retrieval-augmented generator $( \ S 3 . 2 )$ and a memory selector $( \ S \ 3 . 3 )$ . The complete framework and algorithm are illustrated in Figure 2 and Algorithm 1. + +# 3.1 Generation as Memory + +The primary motivation behind our framework stems from the observation that the memory, which is more similar in distribution to the data during inference, is not the training data (38.89 BLEU, as shown in the first row of Table 1). Instead, it is the model’s own output (58.58 BLEU) within the unbounded generation space. One interesting exploration involves directly utilizing the generated output as memory in relation to the primal problem: better memory prompts better generation. + +We conduct experiments on the JRC-Acquis En De dataset. The first row in Table 1 represents conventional retrieval-augmented training with retrieved memory and achieves a 58.58 BLEU score. However, directly incorporating beam output of this trained model as memory (Beam) back into the generation model does not yield any improvements (row 2), despite its higher similarity to the reference compared to the retrieved ones. We hypothesize two potential reasons for this: (1) the retrieval-augmented generator may not generalize effectively in this context due to the + +Table 1: Experiments on the relation between memory quality and the final hypothesis quality, measured by the BLEU score with ground truth translation. The retrieval-augmented translator keeps fixed while the memory is obtained from different sources. + +
Memory SourceMemory QualityHypothesis Quality
Retrieval38.8958.58
Beam58.5858.43
Reference10090.43
Random1.1449.08
+ +memory distribution shift (from 38.89 to 58.58), and (2) the beam memory does not offer any information gain compared to the retrieved one, even it exhibits more overlap with the references. + +To investigate the first hypothesis, we conduct experiments under the oracle and random scenarios by using the reference as memory (Reference) and randomly sampled sentences as memory (Random). The result is shown in Table 1 and it illustrates that a retrieval-augmented generator (trained with retrieved memory) has already learned to discriminate between different memories in both oracle and random scenarios, without updating the model weights. + +![](images/520d5b6545a018750a97ef0612398660acd568140ee11c91fe8800a49ffe5488.jpg) +Figure 2: Overall framework. There are two components in Selfmem, a retrieval-augmented generator (a) and a memory selector (b). For the primal problem, (a) takes source and memory as input to generate candidates for (b). For the dual problem, (b) takes as input source and generated candidates to select memory for (a). + +To evaluate the second conjecture, we first define the token sets of the reference, retrieved memory, and beam memory as $\mathcal { R } , \mathcal { M }$ , and $\boldsymbol { B }$ , respectively. The overlap token set, denoted by $\mathcal { O }$ , is defined as the tokens that overlap with the references in the beam memory but not in the retrieved memory, which is represented as $\mathcal { R } \cap \mathcal { B } - \mathcal { R } \cap \mathcal { M } .$ $\mathcal { O }$ is considered as the additional information provided by the beam memory. Inspired by the confidence analysis of NMT model [58], we compute the set confidence score, $\psi ( \cdot )$ , as follows: + +$$ +\psi ( \cdot ) = { \frac { 1 } { | \cdot | } } \sum _ { y ^ { i } \in \cdot } p ( y _ { i } | x , y _ { < i } ) +$$ + +where $p ( y _ { i } | x , y _ { < i } )$ is defined by the generation model. $\psi ( \cdot )$ measures the confidence with which the generation model generates the tokens. The value of $\psi ( \mathcal { R } )$ is 0.58, while that of $\mathcal { O }$ is 0.76, indicating that the generator is relatively confident in generating tokens in $\mathcal { O }$ , and therefore does not need to resort to external memory [38]. Beam search ranks generated candidates based on $p ( y | x )$ , where the selected memory falls within the confidence region of the generator and consequently provides no information gain. This observation motivates us to select memory according to metrics other than $p ( y | x )$ in the memory selector (§3.3). + +# 3.2 Retrieval-augmented Generator + +Given a text pair $( x , y )$ , where $x = \{ \mathbf { x } _ { 1 } , . . . , \mathbf { x } _ { | x | } \}$ is the source, $y = \{ \mathbf { y } _ { 1 } , . . . , \mathbf { y } _ { | y | } \}$ is the target. They could be (document, summary) in summarization, (context, response) in dialogue generation or (source, target) in machine translation. The retrieval-augmented generation would first use $x$ to retrieve memory $m$ from datastore $\mathbb { D }$ . Then the generator $G _ { \xi } ( x , m )$ , parameterized by $\xi$ , would take both $x$ and $m$ as input to generate the target sentence $y$ . In this paper, following standard practice, we choose the training set as $\mathbb { D } = \{ ( x ^ { i } , y ^ { i } ) \} _ { i = 1 } ^ { | \mathbb { D } | }$ . For LLM as $G _ { \xi }$ , we use the standard in-context learning format to give $( x , y )$ as demonstration example. For tunable generator $G _ { \xi }$ , we only keep the target side of top- $\mathbf { \xi } _ { l }$ retrieval results as memory and we consider two commonly used architectures: Joint-Encoder [29, 87, 41] and Dual-Encoder [92, 8, 17]. + +Joint-Encoder This architecture is the standard encoder-decoder-based model [3, 84]. The input is the concatenation of $x$ and $m$ . The encoder would first map the input into the hidden states $H$ : + +$$ +H = { \mathrm { E n c o d e r } } ( x \ [ { \mathrm { S E P } } ] \ m ) +$$ + +And the decoder would incorporate $H$ by attention mechanism and generate tokens in an autoregressive manner: + +$$ +h ^ { i } = \mathrm { D e c o d e r } ( \mathrm { C r o s s A t t n } ( H ) , y _ { < i } ) \quad P _ { G _ { \xi } } ( \cdot | x , y _ { < i } ) = \mathrm { S o f t m a x } ( h ^ { i } ) +$$ + +Dual-Encoder Instead of treating $x$ and $m$ as a long sequence, this architecture has two encoders, one for $x$ and the other for $m$ . Their outputs are sequentially attended by the decoder with dual cross attention as in [17]: + +$$ +\begin{array} { c } { H _ { x } = \mathrm { S o u r c e E n c o d e r } ( x ) \quad H _ { m } = \mathrm { M e m o r y E n c o d e r } ( m ) } \\ { h ^ { i } = \mathrm { D e c o d e r } ( \mathrm { C r o s s A t t n } ( H _ { x } , H _ { m } ) , y _ { < i } ) } \end{array} +$$ + +We use Transformer [84] as the building block for both architectures and optimize $G _ { \xi }$ with NLL loss: + +$$ +\mathcal { L } _ { \mathrm { n l l } } = - \sum _ { t = 1 } ^ { | y | } \log P _ { G _ { \xi } } ( y _ { t } | x , m , y _ { < t } ) +$$ + +# 3.3 Memory Selector + +The role of memory selector $S _ { \theta } ( x , c )$ , parameterized by $\theta$ , is to select one candidate $c$ from the candidate pool $\mathbb { C }$ generated by $G _ { \xi }$ based on a specific metric $\Delta ( \cdot , \cdot )$ . The chosen candidate $c$ is then utilized as memory $m$ for the subsequent generation round of $G _ { \xi }$ . As discussed in $\ S 3 . 1$ , using $p _ { G _ { \xi } } ( y | x )$ as the metric $\Delta ( \cdot , \cdot )$ would result in falling into the confidence region of $G _ { \xi }$ , leading to no information gain. Moreover, a larger value of $p _ { G _ { \xi } } ( y | x )$ does not necessarily guarantee improved generation quality [59]. Consequently, we define $\Delta ( \cdot , \cdot )$ as model-free metrics that are widely employed for assessing generation quality, such as BLEU for Neural Machine Translation (NMT) and ROUGE for Summarization. Our memory selector takes the concatenation of the source $x$ and candidate $c _ { i }$ as input, and produces a multinomial distribution $p _ { S _ { \theta } } ( \cdot | x )$ over $\mathbb { C }$ . + +In this paper, we focus on the role of the memory selector, $S _ { \theta } ( x , c )$ , which is parameterized by $\theta$ . The objective of this selector is to choose a single candidate $c$ from the candidate pool $\mathbb { C }$ , generated by $G _ { \xi }$ , based on a specific metric, $\Delta ( \cdot , \cdot )$ . + +$$ +p _ { S _ { \theta } } ( c _ { i } | x ) = \frac { \exp ( S _ { \theta } ( x \left[ \mathrm { S E P } \right] c _ { i } ) ) } { \sum _ { j = 1 } ^ { | \mathbb { C } | } \exp ( S _ { \theta } ( x \left[ \mathrm { S E P } \right] c _ { j } ) ) } +$$ + +In accordance with [39], the training goal for $S _ { \theta }$ is to minimize the discrepancy between the $S _ { \theta }$ ’s predictions and the scores determined by $\Delta ( \cdot , \cdot )$ . This divergence is quantified using the KullbackLeibler (KL) divergence. + +$$ +\mathcal { L } _ { \mathrm { k l } } = - \sum _ { i = 1 } ^ { | \mathbb { C } | } p _ { M } ( c _ { i } ) \mathrm { l o g } p _ { S _ { \theta } } ( c _ { i } | x ) \quad \mathrm { w h e r e } \quad p _ { M } ( c _ { i } ) = \frac { \exp ( \Delta ( c _ { i } , y ) / \tau ) } { \sum _ { j = 1 } ^ { | \mathbb { C } | } \exp ( \Delta ( c _ { j } , y ) / \tau ) } +$$ + +$\tau$ is the temperature to control the smoothness of the distribution. At inference, the output of the $S _ { \theta }$ is a $\operatorname { r g m a x } _ { c _ { i } \in \mathbb { C } } { \dot { p } } _ { S _ { \theta } } ( c _ { i } | x )$ . + +# 3.4 Combine Generator and Selector + +We define two generation modes for $G _ { \xi }$ . The first mode, referred to as the hypothesis mode, generates a single output for each input, which is utilized for system evaluation. The second mode, known as the candidate mode, produces $_ \mathrm { N }$ outputs for a given input, and is employed for training $S _ { \theta }$ as well as memory selection. By integrating two modes together, we present the complete framework of our proposed model, Selfmem, as illustrated in Algorithm 1. + +# 4 Experimental Setup + +# 4.1 Dataset + +We assess the performance of Selfmem on three generation tasks, utilizing a total of seven datasets. Translation. We evaluate our framework on JRC-Acquis datasets [82], a collection of parallel + +Require: a dataset $\mathbb { D }$ , a retriever $R$ , a memory selection metric $\Delta ( \cdot , \cdot )$ , a retrieval-augmented +generator $G _ { \xi }$ , and a memory selector $S _ { \theta }$ +1: retrieve memory $\mathbb { M }$ in $\mathbb { D }$ with $R$ +2: train $G _ { \xi }$ with $\mathbb { D }$ and $\mathbb { M }$ (if not LLM) +3: use $G _ { \xi }$ to generate candidate pool $\mathbb { C }$ with $\mathbb { M }$ in candidate mode +4: train ${ \check { S } } _ { \theta }$ on $\mathbb { C }$ with $\Delta ( \cdot , \cdot )$ +5: while not converged in the validation set do +6: $S _ { \theta }$ selects memory from $\mathbb { C }$ as $\mathbb { M }$ +7: $G _ { \xi }$ generates candidate pool $\mathbb { C }$ with $\mathbb { M }$ in candidate mode +8: end while +9: $G _ { \xi }$ generates the final hypothesis with $\mathbb { M }$ in hypothesis mode + +legislative text of European Union Law. It is the benchmark dataset used in translation memoryaugmented NMT task [28, 92, 8, 17]. We choose 4 translation directions, namely, Spanish English $( { \mathrm { E s } } { \mathrm { E n } } )$ , German English $( \mathrm { D e } \mathrm { E n } $ ). Summarization. We evaluate on 2 summarization datasets: 1) XSum [60], extreme summarization, a single-document summarization dataset with highly abstractive articles from British Broadcasting Corporation. 2) BigPatent [73], consisting of 1.3 million records of U.S. patent documents along with human-written abstractive summaries. Dialogue. We experiment on DailyDialog [44], which contains multi-turn dialogs on daily life topics and is used by [13, 4, 103]. The detailed statistics for these datasets can be found in the Appendix A. + +# 4.2 Implementation Details + +We utilize the BM25 algorithm [70] for retrieval purposes. For all tasks, the candidate generation method consists of beam search with a beam width of 50. The number of iterations is determined by the performance on the validation set. For translation, we follow the approach of [93, 8, 17], employing a randomly initialized Transformerbase architecture as $G _ { \xi }$ for trainable small model and XGLM [48] for LLM in-context learning. Evaluation metrics include BLEU, TER, and ${ \mathrm { c h r F } } + +$ obtained from SACREBLEU[66]. The memory selector $S _ { \theta }$ utilizes an XLM- ${ \bf R } _ { b a s e }$ [22] as backbone, with BLEU serving as $\Delta ( \cdot , \cdot )$ . For summarization, we initialize $G _ { \xi }$ with $\mathrm { B A R T _ { b a s e } } [ 4 0 ]$ for BigPatent and employ BRIO [55] for XSum. The evaluation metric comprises ROUGE (R1/2/L) [47]. For dialogue generation, $\mathbf { B A R T _ { b a s e } }$ serves as the backbone for $G _ { \xi }$ . Our dialogue system is evaluated using BLEU (B-1/2) and Distinct (D-1/2) scores [43]. For both dialogue and summarization tasks, we adhere to the methods of [54, 26], adopting $\mathrm { R o B E R T a _ { b a s e } }$ [52] as the backbone for $S _ { \theta }$ The linear combination of $_ { \mathrm { B - } 1 / 2 }$ is chosen as $\Delta ( \cdot , \cdot )$ for Dialogue Generation, while $\mathrm { \mathbf { R } } \mathrm { - } 1 / 2 / \mathrm { L }$ is used for Summarization, following [76]. For further implementation details, please refer to the Appendix B and Appendix C for evaluation metrics. + +# 5 Experimental Results + +# 5.1 Machine Translation + +We select four translation directions and experiment with two generation paradigms: trainable small models and few-shot prompted LLMs [85, 20]. For trainable models, we explore two architectures (joint and dual, as detailed in $\ S 3 . 2 \AA$ . The baselines comprise two types of translation systems: one being the vanilla sequence-to-sequence model [3, 84] without memory augmentation, and the other consisting of retrieval-augmented translation models focusing on memory encoding [28, 92], memory construction [101], memory retrieval [8], and memory diversity [17]. Based on the experimental results2 shown in Table 2, Selfmem significantly enhances the performance of $G _ { \xi }$ across four translation datasets and two different architectures. This is noteworthy, given that the parameters of the $G _ { \xi }$ remain fixed, with the only variable being the input memory. This finding is consistent with the primal problem which posits that improved memory typically leads to better generation results. + +Table 2: Results of translation task on JRC-Acquis measured by BLEU. Models denoted by the same symbol $\times$ and $\dagger .$ ) have the same parameters and only differ in memory as input. The bolded numbers show the SOTA performance and the underlined numbers show the second-best result. $^ *$ denotes the system is significantly better than baselines with $p$ -value $< 0 . 0 5$ tested by [37]. + +
SystemEs-→EnEn→EsDe-→EnEn→De
DevTestDevTestDevTestDevTest
None Memory
RNNsearch [3]55.0259.3450.5450.4850.2049.7444.9443.98
Transformer [84]64.0864.6362.0261.8060.1860.1654.6555.43
Retrieval Memory
SEG-NMT[28]60.2859.3457.6257.2755.6355.3349.2648.80
NMT-pieces [101]63.9764.3061.5061.5660.1060.2655.5455.14
G-TFM [92]66.3766.2162.5062.7661.8561.7257.4356.88
MonoNMT[8]67.7367.4264.1863.8664.4864.6258.7758.42
CMM[17]67.4867.7663.8464.0464.2264.3358.9458.69
Transformerdual*66.8767.1263.1463.5464.0963.3658.6958.06
Transformerunit67.7467.3263.9364.1264.5064.4058.1658.58
Self-Memory
Transformerdual*68.63*69.20*64.12*64.67*65.06*64.98*59.26*59.49*
Transformerunit68.26*68.80*66.07*65.94*65.32*65.65*59.88*60.11*
+ +Table 3: Comparison between retrieval memory and self-memory. The quality of memory and hypothesis is measured by the n-gram overlap with reference (BLEU). All experiments are conducted with Transforme $\mathbf { \dot { j } } \mathbf { o } \mathbf { \dot { i } } \mathbf { n } \mathbf { t }$ on JRC-Acquis. + +
RetrievalSelf
memoryhypothesis memoryhypothesis
En-De→38.8958.5857.9260.11
↑42.5664.4064.3265.65
En-Es→40.6764.1263.5765.94
↑43.0567.3267.7868.80
+ +The dual problem is revealed in Table 3. Self-memory, which essentially represents the model’s own output, exhibits greater similarity with the ground truth and serves as a more effective memory for generating the final output. This observation highlights a key distinction between Selfmem and previous reranking works [39, 68]. Reranking aims to select candidates of higher quality than the beam output, whereas in Selfmem, the chosen candidates serve as memory for the retrieval-augmented generator and do not necessarily need to surpass the quality of the beam hypotheses. + +Table 4: Evaluation results of in-context learning with self-memory. + +
XGLM-1.7BXGLM-4.5BXGLM-7.5B
RandomkNNSelfRandomkNNSelfRandomkNNSelf
En-De↑11.5137.8740.9417.5137.6038.2518.4847.8248.32
27.4251.0051.8830.6248.1248.3633.0355.6555.12
En-Es→23.8746.2048.5631.8348.3749.1729.9753.8654.32
↑25.2951.5553.1332.1648.5549.2235.2257.2557.56
+ +In Table 4, we present the results of LLM with self-memory. We employ XGLM [48] as our backbone generator, with three different sizes ranging from 1.7B to 7.5B. We utilize the recommended prompt as described in [48]. We select three in-context learning examples and report the average scores from three separate runs, taking into account the sensitivity of example selection in ICL [49]. From the table, we first observe a general trend where few-shot translation performance improves as the size of the model increases. Furthermore, we find that more similar translation demonstrations significantly enhance performance across all model sizes (from random, kNN to Self). This suggests that demonstration examples in in-context learning not only act as triggers for model ability but also adhere to the primal problem, where better demonstration example leads to better generation. Also, by comparing the results in Table 2 and Table 4, we can conclude that the cross-lingual LLM with designed examples still falls short of the supervised baselines in this task. + +# 5.2 Summarization + +In this paper, we compare the performance of our trainable model with those of REINA [87], PEGASUS [100], and BART [40]. The results are presented in Table5. Initially, it can be observed that memory has varying impacts on different datasets. The enhancement brought by memory in the BigPatent dataset is significantly larger than that in the XSum dataset. This can be attributed to the inherent characteristics of the BigPatent dataset, which consists of official patent documents that exhibit considerable similarity. Consequently, this greatly improves the summarization quality in accordance with the primal problem. Furthermore, we discovered that self-memory substantially enhances the performance of both BRIO $( + 1 . 2 { \ R } 1 )$ and BART $( + 1 8 . 5 \mathrm { R } 1 ) $ ), achieving state-of-the-art results on both datasets. We selected these baselines for a fair comparison, as they share the same base generator. Due to space constraints, additional comparisons and the confidence region of the SOTA model can be found in the Appendix E. + +Table 5: Results of summarization task on XSum and BigPatent measured by ROUGE. + +
SystemMemoryR-1R-2R-L
XSum
PEGASUSNone47.224.639.3
BRIONone49.125.640.4
REINA (PG)Retrieval48.226.040.2
REINA (B)Retrieval43.221.035.5
REINA (L)Retrieval46.524.138.6
BRIOdual*Retrieval48.626.140.6
BRIOjointRetrieval49.526.541.2
BRIOdual*Self49.226.240.8
BRIOjointtSelf50.326.741.6
+ +
SystemMemoryR-1R-2R-L
BigPatent
PEGASUSNone53.633.243.2
BARTNone44.421.331.0
REINA (B)Retrieval59.542.650.6
REINA (L)Retrieval60.743.351.3
REINA (PG)Retrieval44.621.533.3
BARTdual*Retrieval57.443.349.7
BARTjointtRetrieval59.643.451.0
BARTdual*Self61.244.652.3
BARTjointSelf62.948.159.6
+ +# 5.3 Dialogue Generation + +As demonstrated in Table 6, the self-memory significantly enhances the performance of the retrievalaugmented generator for dialogue generation tasks. By optimizing memory using BLEU as $\Delta ( \cdot , \cdot )$ , the self-memory improves the B-1,2 score over retrieved memory by $3 . 0 8 \ \mathrm { B } \cdot 1$ and $0 . 6 \ \mathbf { B } { - } 2$ on $\mathbf { B A R T _ { j o i n t } }$ . Intriguingly, although Selfmem surpasses the baselines in terms of $_ { \mathrm { B - } 1 / 2 }$ , it falls behind in D-1 and D-2, which can be attributed to the trade-off between BLEU score and Distinct score when evaluating a dialogue system [104]. To address this issue, we opt for D-1,2 as $\Delta ( \cdot , \cdot )$ when optimizing $S _ { \theta }$ , denoted as $\mathbf { B A R T } _ { \mathrm { j o i n t } } \dagger ( \mathbf { D } )$ . The results in Table 6 highlight the remarkable flexibility of Selfmem by directly optimizing memory to achieve the desired attributes for diverse and informative dialogue. + +# 6 Further Analysis + +To gain a deeper insight into Selfmem, we first examine the impact of each key component, namely $G _ { \xi }$ and $S _ { \theta }$ . Subsequently, we perform a detailed token-level analysis of the generated output concerning their frequency in the training set. Experiments are conducted on the JRC-Acquis En De dataset. We also include latency analysis and human evaluation on Appendix F and G. + +Tuning $S _ { \theta }$ We explored various $S _ { \theta }$ by direct selection from the candidate pool based on gold rankings. As shown in Figure 3a, both architectures with enhanced $S _ { \theta }$ significantly outperform the current SOTA performance (60.11 BLEU). Moreover, we assessed the candidate pool quality during this iterative process using an oracle $S _ { \theta }$ , as displayed in Figure 3b. A clear pattern emerges in this boxplot, revealing improvements in the oracle, quartile, average, and minimum scores of the candidate pool. These two experiments jointly clarify the Selfmem’s underlying intuition: a retrieval-augmented generator profits from superior memory, which can be chosen from its own unbounded output, and subsequently, the generator with improved memory produces a higher-quality candidate pool for the next selection round. Consequently, the model lift itself up. + +Table 6: Results of dialogue generation task on DailyDialog measured by B-1/2 and D-1/2. $\mathbf { B A R T _ { j o i n t } }$ (D) denotes the metric $\Delta ( \cdot , \cdot )$ for $S _ { \theta }$ is the average of D-1 and D-2. + +
SystemMemoryB-1B-2D-1D-2
NCM [86]None33.6026.803.0012.80
iVAE [25]None30.9024.902.9025.00
PLATO-2 [5]None34.8025.123.5425.11
DialoFlow [45]None36.1727.674.5627.12
BARTNone20.7211.363.9219.44
BARTdual*Retrieval29.5021.894.7426.01
BARTjointtRetrieval36.7231.556.1335.65
BARTdual*Self33.4322.854.6626.16
BARTjointSelf39.8032.155.8432.16
BARTjoint † (D)Self36.9232.099.1237.05
+ +![](images/e2ff140cd3a0a3d28c8ec3e9f453a47d6086af9a3faa5dab85f106307077b962.jpg) +Figure 3: (a) shows generation quality in the iteration process with different $S _ { \theta }$ in both trainable generator architectures. (b) shows candidates quality in the iteration process with an oracle $S _ { \theta }$ . + +Tuning $G _ { \xi }$ As discussed in $\ S 3 . 1$ , we demonstrated that a trained retrieval-augmented generator, with fixed parameters, possesses the ability to distinguish between "good" and "bad" memory. This observation not only justifies our decision to maintain a fixed generator within our framework but also implies that the $G _ { \xi }$ is not the current bottleneck of the Selfmem. + +![](images/b973e5b8f1f94a7b457be97ac2c4d228ea2a8787fa0754505c1667991affb841.jpg) +Figure 4: 1-gram F1 score sorted by training corpus frequency. + +Frequency Analysis We conduct a comprehensive tokenlevel analysis by computing the 1-gram F1 scores for generated translations and subsequently categorizing the tokens based on their frequency in the training set. The results are depicted in Figure 4. A noticeable pattern emerges, suggesting that the more frequently a model encounters a token during training, the higher the accuracy of the generated output [102]. Moreover, our findings indicate that retrievalaugmented models, particularly those incorporating self-memory augmentation, exhibit superio performance in handling long-tail inputs which are challenges for parametric models [67, 57]. + +# 7 Conclusion + +For the first time, we investigate the fundamental limitation of bounded memory in the current retrieval-augmented literature. We combine the primal and dual problems together and propose Selfmem, a general framework for retrieval-augmented text generation by uplifting generation model with its own output. We conduct comprehensive experiments across various text generation tasks and different generation paradigms, including trainable small model and few-shot prompted LLM. We surpass strong baselines and improve the state-of-the-art performance in serval datasets. We also meticulously investigate each crucial component and pinpoint the existing system bottleneck to guide future research endeavors. + +# Limitations + +We discuss the limitations of our framework as follows: + +(1) Although Selfmem greatly improves the generation quality compared with other retrievalaugmented generation models, it requires more computational resources with respect to the memory selection process. For large dataset with long context (e.g., BigPatent), it would become a more crucial problem considering the quadratic time complexity of transformer architecture. + +(2) This paper proposes a general idea for the retrieval-augmented generation. But we only experiment with transformer-based architecture for both generator and memory selector and the architecture of generator and memory selector keeps the same across all text generation tasks. We believe the task-specific design for the model architecture, training objective and generation methods in different text generation scenarios would further improve the performance. + +# Acknowledgement + +This work was supported by the National Key Research and Development Program of China (No.2021YFC3340304) and National Natural Science Foundation of China (NSFC Grant No.62122089). We appreciate the anonymous reviewers for their helpful comments. Dongyan Zhao and Rui Yan are the corresponding authors. + +# References + +[1] Armen Aghajanyan, Anchit Gupta, Akshat Shrivastava, Xilun Chen, Luke Zettlemoyer, and Sonal Gupta. Muppet: Massive multi-task representations with pre-finetuning. In Proc. of EMNLP, 2021. +[2] Sweta Agrawal, Chunting Zhou, Mike Lewis, Luke Zettlemoyer, and Marjan Ghazvininejad. In-context examples selection for machine translation. CoRR, 2022. +[3] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 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TaskDataset#Train#Dev#Test
TranslationJRC (en ←→ de)663,4872,4542,483
JRC (en ←→ es)653,1272,5332,596
SummarizationBigPatent1,207,22267,06867,072
XSum204,04511,33211,334
DialogueDailyDialog87,1708,0697,740
+ +# B Self Memory Details + +For machine translation tasks, following [93, 8, 17] we use randomly initialize Transformerbase architecture [84] as $G _ { \xi }$ . We use the joint-bpe algorithm [72] and share the parameters between the memory encoder and source encoder for dual encoder architecture. The hyper-parameter setting follows [17] with dropout 0.1, label smoothing 0.1, gradient clipping 1.0, Adafactor [74], warm-up steps 4000, maximum learning rate $4 . 4 \mathrm { e } { - 2 }$ and training epochs 30 for total. The evaluation metrics are BLEU, TER and ${ \mathrm { c h r F } } + +$ from SACREBLEU [66]. The backbone of memory selector $S _ { \theta }$ is XLM- $. { \bf R } _ { b a s e }$ [22] with BLEU as $\Delta ( \cdot , \cdot )$ . The hyper-parameter setting for $S _ { \theta }$ follows [39] with $\tau 0 . 5$ , minmax normalization for candidates ranking, Adam optimizer with max learning rate 5e-5 and polynomial decay scheduler, and classifier dropout 0.2. + +For Summarization, we init the $G _ { \xi }$ with $\mathbf { B A R T _ { b a s e } }$ [40] for BigPatent following [87] and state-of-theart BRIO [55] for XSum. Optimization is based on Adafactor with a maximum learning rate of 5e-3, warm-up steps 10000 and gradient clipping value 1.0. The maximum input length is 512 for XSum and 1024 for BigPatent. The evaluation metric is Rouge (R-1/2/L) [47]. + +For Dialogue Generation, we use $\mathbf { B A R T _ { b a s e } }$ as the backbone for $G _ { \xi }$ on DailyDialog. We tune the hyper-parameters from learning rate $\{ 5 \mathrm { e } { - } 3 , 1 \mathrm { e } { - } 3 , 4 \mathrm { e } { - } 4 \}$ and set dropout 0.1, batch size 64, label smoothing factor 0.1, maximum input length 120 for DailyDialog. Following [4, 13], we evaluate our dialogue system with BLEU (B-1/2) and Distinct (D-1,2) [43]. For both Summarization and Dialogue Generation task, we follow [54, 26] and adopt $\mathrm { R o B E R T a _ { b a s e } }$ [52] as the backbone for $S _ { \theta }$ . We choose the linear combination of B-1/2 as $\Delta ( \cdot , \cdot )$ for Dialogue Generation and R-1/2/L for Summarization following [76]. We tune the hyper-parameters $\tau$ from $\{ 0 . 0 8 , 0 . 2 , 0 . 5 , 0 . 8 \}$ , learning rate from {5e-5,7e-5,2e-4}. The maximum input length for $S _ { \theta }$ is 512 and we truncate tokens from the longer input of source and candidate. + +# C Evaluation Details + +Machine Translation We evaluate our MT system with BLEU, TER and ${ \mathrm { c h r F } } + +$ from SACREBLEU3 [66]. The signatures for BLEU, TER and ${ \mathrm { c h r F } } + +$ are shown in Table 8. + +Table 8: Signature from SACREBLEU. + +
[c]Signature
nrefs:1lcase:mixedleff:noltok:13alsmooth:explversion:2.0.0
nrefs:1lcase:lcltok:tercomlnorm:nolpunct:yeslasian:nolversion:2.0.0
nrefs:1lcase:mixedleff:yeslnc:6lnw:2lspace:nolversion:2.0.0
+ +Summarization We evaluate our Summarization system with standard ROUGE [47] Perl package4 for evaluation. Following [55], we use PTB tokenizer5 for tokenization. And the parameters for ROUGE are " $\mathsf { \Pi } _ { - \mathrm { c } } ^ { \prime } 9 5 \mathsf { \Pi } _ { - \mathrm { r } } 1 0 0 0 \mathsf { \Pi } _ { - \mathrm { n } } 2 \mathsf { \Pi } _ { - \mathrm { m } } \mathsf { " }$ . + +Dialogue Generation Following [27], we evaluate our dialogue system with NLTK BLEU 6 with space as tokenizer and smoothing method1. The Distinction score is from [42]. + +# D More results on translation tasks + +Table 9: Evaluation results on JRC-Acquis En De measured by BLEU, TER and ${ \mathrm { c h r F } } + +$ + +
SystemMemoryBLEU 个chrF++ 个TER
TransformerNone55.4370.3136.35
TransformerdualRetrieval58.0671.5835.41
TransformerjointRetrieval58.5872.2234.39
TransformerdualSelf59.4972.6234.04
TransformerjointSelf60.1173.2532.62
+ +# E More Summarization Baselines + +In this Table 10, we include more baselines on the benchmark dataset XSum and BigPatent. We also report the confidence region of SOTA model for XSum and BigPatent as shown in Table 11. + +Table 10: More baselines on XSum and BigPatent. + +
SystemR-1R-2R-L
XSum
[51]38.816.531.3 37.3
[40]45.122.339.3
[100]47.224.639.4
[54] [55]47.6 49.124.640.4
[87](PG)48.225.640.2
[87](B)43.126.035.5
21.038.6
[87](L)46.524.140.0
[68]48.125.038.8
[69]47.124.1
[16]47.825.039.7
Selfmem50.326.741.6
+ +
SystemR-1 R-2R-L
BigPatent
[100]53.6 33.142.3
[40] 44.421.331.0
[98] 60.642.550.0
[65]38.7 12.334.1
[90] 45.020.339.2
[1] 52.333.542.8
[87] (B) 59.542.650.6
[87] (L) 60.743.351.3
[87] (PG) 44.621.533.3
Selfmem62.9 48.159.6
+ +# F Empirical analysis of latency + +In Table 12, we present empirical results of Selfmem latency, measured in seconds. We compare Selfmem with a retrieval-augmented baseline model across various datasets and computational platforms, including CPU and CUDA. The number of iterations for Selfmem is set to one. All experiments are conducted on the same device, equipped with one NVIDIA A100 GPU and one AMD EPYC 7V13 64-Core Processor. + +Table 11: Confidence region for SOTA model in XSum and BigPatent. + +
SystemROUGE-1/2/L95 % -conf.int
XSum
BRIOjoint50.30.49986 - 0.50602
26.70.26300 - 0.26989
41.60.41231 - 0.41900
BigPatent
BARTjoint62.90.62664 - 0.63080
48.10.47783 - 0.48333
59.60.59401 - 0.59847
+ +Table 12: Generation Latency analysis. + +
NMT XSumBigPatentDailyDialog
Average Input Length87512102471
Average ：Output Length447512716
Retrieval-augmented BaselineCPU 0.971.793.160.32
SelfmemCandidate Generation Memory3.207.5015.001.02
Selection0.500.520.950.14
Hypothesis Generation0.971.793.000.32
×4.80×5.47×6.04×4.63
CUDA
Retrieval-augmented Baseline0.290.440.750.10
SelfmemCandidate Generation Memory Hypothesis Generation0.511.001.720.18
Selection0.010.010.010.01
0.29 ×2.760.44 ×2.990.75 ×3.350.10 ×2.91
+ +# G Human and GPT-4 Evaluation + +We employ both human annotators and GPT-4 (gpt-4-0314) annotators to perform pairwise ranking of the output generated by Selfmem and baseline systems. For GPT-4 annotators, we utilize the prompt from Alpaca Eval 7. We randomly select 50 samples for translation tasks and 20 samples for summarization and dialogue tasks. The win rate of Selfmem versus retrieval-augmented baselines is depicted in Figure 1. + +![](images/50ca523e51f29d99aad67a44b5b7d68d7a0aed4f3d5c8ed8c80cd389713d368f.jpg) +Figure 5: Human and GPT-4 evaluation results. \ No newline at end of file diff --git a/parse/dev/lYNSvp51a7/lYNSvp51a7_content_list.json b/parse/dev/lYNSvp51a7/lYNSvp51a7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..dd75de0626fd5164908182b1ab313072d12925bc --- /dev/null +++ b/parse/dev/lYNSvp51a7/lYNSvp51a7_content_list.json @@ -0,0 +1,1804 @@ +[ + { + "type": "text", + "text": "Lift Yourself Up: Retrieval-augmented Text Generation with Self-Memory ", + "text_level": 1, + "bbox": [ + 235, + 122, + 764, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xin Cheng1 Di Luo2 Xiuying Chen3 Lemao Liu4 Dongyan Zhao1 Rui Yan2 ", + "bbox": [ + 197, + 223, + 797, + 241 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Peking University 2 Remin University of China 3 KAUST 4 Tencent AI Lab chengxin1998@stu.pku.edu.cn ", + "bbox": [ + 330, + 252, + 668, + 296 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 332, + 535, + 348 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "With direct access to human-written reference as memory, retrieval-augmented generation has achieved much progress in a wide range of text generation tasks. Since better memory would typically prompt better generation (we define this as primal problem). The traditional approach for memory retrieval involves selecting memory that exhibits the highest similarity to the input. However, this method is constrained by the quality of the fixed corpus from which memory is retrieved. In this paper, by exploring the duality of the primal problem: better generation also prompts better memory, we propose a novel framework, Selfmem, which addresses this limitation by iteratively employing a retrieval-augmented generator to create an unbounded memory pool and using a memory selector to choose one output as memory for the subsequent generation round. This enables the model to leverage its own output, referred to as self-memory, for improved generation. We evaluate the effectiveness of Selfmem on three distinct text generation tasks: neural machine translation, abstractive text summarization, and dialogue generation, under two generation paradigms: fine-tuned small model and few-shot LLM. Our approach achieves state-of-the-art results in four directions in JRC-Acquis translation dataset, 50.3 ROUGE-1 in XSum, and 62.9 ROUGE-1 in BigPatent, demonstrating the potential of self-memory in enhancing retrieval-augmented generation models. Furthermore, we conduct thorough analyses of each component in the Selfmem framework to identify current system bottlenecks and provide insights for future research1. ", + "bbox": [ + 232, + 366, + 766, + 655 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 681, + 310, + 699 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, retrieval-augmented text generation has attracted growing interest across various fields, including neural machine translation[28, 17, 2], dialogue response generation[81, 6, 46], and language modeling[36, 77, 19]. This innovative generation paradigm initially equips a fine-tuned small model or a large language model (LLM) with access to an external database (typically the training corpus) using information retrieval techniques. Subsequently, the generation process is conducted based on both the input text and the retrieved memory. ", + "bbox": [ + 174, + 714, + 825, + 796 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paradigm, the guiding principle for memory retrieval is to find the memory that exhibits the highest similarity to the current input [36, 96, 49]. This aligns with the human intuition that a more similar demonstration sample typically offers more hints. As demonstrated in Figure 1, for a retrieval-augmented translation model, the memory similarity alone exhibits a strong correlation with the final translation quality, regardless of other factors that may influence translation quality (e.g., polysemy, morphology, and coreference). We define this as the primal problem: better memory prompts better generation. Consequently, numerous studies have focused on how to retrieve better memory, ranging from sparse retrieval to dense retrieval [10, 63], from a fixed retriever to a learnable retriever [41, 8], and from sentence-level memory to more fine-grained token-level memory [36, 35]. ", + "bbox": [ + 174, + 804, + 825, + 873 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 92, + 825, + 147 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, a fundamental limitation exists in all previous works: the memory is retrieved from a fixed corpus and is constrained by the corpus’s quality. Due to the finite retrieval space, bounded memory significantly restricts the potential of memory-augmented generation models [97]. In this paper, we explore the duality of the primal problem, which posits that better generation also prompts better memory. We propose a novel framework called Selfmem, which iteratively employs a retrieval-augmented generator to create an unbounded memory pool and uses a memory selector to choose one output as memory for the subsequent generation round. By combining the primal and dual problem, a retrievalaugmented generation model can elevate itself using its own output, referred to as self-memory. The key insight behind Selfmem is that the text more closely resembling the data distribution during inference is not the training data [87], but the model’s own output. ", + "bbox": [ + 174, + 154, + 529, + 401 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/d932460f60720219128e0be0fd5297215e94face0e4c47c4f2c984e573d265a1.jpg", + "image_caption": [ + "Figure 1: Relation between memory and hypothesis on JRC-Acquis E $_ { 1 \\mathrm { D e } }$ dataset. The hypothesis is generated by a retrievalaugmented translator whose memory is retrieved from the training set. The $\\mathbf { X }$ -axis represents the similarity between memory and the reference. " + ], + "image_footnote": [], + "bbox": [ + 555, + 172, + 808, + 294 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Selfmem consists of two complementary components: ", + "bbox": [ + 174, + 409, + 532, + 422 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "a retrieval-augmented generator and a memory selector. The generator operates under two distinct paradigms: fine-tuning a small model or few-shot prompting an LLM. For the former, we train the generator with labeled data and retrieved memory, while for the latter, we employ a fixed black-box LLM exclusively for inference alongside retrieved in-context learning samples. We then use the generator’s output to train a memory selector based on a specific performance metric. By simply replacing the retrieved memory with unbounded generated memory, we achieve higher-quality generation output (primal problem), which subsequently serves as memory for the next round after being refined by the memory selector (dual problem). ", + "bbox": [ + 174, + 420, + 825, + 532 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To evaluate the efficacy of the Selfmem, we carry out comprehensive experiments in three distinct text generation tasks: neural machine translation, abstractive text summarization, and dialogue generation. We witness substantial enhancements over robust baselines, attaining state-of-the-art outcomes in JRC-Acquis (four directions), XSum (50.3 ROUGE-1), and BigPatent (62.9 ROUGE-1). To gain deeper insights into the Selfmem, we meticulously investigate each crucial component and pinpoint the existing system bottleneck to guide future research endeavors. ", + "bbox": [ + 174, + 540, + 825, + 622 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 640, + 321, + 657 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Retrieval-augmented Text Generation ", + "text_level": 1, + "bbox": [ + 176, + 671, + 477, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Since the world is not a snapshot once the training corpus is collected, we can never expect an ever-large model to capture everything in its parameters, even for LLMs like GPT-4 [62]. Therefore, it is crucial to equip these models with an external memory bank to store additional knowledge or useful demonstration examples for solving various NLP tasks[41, 78, 95]. ", + "bbox": [ + 174, + 696, + 825, + 752 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the translation domain, retrieval techniques have long been employed by the localization industry to enhance human translators’ productivity and consistency even before the advent of machine translation [94]. Early works on machine translation primarily focused on utilizing memory for statistical machine translation (SMT) systems [80, 50]. For neural machine translation (NMT), [28] were the first to use search engines to retrieve memory from the training set and incorporate it with an external memory network. Subsequent research explored various aspects of retrievalaugmented NMT, such as memory encoding methods [92, 93, 31], joint training of retrievers and generators with monolingual data [8], memory granularity [35], and memory diversity [17]. For few-shot LLM generation, strategies for in-context example selection have been proposed to improve translation quality [2]. Furthermore, in-context machine translation has been shown to be effective for on-the-fly adaptation [79]. For dialogue response generation tasks, employing exemplar/template retrieval as an intermediate step has proven advantageous for generating informative responses [89, 91, 6, 7]. In-context learning example retrieval also aids in controllable dialogue [46]. Other applications include abstractive summarization [64, 14, 18, 15], code generation [30], paraphrase generation [34, 83], language modeling [36, 105], counterfactual data generation [24], open domain question answering [12, 33] and semantic parsing [99]. ", + "bbox": [ + 174, + 758, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 161 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Neural Text Reranking ", + "text_level": 1, + "bbox": [ + 176, + 178, + 374, + 193 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By alleviating the discrepancy between training and inference (i.e., exposure bias) and directly optimizing desired metrics, two-stage reranking methods have facilitated significant progress in various text generation tasks. In machine translation, pioneering works by [75] and [61] introduced and popularized discriminative reranking for SMT. In the context of NMT, research has focused on two primary reranking approaches: generative reranking [56, 32, 88] and discriminative reranking [39, 71, 23]. For syntactic parsing, [21] were the first to employ a two-stage reranking method to select outputs from a base parser, while [11] introduced a maximum entropy reranker. In text summarization, RefSum [53] proposed a second-stage summarization framework to address train-test distribution mismatches. SimCLS [54] used pairwise Learning To Rank (LTR) to select candidates with the highest matching scores. SummaReranker [68] adopted a multi-task mixture-of-experts framework to leverage different metrics capturing various aspects of generated candidates. BRIO [55] reused the base model for a second round of fine-tuning with both cross-entropy loss and a candidate-level ranking loss. JGR [76] employed an alternate training paradigm to train the generator and reranker. ", + "bbox": [ + 174, + 203, + 826, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A key limitation of these reranking methods is that they only represent a one-way process, wherein the selected candidates become the system’s final output. In contrast, our framework innovatively utilizes the chosen candidates as memory for the subsequent generation round of a retrieval-augmented generator, which can produce better candidates with enhanced memory. ", + "bbox": [ + 174, + 388, + 825, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methods ", + "text_level": 1, + "bbox": [ + 174, + 463, + 277, + 481 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we begin with a motivating experiment on generation as memory $( \\ S 3 . 1 )$ . Then, we introduce Selfmem, a framework comprising a retrieval-augmented generator $( \\ S 3 . 2 )$ and a memory selector $( \\ S \\ 3 . 3 )$ . The complete framework and algorithm are illustrated in Figure 2 and Algorithm 1. ", + "bbox": [ + 174, + 496, + 825, + 537 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Generation as Memory ", + "text_level": 1, + "bbox": [ + 174, + 554, + 372, + 569 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The primary motivation behind our framework stems from the observation that the memory, which is more similar in distribution to the data during inference, is not the training data (38.89 BLEU, as shown in the first row of Table 1). Instead, it is the model’s own output (58.58 BLEU) within the unbounded generation space. One interesting exploration involves directly utilizing the generated output as memory in relation to the primal problem: better memory prompts better generation. ", + "bbox": [ + 174, + 580, + 825, + 650 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We conduct experiments on the JRC-Acquis En De dataset. The first row in Table 1 represents conventional retrieval-augmented training with retrieved memory and achieves a 58.58 BLEU score. However, directly incorporating beam output of this trained model as memory (Beam) back into the generation model does not yield any improvements (row 2), despite its higher similarity to the reference compared to the retrieved ones. We hypothesize two potential reasons for this: (1) the retrieval-augmented generator may not generalize effectively in this context due to the ", + "bbox": [ + 174, + 656, + 470, + 834 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/d503dce85d49bb6c0672f9d66fe0afc70e358bde163575df3d6baaf9a3a35ad1.jpg", + "table_caption": [ + "Table 1: Experiments on the relation between memory quality and the final hypothesis quality, measured by the BLEU score with ground truth translation. The retrieval-augmented translator keeps fixed while the memory is obtained from different sources. " + ], + "table_footnote": [], + "table_body": "
Memory SourceMemory QualityHypothesis Quality
Retrieval38.8958.58
Beam58.5858.43
Reference10090.43
Random1.1449.08
", + "bbox": [ + 508, + 743, + 792, + 825 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "memory distribution shift (from 38.89 to 58.58), and (2) the beam memory does not offer any information gain compared to the retrieved one, even it exhibits more overlap with the references. ", + "bbox": [ + 174, + 835, + 826, + 863 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To investigate the first hypothesis, we conduct experiments under the oracle and random scenarios by using the reference as memory (Reference) and randomly sampled sentences as memory (Random). The result is shown in Table 1 and it illustrates that a retrieval-augmented generator (trained with retrieved memory) has already learned to discriminate between different memories in both oracle and random scenarios, without updating the model weights. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/520d5b6545a018750a97ef0612398660acd568140ee11c91fe8800a49ffe5488.jpg", + "image_caption": [ + "Figure 2: Overall framework. There are two components in Selfmem, a retrieval-augmented generator (a) and a memory selector (b). For the primal problem, (a) takes source and memory as input to generate candidates for (b). For the dual problem, (b) takes as input source and generated candidates to select memory for (a). " + ], + "image_footnote": [], + "bbox": [ + 187, + 88, + 807, + 275 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 369, + 823, + 398 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To evaluate the second conjecture, we first define the token sets of the reference, retrieved memory, and beam memory as $\\mathcal { R } , \\mathcal { M }$ , and $\\boldsymbol { B }$ , respectively. The overlap token set, denoted by $\\mathcal { O }$ , is defined as the tokens that overlap with the references in the beam memory but not in the retrieved memory, which is represented as $\\mathcal { R } \\cap \\mathcal { B } - \\mathcal { R } \\cap \\mathcal { M } .$ $\\mathcal { O }$ is considered as the additional information provided by the beam memory. Inspired by the confidence analysis of NMT model [58], we compute the set confidence score, $\\psi ( \\cdot )$ , as follows: ", + "bbox": [ + 173, + 404, + 826, + 488 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/57aa37e9d8b2d2f391adc5e012086f8792f0efb265fc88da4fc5d6f9df72b921.jpg", + "text": "$$\n\\psi ( \\cdot ) = { \\frac { 1 } { | \\cdot | } } \\sum _ { y ^ { i } \\in \\cdot } p ( y _ { i } | x , y _ { < i } )\n$$", + "text_format": "latex", + "bbox": [ + 405, + 508, + 593, + 549 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $p ( y _ { i } | x , y _ { < i } )$ is defined by the generation model. $\\psi ( \\cdot )$ measures the confidence with which the generation model generates the tokens. The value of $\\psi ( \\mathcal { R } )$ is 0.58, while that of $\\mathcal { O }$ is 0.76, indicating that the generator is relatively confident in generating tokens in $\\mathcal { O }$ , and therefore does not need to resort to external memory [38]. Beam search ranks generated candidates based on $p ( y | x )$ , where the selected memory falls within the confidence region of the generator and consequently provides no information gain. This observation motivates us to select memory according to metrics other than $p ( y | x )$ in the memory selector (§3.3). ", + "bbox": [ + 173, + 564, + 825, + 662 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Retrieval-augmented Generator ", + "text_level": 1, + "bbox": [ + 174, + 683, + 434, + 698 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given a text pair $( x , y )$ , where $x = \\{ \\mathbf { x } _ { 1 } , . . . , \\mathbf { x } _ { | x | } \\}$ is the source, $y = \\{ \\mathbf { y } _ { 1 } , . . . , \\mathbf { y } _ { | y | } \\}$ is the target. They could be (document, summary) in summarization, (context, response) in dialogue generation or (source, target) in machine translation. The retrieval-augmented generation would first use $x$ to retrieve memory $m$ from datastore $\\mathbb { D }$ . Then the generator $G _ { \\xi } ( x , m )$ , parameterized by $\\xi$ , would take both $x$ and $m$ as input to generate the target sentence $y$ . In this paper, following standard practice, we choose the training set as $\\mathbb { D } = \\{ ( x ^ { i } , y ^ { i } ) \\} _ { i = 1 } ^ { | \\mathbb { D } | }$ . For LLM as $G _ { \\xi }$ , we use the standard in-context learning format to give $( x , y )$ as demonstration example. For tunable generator $G _ { \\xi }$ , we only keep the target side of top- $\\mathbf { \\xi } _ { l }$ retrieval results as memory and we consider two commonly used architectures: Joint-Encoder [29, 87, 41] and Dual-Encoder [92, 8, 17]. ", + "bbox": [ + 173, + 708, + 826, + 838 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Joint-Encoder This architecture is the standard encoder-decoder-based model [3, 84]. The input is the concatenation of $x$ and $m$ . The encoder would first map the input into the hidden states $H$ : ", + "bbox": [ + 173, + 856, + 821, + 885 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/bbbb128d335022cde1b11db1e981f30872c0a75be9166b133999853e0e59a32b.jpg", + "text": "$$\nH = { \\mathrm { E n c o d e r } } ( x \\ [ { \\mathrm { S E P } } ] \\ m )\n$$", + "text_format": "latex", + "bbox": [ + 408, + 896, + 588, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "And the decoder would incorporate $H$ by attention mechanism and generate tokens in an autoregressive manner: ", + "bbox": [ + 171, + 90, + 825, + 119 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f8ab80d89d818f2705518908d2f3c5199c000000768259d461e2b35a22fa745f.jpg", + "text": "$$\nh ^ { i } = \\mathrm { D e c o d e r } ( \\mathrm { C r o s s A t t n } ( H ) , y _ { < i } ) \\quad P _ { G _ { \\xi } } ( \\cdot | x , y _ { < i } ) = \\mathrm { S o f t m a x } ( h ^ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 276, + 119, + 720, + 140 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dual-Encoder Instead of treating $x$ and $m$ as a long sequence, this architecture has two encoders, one for $x$ and the other for $m$ . Their outputs are sequentially attended by the decoder with dual cross attention as in [17]: ", + "bbox": [ + 173, + 148, + 826, + 190 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2c01f680fca335be9ec16faadf2cbb7b2b23239e42e0cf1e38cf4026a34611bd.jpg", + "text": "$$\n\\begin{array} { c } { H _ { x } = \\mathrm { S o u r c e E n c o d e r } ( x ) \\quad H _ { m } = \\mathrm { M e m o r y E n c o d e r } ( m ) } \\\\ { h ^ { i } = \\mathrm { D e c o d e r } ( \\mathrm { C r o s s A t t n } ( H _ { x } , H _ { m } ) , y _ { < i } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 191, + 684, + 231 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use Transformer [84] as the building block for both architectures and optimize $G _ { \\xi }$ with NLL loss: ", + "bbox": [ + 176, + 231, + 825, + 247 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3afaa4e4f7b8981988a849e20acb08b86ae921b046bc3ac8a004497625294890.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { n l l } } = - \\sum _ { t = 1 } ^ { | y | } \\log P _ { G _ { \\xi } } ( y _ { t } | x , m , y _ { < t } )\n$$", + "text_format": "latex", + "bbox": [ + 379, + 250, + 617, + 295 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 Memory Selector ", + "text_level": 1, + "bbox": [ + 174, + 308, + 333, + 321 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The role of memory selector $S _ { \\theta } ( x , c )$ , parameterized by $\\theta$ , is to select one candidate $c$ from the candidate pool $\\mathbb { C }$ generated by $G _ { \\xi }$ based on a specific metric $\\Delta ( \\cdot , \\cdot )$ . The chosen candidate $c$ is then utilized as memory $m$ for the subsequent generation round of $G _ { \\xi }$ . As discussed in $\\ S 3 . 1$ , using $p _ { G _ { \\xi } } ( y | x )$ as the metric $\\Delta ( \\cdot , \\cdot )$ would result in falling into the confidence region of $G _ { \\xi }$ , leading to no information gain. Moreover, a larger value of $p _ { G _ { \\xi } } ( y | x )$ does not necessarily guarantee improved generation quality [59]. Consequently, we define $\\Delta ( \\cdot , \\cdot )$ as model-free metrics that are widely employed for assessing generation quality, such as BLEU for Neural Machine Translation (NMT) and ROUGE for Summarization. Our memory selector takes the concatenation of the source $x$ and candidate $c _ { i }$ as input, and produces a multinomial distribution $p _ { S _ { \\theta } } ( \\cdot | x )$ over $\\mathbb { C }$ . ", + "bbox": [ + 173, + 332, + 825, + 462 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this paper, we focus on the role of the memory selector, $S _ { \\theta } ( x , c )$ , which is parameterized by $\\theta$ . The objective of this selector is to choose a single candidate $c$ from the candidate pool $\\mathbb { C }$ , generated by $G _ { \\xi }$ , based on a specific metric, $\\Delta ( \\cdot , \\cdot )$ . ", + "bbox": [ + 174, + 465, + 826, + 510 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0828132cd133c5aa2a02c80c4031d708b8be08791578f1f20ecc259bb04202eb.jpg", + "text": "$$\np _ { S _ { \\theta } } ( c _ { i } | x ) = \\frac { \\exp ( S _ { \\theta } ( x \\left[ \\mathrm { S E P } \\right] c _ { i } ) ) } { \\sum _ { j = 1 } ^ { | \\mathbb { C } | } \\exp ( S _ { \\theta } ( x \\left[ \\mathrm { S E P } \\right] c _ { j } ) ) }\n$$", + "text_format": "latex", + "bbox": [ + 362, + 527, + 633, + 569 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In accordance with [39], the training goal for $S _ { \\theta }$ is to minimize the discrepancy between the $S _ { \\theta }$ ’s predictions and the scores determined by $\\Delta ( \\cdot , \\cdot )$ . This divergence is quantified using the KullbackLeibler (KL) divergence. ", + "bbox": [ + 174, + 569, + 825, + 611 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8895388070a4aaf22388fc6bf5f1a47c80b01bec6b06f9f111861100d613c9e4.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { k l } } = - \\sum _ { i = 1 } ^ { | \\mathbb { C } | } p _ { M } ( c _ { i } ) \\mathrm { l o g } p _ { S _ { \\theta } } ( c _ { i } | x ) \\quad \\mathrm { w h e r e } \\quad p _ { M } ( c _ { i } ) = \\frac { \\exp ( \\Delta ( c _ { i } , y ) / \\tau ) } { \\sum _ { j = 1 } ^ { | \\mathbb { C } | } \\exp ( \\Delta ( c _ { j } , y ) / \\tau ) }\n$$", + "text_format": "latex", + "bbox": [ + 238, + 613, + 758, + 660 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$\\tau$ is the temperature to control the smoothness of the distribution. At inference, the output of the $S _ { \\theta }$ is a $\\operatorname { r g m a x } _ { c _ { i } \\in \\mathbb { C } } { \\dot { p } } _ { S _ { \\theta } } ( c _ { i } | x )$ . ", + "bbox": [ + 174, + 661, + 826, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 202, + 689, + 233, + 699 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 Combine Generator and Selector ", + "text_level": 1, + "bbox": [ + 174, + 713, + 442, + 729 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define two generation modes for $G _ { \\xi }$ . The first mode, referred to as the hypothesis mode, generates a single output for each input, which is utilized for system evaluation. The second mode, known as the candidate mode, produces $_ \\mathrm { N }$ outputs for a given input, and is employed for training $S _ { \\theta }$ as well as memory selection. By integrating two modes together, we present the complete framework of our proposed model, Selfmem, as illustrated in Algorithm 1. ", + "bbox": [ + 173, + 738, + 825, + 809 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 827, + 372, + 844 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 Dataset ", + "text_level": 1, + "bbox": [ + 174, + 857, + 266, + 872 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We assess the performance of Selfmem on three generation tasks, utilizing a total of seven datasets. Translation. We evaluate our framework on JRC-Acquis datasets [82], a collection of parallel ", + "bbox": [ + 174, + 882, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Require: a dataset $\\mathbb { D }$ , a retriever $R$ , a memory selection metric $\\Delta ( \\cdot , \\cdot )$ , a retrieval-augmented \ngenerator $G _ { \\xi }$ , and a memory selector $S _ { \\theta }$ \n1: retrieve memory $\\mathbb { M }$ in $\\mathbb { D }$ with $R$ \n2: train $G _ { \\xi }$ with $\\mathbb { D }$ and $\\mathbb { M }$ (if not LLM) \n3: use $G _ { \\xi }$ to generate candidate pool $\\mathbb { C }$ with $\\mathbb { M }$ in candidate mode \n4: train ${ \\check { S } } _ { \\theta }$ on $\\mathbb { C }$ with $\\Delta ( \\cdot , \\cdot )$ \n5: while not converged in the validation set do \n6: $S _ { \\theta }$ selects memory from $\\mathbb { C }$ as $\\mathbb { M }$ \n7: $G _ { \\xi }$ generates candidate pool $\\mathbb { C }$ with $\\mathbb { M }$ in candidate mode \n8: end while \n9: $G _ { \\xi }$ generates the final hypothesis with $\\mathbb { M }$ in hypothesis mode ", + "bbox": [ + 176, + 111, + 825, + 266 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "legislative text of European Union Law. It is the benchmark dataset used in translation memoryaugmented NMT task [28, 92, 8, 17]. We choose 4 translation directions, namely, Spanish English $( { \\mathrm { E s } } { \\mathrm { E n } } )$ , German English $( \\mathrm { D e } \\mathrm { E n } $ ). Summarization. We evaluate on 2 summarization datasets: 1) XSum [60], extreme summarization, a single-document summarization dataset with highly abstractive articles from British Broadcasting Corporation. 2) BigPatent [73], consisting of 1.3 million records of U.S. patent documents along with human-written abstractive summaries. Dialogue. We experiment on DailyDialog [44], which contains multi-turn dialogs on daily life topics and is used by [13, 4, 103]. The detailed statistics for these datasets can be found in the Appendix A. ", + "bbox": [ + 173, + 297, + 826, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Implementation Details ", + "text_level": 1, + "bbox": [ + 176, + 428, + 375, + 443 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We utilize the BM25 algorithm [70] for retrieval purposes. For all tasks, the candidate generation method consists of beam search with a beam width of 50. The number of iterations is determined by the performance on the validation set. For translation, we follow the approach of [93, 8, 17], employing a randomly initialized Transformerbase architecture as $G _ { \\xi }$ for trainable small model and XGLM [48] for LLM in-context learning. Evaluation metrics include BLEU, TER, and ${ \\mathrm { c h r F } } + +$ obtained from SACREBLEU[66]. The memory selector $S _ { \\theta }$ utilizes an XLM- ${ \\bf R } _ { b a s e }$ [22] as backbone, with BLEU serving as $\\Delta ( \\cdot , \\cdot )$ . For summarization, we initialize $G _ { \\xi }$ with $\\mathrm { B A R T _ { b a s e } } [ 4 0 ]$ for BigPatent and employ BRIO [55] for XSum. The evaluation metric comprises ROUGE (R1/2/L) [47]. For dialogue generation, $\\mathbf { B A R T _ { b a s e } }$ serves as the backbone for $G _ { \\xi }$ . Our dialogue system is evaluated using BLEU (B-1/2) and Distinct (D-1/2) scores [43]. For both dialogue and summarization tasks, we adhere to the methods of [54, 26], adopting $\\mathrm { R o B E R T a _ { b a s e } }$ [52] as the backbone for $S _ { \\theta }$ The linear combination of $_ { \\mathrm { B - } 1 / 2 }$ is chosen as $\\Delta ( \\cdot , \\cdot )$ for Dialogue Generation, while $\\mathrm { \\mathbf { R } } \\mathrm { - } 1 / 2 / \\mathrm { L }$ is used for Summarization, following [76]. For further implementation details, please refer to the Appendix B and Appendix C for evaluation metrics. ", + "bbox": [ + 174, + 455, + 825, + 648 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Experimental Results ", + "text_level": 1, + "bbox": [ + 174, + 670, + 385, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 Machine Translation ", + "text_level": 1, + "bbox": [ + 174, + 704, + 357, + 718 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We select four translation directions and experiment with two generation paradigms: trainable small models and few-shot prompted LLMs [85, 20]. For trainable models, we explore two architectures (joint and dual, as detailed in $\\ S 3 . 2 \\AA$ . The baselines comprise two types of translation systems: one being the vanilla sequence-to-sequence model [3, 84] without memory augmentation, and the other consisting of retrieval-augmented translation models focusing on memory encoding [28, 92], memory construction [101], memory retrieval [8], and memory diversity [17]. Based on the experimental results2 shown in Table 2, Selfmem significantly enhances the performance of $G _ { \\xi }$ across four translation datasets and two different architectures. This is noteworthy, given that the parameters of the $G _ { \\xi }$ remain fixed, with the only variable being the input memory. This finding is consistent with the primal problem which posits that improved memory typically leads to better generation results. ", + "bbox": [ + 173, + 731, + 825, + 869 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3d712394296b262aa12045fe7e2128e3b707e57b0c9d9837c330f2333d45f149.jpg", + "table_caption": [ + "Table 2: Results of translation task on JRC-Acquis measured by BLEU. Models denoted by the same symbol $\\times$ and $\\dagger .$ ) have the same parameters and only differ in memory as input. The bolded numbers show the SOTA performance and the underlined numbers show the second-best result. $^ *$ denotes the system is significantly better than baselines with $p$ -value $< 0 . 0 5$ tested by [37]. " + ], + "table_footnote": [], + "table_body": "
SystemEs-→EnEn→EsDe-→EnEn→De
DevTestDevTestDevTestDevTest
None Memory
RNNsearch [3]55.0259.3450.5450.4850.2049.7444.9443.98
Transformer [84]64.0864.6362.0261.8060.1860.1654.6555.43
Retrieval Memory
SEG-NMT[28]60.2859.3457.6257.2755.6355.3349.2648.80
NMT-pieces [101]63.9764.3061.5061.5660.1060.2655.5455.14
G-TFM [92]66.3766.2162.5062.7661.8561.7257.4356.88
MonoNMT[8]67.7367.4264.1863.8664.4864.6258.7758.42
CMM[17]67.4867.7663.8464.0464.2264.3358.9458.69
Transformerdual*66.8767.1263.1463.5464.0963.3658.6958.06
Transformerunit67.7467.3263.9364.1264.5064.4058.1658.58
Self-Memory
Transformerdual*68.63*69.20*64.12*64.67*65.06*64.98*59.26*59.49*
Transformerunit68.26*68.80*66.07*65.94*65.32*65.65*59.88*60.11*
", + "bbox": [ + 209, + 154, + 790, + 401 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/46e2cf5fcb73f8f8dc484eaa34a64aaf3d9a9f9a68b88f187f5250b7761c7768.jpg", + "table_caption": [ + "Table 3: Comparison between retrieval memory and self-memory. The quality of memory and hypothesis is measured by the n-gram overlap with reference (BLEU). All experiments are conducted with Transforme $\\mathbf { \\dot { j } } \\mathbf { o } \\mathbf { \\dot { i } } \\mathbf { n } \\mathbf { t }$ on JRC-Acquis. " + ], + "table_footnote": [], + "table_body": "
RetrievalSelf
memoryhypothesis memoryhypothesis
En-De→38.8958.5857.9260.11
↑42.5664.4064.3265.65
En-Es→40.6764.1263.5765.94
↑43.0567.3267.7868.80
", + "bbox": [ + 282, + 464, + 714, + 580 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The dual problem is revealed in Table 3. Self-memory, which essentially represents the model’s own output, exhibits greater similarity with the ground truth and serves as a more effective memory for generating the final output. This observation highlights a key distinction between Selfmem and previous reranking works [39, 68]. Reranking aims to select candidates of higher quality than the beam output, whereas in Selfmem, the chosen candidates serve as memory for the retrieval-augmented generator and do not necessarily need to surpass the quality of the beam hypotheses. ", + "bbox": [ + 173, + 604, + 826, + 689 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d808ef46dcc0836defbafe2e3a62520830ffd2c2ba611d69c175d340ffe7e8c7.jpg", + "table_caption": [ + "Table 4: Evaluation results of in-context learning with self-memory. " + ], + "table_footnote": [], + "table_body": "
XGLM-1.7BXGLM-4.5BXGLM-7.5B
RandomkNNSelfRandomkNNSelfRandomkNNSelf
En-De↑11.5137.8740.9417.5137.6038.2518.4847.8248.32
27.4251.0051.8830.6248.1248.3633.0355.6555.12
En-Es→23.8746.2048.5631.8348.3749.1729.9753.8654.32
↑25.2951.5553.1332.1648.5549.2235.2257.2557.56
", + "bbox": [ + 202, + 724, + 787, + 828 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Table 4, we present the results of LLM with self-memory. We employ XGLM [48] as our backbone generator, with three different sizes ranging from 1.7B to 7.5B. We utilize the recommended prompt as described in [48]. We select three in-context learning examples and report the average scores from three separate runs, taking into account the sensitivity of example selection in ICL [49]. From the table, we first observe a general trend where few-shot translation performance improves as the size of the model increases. Furthermore, we find that more similar translation demonstrations significantly enhance performance across all model sizes (from random, kNN to Self). This suggests that demonstration examples in in-context learning not only act as triggers for model ability but also adhere to the primal problem, where better demonstration example leads to better generation. Also, by comparing the results in Table 2 and Table 4, we can conclude that the cross-lingual LLM with designed examples still falls short of the supervised baselines in this task. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 174 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 Summarization ", + "text_level": 1, + "bbox": [ + 174, + 193, + 320, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we compare the performance of our trainable model with those of REINA [87], PEGASUS [100], and BART [40]. The results are presented in Table5. Initially, it can be observed that memory has varying impacts on different datasets. The enhancement brought by memory in the BigPatent dataset is significantly larger than that in the XSum dataset. This can be attributed to the inherent characteristics of the BigPatent dataset, which consists of official patent documents that exhibit considerable similarity. Consequently, this greatly improves the summarization quality in accordance with the primal problem. Furthermore, we discovered that self-memory substantially enhances the performance of both BRIO $( + 1 . 2 { \\ R } 1 )$ and BART $( + 1 8 . 5 \\mathrm { R } 1 ) $ ), achieving state-of-the-art results on both datasets. We selected these baselines for a fair comparison, as they share the same base generator. Due to space constraints, additional comparisons and the confidence region of the SOTA model can be found in the Appendix E. ", + "bbox": [ + 174, + 218, + 825, + 369 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/28beb855b83301867606a14ac5ba5bd02594f6919a21a865ac18ca0dd546bd30.jpg", + "table_caption": [ + "Table 5: Results of summarization task on XSum and BigPatent measured by ROUGE. " + ], + "table_footnote": [], + "table_body": "
SystemMemoryR-1R-2R-L
XSum
PEGASUSNone47.224.639.3
BRIONone49.125.640.4
REINA (PG)Retrieval48.226.040.2
REINA (B)Retrieval43.221.035.5
REINA (L)Retrieval46.524.138.6
BRIOdual*Retrieval48.626.140.6
BRIOjointRetrieval49.526.541.2
BRIOdual*Self49.226.240.8
BRIOjointtSelf50.326.741.6
", + "bbox": [ + 173, + 406, + 478, + 566 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/1a9ab0c711a95036eb32dab3e53d3034bb678bb5817535c88f02ad3a51937f60.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
SystemMemoryR-1R-2R-L
BigPatent
PEGASUSNone53.633.243.2
BARTNone44.421.331.0
REINA (B)Retrieval59.542.650.6
REINA (L)Retrieval60.743.351.3
REINA (PG)Retrieval44.621.533.3
BARTdual*Retrieval57.443.349.7
BARTjointtRetrieval59.643.451.0
BARTdual*Self61.244.652.3
BARTjointSelf62.948.159.6
", + "bbox": [ + 519, + 409, + 820, + 566 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 Dialogue Generation ", + "text_level": 1, + "bbox": [ + 174, + 594, + 356, + 609 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As demonstrated in Table 6, the self-memory significantly enhances the performance of the retrievalaugmented generator for dialogue generation tasks. By optimizing memory using BLEU as $\\Delta ( \\cdot , \\cdot )$ , the self-memory improves the B-1,2 score over retrieved memory by $3 . 0 8 \\ \\mathrm { B } \\cdot 1$ and $0 . 6 \\ \\mathbf { B } { - } 2$ on $\\mathbf { B A R T _ { j o i n t } }$ . Intriguingly, although Selfmem surpasses the baselines in terms of $_ { \\mathrm { B - } 1 / 2 }$ , it falls behind in D-1 and D-2, which can be attributed to the trade-off between BLEU score and Distinct score when evaluating a dialogue system [104]. To address this issue, we opt for D-1,2 as $\\Delta ( \\cdot , \\cdot )$ when optimizing $S _ { \\theta }$ , denoted as $\\mathbf { B A R T } _ { \\mathrm { j o i n t } } \\dagger ( \\mathbf { D } )$ . The results in Table 6 highlight the remarkable flexibility of Selfmem by directly optimizing memory to achieve the desired attributes for diverse and informative dialogue. ", + "bbox": [ + 173, + 619, + 826, + 731 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Further Analysis ", + "text_level": 1, + "bbox": [ + 174, + 752, + 348, + 768 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To gain a deeper insight into Selfmem, we first examine the impact of each key component, namely $G _ { \\xi }$ and $S _ { \\theta }$ . Subsequently, we perform a detailed token-level analysis of the generated output concerning their frequency in the training set. Experiments are conducted on the JRC-Acquis En De dataset. We also include latency analysis and human evaluation on Appendix F and G. ", + "bbox": [ + 174, + 784, + 825, + 839 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Tuning $S _ { \\theta }$ We explored various $S _ { \\theta }$ by direct selection from the candidate pool based on gold rankings. As shown in Figure 3a, both architectures with enhanced $S _ { \\theta }$ significantly outperform the current SOTA performance (60.11 BLEU). Moreover, we assessed the candidate pool quality during this iterative process using an oracle $S _ { \\theta }$ , as displayed in Figure 3b. A clear pattern emerges in this boxplot, revealing improvements in the oracle, quartile, average, and minimum scores of the candidate pool. These two experiments jointly clarify the Selfmem’s underlying intuition: a retrieval-augmented generator profits from superior memory, which can be chosen from its own unbounded output, and subsequently, the generator with improved memory produces a higher-quality candidate pool for the next selection round. Consequently, the model lift itself up. ", + "bbox": [ + 176, + 856, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a55c042308d464e33049493e5f7e534336696a4bd588792365f841aac839bc10.jpg", + "table_caption": [ + "Table 6: Results of dialogue generation task on DailyDialog measured by B-1/2 and D-1/2. $\\mathbf { B A R T _ { j o i n t } }$ (D) denotes the metric $\\Delta ( \\cdot , \\cdot )$ for $S _ { \\theta }$ is the average of D-1 and D-2. " + ], + "table_footnote": [], + "table_body": "
SystemMemoryB-1B-2D-1D-2
NCM [86]None33.6026.803.0012.80
iVAE [25]None30.9024.902.9025.00
PLATO-2 [5]None34.8025.123.5425.11
DialoFlow [45]None36.1727.674.5627.12
BARTNone20.7211.363.9219.44
BARTdual*Retrieval29.5021.894.7426.01
BARTjointtRetrieval36.7231.556.1335.65
BARTdual*Self33.4322.854.6626.16
BARTjointSelf39.8032.155.8432.16
BARTjoint † (D)Self36.9232.099.1237.05
", + "bbox": [ + 287, + 125, + 709, + 306 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/e2ff140cd3a0a3d28c8ec3e9f453a47d6086af9a3faa5dab85f106307077b962.jpg", + "image_caption": [ + "Figure 3: (a) shows generation quality in the iteration process with different $S _ { \\theta }$ in both trainable generator architectures. (b) shows candidates quality in the iteration process with an oracle $S _ { \\theta }$ . " + ], + "image_footnote": [], + "bbox": [ + 191, + 335, + 803, + 488 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 568, + 825, + 637 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tuning $G _ { \\xi }$ As discussed in $\\ S 3 . 1$ , we demonstrated that a trained retrieval-augmented generator, with fixed parameters, possesses the ability to distinguish between \"good\" and \"bad\" memory. This observation not only justifies our decision to maintain a fixed generator within our framework but also implies that the $G _ { \\xi }$ is not the current bottleneck of the Selfmem. ", + "bbox": [ + 174, + 670, + 578, + 753 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/b973e5b8f1f94a7b457be97ac2c4d228ea2a8787fa0754505c1667991affb841.jpg", + "image_caption": [ + "Figure 4: 1-gram F1 score sorted by training corpus frequency. " + ], + "image_footnote": [], + "bbox": [ + 591, + 674, + 820, + 796 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Frequency Analysis We conduct a comprehensive tokenlevel analysis by computing the 1-gram F1 scores for generated translations and subsequently categorizing the tokens based on their frequency in the training set. The results are depicted in Figure 4. A noticeable pattern emerges, suggesting that the more frequently a model encounters a token during training, the higher the accuracy of the generated output [102]. Moreover, our findings indicate that retrievalaugmented models, particularly those incorporating self-memory augmentation, exhibit superio performance in handling long-tail inputs which are challenges for parametric models [67, 57]. ", + "bbox": [ + 174, + 786, + 580, + 869 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 871, + 818, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 89, + 299, + 106 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For the first time, we investigate the fundamental limitation of bounded memory in the current retrieval-augmented literature. We combine the primal and dual problems together and propose Selfmem, a general framework for retrieval-augmented text generation by uplifting generation model with its own output. We conduct comprehensive experiments across various text generation tasks and different generation paradigms, including trainable small model and few-shot prompted LLM. We surpass strong baselines and improve the state-of-the-art performance in serval datasets. We also meticulously investigate each crucial component and pinpoint the existing system bottleneck to guide future research endeavors. ", + "bbox": [ + 174, + 121, + 825, + 232 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations ", + "text_level": 1, + "bbox": [ + 174, + 252, + 272, + 270 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We discuss the limitations of our framework as follows: ", + "bbox": [ + 174, + 285, + 539, + 299 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(1) Although Selfmem greatly improves the generation quality compared with other retrievalaugmented generation models, it requires more computational resources with respect to the memory selection process. For large dataset with long context (e.g., BigPatent), it would become a more crucial problem considering the quadratic time complexity of transformer architecture. ", + "bbox": [ + 174, + 305, + 825, + 361 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(2) This paper proposes a general idea for the retrieval-augmented generation. But we only experiment with transformer-based architecture for both generator and memory selector and the architecture of generator and memory selector keeps the same across all text generation tasks. We believe the task-specific design for the model architecture, training objective and generation methods in different text generation scenarios would further improve the performance. ", + "bbox": [ + 174, + 367, + 825, + 436 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 457, + 330, + 474 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported by the National Key Research and Development Program of China (No.2021YFC3340304) and National Natural Science Foundation of China (NSFC Grant No.62122089). We appreciate the anonymous reviewers for their helpful comments. Dongyan Zhao and Rui Yan are the corresponding authors. 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TaskDataset#Train#Dev#Test
TranslationJRC (en ←→ de)663,4872,4542,483
JRC (en ←→ es)653,1272,5332,596
SummarizationBigPatent1,207,22267,06867,072
XSum204,04511,33211,334
DialogueDailyDialog87,1708,0697,740
", + "bbox": [ + 267, + 155, + 725, + 270 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B Self Memory Details ", + "text_level": 1, + "bbox": [ + 174, + 309, + 379, + 327 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For machine translation tasks, following [93, 8, 17] we use randomly initialize Transformerbase architecture [84] as $G _ { \\xi }$ . We use the joint-bpe algorithm [72] and share the parameters between the memory encoder and source encoder for dual encoder architecture. The hyper-parameter setting follows [17] with dropout 0.1, label smoothing 0.1, gradient clipping 1.0, Adafactor [74], warm-up steps 4000, maximum learning rate $4 . 4 \\mathrm { e } { - 2 }$ and training epochs 30 for total. The evaluation metrics are BLEU, TER and ${ \\mathrm { c h r F } } + +$ from SACREBLEU [66]. The backbone of memory selector $S _ { \\theta }$ is XLM- $. { \\bf R } _ { b a s e }$ [22] with BLEU as $\\Delta ( \\cdot , \\cdot )$ . The hyper-parameter setting for $S _ { \\theta }$ follows [39] with $\\tau 0 . 5$ , minmax normalization for candidates ranking, Adam optimizer with max learning rate 5e-5 and polynomial decay scheduler, and classifier dropout 0.2. ", + "bbox": [ + 173, + 343, + 825, + 469 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For Summarization, we init the $G _ { \\xi }$ with $\\mathbf { B A R T _ { b a s e } }$ [40] for BigPatent following [87] and state-of-theart BRIO [55] for XSum. Optimization is based on Adafactor with a maximum learning rate of 5e-3, warm-up steps 10000 and gradient clipping value 1.0. The maximum input length is 512 for XSum and 1024 for BigPatent. The evaluation metric is Rouge (R-1/2/L) [47]. ", + "bbox": [ + 174, + 474, + 825, + 531 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "For Dialogue Generation, we use $\\mathbf { B A R T _ { b a s e } }$ as the backbone for $G _ { \\xi }$ on DailyDialog. We tune the hyper-parameters from learning rate $\\{ 5 \\mathrm { e } { - } 3 , 1 \\mathrm { e } { - } 3 , 4 \\mathrm { e } { - } 4 \\}$ and set dropout 0.1, batch size 64, label smoothing factor 0.1, maximum input length 120 for DailyDialog. Following [4, 13], we evaluate our dialogue system with BLEU (B-1/2) and Distinct (D-1,2) [43]. For both Summarization and Dialogue Generation task, we follow [54, 26] and adopt $\\mathrm { R o B E R T a _ { b a s e } }$ [52] as the backbone for $S _ { \\theta }$ . We choose the linear combination of B-1/2 as $\\Delta ( \\cdot , \\cdot )$ for Dialogue Generation and R-1/2/L for Summarization following [76]. We tune the hyper-parameters $\\tau$ from $\\{ 0 . 0 8 , 0 . 2 , 0 . 5 , 0 . 8 \\}$ , learning rate from {5e-5,7e-5,2e-4}. The maximum input length for $S _ { \\theta }$ is 512 and we truncate tokens from the longer input of source and candidate. ", + "bbox": [ + 173, + 537, + 825, + 662 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C Evaluation Details ", + "text_level": 1, + "bbox": [ + 176, + 686, + 362, + 704 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Machine Translation We evaluate our MT system with BLEU, TER and ${ \\mathrm { c h r F } } + +$ from SACREBLEU3 [66]. The signatures for BLEU, TER and ${ \\mathrm { c h r F } } + +$ are shown in Table 8. ", + "bbox": [ + 174, + 722, + 823, + 751 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/991c4e62d96ef968486f77a266c094cf6834d5929faaa717b1946b20d425dde0.jpg", + "table_caption": [ + "Table 8: Signature from SACREBLEU. " + ], + "table_footnote": [], + "table_body": "
[c]Signature
nrefs:1lcase:mixedleff:noltok:13alsmooth:explversion:2.0.0
nrefs:1lcase:lcltok:tercomlnorm:nolpunct:yeslasian:nolversion:2.0.0
nrefs:1lcase:mixedleff:yeslnc:6lnw:2lspace:nolversion:2.0.0
", + "bbox": [ + 266, + 791, + 727, + 864 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Summarization We evaluate our Summarization system with standard ROUGE [47] Perl package4 for evaluation. Following [55], we use PTB tokenizer5 for tokenization. And the parameters for ROUGE are \" $\\mathsf { \\Pi } _ { - \\mathrm { c } } ^ { \\prime } 9 5 \\mathsf { \\Pi } _ { - \\mathrm { r } } 1 0 0 0 \\mathsf { \\Pi } _ { - \\mathrm { n } } 2 \\mathsf { \\Pi } _ { - \\mathrm { m } } \\mathsf { \" }$ . ", + "bbox": [ + 174, + 90, + 823, + 132 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Dialogue Generation Following [27], we evaluate our dialogue system with NLTK BLEU 6 with space as tokenizer and smoothing method1. The Distinction score is from [42]. ", + "bbox": [ + 174, + 147, + 823, + 176 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D More results on translation tasks ", + "text_level": 1, + "bbox": [ + 173, + 195, + 485, + 213 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/83483d135ef2d23de94eeaee50fad4cc4cb1fbc87d3212ee3d9060838a04eb37.jpg", + "table_caption": [ + "Table 9: Evaluation results on JRC-Acquis En De measured by BLEU, TER and ${ \\mathrm { c h r F } } + +$ " + ], + "table_footnote": [], + "table_body": "
SystemMemoryBLEU 个chrF++ 个TER
TransformerNone55.4370.3136.35
TransformerdualRetrieval58.0671.5835.41
TransformerjointRetrieval58.5872.2234.39
TransformerdualSelf59.4972.6234.04
TransformerjointSelf60.1173.2532.62
", + "bbox": [ + 285, + 252, + 712, + 359 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "E More Summarization Baselines ", + "text_level": 1, + "bbox": [ + 174, + 388, + 470, + 406 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this Table 10, we include more baselines on the benchmark dataset XSum and BigPatent. We also report the confidence region of SOTA model for XSum and BigPatent as shown in Table 11. ", + "bbox": [ + 173, + 420, + 826, + 449 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/6a84a4498b669152dcbeb6e25a41a3f10e922a3e577e668a2c80b5a474d20597.jpg", + "table_caption": [ + "Table 10: More baselines on XSum and BigPatent. " + ], + "table_footnote": [], + "table_body": "
SystemR-1R-2R-L
XSum
[51]38.816.531.3 37.3
[40]45.122.339.3
[100]47.224.639.4
[54] [55]47.6 49.124.640.4
[87](PG)48.225.640.2
[87](B)43.126.035.5
21.038.6
[87](L)46.524.140.0
[68]48.125.038.8
[69]47.124.1
[16]47.825.039.7
Selfmem50.326.741.6
", + "bbox": [ + 210, + 484, + 441, + 707 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/4e3ae66ddbdc3402231544a0817f983ef023023e1e873b71652326e981001ae2.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
SystemR-1 R-2R-L
BigPatent
[100]53.6 33.142.3
[40] 44.421.331.0
[98] 60.642.550.0
[65]38.7 12.334.1
[90] 45.020.339.2
[1] 52.333.542.8
[87] (B) 59.542.650.6
[87] (L) 60.743.351.3
[87] (PG) 44.621.533.3
Selfmem62.9 48.159.6
", + "bbox": [ + 560, + 497, + 792, + 694 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "F Empirical analysis of latency ", + "text_level": 1, + "bbox": [ + 174, + 734, + 447, + 752 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In Table 12, we present empirical results of Selfmem latency, measured in seconds. We compare Selfmem with a retrieval-augmented baseline model across various datasets and computational platforms, including CPU and CUDA. The number of iterations for Selfmem is set to one. All experiments are conducted on the same device, equipped with one NVIDIA A100 GPU and one AMD EPYC 7V13 64-Core Processor. ", + "bbox": [ + 173, + 765, + 825, + 835 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/b590bdd913e81c6866a1160a66948e81efa9b02187d318e52ccf203ff5ad6572.jpg", + "table_caption": [ + "Table 11: Confidence region for SOTA model in XSum and BigPatent. " + ], + "table_footnote": [], + "table_body": "
SystemROUGE-1/2/L95 % -conf.int
XSum
BRIOjoint50.30.49986 - 0.50602
26.70.26300 - 0.26989
41.60.41231 - 0.41900
BigPatent
BARTjoint62.90.62664 - 0.63080
48.10.47783 - 0.48333
59.60.59401 - 0.59847
", + "bbox": [ + 325, + 199, + 673, + 359 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/09ed51245fa5f48106d84c944f28788f086ae897a6c425aebe2c49a2f3d87319.jpg", + "table_caption": [ + "Table 12: Generation Latency analysis. " + ], + "table_footnote": [], + "table_body": "
NMT XSumBigPatentDailyDialog
Average Input Length87512102471
Average ：Output Length447512716
Retrieval-augmented BaselineCPU 0.971.793.160.32
SelfmemCandidate Generation Memory3.207.5015.001.02
Selection0.500.520.950.14
Hypothesis Generation0.971.793.000.32
×4.80×5.47×6.04×4.63
CUDA
Retrieval-augmented Baseline0.290.440.750.10
SelfmemCandidate Generation Memory Hypothesis Generation0.511.001.720.18
Selection0.010.010.010.01
0.29 ×2.760.44 ×2.990.75 ×3.350.10 ×2.91
", + "bbox": [ + 225, + 570, + 772, + 820 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G Human and GPT-4 Evaluation ", + "text_level": 1, + "bbox": [ + 174, + 88, + 465, + 106 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We employ both human annotators and GPT-4 (gpt-4-0314) annotators to perform pairwise ranking of the output generated by Selfmem and baseline systems. For GPT-4 annotators, we utilize the prompt from Alpaca Eval 7. We randomly select 50 samples for translation tasks and 20 samples for summarization and dialogue tasks. The win rate of Selfmem versus retrieval-augmented baselines is depicted in Figure 1. ", + "bbox": [ + 173, + 119, + 826, + 190 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/50ca523e51f29d99aad67a44b5b7d68d7a0aed4f3d5c8ed8c80cd389713d368f.jpg", + "image_caption": [ + "Figure 5: Human and GPT-4 evaluation results. " + ], + "image_footnote": [], + "bbox": [ + 258, + 208, + 733, + 361 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/dev/lYNSvp51a7/lYNSvp51a7_middle.json b/parse/dev/lYNSvp51a7/lYNSvp51a7_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..50f6cb9338a70c75469e5064178eec32c93ca3a1 --- /dev/null +++ b/parse/dev/lYNSvp51a7/lYNSvp51a7_middle.json @@ -0,0 +1,47984 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 144, + 97, + 468, + 137 + ], + "lines": [ + { + "bbox": [ + 141, + 96, + 469, + 119 + ], + "spans": [ + { + "bbox": [ + 141, + 96, + 469, + 119 + ], + "score": 1.0, + "content": "Lift Yourself Up: Retrieval-augmented Text", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 194, + 116, + 417, + 139 + ], + "spans": [ + { + "bbox": [ + 194, + 116, + 417, + 139 + ], + "score": 1.0, + "content": "Generation with Self-Memory", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 121, + 177, + 488, + 191 + ], + "lines": [ + { + "bbox": [ + 119, + 176, + 491, + 194 + ], + "spans": [ + { + "bbox": [ + 119, + 176, + 491, + 194 + ], + "score": 1.0, + "content": "Xin Cheng1 Di Luo2 Xiuying Chen3 Lemao Liu4 Dongyan Zhao1 Rui Yan2", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 202, + 200, + 409, + 235 + ], + "lines": [ + { + "bbox": [ + 201, + 199, + 410, + 214 + ], + "spans": [ + { + "bbox": [ + 201, + 199, + 410, + 214 + ], + "score": 1.0, + "content": "1 Peking University 2 Remin University of China", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 242, + 210, + 368, + 225 + ], + "spans": [ + { + "bbox": [ + 242, + 210, + 368, + 225 + ], + "score": 1.0, + "content": "3 KAUST 4 Tencent AI Lab", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 242, + 223, + 369, + 237 + ], + "spans": [ + { + "bbox": [ + 242, + 223, + 369, + 237 + ], + "score": 1.0, + "content": "chengxin1998@stu.pku.edu.cn", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 283, + 263, + 328, + 276 + ], + "lines": [ + { + "bbox": [ + 282, + 263, + 330, + 277 + ], + "spans": [ + { + "bbox": [ + 282, + 263, + 330, + 277 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 142, + 290, + 469, + 519 + ], + "lines": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 302 + ], + "score": 1.0, + "content": "With direct access to human-written reference as memory, retrieval-augmented", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 300, + 471, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 471, + 313 + ], + "score": 1.0, + "content": "generation has achieved much progress in a wide range of text generation tasks.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 311, + 470, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 470, + 323 + ], + "score": 1.0, + "content": "Since better memory would typically prompt better generation (we define this as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 335 + ], + "score": 1.0, + "content": "primal problem). The traditional approach for memory retrieval involves selecting", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "memory that exhibits the highest similarity to the input. However, this method", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 470, + 356 + ], + "score": 1.0, + "content": "is constrained by the quality of the fixed corpus from which memory is retrieved.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 367 + ], + "score": 1.0, + "content": "In this paper, by exploring the duality of the primal problem: better generation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "also prompts better memory, we propose a novel framework, Selfmem, which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 377, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 377, + 469, + 389 + ], + "score": 1.0, + "content": "addresses this limitation by iteratively employing a retrieval-augmented generator", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 388, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 399 + ], + "score": 1.0, + "content": "to create an unbounded memory pool and using a memory selector to choose one", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 399, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 469, + 411 + ], + "score": 1.0, + "content": "output as memory for the subsequent generation round. 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This innovative generation paradigm initially equips a fine-tuned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "small model or a large language model (LLM) with access to an external database (typically the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "training corpus) using information retrieval techniques. 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The traditional approach for memory retrieval involves selecting", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 345 + ], + "score": 1.0, + "content": "memory that exhibits the highest similarity to the input. 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This enables the model", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 410, + 470, + 422 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 470, + 422 + ], + "score": 1.0, + "content": "to leverage its own output, referred to as self-memory, for improved generation.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 420, + 470, + 432 + ], + "spans": [ + { + "bbox": [ + 141, + 420, + 470, + 432 + ], + "score": 1.0, + "content": "We evaluate the effectiveness of Selfmem on three distinct text generation tasks:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 432, + 470, + 444 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 470, + 444 + ], + "score": 1.0, + "content": "neural machine translation, abstractive text summarization, and dialogue generation,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 443, + 471, + 454 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 471, + 454 + ], + "score": 1.0, + "content": "under two generation paradigms: fine-tuned small model and few-shot LLM.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "spans": [ + { + "bbox": [ + 141, + 453, + 469, + 465 + ], + "score": 1.0, + "content": "Our approach achieves state-of-the-art results in four directions in JRC-Acquis", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 463, + 471, + 477 + ], + "spans": [ + { + "bbox": [ + 141, + 463, + 471, + 477 + ], + "score": 1.0, + "content": "translation dataset, 50.3 ROUGE-1 in XSum, and 62.9 ROUGE-1 in BigPatent,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 475, + 470, + 487 + ], + "spans": [ + { + "bbox": [ + 142, + 475, + 470, + 487 + ], + "score": 1.0, + "content": "demonstrating the potential of self-memory in enhancing retrieval-augmented", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 469, + 498 + ], + "score": 1.0, + "content": "generation models. 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As demonstrated in Figure 1, for a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "retrieval-augmented translation model, the memory similarity alone exhibits a strong correlation with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 678, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 695 + ], + "score": 1.0, + "content": "the final translation quality, regardless of other factors that may influence translation quality (e.g.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "polysemy, morphology, and coreference). 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Consequently, numerous studies have focused on how to retrieve better", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 96, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 96, + 505, + 106 + ], + "score": 1.0, + "content": "memory, ranging from sparse retrieval to dense retrieval [10, 63], from a fixed retriever to a learnable", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "retriever [41, 8], and from sentence-level memory to more fine-grained token-level memory [36, 35].", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 324, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 325, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 325, + 134 + ], + "score": 1.0, + "content": "However, a fundamental limitation exists in all previous", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "score": 1.0, + "content": "works: the memory is retrieved from a fixed corpus", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 325, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 325, + 156 + ], + "score": 1.0, + "content": "and is constrained by the corpus’s quality. Due to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 325, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 325, + 167 + ], + "score": 1.0, + "content": "finite retrieval space, bounded memory significantly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 325, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 325, + 178 + ], + "score": 1.0, + "content": "restricts the potential of memory-augmented generation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 325, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 325, + 189 + ], + "score": 1.0, + "content": "models [97]. 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The key", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 286, + 325, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 325, + 298 + ], + "score": 1.0, + "content": "insight behind Selfmem is that the text more closely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 296, + 325, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 325, + 308 + ], + "score": 1.0, + "content": "resembling the data distribution during inference is not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 309, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 309, + 320 + ], + "score": 1.0, + "content": "the training data [87], but the model’s own output.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 12.5 + }, + { + "type": "image", + "bbox": [ + 340, + 137, + 495, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 340, + 137, + 495, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 340, + 137, + 495, + 233 + ], + "spans": [ + { + "bbox": [ + 340, + 137, + 495, + 233 + ], + "score": 0.969, + "type": "image", + "image_path": "d932460f60720219128e0be0fd5297215e94face0e4c47c4f2c984e573d265a1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 340, + 137, + 495, + 149.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 340, + 149.0, + 495, + 161.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 340, + 161.0, + 495, + 173.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 340, + 173.0, + 495, + 185.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 340, + 185.0, + 495, + 197.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 340, + 197.0, + 495, + 209.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 340, + 209.0, + 495, + 221.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 340, + 221.0, + 495, + 233.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 333, + 241, + 505, + 317 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 331, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 331, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "Figure 1: Relation between memory and hy-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 331, + 253, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 331, + 253, + 446, + 263 + ], + "score": 1.0, + "content": "pothesis on JRC-Acquis E", + "type": "text" + }, + { + "bbox": [ + 446, + 253, + 472, + 262 + ], + "score": 0.27, + "content": "_ { 1 \\mathrm { D e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 253, + 506, + 263 + ], + "score": 1.0, + "content": "dataset.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 331, + 262, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 331, + 262, + 507, + 275 + ], + "score": 1.0, + "content": "The hypothesis is generated by a retrieval-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 331, + 274, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 331, + 274, + 506, + 286 + ], + "score": 1.0, + "content": "augmented translator whose memory is re-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 331, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 331, + 285, + 475, + 297 + ], + "score": 1.0, + "content": "trieved from the training set. The", + "type": "text" + }, + { + "bbox": [ + 475, + 285, + 484, + 295 + ], + "score": 0.34, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 330, + 294, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 330, + 294, + 505, + 309 + ], + "score": 1.0, + "content": "represents the similarity between memory", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 332, + 306, + 406, + 318 + ], + "spans": [ + { + "bbox": [ + 332, + 306, + 406, + 318 + ], + "score": 1.0, + "content": "and the reference.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + } + ], + "index": 29.25 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 326, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 327, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 327, + 338 + ], + "score": 1.0, + "content": "Selfmem consists of two complementary components:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "a retrieval-augmented generator and a memory selector. The generator operates under two distinct", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "paradigms: fine-tuning a small model or few-shot prompting an LLM. For the former, we train the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "generator with labeled data and retrieved memory, while for the latter, we employ a fixed black-box", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "LLM exclusively for inference alongside retrieved in-context learning samples. We then use the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "generator’s output to train a memory selector based on a specific performance metric. By simply", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "replacing the retrieved memory with unbounded generated memory, we achieve higher-quality", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "score": 1.0, + "content": "generation output (primal problem), which subsequently serves as memory for the next round after", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 411, + 321, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 321, + 423 + ], + "score": 1.0, + "content": "being refined by the memory selector (dual problem).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "To evaluate the efficacy of the Selfmem, we carry out comprehensive experiments in three distinct text", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 438, + 507, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 507, + 451 + ], + "score": 1.0, + "content": "generation tasks: neural machine translation, abstractive text summarization, and dialogue generation.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "We witness substantial enhancements over robust baselines, attaining state-of-the-art outcomes in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "JRC-Acquis (four directions), XSum (50.3 ROUGE-1), and BigPatent (62.9 ROUGE-1). To gain", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "deeper insights into the Selfmem, we meticulously investigate each crucial component and pinpoint", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 482, + 371, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 371, + 495 + ], + "score": 1.0, + "content": "the existing system bottleneck to guide future research endeavors.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5 + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 197, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 198, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 198, + 523 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 292, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 293, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 293, + 546 + ], + "score": 1.0, + "content": "2.1 Retrieval-augmented Text Generation", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "Since the world is not a snapshot once the training corpus is collected, we can never expect an", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 564, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 507, + 576 + ], + "score": 1.0, + "content": "ever-large model to capture everything in its parameters, even for LLMs like GPT-4 [62]. Therefore,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "it is crucial to equip these models with an external memory bank to store additional knowledge or", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 585, + 403, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 403, + 597 + ], + "score": 1.0, + "content": "useful demonstration examples for solving various NLP tasks[41, 78, 95].", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 55.5 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "In the translation domain, retrieval techniques have long been employed by the localization industry", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "to enhance human translators’ productivity and consistency even before the advent of machine", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "translation [94]. Early works on machine translation primarily focused on utilizing memory for", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "statistical machine translation (SMT) systems [80, 50]. For neural machine translation (NMT),", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "[28] were the first to use search engines to retrieve memory from the training set and incorporate", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "it with an external memory network. Subsequent research explored various aspects of retrieval-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 668, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 680 + ], + "score": 1.0, + "content": "augmented NMT, such as memory encoding methods [92, 93, 31], joint training of retrievers and", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "generators with monolingual data [8], memory granularity [35], and memory diversity [17]. For", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "few-shot LLM generation, strategies for in-context example selection have been proposed to improve", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "translation quality [2]. Furthermore, in-context machine translation has been shown to be effective", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "for on-the-fly adaptation [79]. For dialogue response generation tasks, employing exemplar/template", + "type": "text" + } + ], + "index": 68 + } + ], + "index": 63 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 73, + 505, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 72, + 507, + 118 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 122, + 324, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 122, + 325, + 134 + ], + "spans": [ + { + "bbox": [ + 106, + 122, + 325, + 134 + ], + "score": 1.0, + "content": "However, a fundamental limitation exists in all previous", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 325, + 145 + ], + "score": 1.0, + "content": "works: the memory is retrieved from a fixed corpus", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 325, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 325, + 156 + ], + "score": 1.0, + "content": "and is constrained by the corpus’s quality. Due to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 325, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 325, + 167 + ], + "score": 1.0, + "content": "finite retrieval space, bounded memory significantly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 325, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 325, + 178 + ], + "score": 1.0, + "content": "restricts the potential of memory-augmented generation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 177, + 325, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 325, + 189 + ], + "score": 1.0, + "content": "models [97]. In this paper, we explore the duality of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 188, + 325, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 325, + 200 + ], + "score": 1.0, + "content": "primal problem, which posits that better generation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 198, + 325, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 325, + 210 + ], + "score": 1.0, + "content": "also prompts better memory. We propose a novel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 209, + 325, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 325, + 221 + ], + "score": 1.0, + "content": "framework called Selfmem, which iteratively employs", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 325, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 325, + 232 + ], + "score": 1.0, + "content": "a retrieval-augmented generator to create an unbounded", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 232, + 325, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 325, + 243 + ], + "score": 1.0, + "content": "memory pool and uses a memory selector to choose one", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 243, + 326, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 326, + 253 + ], + "score": 1.0, + "content": "output as memory for the subsequent generation round.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 326, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 326, + 265 + ], + "score": 1.0, + "content": "By combining the primal and dual problem, a retrieval-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 264, + 326, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 326, + 276 + ], + "score": 1.0, + "content": "augmented generation model can elevate itself using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 274, + 325, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 325, + 287 + ], + "score": 1.0, + "content": "its own output, referred to as self-memory. The key", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 286, + 325, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 325, + 298 + ], + "score": 1.0, + "content": "insight behind Selfmem is that the text more closely", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 296, + 325, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 325, + 308 + ], + "score": 1.0, + "content": "resembling the data distribution during inference is not", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 307, + 309, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 309, + 320 + ], + "score": 1.0, + "content": "the training data [87], but the model’s own output.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 122, + 326, + 320 + ] + }, + { + "type": "image", + "bbox": [ + 340, + 137, + 495, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 340, + 137, + 495, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 340, + 137, + 495, + 233 + ], + "spans": [ + { + "bbox": [ + 340, + 137, + 495, + 233 + ], + "score": 0.969, + "type": "image", + "image_path": "d932460f60720219128e0be0fd5297215e94face0e4c47c4f2c984e573d265a1.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 340, + 137, + 495, + 149.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 340, + 149.0, + 495, + 161.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 340, + 161.0, + 495, + 173.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 340, + 173.0, + 495, + 185.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 340, + 185.0, + 495, + 197.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 340, + 197.0, + 495, + 209.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 340, + 209.0, + 495, + 221.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 340, + 221.0, + 495, + 233.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 333, + 241, + 505, + 317 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 331, + 241, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 331, + 241, + 506, + 253 + ], + "score": 1.0, + "content": "Figure 1: Relation between memory and hy-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 331, + 253, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 331, + 253, + 446, + 263 + ], + "score": 1.0, + "content": "pothesis on JRC-Acquis E", + "type": "text" + }, + { + "bbox": [ + 446, + 253, + 472, + 262 + ], + "score": 0.27, + "content": "_ { 1 \\mathrm { D e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 253, + 506, + 263 + ], + "score": 1.0, + "content": "dataset.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 331, + 262, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 331, + 262, + 507, + 275 + ], + "score": 1.0, + "content": "The hypothesis is generated by a retrieval-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 331, + 274, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 331, + 274, + 506, + 286 + ], + "score": 1.0, + "content": "augmented translator whose memory is re-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 331, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 331, + 285, + 475, + 297 + ], + "score": 1.0, + "content": "trieved from the training set. The", + "type": "text" + }, + { + "bbox": [ + 475, + 285, + 484, + 295 + ], + "score": 0.34, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 330, + 294, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 330, + 294, + 505, + 309 + ], + "score": 1.0, + "content": "represents the similarity between memory", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 332, + 306, + 406, + 318 + ], + "spans": [ + { + "bbox": [ + 332, + 306, + 406, + 318 + ], + "score": 1.0, + "content": "and the reference.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + } + ], + "index": 29.25 + }, + { + "type": "text", + "bbox": [ + 107, + 324, + 326, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 327, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 327, + 338 + ], + "score": 1.0, + "content": "Selfmem consists of two complementary components:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 322, + 327, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 333, + 505, + 422 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 347 + ], + "score": 1.0, + "content": "a retrieval-augmented generator and a memory selector. The generator operates under two distinct", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "paradigms: fine-tuning a small model or few-shot prompting an LLM. For the former, we train the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "generator with labeled data and retrieved memory, while for the latter, we employ a fixed black-box", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "LLM exclusively for inference alongside retrieved in-context learning samples. We then use the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "generator’s output to train a memory selector based on a specific performance metric. By simply", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "replacing the retrieved memory with unbounded generated memory, we achieve higher-quality", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "score": 1.0, + "content": "generation output (primal problem), which subsequently serves as memory for the next round after", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 411, + 321, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 321, + 423 + ], + "score": 1.0, + "content": "being refined by the memory selector (dual problem).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 335, + 506, + 423 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "To evaluate the efficacy of the Selfmem, we carry out comprehensive experiments in three distinct text", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 438, + 507, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 507, + 451 + ], + "score": 1.0, + "content": "generation tasks: neural machine translation, abstractive text summarization, and dialogue generation.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "We witness substantial enhancements over robust baselines, attaining state-of-the-art outcomes in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 473 + ], + "score": 1.0, + "content": "JRC-Acquis (four directions), XSum (50.3 ROUGE-1), and BigPatent (62.9 ROUGE-1). To gain", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "deeper insights into the Selfmem, we meticulously investigate each crucial component and pinpoint", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 482, + 371, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 371, + 495 + ], + "score": 1.0, + "content": "the existing system bottleneck to guide future research endeavors.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 427, + 507, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 507, + 197, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 198, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 198, + 523 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "title", + "bbox": [ + 108, + 532, + 292, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 293, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 293, + 546 + ], + "score": 1.0, + "content": "2.1 Retrieval-augmented Text Generation", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 552, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "Since the world is not a snapshot once the training corpus is collected, we can never expect an", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 564, + 507, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 507, + 576 + ], + "score": 1.0, + "content": "ever-large model to capture everything in its parameters, even for LLMs like GPT-4 [62]. Therefore,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "it is crucial to equip these models with an external memory bank to store additional knowledge or", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 585, + 403, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 403, + 597 + ], + "score": 1.0, + "content": "useful demonstration examples for solving various NLP tasks[41, 78, 95].", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 55.5, + "bbox_fs": [ + 105, + 552, + 507, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "In the translation domain, retrieval techniques have long been employed by the localization industry", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "to enhance human translators’ productivity and consistency even before the advent of machine", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "translation [94]. Early works on machine translation primarily focused on utilizing memory for", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "statistical machine translation (SMT) systems [80, 50]. For neural machine translation (NMT),", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "[28] were the first to use search engines to retrieve memory from the training set and incorporate", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "it with an external memory network. Subsequent research explored various aspects of retrieval-", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 106, + 668, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 680 + ], + "score": 1.0, + "content": "augmented NMT, such as memory encoding methods [92, 93, 31], joint training of retrievers and", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "generators with monolingual data [8], memory granularity [35], and memory diversity [17]. For", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "few-shot LLM generation, strategies for in-context example selection have been proposed to improve", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "translation quality [2]. Furthermore, in-context machine translation has been shown to be effective", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "for on-the-fly adaptation [79]. For dialogue response generation tasks, employing exemplar/template", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "retrieval as an intermediate step has proven advantageous for generating informative responses [89,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "91, 6, 7]. In-context learning example retrieval also aids in controllable dialogue [46]. Other", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "applications include abstractive summarization [64, 14, 18, 15], code generation [30], paraphrase", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "generation [34, 83], language modeling [36, 105], counterfactual data generation [24], open domain", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 115, + 328, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 328, + 130 + ], + "score": 1.0, + "content": "question answering [12, 33] and semantic parsing [99].", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 63, + "bbox_fs": [ + 105, + 602, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "retrieval as an intermediate step has proven advantageous for generating informative responses [89,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "91, 6, 7]. In-context learning example retrieval also aids in controllable dialogue [46]. 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In machine translation, pioneering works by [75] and [61] introduced", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "and popularized discriminative reranking for SMT. In the context of NMT, research has focused on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "two primary reranking approaches: generative reranking [56, 32, 88] and discriminative reranking [39,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "score": 1.0, + "content": "71, 23]. 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SimCLS [54] used pairwise Learning To Rank (LTR) to select candidates with the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "highest matching scores. SummaReranker [68] adopted a multi-task mixture-of-experts framework", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "to leverage different metrics capturing various aspects of generated candidates. BRIO [55] reused", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "the base model for a second round of fine-tuning with both cross-entropy loss and a candidate-level", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 504, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 504, + 305 + ], + "score": 1.0, + "content": "ranking loss. JGR [76] employed an alternate training paradigm to train the generator and reranker.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 161, + 506, + 305 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 505, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "A key limitation of these reranking methods is that they only represent a one-way process, wherein the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 331 + ], + "score": 1.0, + "content": "selected candidates become the system’s final output. In contrast, our framework innovatively utilizes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "the chosen candidates as memory for the subsequent generation round of a retrieval-augmented", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 395, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 395, + 354 + ], + "score": 1.0, + "content": "generator, which can produce better candidates with enhanced memory.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 308, + 505, + 354 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 170, + 381 + ], + "lines": [ + { + "bbox": [ + 104, + 366, + 172, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 366, + 172, + 384 + ], + "score": 1.0, + "content": "3 Methods", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 434, + 406 + ], + "score": 1.0, + "content": "In this section, we begin with a motivating experiment on generation as memory", + "type": "text" + }, + { + "bbox": [ + 434, + 394, + 460, + 405 + ], + "score": 0.4, + "content": "( \\ S 3 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 392, + 505, + 406 + ], + "score": 1.0, + "content": ". Then, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 419, + 416 + ], + "score": 1.0, + "content": "introduce Selfmem, a framework comprising a retrieval-augmented generator", + "type": "text" + }, + { + "bbox": [ + 419, + 405, + 446, + 415 + ], + "score": 0.45, + "content": "( \\ S 3 . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "and a memory", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 141, + 427 + ], + "score": 1.0, + "content": "selector", + "type": "text" + }, + { + "bbox": [ + 142, + 415, + 168, + 426 + ], + "score": 0.31, + "content": "( \\ S \\ 3 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 415, + 506, + 427 + ], + "score": 1.0, + "content": ". The complete framework and algorithm are illustrated in Figure 2 and Algorithm 1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 392, + 506, + 427 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 439, + 228, + 451 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 230, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 230, + 455 + ], + "score": 1.0, + "content": "3.1 Generation as Memory", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 460, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "The primary motivation behind our framework stems from the observation that the memory, which is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "more similar in distribution to the data during inference, is not the training data (38.89 BLEU, as", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "shown in the first row of Table 1). Instead, it is the model’s own output (58.58 BLEU) within the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "unbounded generation space. One interesting exploration involves directly utilizing the generated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 484, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 484, + 517 + ], + "score": 1.0, + "content": "output as memory in relation to the primal problem: better memory prompts better generation.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 460, + 506, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 288, + 661 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 288, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 288, + 532 + ], + "score": 1.0, + "content": "We conduct experiments on the JRC-Acquis", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 531, + 288, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 118, + 541 + ], + "score": 1.0, + "content": "En", + "type": "text" + }, + { + "bbox": [ + 118, + 532, + 129, + 541 + ], + "score": 0.27, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 531, + 288, + 541 + ], + "score": 1.0, + "content": "De dataset. 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However, directly in-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 575, + 288, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 288, + 585 + ], + "score": 1.0, + "content": "corporating beam output of this trained model", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 586, + 289, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 289, + 597 + ], + "score": 1.0, + "content": "as memory (Beam) back into the generation", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 597, + 289, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 289, + 608 + ], + "score": 1.0, + "content": "model does not yield any improvements (row", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 607, + 289, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 289, + 619 + ], + "score": 1.0, + "content": "2), despite its higher similarity to the reference", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 618, + 289, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 289, + 630 + ], + "score": 1.0, + "content": "compared to the retrieved ones. We hypoth-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 630, + 288, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 288, + 640 + ], + "score": 1.0, + "content": "esize two potential reasons for this: (1) the", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 640, + 289, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 289, + 652 + ], + "score": 1.0, + "content": "retrieval-augmented generator may not gen-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 650, + 289, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 289, + 663 + ], + "score": 1.0, + "content": "eralize effectively in this context due to the", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 520, + 289, + 663 + ] + }, + { + "type": "table", + "bbox": [ + 311, + 589, + 485, + 654 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 295, + 523, + 505, + 578 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 295, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 295, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "Table 1: Experiments on the relation between mem-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 294, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 294, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "ory quality and the final hypothesis quality, measured", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 294, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 294, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "by the BLEU score with ground truth translation. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 295, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 295, + 556, + 505, + 568 + ], + "score": 1.0, + "content": "retrieval-augmented translator keeps fixed while the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 294, + 567, + 471, + 579 + ], + "spans": [ + { + "bbox": [ + 294, + 567, + 471, + 579 + ], + "score": 1.0, + "content": "memory is obtained from different sources.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "table_body", + "bbox": [ + 311, + 589, + 485, + 654 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 311, + 589, + 485, + 654 + ], + "spans": [ + { + "bbox": [ + 311, + 589, + 485, + 654 + ], + "score": 0.971, + "html": "
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There are two components in Selfmem, a retrieval-augmented genera-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "tor (a) and a memory selector (b). For the primal problem, (a) takes source and memory as input to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "generate candidates for (b). 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Inspired by the confidence analysis of NMT model [58], we compute the set", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 376, + 246, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 178, + 387 + ], + "score": 1.0, + "content": "confidence score,", + "type": "text" + }, + { + "bbox": [ + 178, + 376, + 196, + 388 + ], + "score": 0.9, + "content": "\\psi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 376, + 246, + 387 + ], + "score": 1.0, + "content": ", as follows:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 403, + 363, + 435 + ], + "lines": [ + { + "bbox": [ + 248, + 403, + 363, + 435 + ], + "spans": [ + { + "bbox": [ + 248, + 403, + 363, + 435 + ], + "score": 0.94, + "content": "\\psi ( \\cdot ) = { \\frac { 1 } { | \\cdot | } } \\sum _ { y ^ { i } \\in \\cdot } p ( y _ { i } | x , y _ { < i } )", + "type": "interline_equation", + "image_path": "57aa37e9d8b2d2f391adc5e012086f8792f0efb265fc88da4fc5d6f9df72b921.jpg" + } + ] + } + ], + "index": 15.5, + "virtual_lines": [ + { + "bbox": [ + 248, + 403, + 363, + 419.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 248, + 419.0, + 363, + 435.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 447, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 133, + 461 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 448, + 182, + 460 + ], + "score": 0.93, + "content": "p ( y _ { i } | x , y _ { < i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 447, + 325, + 461 + ], + "score": 1.0, + "content": "is defined by the generation model.", + "type": "text" + }, + { + "bbox": [ + 325, + 448, + 343, + 459 + ], + "score": 0.89, + "content": "\\psi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "measures the confidence with which the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 313, + 472 + ], + "score": 1.0, + "content": "generation model generates the tokens. The value of", + "type": "text" + }, + { + "bbox": [ + 314, + 459, + 337, + 471 + ], + "score": 0.92, + "content": "\\psi ( \\mathcal { R } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 458, + 422, + 472 + ], + "score": 1.0, + "content": "is 0.58, while that of", + "type": "text" + }, + { + "bbox": [ + 422, + 459, + 432, + 469 + ], + "score": 0.82, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 458, + 506, + 472 + ], + "score": 1.0, + "content": "is 0.76, indicating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 366, + 482 + ], + "score": 1.0, + "content": "that the generator is relatively confident in generating tokens in", + "type": "text" + }, + { + "bbox": [ + 366, + 470, + 376, + 479 + ], + "score": 0.77, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 470, + 505, + 482 + ], + "score": 1.0, + "content": ", and therefore does not need to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 432, + 493 + ], + "score": 1.0, + "content": "resort to external memory [38]. Beam search ranks generated candidates based on", + "type": "text" + }, + { + "bbox": [ + 433, + 480, + 460, + 493 + ], + "score": 0.92, + "content": "p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ", where the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "selected memory falls within the confidence region of the generator and consequently provides no", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "information gain. This observation motivates us to select memory according to metrics other than", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 513, + 258, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 134, + 525 + ], + "score": 0.91, + "content": "p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 513, + 258, + 525 + ], + "score": 1.0, + "content": "in the memory selector (§3.3).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 107, + 541, + 266, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 268, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 268, + 555 + ], + "score": 1.0, + "content": "3.2 Retrieval-augmented Generator", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 175, + 576 + ], + "score": 1.0, + "content": "Given a text pair", + "type": "text" + }, + { + "bbox": [ + 176, + 563, + 199, + 574 + ], + "score": 0.91, + "content": "( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 560, + 230, + 576 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 231, + 563, + 302, + 575 + ], + "score": 0.93, + "content": "x = \\{ \\mathbf { x } _ { 1 } , . . . , \\mathbf { x } _ { | x | } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 560, + 358, + 576 + ], + "score": 1.0, + "content": "is the source,", + "type": "text" + }, + { + "bbox": [ + 358, + 562, + 429, + 576 + ], + "score": 0.92, + "content": "y = \\{ \\mathbf { y } _ { 1 } , . . . , \\mathbf { y } _ { | y | } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 560, + 506, + 576 + ], + "score": 1.0, + "content": "is the target. 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There are two components in Selfmem, a retrieval-augmented genera-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 248 + ], + "score": 1.0, + "content": "tor (a) and a memory selector (b). For the primal problem, (a) takes source and memory as input to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "generate candidates for (b). 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Evaluation metrics include BLEU, TER, and", + "type": "text" + }, + { + "bbox": [ + 474, + 405, + 506, + 415 + ], + "score": 0.7, + "content": "{ \\mathrm { c h r F } } + +", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 342, + 428 + ], + "score": 1.0, + "content": "obtained from SACREBLEU[66]. The memory selector", + "type": "text" + }, + { + "bbox": [ + 342, + 416, + 353, + 426 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 415, + 427, + 428 + ], + "score": 1.0, + "content": "utilizes an XLM-", + "type": "text" + }, + { + "bbox": [ + 428, + 416, + 450, + 426 + ], + "score": 0.8, + "content": "{ \\bf R } _ { b a s e }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "[22] as back-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 426, + 504, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 229, + 439 + ], + "score": 1.0, + "content": "bone, with BLEU serving as", + "type": "text" + }, + { + "bbox": [ + 230, + 426, + 257, + 438 + ], + "score": 0.94, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 426, + 409, + 439 + ], + "score": 1.0, + "content": ". 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For both dialogue and summariza-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 469, + 503, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 337, + 483 + ], + "score": 1.0, + "content": "tion tasks, we adhere to the methods of [54, 26], adopting", + "type": "text" + }, + { + "bbox": [ + 337, + 470, + 391, + 482 + ], + "score": 0.79, + "content": "\\mathrm { R o B E R T a _ { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 469, + 491, + 483 + ], + "score": 1.0, + "content": "[52] as the backbone for", + "type": "text" + }, + { + "bbox": [ + 491, + 470, + 503, + 481 + ], + "score": 0.88, + "content": "S _ { \\theta }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 212, + 493 + ], + "score": 1.0, + "content": "The linear combination of", + "type": "text" + }, + { + "bbox": [ + 212, + 481, + 236, + 492 + ], + "score": 0.38, + "content": "_ { \\mathrm { B - } 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 481, + 286, + 493 + ], + "score": 1.0, + "content": "is chosen as", + "type": "text" + }, + { + "bbox": [ + 287, + 481, + 314, + 493 + ], + "score": 0.92, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 481, + 441, + 493 + ], + "score": 1.0, + "content": "for Dialogue Generation, while", + "type": "text" + }, + { + "bbox": [ + 441, + 481, + 474, + 492 + ], + "score": 0.48, + "content": "\\mathrm { \\mathbf { R } } \\mathrm { - } 1 / 2 / \\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "is used", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "for Summarization, following [76]. 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The baselines comprise two types of translation systems: one", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "being the vanilla sequence-to-sequence model [3, 84] without memory augmentation, and the other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "score": 1.0, + "content": "consisting of retrieval-augmented translation models focusing on memory encoding [28, 92], memory", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "construction [101], memory retrieval [8], and memory diversity [17]. 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This is noteworthy, given that the parameters of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 504, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 121, + 679 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 667, + 135, + 679 + ], + "score": 0.89, + "content": "G _ { \\xi }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 666, + 504, + 679 + ], + "score": 1.0, + "content": "remain fixed, with the only variable being the input memory. 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For dialogue generation,", + "type": "text" + }, + { + "bbox": [ + 259, + 448, + 298, + 459 + ], + "score": 0.77, + "content": "\\mathbf { B A R T _ { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 448, + 404, + 461 + ], + "score": 1.0, + "content": "serves as the backbone for", + "type": "text" + }, + { + "bbox": [ + 404, + 448, + 417, + 460 + ], + "score": 0.88, + "content": "G _ { \\xi }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 448, + 505, + 461 + ], + "score": 1.0, + "content": ". Our dialogue system", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "is evaluated using BLEU (B-1/2) and Distinct (D-1/2) scores [43]. For both dialogue and summariza-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 469, + 503, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 337, + 483 + ], + "score": 1.0, + "content": "tion tasks, we adhere to the methods of [54, 26], adopting", + "type": "text" + }, + { + "bbox": [ + 337, + 470, + 391, + 482 + ], + "score": 0.79, + "content": "\\mathrm { R o B E R T a _ { b a s e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 469, + 491, + 483 + ], + "score": 1.0, + "content": "[52] as the backbone for", + "type": "text" + }, + { + "bbox": [ + 491, + 470, + 503, + 481 + ], + "score": 0.88, + "content": "S _ { \\theta }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 212, + 493 + ], + "score": 1.0, + "content": "The linear combination of", + "type": "text" + }, + { + "bbox": [ + 212, + 481, + 236, + 492 + ], + "score": 0.38, + "content": "_ { \\mathrm { B - } 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 481, + 286, + 493 + ], + "score": 1.0, + "content": "is chosen as", + "type": "text" + }, + { + "bbox": [ + 287, + 481, + 314, + 493 + ], + "score": 0.92, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 481, + 441, + 493 + ], + "score": 1.0, + "content": "for Dialogue Generation, while", + "type": "text" + }, + { + "bbox": [ + 441, + 481, + 474, + 492 + ], + "score": 0.48, + "content": "\\mathrm { \\mathbf { R } } \\mathrm { - } 1 / 2 / \\mathrm { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "is used", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "for Summarization, following [76]. For further implementation details, please refer to the Appendix", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 502, + 275, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 275, + 515 + ], + "score": 1.0, + "content": "B and Appendix C for evaluation metrics.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 360, + 506, + 515 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 531, + 236, + 545 + ], + "lines": [ + { + "bbox": [ + 104, + 531, + 237, + 548 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 237, + 548 + ], + "score": 1.0, + "content": "5 Experimental Results", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 219, + 569 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 220, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 220, + 570 + ], + "score": 1.0, + "content": "5.1 Machine Translation", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 505, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "We select four translation directions and experiment with two generation paradigms: trainable small", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "models and few-shot prompted LLMs [85, 20]. For trainable models, we explore two architectures", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 226, + 615 + ], + "score": 1.0, + "content": "(joint and dual, as detailed in", + "type": "text" + }, + { + "bbox": [ + 227, + 601, + 247, + 612 + ], + "score": 0.83, + "content": "\\ S 3 . 2 \\AA", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 600, + 506, + 615 + ], + "score": 1.0, + "content": ". The baselines comprise two types of translation systems: one", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "being the vanilla sequence-to-sequence model [3, 84] without memory augmentation, and the other", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "score": 1.0, + "content": "consisting of retrieval-augmented translation models focusing on memory encoding [28, 92], memory", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "construction [101], memory retrieval [8], and memory diversity [17]. Based on the experimental", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 440, + 657 + ], + "score": 1.0, + "content": "results2 shown in Table 2, Selfmem significantly enhances the performance of", + "type": "text" + }, + { + "bbox": [ + 441, + 645, + 454, + 657 + ], + "score": 0.89, + "content": "G _ { \\xi }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "across four", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "translation datasets and two different architectures. This is noteworthy, given that the parameters of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 504, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 121, + 679 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 667, + 135, + 679 + ], + "score": 0.89, + "content": "G _ { \\xi }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 666, + 504, + 679 + ], + "score": 1.0, + "content": "remain fixed, with the only variable being the input memory. This finding is consistent with", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 677, + 503, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 503, + 690 + ], + "score": 1.0, + "content": "the primal problem which posits that improved memory typically leads to better generation results.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 578, + 506, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 122, + 484, + 318 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 506, + 115 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 506, + 83 + ], + "score": 1.0, + "content": "Table 2: Results of translation task on JRC-Acquis measured by BLEU. Models denoted by the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 82, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 165, + 93 + ], + "score": 1.0, + "content": "same symbol", + "type": "text" + }, + { + "bbox": [ + 166, + 83, + 173, + 92 + ], + "score": 0.63, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 82, + 190, + 93 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 197, + 93 + ], + "score": 0.32, + "content": "\\dagger .", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 82, + 506, + 93 + ], + "score": 1.0, + "content": ") have the same parameters and only differ in memory as input. The bolded", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 497, + 104 + ], + "score": 1.0, + "content": "numbers show the SOTA performance and the underlined numbers show the second-best result.", + "type": "text" + }, + { + "bbox": [ + 497, + 94, + 505, + 102 + ], + "score": 0.71, + "content": "^ *", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 103, + 469, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 348, + 116 + ], + "score": 1.0, + "content": "denotes the system is significantly better than baselines with", + "type": "text" + }, + { + "bbox": [ + 349, + 104, + 355, + 115 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 103, + 381, + 116 + ], + "score": 1.0, + "content": "-value", + "type": "text" + }, + { + "bbox": [ + 381, + 103, + 408, + 114 + ], + "score": 0.82, + "content": "< 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 103, + 469, + 116 + ], + "score": 1.0, + "content": "tested by [37].", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 122, + 484, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 122, + 484, + 318 + ], + "spans": [ + { + "bbox": [ + 128, + 122, + 484, + 318 + ], + "score": 0.982, + "html": "
SystemEs-→EnEn→EsDe-→EnEn→De
DevTestDevTestDevTestDevTest
None Memory
RNNsearch [3]55.0259.3450.5450.4850.2049.7444.9443.98
Transformer [84]64.0864.6362.0261.8060.1860.1654.6555.43
Retrieval Memory
SEG-NMT[28]60.2859.3457.6257.2755.6355.3349.2648.80
NMT-pieces [101]63.9764.3061.5061.5660.1060.2655.5455.14
G-TFM [92]66.3766.2162.5062.7661.8561.7257.4356.88
MonoNMT[8]67.7367.4264.1863.8664.4864.6258.7758.42
CMM[17]67.4867.7663.8464.0464.2264.3358.9458.69
Transformerdual*66.8767.1263.1463.5464.0963.3658.6958.06
Transformerunit67.7467.3263.9364.1264.5064.4058.1658.58
Self-Memory
Transformerdual*68.63*69.20*64.12*64.67*65.06*64.98*59.26*59.49*
Transformerunit68.26*68.80*66.07*65.94*65.32*65.65*59.88*60.11*
", + "type": "table", + "image_path": "3d712394296b262aa12045fe7e2128e3b707e57b0c9d9837c330f2333d45f149.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 128, + 122, + 484, + 187.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 128, + 187.33333333333331, + 484, + 252.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 128, + 252.66666666666663, + 484, + 317.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "table", + "bbox": [ + 173, + 368, + 437, + 460 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 329, + 504, + 363 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "Table 3: Comparison between retrieval memory and self-memory. The quality of memory and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "hypothesis is measured by the n-gram overlap with reference (BLEU). All experiments are conducted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 351, + 261, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 175, + 364 + ], + "score": 1.0, + "content": "with Transforme", + "type": "text" + }, + { + "bbox": [ + 176, + 353, + 190, + 363 + ], + "score": 0.65, + "content": "\\mathbf { \\dot { j } } \\mathbf { o } \\mathbf { \\dot { i } } \\mathbf { n } \\mathbf { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 351, + 261, + 364 + ], + "score": 1.0, + "content": "on JRC-Acquis.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 173, + 368, + 437, + 460 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 173, + 368, + 437, + 460 + ], + "spans": [ + { + "bbox": [ + 173, + 368, + 437, + 460 + ], + "score": 0.98, + "html": "
RetrievalSelf
memoryhypothesis memoryhypothesis
En-De→38.8958.5857.9260.11
↑42.5664.4064.3265.65
En-Es→40.6764.1263.5765.94
↑43.0567.3267.7868.80
", + "type": "table", + "image_path": "46e2cf5fcb73f8f8dc484eaa34a64aaf3d9a9f9a68b88f187f5250b7761c7768.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 173, + 368, + 437, + 398.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 398.6666666666667, + 437, + 429.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 429.33333333333337, + 437, + 460.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 506, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "The dual problem is revealed in Table 3. Self-memory, which essentially represents the model’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "own output, exhibits greater similarity with the ground truth and serves as a more effective memory", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "for generating the final output. This observation highlights a key distinction between Selfmem and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 514, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 525 + ], + "score": 1.0, + "content": "previous reranking works [39, 68]. Reranking aims to select candidates of higher quality than the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "beam output, whereas in Selfmem, the chosen candidates serve as memory for the retrieval-augmented", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 535, + 445, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 445, + 547 + ], + "score": 1.0, + "content": "generator and do not necessarily need to surpass the quality of the beam hypotheses.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 124, + 574, + 482, + 656 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 169, + 557, + 442, + 569 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 168, + 555, + 442, + 571 + ], + "spans": [ + { + "bbox": [ + 168, + 555, + 442, + 571 + ], + "score": 1.0, + "content": "Table 4: Evaluation results of in-context learning with self-memory.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "table_body", + "bbox": [ + 124, + 574, + 482, + 656 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 124, + 574, + 482, + 656 + ], + "spans": [ + { + "bbox": [ + 124, + 574, + 482, + 656 + ], + "score": 0.983, + "html": "
XGLM-1.7BXGLM-4.5BXGLM-7.5B
RandomkNNSelfRandomkNNSelfRandomkNNSelf
En-De↑11.5137.8740.9417.5137.6038.2518.4847.8248.32
27.4251.0051.8830.6248.1248.3633.0355.6555.12
En-Es→23.8746.2048.5631.8348.3749.1729.9753.8654.32
↑25.2951.5553.1332.1648.5549.2235.2257.2557.56
", + "type": "table", + "image_path": "d808ef46dcc0836defbafe2e3a62520830ffd2c2ba611d69c175d340ffe7e8c7.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 124, + 574, + 482, + 601.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 124, + 601.3333333333334, + 482, + 628.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 124, + 628.6666666666667, + 482, + 656.0000000000001 + ], + "spans": [], + "index": 22 + } + ] + } + ], + "index": 20.0 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "In Table 4, we present the results of LLM with self-memory. We employ XGLM [48] as our backbone", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 692 + ], + "score": 1.0, + "content": "generator, with three different sizes ranging from 1.7B to 7.5B. We utilize the recommended prompt", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "as described in [48]. We select three in-context learning examples and report the average scores", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "from three separate runs, taking into account the sensitivity of example selection in ICL [49]. From", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "the table, we first observe a general trend where few-shot translation performance improves as the", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 128, + 122, + 484, + 318 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 506, + 115 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 70, + 506, + 83 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 506, + 83 + ], + "score": 1.0, + "content": "Table 2: Results of translation task on JRC-Acquis measured by BLEU. Models denoted by the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 82, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 165, + 93 + ], + "score": 1.0, + "content": "same symbol", + "type": "text" + }, + { + "bbox": [ + 166, + 83, + 173, + 92 + ], + "score": 0.63, + "content": "\\times", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 82, + 190, + 93 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 197, + 93 + ], + "score": 0.32, + "content": "\\dagger .", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 82, + 506, + 93 + ], + "score": 1.0, + "content": ") have the same parameters and only differ in memory as input. The bolded", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 497, + 104 + ], + "score": 1.0, + "content": "numbers show the SOTA performance and the underlined numbers show the second-best result.", + "type": "text" + }, + { + "bbox": [ + 497, + 94, + 505, + 102 + ], + "score": 0.71, + "content": "^ *", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 103, + 469, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 348, + 116 + ], + "score": 1.0, + "content": "denotes the system is significantly better than baselines with", + "type": "text" + }, + { + "bbox": [ + 349, + 104, + 355, + 115 + ], + "score": 0.77, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 103, + 381, + 116 + ], + "score": 1.0, + "content": "-value", + "type": "text" + }, + { + "bbox": [ + 381, + 103, + 408, + 114 + ], + "score": 0.82, + "content": "< 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 103, + 469, + 116 + ], + "score": 1.0, + "content": "tested by [37].", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "table_body", + "bbox": [ + 128, + 122, + 484, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 122, + 484, + 318 + ], + "spans": [ + { + "bbox": [ + 128, + 122, + 484, + 318 + ], + "score": 0.982, + "html": "
SystemEs-→EnEn→EsDe-→EnEn→De
DevTestDevTestDevTestDevTest
None Memory
RNNsearch [3]55.0259.3450.5450.4850.2049.7444.9443.98
Transformer [84]64.0864.6362.0261.8060.1860.1654.6555.43
Retrieval Memory
SEG-NMT[28]60.2859.3457.6257.2755.6355.3349.2648.80
NMT-pieces [101]63.9764.3061.5061.5660.1060.2655.5455.14
G-TFM [92]66.3766.2162.5062.7661.8561.7257.4356.88
MonoNMT[8]67.7367.4264.1863.8664.4864.6258.7758.42
CMM[17]67.4867.7663.8464.0464.2264.3358.9458.69
Transformerdual*66.8767.1263.1463.5464.0963.3658.6958.06
Transformerunit67.7467.3263.9364.1264.5064.4058.1658.58
Self-Memory
Transformerdual*68.63*69.20*64.12*64.67*65.06*64.98*59.26*59.49*
Transformerunit68.26*68.80*66.07*65.94*65.32*65.65*59.88*60.11*
", + "type": "table", + "image_path": "3d712394296b262aa12045fe7e2128e3b707e57b0c9d9837c330f2333d45f149.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 128, + 122, + 484, + 187.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 128, + 187.33333333333331, + 484, + 252.66666666666663 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 128, + 252.66666666666663, + 484, + 317.99999999999994 + ], + "spans": [], + "index": 6 + } + ] + } + ], + "index": 3.25 + }, + { + "type": "table", + "bbox": [ + 173, + 368, + 437, + 460 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 108, + 329, + 504, + 363 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 342 + ], + "score": 1.0, + "content": "Table 3: Comparison between retrieval memory and self-memory. The quality of memory and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "hypothesis is measured by the n-gram overlap with reference (BLEU). All experiments are conducted", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 351, + 261, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 175, + 364 + ], + "score": 1.0, + "content": "with Transforme", + "type": "text" + }, + { + "bbox": [ + 176, + 353, + 190, + 363 + ], + "score": 0.65, + "content": "\\mathbf { \\dot { j } } \\mathbf { o } \\mathbf { \\dot { i } } \\mathbf { n } \\mathbf { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 351, + 261, + 364 + ], + "score": 1.0, + "content": "on JRC-Acquis.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "table_body", + "bbox": [ + 173, + 368, + 437, + 460 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 173, + 368, + 437, + 460 + ], + "spans": [ + { + "bbox": [ + 173, + 368, + 437, + 460 + ], + "score": 0.98, + "html": "
RetrievalSelf
memoryhypothesis memoryhypothesis
En-De→38.8958.5857.9260.11
↑42.5664.4064.3265.65
En-Es→40.6764.1263.5765.94
↑43.0567.3267.7868.80
", + "type": "table", + "image_path": "46e2cf5fcb73f8f8dc484eaa34a64aaf3d9a9f9a68b88f187f5250b7761c7768.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 173, + 368, + 437, + 398.6666666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 173, + 398.6666666666667, + 437, + 429.33333333333337 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 429.33333333333337, + 437, + 460.00000000000006 + ], + "spans": [], + "index": 12 + } + ] + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 506, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "The dual problem is revealed in Table 3. Self-memory, which essentially represents the model’s", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "own output, exhibits greater similarity with the ground truth and serves as a more effective memory", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "for generating the final output. This observation highlights a key distinction between Selfmem and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 514, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 525 + ], + "score": 1.0, + "content": "previous reranking works [39, 68]. Reranking aims to select candidates of higher quality than the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "beam output, whereas in Selfmem, the chosen candidates serve as memory for the retrieval-augmented", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 535, + 445, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 445, + 547 + ], + "score": 1.0, + "content": "generator and do not necessarily need to surpass the quality of the beam hypotheses.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 480, + 506, + 547 + ] + }, + { + "type": "table", + "bbox": [ + 124, + 574, + 482, + 656 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 169, + 557, + 442, + 569 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 168, + 555, + 442, + 571 + ], + "spans": [ + { + "bbox": [ + 168, + 555, + 442, + 571 + ], + "score": 1.0, + "content": "Table 4: Evaluation results of in-context learning with self-memory.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "table_body", + "bbox": [ + 124, + 574, + 482, + 656 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 124, + 574, + 482, + 656 + ], + "spans": [ + { + "bbox": [ + 124, + 574, + 482, + 656 + ], + "score": 0.983, + "html": "
XGLM-1.7BXGLM-4.5BXGLM-7.5B
RandomkNNSelfRandomkNNSelfRandomkNNSelf
En-De↑11.5137.8740.9417.5137.6038.2518.4847.8248.32
27.4251.0051.8830.6248.1248.3633.0355.6555.12
En-Es→23.8746.2048.5631.8348.3749.1729.9753.8654.32
↑25.2951.5553.1332.1648.5549.2235.2257.2557.56
", + "type": "table", + "image_path": "d808ef46dcc0836defbafe2e3a62520830ffd2c2ba611d69c175d340ffe7e8c7.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 124, + 574, + 482, + 601.3333333333334 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 124, + 601.3333333333334, + 482, + 628.6666666666667 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 124, + 628.6666666666667, + 482, + 656.0000000000001 + ], + "spans": [], + "index": 22 + } + ] + } + ], + "index": 20.0 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "In Table 4, we present the results of LLM with self-memory. We employ XGLM [48] as our backbone", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 677, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 692 + ], + "score": 1.0, + "content": "generator, with three different sizes ranging from 1.7B to 7.5B. We utilize the recommended prompt", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "as described in [48]. We select three in-context learning examples and report the average scores", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "from three separate runs, taking into account the sensitivity of example selection in ICL [49]. From", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "the table, we first observe a general trend where few-shot translation performance improves as the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "size of the model increases. Furthermore, we find that more similar translation demonstrations", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "score": 1.0, + "content": "significantly enhance performance across all model sizes (from random, kNN to Self). This suggests", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "that demonstration examples in in-context learning not only act as triggers for model ability but also", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "adhere to the primal problem, where better demonstration example leads to better generation. Also,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "by comparing the results in Table 2 and Table 4, we can conclude that the cross-lingual LLM with", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 401, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 401, + 140 + ], + "score": 1.0, + "content": "designed examples still falls short of the supervised baselines in this task.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 667, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 84 + ], + "score": 1.0, + "content": "size of the model increases. Furthermore, we find that more similar translation demonstrations", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 97 + ], + "score": 1.0, + "content": "significantly enhance performance across all model sizes (from random, kNN to Self). This suggests", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "that demonstration examples in in-context learning not only act as triggers for model ability but also", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "adhere to the primal problem, where better demonstration example leads to better generation. Also,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "by comparing the results in Table 2 and Table 4, we can conclude that the cross-lingual LLM with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 401, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 401, + 140 + ], + "score": 1.0, + "content": "designed examples still falls short of the supervised baselines in this task.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 196, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 197, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 197, + 165 + ], + "score": 1.0, + "content": "5.2 Summarization", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "score": 1.0, + "content": "In this paper, we compare the performance of our trainable model with those of REINA [87],", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "PEGASUS [100], and BART [40]. The results are presented in Table5. Initially, it can be observed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "that memory has varying impacts on different datasets. The enhancement brought by memory in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "BigPatent dataset is significantly larger than that in the XSum dataset. This can be attributed to the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "inherent characteristics of the BigPatent dataset, which consists of official patent documents that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "exhibit considerable similarity. Consequently, this greatly improves the summarization quality in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "score": 1.0, + "content": "accordance with the primal problem. Furthermore, we discovered that self-memory substantially", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 269, + 262 + ], + "score": 1.0, + "content": "enhances the performance of both BRIO", + "type": "text" + }, + { + "bbox": [ + 270, + 249, + 308, + 261 + ], + "score": 0.71, + "content": "( + 1 . 2 { \\ R } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 249, + 356, + 262 + ], + "score": 1.0, + "content": "and BART", + "type": "text" + }, + { + "bbox": [ + 356, + 249, + 397, + 260 + ], + "score": 0.74, + "content": "( + 1 8 . 5 \\mathrm { R } 1 ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "), achieving state-of-the-art", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "results on both datasets. 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By optimizing memory using BLEU as", + "type": "text" + }, + { + "bbox": [ + 475, + 503, + 502, + 515 + ], + "score": 0.91, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 502, + 506, + 515 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 398, + 526 + ], + "score": 1.0, + "content": "the self-memory improves the B-1,2 score over retrieved memory by", + "type": "text" + }, + { + "bbox": [ + 398, + 514, + 437, + 524 + ], + "score": 0.29, + "content": "3 . 0 8 \\ \\mathrm { B } \\cdot 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 514, + 457, + 526 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 514, + 490, + 524 + ], + "score": 0.37, + "content": "0 . 6 \\ \\mathbf { B } { - } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 146, + 537 + ], + "score": 0.81, + "content": "\\mathbf { B A R T _ { j o i n t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 524, + 412, + 537 + ], + "score": 1.0, + "content": ". Intriguingly, although Selfmem surpasses the baselines in terms of", + "type": "text" + }, + { + "bbox": [ + 412, + 525, + 435, + 535 + ], + "score": 0.31, + "content": "_ { \\mathrm { B - } 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ", it falls behind in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "D-1 and D-2, which can be attributed to the trade-off between BLEU score and Distinct score when", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 408, + 560 + ], + "score": 1.0, + "content": "evaluating a dialogue system [104]. To address this issue, we opt for D-1,2 as", + "type": "text" + }, + { + "bbox": [ + 409, + 547, + 436, + 558 + ], + "score": 0.92, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "when optimizing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 118, + 568 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 557, + 166, + 570 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 167, + 558, + 224, + 569 + ], + "score": 0.79, + "content": "\\mathbf { B A R T } _ { \\mathrm { j o i n t } } \\dagger ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 557, + 506, + 570 + ], + "score": 1.0, + "content": ". The results in Table 6 highlight the remarkable flexibility of Selfmem", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "by directly optimizing memory to achieve the desired attributes for diverse and informative dialogue.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 213, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 214, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 214, + 612 + ], + "score": 1.0, + "content": "6 Further Analysis", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 490, + 635 + ], + "score": 1.0, + "content": "To gain a deeper insight into Selfmem, we first examine the impact of each key component, namely", + "type": "text" + }, + { + "bbox": [ + 491, + 622, + 504, + 634 + ], + "score": 0.9, + "content": "G _ { \\xi }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 631, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 123, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 632, + 135, + 643 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 631, + 506, + 646 + ], + "score": 1.0, + "content": ". Subsequently, we perform a detailed token-level analysis of the generated output concerning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 507, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 451, + 656 + ], + "score": 1.0, + "content": "their frequency in the training set. Experiments are conducted on the JRC-Acquis En", + "type": "text" + }, + { + "bbox": [ + 451, + 644, + 461, + 653 + ], + "score": 0.41, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 643, + 507, + 656 + ], + "score": 1.0, + "content": "De dataset.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 419, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 419, + 666 + ], + "score": 1.0, + "content": "We also include latency analysis and human evaluation on Appendix F and G.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 140, + 690 + ], + "score": 1.0, + "content": "Tuning", + "type": "text" + }, + { + "bbox": [ + 141, + 678, + 153, + 689 + ], + "score": 0.86, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 677, + 249, + 690 + ], + "score": 1.0, + "content": "We explored various", + "type": "text" + }, + { + "bbox": [ + 249, + 678, + 261, + 689 + ], + "score": 0.89, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "by direct selection from the candidate pool based on gold", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 389, + 701 + ], + "score": 1.0, + "content": "rankings. As shown in Figure 3a, both architectures with enhanced", + "type": "text" + }, + { + "bbox": [ + 389, + 689, + 401, + 700 + ], + "score": 0.88, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "significantly outperform", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "the current SOTA performance (60.11 BLEU). 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A clear pattern emerges", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 138 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 73, + 506, + 140 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 196, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 151, + 197, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 197, + 165 + ], + "score": 1.0, + "content": "5.2 Summarization", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 185 + ], + "score": 1.0, + "content": "In this paper, we compare the performance of our trainable model with those of REINA [87],", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "PEGASUS [100], and BART [40]. The results are presented in Table5. Initially, it can be observed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "that memory has varying impacts on different datasets. The enhancement brought by memory in the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 218 + ], + "score": 1.0, + "content": "BigPatent dataset is significantly larger than that in the XSum dataset. This can be attributed to the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "inherent characteristics of the BigPatent dataset, which consists of official patent documents that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "exhibit considerable similarity. Consequently, this greatly improves the summarization quality in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 252 + ], + "score": 1.0, + "content": "accordance with the primal problem. Furthermore, we discovered that self-memory substantially", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 269, + 262 + ], + "score": 1.0, + "content": "enhances the performance of both BRIO", + "type": "text" + }, + { + "bbox": [ + 270, + 249, + 308, + 261 + ], + "score": 0.71, + "content": "( + 1 . 2 { \\ R } 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 249, + 356, + 262 + ], + "score": 1.0, + "content": "and BART", + "type": "text" + }, + { + "bbox": [ + 356, + 249, + 397, + 260 + ], + "score": 0.74, + "content": "( + 1 8 . 5 \\mathrm { R } 1 ) ", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "), achieving state-of-the-art", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "results on both datasets. We selected these baselines for a fair comparison, as they share the same", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 272, + 504, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 504, + 283 + ], + "score": 1.0, + "content": "base generator. 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SystemMemoryR-1R-2R-L
BigPatent
PEGASUSNone53.633.243.2
BARTNone44.421.331.0
REINA (B)Retrieval59.542.650.6
REINA (L)Retrieval60.743.351.3
REINA (PG)Retrieval44.621.533.3
BARTdual*Retrieval57.443.349.7
BARTjointtRetrieval59.643.451.0
BARTdual*Self61.244.652.3
BARTjointSelf62.948.159.6
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By optimizing memory using BLEU as", + "type": "text" + }, + { + "bbox": [ + 475, + 503, + 502, + 515 + ], + "score": 0.91, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 502, + 506, + 515 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 514, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 398, + 526 + ], + "score": 1.0, + "content": "the self-memory improves the B-1,2 score over retrieved memory by", + "type": "text" + }, + { + "bbox": [ + 398, + 514, + 437, + 524 + ], + "score": 0.29, + "content": "3 . 0 8 \\ \\mathrm { B } \\cdot 1", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 514, + 457, + 526 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 514, + 490, + 524 + ], + "score": 0.37, + "content": "0 . 6 \\ \\mathbf { B } { - } 2", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 514, + 506, + 526 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 146, + 537 + ], + "score": 0.81, + "content": "\\mathbf { B A R T _ { j o i n t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 524, + 412, + 537 + ], + "score": 1.0, + "content": ". Intriguingly, although Selfmem surpasses the baselines in terms of", + "type": "text" + }, + { + "bbox": [ + 412, + 525, + 435, + 535 + ], + "score": 0.31, + "content": "_ { \\mathrm { B - } 1 / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 524, + 506, + 537 + ], + "score": 1.0, + "content": ", it falls behind in", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "D-1 and D-2, which can be attributed to the trade-off between BLEU score and Distinct score when", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 408, + 560 + ], + "score": 1.0, + "content": "evaluating a dialogue system [104]. To address this issue, we opt for D-1,2 as", + "type": "text" + }, + { + "bbox": [ + 409, + 547, + 436, + 558 + ], + "score": 0.92, + "content": "\\Delta ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "when optimizing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 118, + 568 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 557, + 166, + 570 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 167, + 558, + 224, + 569 + ], + "score": 0.79, + "content": "\\mathbf { B A R T } _ { \\mathrm { j o i n t } } \\dagger ( \\mathbf { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 557, + 506, + 570 + ], + "score": 1.0, + "content": ". The results in Table 6 highlight the remarkable flexibility of Selfmem", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 581 + ], + "score": 1.0, + "content": "by directly optimizing memory to achieve the desired attributes for diverse and informative dialogue.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 491, + 507, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 213, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 214, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 214, + 612 + ], + "score": 1.0, + "content": "6 Further Analysis", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 504, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 490, + 635 + ], + "score": 1.0, + "content": "To gain a deeper insight into Selfmem, we first examine the impact of each key component, namely", + "type": "text" + }, + { + "bbox": [ + 491, + 622, + 504, + 634 + ], + "score": 0.9, + "content": "G _ { \\xi }", + "type": "inline_equation" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 631, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 123, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 632, + 135, + 643 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 631, + 506, + 646 + ], + "score": 1.0, + "content": ". Subsequently, we perform a detailed token-level analysis of the generated output concerning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 507, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 451, + 656 + ], + "score": 1.0, + "content": "their frequency in the training set. Experiments are conducted on the JRC-Acquis En", + "type": "text" + }, + { + "bbox": [ + 451, + 644, + 461, + 653 + ], + "score": 0.41, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 643, + 507, + 656 + ], + "score": 1.0, + "content": "De dataset.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 654, + 419, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 419, + 666 + ], + "score": 1.0, + "content": "We also include latency analysis and human evaluation on Appendix F and G.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 619, + 507, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 140, + 690 + ], + "score": 1.0, + "content": "Tuning", + "type": "text" + }, + { + "bbox": [ + 141, + 678, + 153, + 689 + ], + "score": 0.86, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 677, + 249, + 690 + ], + "score": 1.0, + "content": "We explored various", + "type": "text" + }, + { + "bbox": [ + 249, + 678, + 261, + 689 + ], + "score": 0.89, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "by direct selection from the candidate pool based on gold", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 389, + 701 + ], + "score": 1.0, + "content": "rankings. As shown in Figure 3a, both architectures with enhanced", + "type": "text" + }, + { + "bbox": [ + 389, + 689, + 401, + 700 + ], + "score": 0.88, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "significantly outperform", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "the current SOTA performance (60.11 BLEU). Moreover, we assessed the candidate pool quality", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 284, + 724 + ], + "score": 1.0, + "content": "during this iterative process using an oracle", + "type": "text" + }, + { + "bbox": [ + 284, + 712, + 295, + 722 + ], + "score": 0.87, + "content": "S _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 710, + 506, + 724 + ], + "score": 1.0, + "content": ", as displayed in Figure 3b. A clear pattern emerges", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 463 + ], + "score": 1.0, + "content": "in this boxplot, revealing improvements in the oracle, quartile, average, and minimum scores of", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "the candidate pool. These two experiments jointly clarify the Selfmem’s underlying intuition: a", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "retrieval-augmented generator profits from superior memory, which can be chosen from its own", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "unbounded output, and subsequently, the generator with improved memory produces a higher-quality", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 493, + 434, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 434, + 507 + ], + "score": 1.0, + "content": "candidate pool for the next selection round. 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SystemMemoryB-1B-2D-1D-2
NCM [86]None33.6026.803.0012.80
iVAE [25]None30.9024.902.9025.00
PLATO-2 [5]None34.8025.123.5425.11
DialoFlow [45]None36.1727.674.5627.12
BARTNone20.7211.363.9219.44
BARTdual*Retrieval29.5021.894.7426.01
BARTjointtRetrieval36.7231.556.1335.65
BARTdual*Self33.4322.854.6626.16
BARTjointSelf39.8032.155.8432.16
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We appreciate the anonymous reviewers for their helpful comments. 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We appreciate the anonymous reviewers for their helpful comments. 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JRC (en ←→ es)653,1272,5332,596
SummarizationBigPatent1,207,22267,06867,072
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[c]Signature
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nrefs:1lcase:lcltok:tercomlnorm:nolpunct:yeslasian:nolversion:2.0.0
nrefs:1lcase:mixedleff:yeslnc:6lnw:2lspace:nolversion:2.0.0
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TaskDataset#Train#Dev#Test
TranslationJRC (en ←→ de)663,4872,4542,483
JRC (en ←→ es)653,1272,5332,596
SummarizationBigPatent1,207,22267,06867,072
XSum204,04511,33211,334
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The signatures for BLEU, TER and", + "type": "text" + }, + { + "bbox": [ + 307, + 584, + 339, + 594 + ], + "score": 0.49, + "content": "{ \\mathrm { c h r F } } + +", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 582, + 426, + 595 + ], + "score": 1.0, + "content": "are shown in Table 8.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 572, + 506, + 595 + ] + }, + { + "type": "table", + "bbox": [ + 163, + 627, + 445, + 685 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 226, + 609, + 384, + 621 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 225, + 609, + 385, + 622 + ], + "spans": [ + { + "bbox": [ + 225, + 609, + 385, + 622 + ], + "score": 1.0, + "content": "Table 8: Signature from SACREBLEU.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "table_body", + "bbox": [ + 163, + 627, + 445, + 685 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 163, + 627, + 445, + 685 + ], + "spans": [ + { + "bbox": [ + 163, + 627, + 445, + 685 + ], + "score": 0.967, + "html": "
[c]Signature
nrefs:1lcase:mixedleff:noltok:13alsmooth:explversion:2.0.0
nrefs:1lcase:lcltok:tercomlnorm:nolpunct:yeslasian:nolversion:2.0.0
nrefs:1lcase:mixedleff:yeslnc:6lnw:2lspace:nolversion:2.0.0
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SystemMemoryBLEU 个chrF++ 个TER
TransformerNone55.4370.3136.35
TransformerdualRetrieval58.0671.5835.41
TransformerjointRetrieval58.5872.2234.39
TransformerdualSelf59.4972.6234.04
TransformerjointSelf60.1173.2532.62
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SystemR-1R-2R-L
XSum
[51]38.816.531.3 37.3
[40]45.122.339.3
[100]47.224.639.4
[54] [55]47.6 49.124.640.4
[87](PG)48.225.640.2
[87](B)43.126.035.5
21.038.6
[87](L)46.524.140.0
[68]48.125.038.8
[69]47.124.1
[16]47.825.039.7
Selfmem50.326.741.6
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SystemR-1 R-2R-L
BigPatent
[100]53.6 33.142.3
[40] 44.421.331.0
[98] 60.642.550.0
[65]38.7 12.334.1
[90] 45.020.339.2
[1] 52.333.542.8
[87] (B) 59.542.650.6
[87] (L) 60.743.351.3
[87] (PG) 44.621.533.3
Selfmem62.9 48.159.6
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The Distinction score is from [42].", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 117, + 506, + 141 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 155, + 297, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 297, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 297, + 171 + ], + "score": 1.0, + "content": "D More results on translation tasks", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "table", + "bbox": [ + 175, + 200, + 436, + 285 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 121, + 183, + 488, + 195 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 182, + 486, + 197 + ], + "spans": [ + { + "bbox": [ + 120, + 182, + 309, + 197 + ], + "score": 1.0, + "content": "Table 9: Evaluation results on JRC-Acquis En", + "type": "text" + }, + { + "bbox": [ + 310, + 185, + 320, + 193 + ], + "score": 0.29, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 182, + 455, + 197 + ], + "score": 1.0, + "content": "De measured by BLEU, TER and", + "type": "text" + }, + { + "bbox": [ + 455, + 183, + 486, + 194 + ], + "score": 0.25, + "content": "{ \\mathrm { c h r F } } + +", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 175, + 200, + 436, + 285 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 175, + 200, + 436, + 285 + ], + "spans": [ + { + "bbox": [ + 175, + 200, + 436, + 285 + ], + "score": 0.98, + "html": "
SystemMemoryBLEU 个chrF++ 个TER
TransformerNone55.4370.3136.35
TransformerdualRetrieval58.0671.5835.41
TransformerjointRetrieval58.5872.2234.39
TransformerdualSelf59.4972.6234.04
TransformerjointSelf60.1173.2532.62
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SystemR-1R-2R-L
XSum
[51]38.816.531.3 37.3
[40]45.122.339.3
[100]47.224.639.4
[54] [55]47.6 49.124.640.4
[87](PG)48.225.640.2
[87](B)43.126.035.5
21.038.6
[87](L)46.524.140.0
[68]48.125.038.8
[69]47.124.1
[16]47.825.039.7
Selfmem50.326.741.6
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SystemR-1 R-2R-L
BigPatent
[100]53.6 33.142.3
[40] 44.421.331.0
[98] 60.642.550.0
[65]38.7 12.334.1
[90] 45.020.339.2
[1] 52.333.542.8
[87] (B) 59.542.650.6
[87] (L) 60.743.351.3
[87] (PG) 44.621.533.3
Selfmem62.9 48.159.6
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SystemROUGE-1/2/L95 % -conf.int
XSum
BRIOjoint50.30.49986 - 0.50602
26.70.26300 - 0.26989
41.60.41231 - 0.41900
BigPatent
BARTjoint62.90.62664 - 0.63080
48.10.47783 - 0.48333
59.60.59401 - 0.59847
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NMT XSumBigPatentDailyDialog
Average Input Length87512102471
Average ：Output Length447512716
Retrieval-augmented BaselineCPU 0.971.793.160.32
SelfmemCandidate Generation Memory3.207.5015.001.02
Selection0.500.520.950.14
Hypothesis Generation0.971.793.000.32
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CUDA
Retrieval-augmented Baseline0.290.440.750.10
SelfmemCandidate Generation Memory Hypothesis Generation0.511.001.720.18
Selection0.010.010.010.01
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SystemROUGE-1/2/L95 % -conf.int
XSum
BRIOjoint50.30.49986 - 0.50602
26.70.26300 - 0.26989
41.60.41231 - 0.41900
BigPatent
BARTjoint62.90.62664 - 0.63080
48.10.47783 - 0.48333
59.60.59401 - 0.59847
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Average Input Length87512102471
Average ：Output Length447512716
Retrieval-augmented BaselineCPU 0.971.793.160.32
SelfmemCandidate Generation Memory3.207.5015.001.02
Selection0.500.520.950.14
Hypothesis Generation0.971.793.000.32
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CUDA
Retrieval-augmented Baseline0.290.440.750.10
SelfmemCandidate Generation Memory Hypothesis Generation0.511.001.720.18
Selection0.010.010.010.01
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Require: a dataset D,a retriever R, a memory selection metric △(*,-)，,a retrieval-augmented
1:retrieve memory M in D with R generator Gg,and a memory selector Sθ
2:train Gg with D and M (if not LLM)
3: use Gg to generate candidate pool C with M in candidate mode
4:train Sθ on C with △(·,·)
5:while not converged in the validation set do
6:Se selects memory from C as MI
7:Gε generates candidate pool C with M in candidate mode
8:end while
9:Gε generates the final hypothesis with M in hypothesis mode
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XGLM-1.7BXGLM-4.5BXGLM-7.5B
RandomkNNSelfRandomkNNSelfRandomkNNSelf
En-De↑11.5137.8740.9417.5137.6038.2518.4847.8248.32
27.4251.0051.8830.6248.1248.3633.0355.6555.12
En-Es→23.8746.2048.5631.8348.3749.1729.9753.8654.32
↑25.2951.5553.1332.1648.5549.2235.2257.2557.56
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SystemEs-→EnEn→EsDe-→EnEn→De
DevTestDevTestDevTestDevTest
None Memory
RNNsearch [3]55.0259.3450.5450.4850.2049.7444.9443.98
Transformer [84]64.0864.6362.0261.8060.1860.1654.6555.43
Retrieval Memory
SEG-NMT[28]60.2859.3457.6257.2755.6355.3349.2648.80
NMT-pieces [101]63.9764.3061.5061.5660.1060.2655.5455.14
G-TFM [92]66.3766.2162.5062.7661.8561.7257.4356.88
MonoNMT[8]67.7367.4264.1863.8664.4864.6258.7758.42
CMM[17]67.4867.7663.8464.0464.2264.3358.9458.69
Transformerdual*66.8767.1263.1463.5464.0963.3658.6958.06
Transformerunit67.7467.3263.9364.1264.5064.4058.1658.58
Self-Memory
Transformerdual*68.63*69.20*64.12*64.67*65.06*64.98*59.26*59.49*
Transformerunit68.26*68.80*66.07*65.94*65.32*65.65*59.88*60.11*
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RetrievalSelf
memoryhypothesis memoryhypothesis
En-De→38.8958.5857.9260.11
↑42.5664.4064.3265.65
En-Es→40.6764.1263.5765.94
↑43.0567.3267.7868.80
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SystemMemoryR-1R-2R-L
XSum
PEGASUSNone47.224.639.3
BRIONone49.125.640.4
REINA (PG)Retrieval48.226.040.2
REINA (B)Retrieval43.221.035.5
REINA (L)Retrieval46.524.138.6
BRIOdual*Retrieval48.626.140.6
BRIOjointRetrieval49.526.541.2
BRIOdual*Self49.226.240.8
BRIOjointtSelf50.326.741.6
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SystemMemoryR-1R-2R-L
BigPatent
PEGASUSNone53.633.243.2
BARTNone44.421.331.0
REINA (B)Retrieval59.542.650.6
REINA (L)Retrieval60.743.351.3
REINA (PG)Retrieval44.621.533.3
BARTdual*Retrieval57.443.349.7
BARTjointtRetrieval59.643.451.0
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SystemMemoryB-1B-2D-1D-2
NCM [86]None33.6026.803.0012.80
iVAE [25]None30.9024.902.9025.00
PLATO-2 [5]None34.8025.123.5425.11
DialoFlow [45]None36.1727.674.5627.12
BARTNone20.7211.363.9219.44
BARTdual*Retrieval29.5021.894.7426.01
BARTjointtRetrieval36.7231.556.1335.65
BARTdual*Self33.4322.854.6626.16
BARTjointSelf39.8032.155.8432.16
BARTjoint † (D)Self36.9232.099.1237.05
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TaskDataset#Train#Dev#Test
TranslationJRC (en ←→ de)663,4872,4542,483
JRC (en ←→ es)653,1272,5332,596
SummarizationBigPatent1,207,22267,06867,072
XSum204,04511,33211,334
DialogueDailyDialog87,1708,0697,740
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[c]Signature
nrefs:1lcase:mixedleff:noltok:13alsmooth:explversion:2.0.0
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NMT XSumBigPatentDailyDialog
Average Input Length87512102471
Average ：Output Length447512716
Retrieval-augmented BaselineCPU 0.971.793.160.32
SelfmemCandidate Generation Memory3.207.5015.001.02
Selection0.500.520.950.14
Hypothesis Generation0.971.793.000.32
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CUDA
Retrieval-augmented Baseline0.290.440.750.10
SelfmemCandidate Generation Memory Hypothesis Generation0.511.001.720.18
Selection0.010.010.010.01
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XSum
BRIOjoint50.30.49986 - 0.50602
26.70.26300 - 0.26989
41.60.41231 - 0.41900
BigPatent
BARTjoint62.90.62664 - 0.63080
48.10.47783 - 0.48333
59.60.59401 - 0.59847
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The regret definitions", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 93, + 292, + 470, + 305 + ], + "spans": [ + { + "bbox": [ + 93, + 295, + 99, + 304 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 141, + 292, + 470, + 305 + ], + "score": 1.0, + "content": "build upon an equivalent transformation of the multi-objective dynamic regret", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 93, + 302, + 471, + 317 + ], + "spans": [ + { + "bbox": [ + 93, + 305, + 99, + 314 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 140, + 302, + 471, + 317 + ], + "score": 1.0, + "content": "based on the commonly used Pareto suboptimality gap metric in zero-order multi-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 92, + 314, + 470, + 326 + ], + "spans": [ + { + "bbox": [ + 92, + 316, + 100, + 326 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 142, + 314, + 470, + 326 + ], + "score": 1.0, + "content": "objective bandits, making it amenable to be optimized via first-order iterative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 92, + 324, + 470, + 338 + ], + "spans": [ + { + "bbox": [ + 92, + 327, + 99, + 336 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 141, + 324, + 470, + 338 + ], + "score": 1.0, + "content": "methods. To motivate the algorithm design, we give an explicit example in which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 93, + 335, + 470, + 349 + ], + "spans": [ + { + "bbox": [ + 93, + 338, + 100, + 347 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 141, + 335, + 470, + 349 + ], + "score": 1.0, + "content": "equipping OMD with the vanilla min-norm solver for gradient composition will", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 92, + 346, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 92, + 349, + 99, + 358 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 140, + 346, + 470, + 359 + ], + "score": 1.0, + "content": "incur a linear regret, which shows that only regularizing the iterates, as in single-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 90, + 357, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 90, + 360, + 99, + 369 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "score": 1.0, + "content": "objective online learning, is not enough to guarantee sublinear regrets in the multi-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 89, + 368, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 89, + 370, + 100, + 381 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 142, + 368, + 470, + 381 + ], + "score": 1.0, + "content": "objective setting. To resolve this issue, we propose a novel min-regularized-norm", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 89, + 379, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 89, + 381, + 100, + 392 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "score": 1.0, + "content": "solver that regularizes the composite weights. Combining min-regularized-norm", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 89, + 389, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 89, + 392, + 100, + 402 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 140, + 389, + 470, + 403 + ], + "score": 1.0, + "content": "with OMD results in the Doubly Regularized Online Mirror Multiple Descent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 90, + 401, + 469, + 413 + ], + "spans": [ + { + "bbox": [ + 90, + 404, + 99, + 413 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 142, + 401, + 469, + 413 + ], + "score": 1.0, + "content": "algorithm. We further derive both the static and dynamic regret bounds for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 89, + 412, + 470, + 424 + ], + "spans": [ + { + "bbox": [ + 89, + 414, + 100, + 424 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 141, + 412, + 470, + 424 + ], + "score": 1.0, + "content": "proposed algorithm, each of which matches the corresponding optimal bound in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 423, + 470, + 435 + ], + "spans": [ + { + "bbox": [ + 89, + 425, + 100, + 435 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 141, + 423, + 470, + 434 + ], + "score": 1.0, + "content": "single-objective setting. Extensive experiments on both simulation and real-world", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 90, + 434, + 379, + 446 + ], + "spans": [ + { + "bbox": [ + 90, + 436, + 99, + 446 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 142, + 434, + 379, + 446 + ], + "score": 1.0, + "content": "datasets verify the effectiveness of the proposed algorithm.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 91, + 471, + 191, + 484 + ], + "lines": [ + { + "bbox": [ + 87, + 470, + 192, + 487 + ], + "spans": [ + { + "bbox": [ + 87, + 470, + 192, + 487 + ], + "score": 1.0, + "content": "18 1 Introduction", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 90, + 497, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 90, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 90, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "Traditional optimization methods for machine learning are usually designed to optimize a single", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 89, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "objective. However, in many real-world applications, we are often required to optimize multiple", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 88, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 88, + 522, + 99, + 531 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "correlated objectives concurrently. For example, in autonomous driving [12, 20], the self-driving", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 88, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 88, + 533, + 100, + 542 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "vehicles need to solve multiple tasks such as self-localization and object identification at the same", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 89, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 89, + 543, + 100, + 553 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "time. In online advertising [21, 22], advertisers need to determine the exposure of items to different", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 88, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 88, + 555, + 100, + 564 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "users to maximize both the Click-Through Rate (CTR) and the Post-Click Conversion Rate (CVR).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 561, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 104, + 561, + 506, + 577 + ], + "score": 1.0, + "content": "In many multi-objective scenarios, the objectives may conflict with each other [15]. Hence, there may", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 89, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "not exist any single solution that optimizes all the objectives simultaneously. For example, in online", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 89, + 586, + 491, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 586, + 491, + 597 + ], + "score": 1.0, + "content": "advertising, merely optimizing CTR or CVR will degrade the performance of the other [21, 22].", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 90, + 601, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 89, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 89, + 604, + 100, + 613 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "Multi-objective optimization (MOO) [23, 6] is concerned with optimizing multiple conflicting", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 89, + 614, + 99, + 624 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "objectives simultaneously. It seeks Pareto optimality, where no single objective can be improved", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 622, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 89, + 626, + 99, + 635 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "score": 1.0, + "content": "without hurting the performance of the others. Many different methods for MOO have been proposed,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 88, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 88, + 636, + 99, + 645 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "including evolutionary methods [26, 39], scalarization methods [9], and gradient-based iterative", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 88, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 88, + 647, + 99, + 657 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "methods [7]. 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These methods", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 89, + 680, + 100, + 689 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "compute a composite gradient based on the gradient information of all the individual objectives", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 89, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "and then apply the composite gradient to update the model parameters. 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The regret definitions", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 292, + 470, + 305 + ], + "spans": [ + { + "bbox": [ + 93, + 295, + 99, + 304 + ], + "score": 1.0, + "content": "4", + "type": "text" + }, + { + "bbox": [ + 141, + 292, + 470, + 305 + ], + "score": 1.0, + "content": "build upon an equivalent transformation of the multi-objective dynamic regret", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 302, + 471, + 317 + ], + "spans": [ + { + "bbox": [ + 93, + 305, + 99, + 314 + ], + "score": 1.0, + "content": "5", + "type": "text" + }, + { + "bbox": [ + 140, + 302, + 471, + 317 + ], + "score": 1.0, + "content": "based on the commonly used Pareto suboptimality gap metric in zero-order multi-", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 314, + 470, + 326 + ], + "spans": [ + { + "bbox": [ + 92, + 316, + 100, + 326 + ], + "score": 1.0, + "content": "6", + "type": "text" + }, + { + "bbox": [ + 142, + 314, + 470, + 326 + ], + "score": 1.0, + "content": "objective bandits, making it amenable to be optimized via first-order iterative", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 324, + 470, + 338 + ], + "spans": [ + { + "bbox": [ + 92, + 327, + 99, + 336 + ], + "score": 1.0, + "content": "7", + "type": "text" + }, + { + "bbox": [ + 141, + 324, + 470, + 338 + ], + "score": 1.0, + "content": "methods. To motivate the algorithm design, we give an explicit example in which", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 335, + 470, + 349 + ], + "spans": [ + { + "bbox": [ + 93, + 338, + 100, + 347 + ], + "score": 1.0, + "content": "8", + "type": "text" + }, + { + "bbox": [ + 141, + 335, + 470, + 349 + ], + "score": 1.0, + "content": "equipping OMD with the vanilla min-norm solver for gradient composition will", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 346, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 92, + 349, + 99, + 358 + ], + "score": 1.0, + "content": "9", + "type": "text" + }, + { + "bbox": [ + 140, + 346, + 470, + 359 + ], + "score": 1.0, + "content": "incur a linear regret, which shows that only regularizing the iterates, as in single-", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 357, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 90, + 360, + 99, + 369 + ], + "score": 1.0, + "content": "10", + "type": "text" + }, + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "score": 1.0, + "content": "objective online learning, is not enough to guarantee sublinear regrets in the multi-", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 368, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 89, + 370, + 100, + 381 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 142, + 368, + 470, + 381 + ], + "score": 1.0, + "content": "objective setting. To resolve this issue, we propose a novel min-regularized-norm", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 379, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 89, + 381, + 100, + 392 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "score": 1.0, + "content": "solver that regularizes the composite weights. Combining min-regularized-norm", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 389, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 89, + 392, + 100, + 402 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 140, + 389, + 470, + 403 + ], + "score": 1.0, + "content": "with OMD results in the Doubly Regularized Online Mirror Multiple Descent", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 401, + 469, + 413 + ], + "spans": [ + { + "bbox": [ + 90, + 404, + 99, + 413 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 142, + 401, + 469, + 413 + ], + "score": 1.0, + "content": "algorithm. We further derive both the static and dynamic regret bounds for the", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 412, + 470, + 424 + ], + "spans": [ + { + "bbox": [ + 89, + 414, + 100, + 424 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 141, + 412, + 470, + 424 + ], + "score": 1.0, + "content": "proposed algorithm, each of which matches the corresponding optimal bound in the", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 423, + 470, + 435 + ], + "spans": [ + { + "bbox": [ + 89, + 425, + 100, + 435 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 141, + 423, + 470, + 434 + ], + "score": 1.0, + "content": "single-objective setting. Extensive experiments on both simulation and real-world", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 434, + 379, + 446 + ], + "spans": [ + { + "bbox": [ + 90, + 436, + 99, + 446 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 142, + 434, + 379, + 446 + ], + "score": 1.0, + "content": "datasets verify the effectiveness of the proposed algorithm.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + } + ], + "index": 14, + "bbox_fs": [ + 89, + 259, + 471, + 446 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 471, + 191, + 484 + ], + "lines": [ + { + "bbox": [ + 87, + 470, + 192, + 487 + ], + "spans": [ + { + "bbox": [ + 87, + 470, + 192, + 487 + ], + "score": 1.0, + "content": "18 1 Introduction", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "index", + "bbox": [ + 90, + 497, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 90, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 90, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "Traditional optimization methods for machine learning are usually designed to optimize a single", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "objective. However, in many real-world applications, we are often required to optimize multiple", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 518, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 88, + 522, + 99, + 531 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 506, + 534 + ], + "score": 1.0, + "content": "correlated objectives concurrently. For example, in autonomous driving [12, 20], the self-driving", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 88, + 533, + 100, + 542 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "vehicles need to solve multiple tasks such as self-localization and object identification at the same", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 89, + 543, + 100, + 553 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "time. In online advertising [21, 22], advertisers need to determine the exposure of items to different", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 88, + 555, + 100, + 564 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "users to maximize both the Click-Through Rate (CTR) and the Post-Click Conversion Rate (CVR).", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 561, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 104, + 561, + 506, + 577 + ], + "score": 1.0, + "content": "In many multi-objective scenarios, the objectives may conflict with each other [15]. Hence, there may", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 89, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "not exist any single solution that optimizes all the objectives simultaneously. For example, in online", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 586, + 491, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 586, + 491, + 597 + ], + "score": 1.0, + "content": "advertising, merely optimizing CTR or CVR will degrade the performance of the other [21, 22].", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 89, + 604, + 100, + 613 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "Multi-objective optimization (MOO) [23, 6] is concerned with optimizing multiple conflicting", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 89, + 614, + 99, + 624 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 106, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "objectives simultaneously. It seeks Pareto optimality, where no single objective can be improved", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 622, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 89, + 626, + 99, + 635 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "score": 1.0, + "content": "without hurting the performance of the others. Many different methods for MOO have been proposed,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 88, + 636, + 99, + 645 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "including evolutionary methods [26, 39], scalarization methods [9], and gradient-based iterative", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 88, + 647, + 99, + 657 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 106, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "methods [7]. Recently, the Multiple Gradient Descent Algorithm (MGDA) and its variants have been", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 88, + 658, + 100, + 668 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "introduced to the training of multi-task deep neural networks and achieved great empirical success", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 88, + 669, + 100, + 678 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "[29], making them regain a significant amount of research interest [17, 33, 18]. These methods", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 89, + 680, + 100, + 689 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "compute a composite gradient based on the gradient information of all the individual objectives", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 89, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "and then apply the composite gradient to update the model parameters. The composite weights are", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 698, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "37", + "type": "text" + }, + { + "bbox": [ + 104, + 698, + 507, + 713 + ], + "score": 1.0, + "content": "determined by a min-norm solver [7] which yields a common descent direction of all the objectives.", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "However, compared to the increasingly wide application prospect, the gradient-based iterative", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 89, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "algorithms are relatively understudied, especially for the online learning setting. Multi-objective", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 95, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 506, + 106 + ], + "score": 1.0, + "content": "online learning is of essential importance due to reasons in two folds. First, due to the data explosion in", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 88, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "many real-world scenarios such as web applications, making in-time predictions requires performing", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 88, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "online learning. Second, the theoretical investigation of multi-objective online learning will lay a solid", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 89, + 129, + 99, + 138 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "foundation for the design of new optimizers for multi-task deep neural networks. This is analogous to", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 89, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "the single-objective setting, where nearly all the optimizers for training DNNs are initially analyzed", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 149, + 407, + 161 + ], + "spans": [ + { + "bbox": [ + 88, + 151, + 100, + 161 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 149, + 407, + 161 + ], + "score": 1.0, + "content": "in the online setting, such as AdaGrad [8], Adam [16], and AMSGrad [28].", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 89, + 168, + 100, + 177 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "In this paper, we give a systematic study of multi-objective online learning. To begin with, we", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 89, + 178, + 100, + 188 + ], + "score": 1.0, + "content": "47", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "formulate the framework of Multi-Objective Online Convex Optimization (MO-OCO). The first", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 187, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 89, + 190, + 100, + 199 + ], + "score": 1.0, + "content": "48", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 187, + 507, + 201 + ], + "score": 1.0, + "content": "major challenge is the lack of regret definitions in the multi-objective setting. To tackle this challenge,", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 89, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "49", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "we need appropriate discrepancy metrics that can be used in the regret definitions, which evaluate the", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 89, + 212, + 100, + 221 + ], + "score": 1.0, + "content": "50", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "gap between any two vector losses by producing scalar values. Intuitively, the Pareto suboptimality", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 219, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 89, + 222, + 100, + 232 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 219, + 506, + 235 + ], + "score": 1.0, + "content": "gap (PSG) metric, which is frequently used in zero-order multi-objective bandits [30, 19], is a very", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 231, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 89, + 233, + 100, + 243 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 231, + 507, + 244 + ], + "score": 1.0, + "content": "promising candidate. It can yield scalarized distances from any vector loss to a given comparator set.", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 241, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 89, + 244, + 100, + 253 + ], + "score": 1.0, + "content": "53", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "score": 1.0, + "content": "We can thus define the multi-objective regret by simply plugging in PSG as the discrepancy metric.", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 89, + 255, + 100, + 264 + ], + "score": 1.0, + "content": "54", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "However, as a metric designed purely from the geometric view, PSG is intrinsically difficult to be", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 89, + 266, + 100, + 275 + ], + "score": 1.0, + "content": "55", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "optimized directly via gradient-based iterative methods. To resolve this problem, for the PSG-based", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 89, + 277, + 100, + 286 + ], + "score": 1.0, + "content": "56", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "multi-objective dynamic regret, we derive its equivalent unconstrained max-min form via a highly", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 89, + 288, + 100, + 297 + ], + "score": 1.0, + "content": "57", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "non-trivial transformation. This form is intuitive to the design of first-order multi-objective online", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 89, + 298, + 100, + 308 + ], + "score": 1.0, + "content": "58", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "algorithms, indicating that we should select a convex combination of the gradients at each round.", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 89, + 309, + 100, + 319 + ], + "score": 1.0, + "content": "59", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "Unfortunately, for the PSG-based static variant, such an equivalence does not exist. To remedy this", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 89, + 321, + 100, + 330 + ], + "score": 1.0, + "content": "60", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "issue, we make extensions of the dynamic variant by fixing the comparator set and the composite", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 329, + 435, + 343 + ], + "spans": [ + { + "bbox": [ + 90, + 332, + 99, + 340 + ], + "score": 1.0, + "content": "61", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 329, + 435, + 343 + ], + "score": 1.0, + "content": "weights, which yields an appropriate definition of the multi-objective static regret.", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 89, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "62", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "Based on the MO-OCO framework, we develop a novel multi-objective online algorithm termed", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 89, + 358, + 99, + 369 + ], + "score": 1.0, + "content": "63", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "Doubly Regularized Online Mirror Multiple Descent. The key module of the algorithm is the gradient", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 89, + 370, + 99, + 379 + ], + "score": 1.0, + "content": "64", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "composition scheme, which calculates a composite gradient in the form of a convex combination of", + "type": "text", + "cross_page": true + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 89, + 380, + 99, + 390 + ], + "score": 1.0, + "content": "65", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "the gradients of all objectives. Intuitively, the most direct way to determine the composite weights is", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 88, + 391, + 99, + 401 + ], + "score": 1.0, + "content": "66", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "to apply the min-norm solver [7] commonly used in offline multi-objective optimization. However,", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 89, + 402, + 99, + 412 + ], + "score": 1.0, + "content": "67", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 399, + 506, + 414 + ], + "score": 1.0, + "content": "directly applying min-norm is not workable in the online setting. Specifically, the composite weights", + "type": "text", + "cross_page": true + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 88, + 413, + 99, + 423 + ], + "score": 1.0, + "content": "68", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "in min-norm are only determined by the gradients at the current round. In the online setting, since", + "type": "text", + "cross_page": true + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 420, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 89, + 424, + 99, + 434 + ], + "score": 1.0, + "content": "69", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 420, + 506, + 437 + ], + "score": 1.0, + "content": "the gradients can be adversarial, they may result in undesired composite weights, further producing", + "type": "text", + "cross_page": true + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 88, + 435, + 99, + 445 + ], + "score": 1.0, + "content": "70", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "a composite gradient that reversely optimizes the loss. To rigorously verify this point, we give a", + "type": "text", + "cross_page": true + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 88, + 445, + 100, + 456 + ], + "score": 1.0, + "content": "71", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "showcase in which equipping OMD with vanilla min-norm even incurs a linear regret, showing that", + "type": "text", + "cross_page": true + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 89, + 456, + 99, + 466 + ], + "score": 1.0, + "content": "72", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "score": 1.0, + "content": "only regularizing the iterate, as in OMD, is not enough to guarantee sublinear regrets in the multi-", + "type": "text", + "cross_page": true + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 89, + 468, + 99, + 477 + ], + "score": 1.0, + "content": "73", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "objective setting. To fix this issue, we devise a novel min-regularized-norm solver with an explicit", + "type": "text", + "cross_page": true + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 477, + 492, + 489 + ], + "spans": [ + { + "bbox": [ + 89, + 479, + 99, + 488 + ], + "score": 1.0, + "content": "74", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 477, + 492, + 489 + ], + "score": 1.0, + "content": "regularization on composite weights. Equipping it with OMD results in our proposed algorithm.", + "type": "text", + "cross_page": true + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 90, + 495, + 100, + 504 + ], + "score": 1.0, + "content": "75", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "We then conduct the theoretical analysis for our proposed algorithm. We derive a multi-objective static", + "type": "text", + "cross_page": true + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 503, + 510, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "76", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 101, + 503, + 161, + 522 + ], + "score": 1.0, + "content": "regret bound", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 161, + 505, + 194, + 518 + ], + "score": 0.94, + "content": "O ( \\sqrt { T } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 194, + 503, + 378, + 522 + ], + "score": 1.0, + "content": "and a multi-objective dynamic regret bound", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 378, + 504, + 434, + 519 + ], + "score": 0.93, + "content": "O ( V _ { T } ^ { 1 / 3 } T ^ { 2 / 3 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 435, + 503, + 510, + 522 + ], + "score": 1.0, + "content": "for DR-OMMD.", + "type": "text", + "cross_page": true + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 100, + 528 + ], + "score": 1.0, + "content": "77", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "Both bounds match the optimal bounds in the single-objective setting [11, 34]. Our analysis also", + "type": "text", + "cross_page": true + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 527, + 482, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 100, + 539 + ], + "score": 1.0, + "content": "78", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 527, + 482, + 541 + ], + "score": 1.0, + "content": "shows that DR-OMMD attains a lower regret than linearization with fixed composite weights.", + "type": "text", + "cross_page": true + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 89, + 546, + 99, + 555 + ], + "score": 1.0, + "content": "79", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "To evaluate the effectiveness of DR-OMMD, we conduct extensive experiments on both simulation", + "type": "text", + "cross_page": true + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 89, + 557, + 100, + 567 + ], + "score": 1.0, + "content": "80", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "datasets and real-world datasets. We first elaborate simulation experiments, in which we find", + "type": "text", + "cross_page": true + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 88, + 568, + 99, + 578 + ], + "score": 1.0, + "content": "81", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "that DR-OMMD attains lower regret than vanilla min-norm and linearization, which verifies the", + "type": "text", + "cross_page": true + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 88, + 580, + 100, + 589 + ], + "score": 1.0, + "content": "82", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "superiority of the min-regularized-norm solver. We then realize adaptive regularization via multi-", + "type": "text", + "cross_page": true + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 89, + 591, + 100, + 600 + ], + "score": 1.0, + "content": "83", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "objective optimization on real-world datasets, and find that adaptive regularization with DR-OMMD", + "type": "text", + "cross_page": true + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 600, + 364, + 611 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 100, + 611 + ], + "score": 1.0, + "content": "84", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 600, + 364, + 611 + ], + "score": 1.0, + "content": "significantly outperforms fixed regularization with linearization.", + "type": "text", + "cross_page": true + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 89, + 617, + 100, + 627 + ], + "score": 1.0, + "content": "85", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "In summary, in this paper, we give the first systematic study of multi-objective online learning, which", + "type": "text", + "cross_page": true + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 89, + 629, + 100, + 638 + ], + "score": 1.0, + "content": "86", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "encompasses a novel framework, a new algorithm, and corresponding non-trivial theoretical analysis.", + "type": "text", + "cross_page": true + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 89, + 639, + 100, + 649 + ], + "score": 1.0, + "content": "87", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "We believe that this work paves the way for future research on more advanced multiple-objective", + "type": "text", + "cross_page": true + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 646, + 507, + 663 + ], + "spans": [ + { + "bbox": [ + 89, + 650, + 100, + 660 + ], + "score": 1.0, + "content": "88", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 646, + 507, + 663 + ], + "score": 1.0, + "content": "optimization algorithms, which may inspire the design of new optimizers for multi-task deep learning.", + "type": "text", + "cross_page": true + } + ], + "index": 50, + "is_list_start_line": true + } + ], + "index": 28, + "bbox_fs": [ + 88, + 497, + 506, + 597 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 601, + 505, + 711 + ], + "lines": [], + "index": 37.5, + "bbox_fs": [ + 88, + 600, + 507, + 713 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 90, + 73, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 89, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 89, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 104, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "However, compared to the increasingly wide application prospect, the gradient-based iterative", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 89, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 89, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "39", + "type": "text" + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "algorithms are relatively understudied, especially for the online learning setting. Multi-objective", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 89, + 95, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 89, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 95, + 506, + 106 + ], + "score": 1.0, + "content": "online learning is of essential importance due to reasons in two folds. First, due to the data explosion in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 88, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 88, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "many real-world scenarios such as web applications, making in-time predictions requires performing", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 88, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 88, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "online learning. Second, the theoretical investigation of multi-objective online learning will lay a solid", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 89, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 89, + 129, + 99, + 138 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "foundation for the design of new optimizers for multi-task deep neural networks. This is analogous to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 89, + 138, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 89, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 106, + 138, + 506, + 150 + ], + "score": 1.0, + "content": "the single-objective setting, where nearly all the optimizers for training DNNs are initially analyzed", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 88, + 149, + 407, + 161 + ], + "spans": [ + { + "bbox": [ + 88, + 151, + 100, + 161 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 149, + 407, + 161 + ], + "score": 1.0, + "content": "in the online setting, such as AdaGrad [8], Adam [16], and AMSGrad [28].", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 89, + 165, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 89, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 89, + 168, + 100, + 177 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "In this paper, we give a systematic study of multi-objective online learning. To begin with, we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 89, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 89, + 178, + 100, + 188 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "formulate the framework of Multi-Objective Online Convex Optimization (MO-OCO). The first", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 89, + 187, + 507, + 201 + ], + "spans": [ + { + "bbox": [ + 89, + 190, + 100, + 199 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 187, + 507, + 201 + ], + "score": 1.0, + "content": "major challenge is the lack of regret definitions in the multi-objective setting. To tackle this challenge,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 89, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 89, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "49", + "type": "text" + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "we need appropriate discrepancy metrics that can be used in the regret definitions, which evaluate the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 89, + 209, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 89, + 212, + 100, + 221 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 104, + 209, + 506, + 223 + ], + "score": 1.0, + "content": "gap between any two vector losses by producing scalar values. Intuitively, the Pareto suboptimality", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 89, + 219, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 89, + 222, + 100, + 232 + ], + "score": 1.0, + "content": "51", + "type": "text" + }, + { + "bbox": [ + 104, + 219, + 506, + 235 + ], + "score": 1.0, + "content": "gap (PSG) metric, which is frequently used in zero-order multi-objective bandits [30, 19], is a very", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 89, + 231, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 89, + 233, + 100, + 243 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 105, + 231, + 507, + 244 + ], + "score": 1.0, + "content": "promising candidate. It can yield scalarized distances from any vector loss to a given comparator set.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 89, + 241, + 507, + 255 + ], + "spans": [ + { + "bbox": [ + 89, + 244, + 100, + 253 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 507, + 255 + ], + "score": 1.0, + "content": "We can thus define the multi-objective regret by simply plugging in PSG as the discrepancy metric.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 89, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 89, + 255, + 100, + 264 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "However, as a metric designed purely from the geometric view, PSG is intrinsically difficult to be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 89, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 89, + 266, + 100, + 275 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 105, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "optimized directly via gradient-based iterative methods. To resolve this problem, for the PSG-based", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 89, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 89, + 277, + 100, + 286 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "multi-objective dynamic regret, we derive its equivalent unconstrained max-min form via a highly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 89, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 89, + 288, + 100, + 297 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "non-trivial transformation. This form is intuitive to the design of first-order multi-objective online", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 89, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 89, + 298, + 100, + 308 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "algorithms, indicating that we should select a convex combination of the gradients at each round.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 89, + 309, + 100, + 319 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "Unfortunately, for the PSG-based static variant, such an equivalence does not exist. To remedy this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 89, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 89, + 321, + 100, + 330 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "issue, we make extensions of the dynamic variant by fixing the comparator set and the composite", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 90, + 329, + 435, + 343 + ], + "spans": [ + { + "bbox": [ + 90, + 332, + 99, + 340 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 105, + 329, + 435, + 343 + ], + "score": 1.0, + "content": "weights, which yields an appropriate definition of the multi-objective static regret.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 89, + 345, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 89, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 89, + 348, + 99, + 357 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 106, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "Based on the MO-OCO framework, we develop a novel multi-objective online algorithm termed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 89, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 89, + 358, + 99, + 369 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "Doubly Regularized Online Mirror Multiple Descent. The key module of the algorithm is the gradient", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 89, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 89, + 370, + 99, + 379 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "score": 1.0, + "content": "composition scheme, which calculates a composite gradient in the form of a convex combination of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 89, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 89, + 380, + 99, + 390 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 106, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "the gradients of all objectives. Intuitively, the most direct way to determine the composite weights is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 88, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 88, + 391, + 99, + 401 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 389, + 506, + 402 + ], + "score": 1.0, + "content": "to apply the min-norm solver [7] commonly used in offline multi-objective optimization. However,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 89, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 89, + 402, + 99, + 412 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 104, + 399, + 506, + 414 + ], + "score": 1.0, + "content": "directly applying min-norm is not workable in the online setting. Specifically, the composite weights", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 88, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 88, + 413, + 99, + 423 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 104, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "in min-norm are only determined by the gradients at the current round. In the online setting, since", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 420, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 89, + 424, + 99, + 434 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 105, + 420, + 506, + 437 + ], + "score": 1.0, + "content": "the gradients can be adversarial, they may result in undesired composite weights, further producing", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 88, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 88, + 435, + 99, + 445 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "a composite gradient that reversely optimizes the loss. To rigorously verify this point, we give a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 88, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 88, + 445, + 100, + 456 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "showcase in which equipping OMD with vanilla min-norm even incurs a linear regret, showing that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 89, + 456, + 99, + 466 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "score": 1.0, + "content": "only regularizing the iterate, as in OMD, is not enough to guarantee sublinear regrets in the multi-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 89, + 468, + 99, + 477 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "objective setting. To fix this issue, we devise a novel min-regularized-norm solver with an explicit", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 89, + 477, + 492, + 489 + ], + "spans": [ + { + "bbox": [ + 89, + 479, + 99, + 488 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 105, + 477, + 492, + 489 + ], + "score": 1.0, + "content": "regularization on composite weights. Equipping it with OMD results in our proposed algorithm.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 90, + 493, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 90, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 90, + 495, + 100, + 504 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "We then conduct the theoretical analysis for our proposed algorithm. We derive a multi-objective static", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 89, + 503, + 510, + 522 + ], + "spans": [ + { + "bbox": [ + 89, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 101, + 503, + 161, + 522 + ], + "score": 1.0, + "content": "regret bound", + "type": "text" + }, + { + "bbox": [ + 161, + 505, + 194, + 518 + ], + "score": 0.94, + "content": "O ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 503, + 378, + 522 + ], + "score": 1.0, + "content": "and a multi-objective dynamic regret bound", + "type": "text" + }, + { + "bbox": [ + 378, + 504, + 434, + 519 + ], + "score": 0.93, + "content": "O ( V _ { T } ^ { 1 / 3 } T ^ { 2 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 503, + 510, + 522 + ], + "score": 1.0, + "content": "for DR-OMMD.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 100, + 528 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "Both bounds match the optimal bounds in the single-objective setting [11, 34]. Our analysis also", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 527, + 482, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 100, + 539 + ], + "score": 1.0, + "content": "78", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 482, + 541 + ], + "score": 1.0, + "content": "shows that DR-OMMD attains a lower regret than linearization with fixed composite weights.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 90, + 544, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 89, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 89, + 546, + 99, + 555 + ], + "score": 1.0, + "content": "79", + "type": "text" + }, + { + "bbox": [ + 105, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "To evaluate the effectiveness of DR-OMMD, we conduct extensive experiments on both simulation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 89, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 89, + 557, + 100, + 567 + ], + "score": 1.0, + "content": "80", + "type": "text" + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "datasets and real-world datasets. We first elaborate simulation experiments, in which we find", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 88, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 88, + 568, + 99, + 578 + ], + "score": 1.0, + "content": "81", + "type": "text" + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "that DR-OMMD attains lower regret than vanilla min-norm and linearization, which verifies the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 88, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 88, + 580, + 100, + 589 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "superiority of the min-regularized-norm solver. We then realize adaptive regularization via multi-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 89, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 89, + 591, + 100, + 600 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "objective optimization on real-world datasets, and find that adaptive regularization with DR-OMMD", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 89, + 600, + 364, + 611 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 100, + 611 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 105, + 600, + 364, + 611 + ], + "score": 1.0, + "content": "significantly outperforms fixed regularization with linearization.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 90, + 615, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 89, + 615, + 506, + 629 + ], + 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As a meta-algorithm, by instantiating", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 381 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "different regularization functions, OMD can induce two important algorithms, i.e., Online Gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 378, + 341, + 393 + ], + "spans": [ + { + "bbox": [ + 86, + 382, + 99, + 391 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 341, + 393 + ], + "score": 1.0, + "content": "Descent [38, 13] and Online Exponentiated Gradient [11].", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 93, + 403, + 257, + 416 + ], + "lines": [ + { + "bbox": [ + 90, + 402, + 258, + 418 + ], + "spans": [ + { + "bbox": [ + 90, + 402, + 258, + 418 + ], + "score": 1.0, + "content": "11 2.2 Multi-Objective Optimization", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 86, + 423, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 86, + 422, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 86, + 425, + 100, + 436 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 104, + 422, + 506, + 438 + ], + "score": 1.0, + "content": "Multiple-objective optimization (MOO) is concerned with solving the problems of optimizing", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 85, + 437, + 100, + 447 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "multiple objectives simultaneously [39, 29]. In general, since different objectives may conflict with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 86, + 448, + 100, + 457 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "each other, there is no single solution that can optimize all the objectives at the same time. Instead,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 86, + 459, + 100, + 468 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "MOO seeks to find solutions that achieve Pareto optimality. Next, we exposit Pareto optimality and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 85, + 469, + 101, + 479 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 104, + 466, + 340, + 481 + ], + "score": 1.0, + "content": "related definitions more formally using a vector-valued loss", + "type": "text" + }, + { + "bbox": [ + 340, + 467, + 423, + 479 + ], + "score": 0.93, + "content": "H = ( h ^ { 1 } , \\ldots , \\overline { { { h ^ { m } } } } ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "as objectives, where", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 478, + 402, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 480, + 100, + 490 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 106, + 479, + 135, + 490 + ], + "score": 0.89, + "content": "m \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 478, + 153, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 478, + 202, + 489 + ], + "score": 0.85, + "content": "h ^ { i } : { \\mathcal { K } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 478, + 206, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 479, + 268, + 491 + ], + "score": 0.89, + "content": "i \\in \\{ 1 , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 478, + 272, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 272, + 479, + 302, + 490 + ], + "score": 0.84, + "content": "\\kappa \\subset \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 478, + 329, + 491 + ], + "score": 1.0, + "content": ", is the", + "type": "text" + }, + { + "bbox": [ + 329, + 479, + 334, + 488 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 478, + 402, + 491 + ], + "score": 1.0, + "content": "-th loss function.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 101, + 493, + 503, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 361, + 505 + ], + "score": 1.0, + "content": "Definition 2.1 (Pareto optimality). (a) For any two solutions", + "type": "text" + }, + { + "bbox": [ + 361, + 493, + 401, + 504 + ], + "score": 0.93, + "content": "x , x ^ { \\prime } \\in \\mathcal { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 492, + 452, + 505 + ], + "score": 1.0, + "content": ", we say that", + "type": "text" + }, + { + "bbox": [ + 453, + 495, + 460, + 503 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "dominates", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 116, + 514 + ], + "score": 0.84, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 503, + 169, + 516 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 169, + 504, + 201, + 514 + ], + "score": 0.91, + "content": "\\boldsymbol { x } \\prec \\boldsymbol { x } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 503, + 216, + 516 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 216, + 504, + 249, + 514 + ], + "score": 0.9, + "content": "x ^ { \\prime } \\succ x", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 503, + 264, + 516 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 264, + 503, + 330, + 516 + ], + "score": 0.93, + "content": "h ^ { i } ( x ) \\leq h ^ { i } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 503, + 361, + 516 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 361, + 505, + 366, + 514 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 503, + 458, + 516 + ], + "score": 1.0, + "content": ", and there exists one", + "type": "text" + }, + { + "bbox": [ + 458, + 505, + 463, + 514 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 514, + 480, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 168, + 527 + ], + "score": 0.92, + "content": "h ^ { i } ( x ) < h ^ { i } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 514, + 263, + 527 + ], + "score": 1.0, + "content": "; otherwise, we say that", + "type": "text" + }, + { + "bbox": [ + 264, + 517, + 271, + 525 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 514, + 347, + 527 + ], + "score": 1.0, + "content": "does not dominate", + "type": "text" + }, + { + "bbox": [ + 348, + 515, + 357, + 525 + ], + "score": 0.85, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 514, + 406, + 527 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 406, + 515, + 434, + 526 + ], + "score": 0.92, + "content": "x \\not \\prec x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 514, + 446, + 527 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 447, + 515, + 475, + 526 + ], + "score": 0.91, + "content": "x ^ { \\prime } \\nsimeq x", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 514, + 480, + 527 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 98, + 526, + 489, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 489, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 165, + 539 + ], + "score": 1.0, + "content": "(b) A solution", + "type": "text" + }, + { + "bbox": [ + 165, + 526, + 197, + 536 + ], + "score": 0.92, + "content": "x ^ { * } \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 524, + 476, + 539 + ], + "score": 1.0, + "content": "is called Pareto optimal if it is not dominated by any other solution in", + "type": "text" + }, + { + "bbox": [ + 477, + 526, + 485, + 536 + ], + "score": 0.82, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 524, + 489, + 539 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 87, + 545, + 504, + 579 + ], + "lines": [ + { + "bbox": [ + 87, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 87, + 547, + 100, + 556 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "There may exist multiple Pareto optimal solutions. For example, it is easy to show that the optimizer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 87, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 87, + 559, + 100, + 567 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 222, + 570 + ], + "score": 1.0, + "content": "of any single objective, i.e.,", + "type": "text" + }, + { + "bbox": [ + 222, + 556, + 386, + 569 + ], + "score": 0.91, + "content": "x _ { i } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { K } } h ^ { i } ( x ) , i \\in \\{ \\bar { 1 } , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 556, + 505, + 570 + ], + "score": 1.0, + "content": ", is Pareto optimal. Different", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 87, + 568, + 419, + 580 + ], + "spans": [ + { + "bbox": [ + 87, + 569, + 100, + 579 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 104, + 568, + 419, + 580 + ], + "score": 1.0, + "content": "Pareto optimal solutions reflect different trade-offs among the objectives [17].", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 94, + 581, + 496, + 605 + ], + "lines": [ + { + "bbox": [ + 90, + 581, + 474, + 594 + ], + "spans": [ + { + "bbox": [ + 90, + 581, + 438, + 594 + ], + "score": 1.0, + "content": "25 Definition 2.2 (Pareto front). (a) All Pareto optimal solutions form the Pareto set", + "type": "text" + }, + { + "bbox": [ + 438, + 582, + 470, + 593 + ], + "score": 0.91, + "content": "{ \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 581, + 474, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 90, + 592, + 495, + 606 + ], + "spans": [ + { + "bbox": [ + 90, + 592, + 176, + 606 + ], + "score": 1.0, + "content": "26 (b) The image of", + "type": "text" + }, + { + "bbox": [ + 177, + 593, + 208, + 605 + ], + "score": 0.93, + "content": "{ \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 592, + 366, + 606 + ], + "score": 1.0, + "content": "constitutes the Pareto front, denoted as", + "type": "text" + }, + { + "bbox": [ + 366, + 592, + 495, + 605 + ], + "score": 0.9, + "content": "\\mathcal { P } ( H ) = \\{ H ( x ) \\mid x \\in \\mathcal { P } _ { K } ( H ) \\} .", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 86, + 612, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "127", + "type": "text" + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Now that we have established the notion of optimality in MOO, we proceed to introduce the metrics", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 86, + 625, + 100, + 635 + ], + "score": 1.0, + "content": "128", + "type": "text" + }, + { + "bbox": [ + 105, + 623, + 320, + 636 + ], + "score": 1.0, + "content": "that measure the discrepancy of an arbitrary solution", + "type": "text" + }, + { + "bbox": [ + 320, + 624, + 347, + 634 + ], + "score": 0.91, + "content": "x \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "from being optimal. Recall that, in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 105, + 634, + 328, + 647 + ], + "score": 1.0, + "content": "single-objective setting with merely one loss function", + "type": "text" + }, + { + "bbox": [ + 328, + 635, + 376, + 645 + ], + "score": 0.91, + "content": "h : \\mathcal { Q } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 634, + 408, + 647 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 408, + 635, + 439, + 645 + ], + "score": 0.91, + "content": "\\mathcal { Q } \\subset \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 634, + 475, + 647 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 475, + 635, + 502, + 645 + ], + "score": 0.91, + "content": "z \\in \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 634, + 506, + 647 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 104, + 644, + 158, + 658 + ], + "score": 1.0, + "content": "the loss gap", + "type": "text" + }, + { + "bbox": [ + 158, + 645, + 253, + 657 + ], + "score": 0.9, + "content": "h ( z ) - \\mathrm { { m i n } } _ { z ^ { \\prime \\prime } \\in \\mathcal { Q } } h ( z ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "is directly the discrepancy measure. However, in MOO with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 655, + 508, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 667 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 104, + 655, + 221, + 669 + ], + "score": 1.0, + "content": "more than one loss, for any", + "type": "text" + }, + { + "bbox": [ + 221, + 657, + 249, + 666 + ], + "score": 0.87, + "content": "x \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 655, + 305, + 669 + ], + "score": 1.0, + "content": ", the loss gap", + "type": "text" + }, + { + "bbox": [ + 305, + 656, + 368, + 668 + ], + "score": 0.94, + "content": "H ( x ) - \\overline { { H ( x ^ { \\prime \\prime } ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 655, + 400, + 669 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 401, + 656, + 456, + 668 + ], + "score": 0.93, + "content": "x ^ { \\prime \\prime } \\in { \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 655, + 508, + 669 + ], + "score": 1.0, + "content": ", is a vector.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 85, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 104, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Intuitionally, the desired discrepancy metric shall scalarize the vector-valued loss gap and yield", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 676, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 676, + 506, + 692 + ], + "score": 1.0, + "content": "the value 0 for any Pareto optimal solution. In general, there are two commonly used discrepancy", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 100, + 700 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 104, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "metrics in MOO, i.e. Pareto suboptimality gap (PSG) [30] and Hypervolume (HV) [4]. As HV is a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 85, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "volume-based metric, it is more difficult to optimize or analyze via iterative algorithms [36]. Hence", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 86, + 711, + 495, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 495, + 723 + ], + "score": 1.0, + "content": "in this paper, we adopt PSG, which has been extensively used in multi-objective bandits [30, 19].", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 92, + 73, + 252, + 84 + ], + "lines": [ + { + "bbox": [ + 87, + 70, + 254, + 87 + ], + "spans": [ + { + "bbox": [ + 87, + 70, + 254, + 87 + ], + "score": 1.0, + "content": "92 2.1 Online Convex Optimization", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 89, + 92, + 505, + 159 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 89, + 92, + 506, + 160 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 162, + 401, + 186 + ], + "lines": [ + { + "bbox": [ + 209, + 162, + 401, + 186 + ], + "spans": [ + { + "bbox": [ + 209, + 162, + 401, + 186 + ], + "score": 0.93, + "content": "R _ { S } ( T ) = \\sum _ { t = 1 } ^ { T } f _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x ^ { * } \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } f _ { t } ( x ^ { * } ) .", + "type": "interline_equation", + "image_path": "fbff3e00487c576616ab7a128d7bdf67f62a72d02e7057df9fe3abb4b171d697.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 209, + 162, + 401, + 186 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 190, + 506, + 224 + ], + "lines": [ + { + "bbox": [ + 89, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 89, + 192, + 100, + 201 + ], + "score": 1.0, + "content": "99", + "type": "text" + }, + { + "bbox": [ + 104, + 190, + 506, + 203 + ], + "score": 1.0, + "content": "Note that the above regret is the static regret [10], which compares the learner’s cumulative loss", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 201, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 87, + 204, + 99, + 212 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "with that of a fixed decision. 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As a meta-algorithm, by instantiating", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 86, + 370, + 100, + 381 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "different regularization functions, OMD can induce two important algorithms, i.e., Online Gradient", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 378, + 341, + 393 + ], + "spans": [ + { + "bbox": [ + 86, + 382, + 99, + 391 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 341, + 393 + ], + "score": 1.0, + "content": "Descent [38, 13] and Online Exponentiated Gradient [11].", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + } + ], + "index": 18.5, + "bbox_fs": [ + 86, + 346, + 505, + 393 + ] + }, + { + "type": "title", + "bbox": [ + 93, + 403, + 257, + 416 + ], + "lines": [ + { + "bbox": [ + 90, + 402, + 258, + 418 + ], + "spans": [ + { + "bbox": [ + 90, + 402, + 258, + 418 + ], + "score": 1.0, + "content": "11 2.2 Multi-Objective Optimization", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "index", + "bbox": [ + 86, + 423, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 86, + 422, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 86, + 425, + 100, + 436 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 104, + 422, + 506, + 438 + ], + "score": 1.0, + "content": "Multiple-objective optimization (MOO) is concerned with solving the problems of optimizing", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 85, + 437, + 100, + 447 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "multiple objectives simultaneously [39, 29]. In general, since different objectives may conflict with", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 86, + 448, + 100, + 457 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "each other, there is no single solution that can optimize all the objectives at the same time. Instead,", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 86, + 459, + 100, + 468 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "MOO seeks to find solutions that achieve Pareto optimality. Next, we exposit Pareto optimality and", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 85, + 469, + 101, + 479 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 104, + 466, + 340, + 481 + ], + "score": 1.0, + "content": "related definitions more formally using a vector-valued loss", + "type": "text" + }, + { + "bbox": [ + 340, + 467, + 423, + 479 + ], + "score": 0.93, + "content": "H = ( h ^ { 1 } , \\ldots , \\overline { { { h ^ { m } } } } ) ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "as objectives, where", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 478, + 402, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 480, + 100, + 490 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 106, + 479, + 135, + 490 + ], + "score": 0.89, + "content": "m \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 478, + 153, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 153, + 478, + 202, + 489 + ], + "score": 0.85, + "content": "h ^ { i } : { \\mathcal { K } } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 478, + 206, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 206, + 479, + 268, + 491 + ], + "score": 0.89, + "content": "i \\in \\{ 1 , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 478, + 272, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 272, + 479, + 302, + 490 + ], + "score": 0.84, + "content": "\\kappa \\subset \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 478, + 329, + 491 + ], + "score": 1.0, + "content": ", is the", + "type": "text" + }, + { + "bbox": [ + 329, + 479, + 334, + 488 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 478, + 402, + 491 + ], + "score": 1.0, + "content": "-th loss function.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + } + ], + "index": 24.5, + "bbox_fs": [ + 85, + 422, + 507, + 491 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 493, + 503, + 526 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 361, + 505 + ], + "score": 1.0, + "content": "Definition 2.1 (Pareto optimality). (a) For any two solutions", + "type": "text" + }, + { + "bbox": [ + 361, + 493, + 401, + 504 + ], + "score": 0.93, + "content": "x , x ^ { \\prime } \\in \\mathcal { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 492, + 452, + 505 + ], + "score": 1.0, + "content": ", we say that", + "type": "text" + }, + { + "bbox": [ + 453, + 495, + 460, + 503 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "dominates", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 116, + 514 + ], + "score": 0.84, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 503, + 169, + 516 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 169, + 504, + 201, + 514 + ], + "score": 0.91, + "content": "\\boldsymbol { x } \\prec \\boldsymbol { x } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 503, + 216, + 516 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 216, + 504, + 249, + 514 + ], + "score": 0.9, + "content": "x ^ { \\prime } \\succ x", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 503, + 264, + 516 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 264, + 503, + 330, + 516 + ], + "score": 0.93, + "content": "h ^ { i } ( x ) \\leq h ^ { i } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 503, + 361, + 516 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 361, + 505, + 366, + 514 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 503, + 458, + 516 + ], + "score": 1.0, + "content": ", and there exists one", + "type": "text" + }, + { + "bbox": [ + 458, + 505, + 463, + 514 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 514, + 480, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 514, + 168, + 527 + ], + "score": 0.92, + "content": "h ^ { i } ( x ) < h ^ { i } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 514, + 263, + 527 + ], + "score": 1.0, + "content": "; otherwise, we say that", + "type": "text" + }, + { + "bbox": [ + 264, + 517, + 271, + 525 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 514, + 347, + 527 + ], + "score": 1.0, + "content": "does not dominate", + "type": "text" + }, + { + "bbox": [ + 348, + 515, + 357, + 525 + ], + "score": 0.85, + "content": "x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 514, + 406, + 527 + ], + "score": 1.0, + "content": ", denoted as", + "type": "text" + }, + { + "bbox": [ + 406, + 515, + 434, + 526 + ], + "score": 0.92, + "content": "x \\not \\prec x ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 514, + 446, + 527 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 447, + 515, + 475, + 526 + ], + "score": 0.91, + "content": "x ^ { \\prime } \\nsimeq x", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 514, + 480, + 527 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 492, + 506, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 98, + 526, + 489, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 489, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 165, + 539 + ], + "score": 1.0, + "content": "(b) A solution", + "type": "text" + }, + { + "bbox": [ + 165, + 526, + 197, + 536 + ], + "score": 0.92, + "content": "x ^ { * } \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 524, + 476, + 539 + ], + "score": 1.0, + "content": "is called Pareto optimal if it is not dominated by any other solution in", + "type": "text" + }, + { + "bbox": [ + 477, + 526, + 485, + 536 + ], + "score": 0.82, + "content": "\\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 524, + 489, + 539 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 524, + 489, + 539 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 545, + 504, + 579 + ], + "lines": [ + { + "bbox": [ + 87, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 87, + 547, + 100, + 556 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 104, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "There may exist multiple Pareto optimal solutions. For example, it is easy to show that the optimizer", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 87, + 559, + 100, + 567 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 104, + 556, + 222, + 570 + ], + "score": 1.0, + "content": "of any single objective, i.e.,", + "type": "text" + }, + { + "bbox": [ + 222, + 556, + 386, + 569 + ], + "score": 0.91, + "content": "x _ { i } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { K } } h ^ { i } ( x ) , i \\in \\{ \\bar { 1 } , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 556, + 505, + 570 + ], + "score": 1.0, + "content": ", is Pareto optimal. Different", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 568, + 419, + 580 + ], + "spans": [ + { + "bbox": [ + 87, + 569, + 100, + 579 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 104, + 568, + 419, + 580 + ], + "score": 1.0, + "content": "Pareto optimal solutions reflect different trade-offs among the objectives [17].", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 581, + 474, + 594 + ], + "spans": [ + { + "bbox": [ + 90, + 581, + 438, + 594 + ], + "score": 1.0, + "content": "25 Definition 2.2 (Pareto front). (a) All Pareto optimal solutions form the Pareto set", + "type": "text" + }, + { + "bbox": [ + 438, + 582, + 470, + 593 + ], + "score": 0.91, + "content": "{ \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 581, + 474, + 594 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 592, + 495, + 606 + ], + "spans": [ + { + "bbox": [ + 90, + 592, + 176, + 606 + ], + "score": 1.0, + "content": "26 (b) The image of", + "type": "text" + }, + { + "bbox": [ + 177, + 593, + 208, + 605 + ], + "score": 0.93, + "content": "{ \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 592, + 366, + 606 + ], + "score": 1.0, + "content": "constitutes the Pareto front, denoted as", + "type": "text" + }, + { + "bbox": [ + 366, + 592, + 495, + 605 + ], + "score": 0.9, + "content": "\\mathcal { P } ( H ) = \\{ H ( x ) \\mid x \\in \\mathcal { P } _ { K } ( H ) \\} .", + "type": "inline_equation" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "127", + "type": "text" + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "Now that we have established the notion of optimality in MOO, we proceed to introduce the metrics", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 86, + 625, + 100, + 635 + ], + "score": 1.0, + "content": "128", + "type": "text" + }, + { + "bbox": [ + 105, + 623, + 320, + 636 + ], + "score": 1.0, + "content": "that measure the discrepancy of an arbitrary solution", + "type": "text" + }, + { + "bbox": [ + 320, + 624, + 347, + 634 + ], + "score": 0.91, + "content": "x \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "from being optimal. Recall that, in the", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 634, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 86, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 105, + 634, + 328, + 647 + ], + "score": 1.0, + "content": "single-objective setting with merely one loss function", + "type": "text" + }, + { + "bbox": [ + 328, + 635, + 376, + 645 + ], + "score": 0.91, + "content": "h : \\mathcal { Q } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 634, + 408, + 647 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 408, + 635, + 439, + 645 + ], + "score": 0.91, + "content": "\\mathcal { Q } \\subset \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 634, + 475, + 647 + ], + "score": 1.0, + "content": ", for any", + "type": "text" + }, + { + "bbox": [ + 475, + 635, + 502, + 645 + ], + "score": 0.91, + "content": "z \\in \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 634, + 506, + 647 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 104, + 644, + 158, + 658 + ], + "score": 1.0, + "content": "the loss gap", + "type": "text" + }, + { + "bbox": [ + 158, + 645, + 253, + 657 + ], + "score": 0.9, + "content": "h ( z ) - \\mathrm { { m i n } } _ { z ^ { \\prime \\prime } \\in \\mathcal { Q } } h ( z ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "is directly the discrepancy measure. However, in MOO with", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 655, + 508, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 667 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 104, + 655, + 221, + 669 + ], + "score": 1.0, + "content": "more than one loss, for any", + "type": "text" + }, + { + "bbox": [ + 221, + 657, + 249, + 666 + ], + "score": 0.87, + "content": "x \\in \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 655, + 305, + 669 + ], + "score": 1.0, + "content": ", the loss gap", + "type": "text" + }, + { + "bbox": [ + 305, + 656, + 368, + 668 + ], + "score": 0.94, + "content": "H ( x ) - \\overline { { H ( x ^ { \\prime \\prime } ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 655, + 400, + 669 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 401, + 656, + 456, + 668 + ], + "score": 0.93, + "content": "x ^ { \\prime \\prime } \\in { \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 655, + 508, + 669 + ], + "score": 1.0, + "content": ", is a vector.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 85, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 104, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Intuitionally, the desired discrepancy metric shall scalarize the vector-valued loss gap and yield", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 676, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 676, + 506, + 692 + ], + "score": 1.0, + "content": "the value 0 for any Pareto optimal solution. In general, there are two commonly used discrepancy", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 100, + 700 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 104, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "metrics in MOO, i.e. Pareto suboptimality gap (PSG) [30] and Hypervolume (HV) [4]. As HV is a", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "volume-based metric, it is more difficult to optimize or analyze via iterative algorithms [36]. Hence", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 495, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 495, + 723 + ], + "score": 1.0, + "content": "in this paper, we adopt PSG, which has been extensively used in multi-objective bandits [30, 19].", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "137", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 73, + 318, + 86 + ], + "score": 1.0, + "content": "Definition 2.3 (Pareto suboptimality gap). 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This is a min-norm solver which finds the weights in the simplex that yields the minimum", + "type": "text" + }, + { + "bbox": [ + 492, + 317, + 504, + 327 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 85, + 329, + 100, + 339 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "norm of the composite gradient. Thus MGDA is also called the min-norm method. Existing works", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 337, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 86, + 340, + 100, + 349 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 337, + 507, + 351 + ], + "score": 1.0, + "content": "[7, 29] have shown that MGDA is guaranteed to decrease all the objectives simultaneously until it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 349, + 486, + 361 + ], + "spans": [ + { + "bbox": [ + 86, + 351, + 100, + 360 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 349, + 384, + 361 + ], + "score": 1.0, + "content": "reaches a Pareto optimal decision (under the convex setting where all", + "type": "text" + }, + { + "bbox": [ + 384, + 349, + 394, + 359 + ], + "score": 0.87, + "content": "h ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 349, + 486, + 361 + ], + "score": 1.0, + "content": "are convex functions).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 97, + 376, + 356, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 358, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 358, + 395 + ], + "score": 1.0, + "content": "3 Multi-Objective Online Convex Optimization", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 102, + 401, + 504, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "In this section, we formally formulate the framework of multi-objective optimization in the online", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 404, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 404, + 425 + ], + "score": 1.0, + "content": "setting, termed Multi-Objective Online Convex Optimization (MO-OCO).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 104, + 429, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "Framework overview. We tailor the famous online convex optimization (OCO) framework to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "multi-objective setting, which can be viewed as a repeated game between an online learner and the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 270, + 464 + ], + "score": 1.0, + "content": "adversarial environment. At each round", + "type": "text" + }, + { + "bbox": [ + 270, + 451, + 330, + 463 + ], + "score": 0.93, + "content": "t \\in \\{ 1 , \\ldots , T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 451, + 464, + 464 + ], + "score": 1.0, + "content": ", the learner generates a decision", + "type": "text" + }, + { + "bbox": [ + 464, + 452, + 474, + 462 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "from a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 249, + 474 + ], + "score": 1.0, + "content": "given convex compact decision set", + "type": "text" + }, + { + "bbox": [ + 249, + 462, + 285, + 472 + ], + "score": 0.9, + "content": "\\mathcal { X } \\subset \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 461, + 506, + 474 + ], + "score": 1.0, + "content": ". Then the adversary replies the decision with a vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 163, + 486 + ], + "score": 1.0, + "content": "loss function", + "type": "text" + }, + { + "bbox": [ + 163, + 473, + 241, + 484 + ], + "score": 0.93, + "content": "F _ { t } ( \\bar { x } ) : \\mathcal { X } \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 471, + 287, + 486 + ], + "score": 1.0, + "content": ", where its", + "type": "text" + }, + { + "bbox": [ + 287, + 474, + 292, + 483 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 471, + 354, + 486 + ], + "score": 1.0, + "content": "-th component", + "type": "text" + }, + { + "bbox": [ + 354, + 473, + 424, + 484 + ], + "score": 0.93, + "content": "f _ { t } ^ { i } ( x ) \\ \\bar { : } \\ x \\ \\to \\ \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 471, + 487, + 486 + ], + "score": 1.0, + "content": "belongs to the", + "type": "text" + }, + { + "bbox": [ + 488, + 474, + 492, + 483 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "-th", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 279, + 497 + ], + "score": 1.0, + "content": "objective, and the learner suffers the loss", + "type": "text" + }, + { + "bbox": [ + 279, + 484, + 335, + 496 + ], + "score": 0.92, + "content": "F _ { t } ( x _ { t } ) \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 482, + 506, + 497 + ], + "score": 1.0, + "content": ". 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For", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 86, + 540, + 100, + 550 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 105, + 537, + 184, + 551 + ], + "score": 1.0, + "content": "the static regret, all", + "type": "text" + }, + { + "bbox": [ + 184, + 540, + 194, + 549 + ], + "score": 0.84, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 537, + 381, + 551 + ], + "score": 1.0, + "content": "are identically set as the fixed optimal decision", + "type": "text" + }, + { + "bbox": [ + 381, + 539, + 393, + 548 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "w.r.t. all losses in hindsight,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 549, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 104, + 549, + 124, + 568 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 124, + 549, + 268, + 564 + ], + "score": 0.89, + "content": "\\begin{array} { r } { z _ { t } \\equiv x ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } f _ { t } ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 549, + 390, + 568 + ], + "score": 1.0, + "content": ". For the dynamic regret, each", + "type": "text" + }, + { + "bbox": [ + 390, + 552, + 400, + 562 + ], + "score": 0.82, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 549, + 507, + 568 + ], + "score": 1.0, + "content": "is selected as the optimal", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 562, + 472, + 576 + ], + "spans": [ + { + "bbox": [ + 86, + 564, + 100, + 573 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 562, + 142, + 576 + ], + "score": 1.0, + "content": "decision", + "type": "text" + }, + { + "bbox": [ + 142, + 563, + 153, + 574 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 562, + 267, + 576 + ], + "score": 1.0, + "content": "w.r.t. the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 267, + 563, + 277, + 573 + ], + "score": 0.87, + "content": "f _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 562, + 350, + 576 + ], + "score": 1.0, + "content": "at that round, i.e.,", + "type": "text" + }, + { + "bbox": [ + 351, + 562, + 469, + 574 + ], + "score": 0.9, + "content": "z _ { t } = x _ { t } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } f _ { t } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 562, + 472, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 87, + 579, + 506, + 702 + ], + "lines": [ + { + "bbox": [ + 86, + 572, + 509, + 601 + ], + "spans": [ + { + "bbox": [ + 86, + 582, + 100, + 593 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 101, + 572, + 324, + 601 + ], + "score": 1.0, + "content": "In analogy, we can define the multi-objective regret as", + "type": "text" + }, + { + "bbox": [ + 325, + 579, + 399, + 594 + ], + "score": 0.93, + "content": "\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 572, + 450, + 601 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 450, + 581, + 463, + 592 + ], + "score": 0.88, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 572, + 509, + 601 + ], + "score": 1.0, + "content": "compares", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 86, + 593, + 99, + 604 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 323, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ". However, in general, no single decision can", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 86, + 605, + 99, + 615 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 603, + 403, + 617 + ], + "score": 1.0, + "content": "optimize all the objectives at the same time. Hence, it is natural to compare", + "type": "text" + }, + { + "bbox": [ + 403, + 605, + 414, + 614 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "with a group of Pareto", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 613, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 86, + 615, + 100, + 627 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 312, + 628 + ], + "score": 1.0, + "content": "optimal decisions, which constitute a comparator set", + "type": "text" + }, + { + "bbox": [ + 312, + 614, + 344, + 625 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { t } \\subset \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 613, + 493, + 628 + ], + "score": 1.0, + "content": ". To measure the discrepancy between", + "type": "text" + }, + { + "bbox": [ + 493, + 616, + 504, + 625 + ], + "score": 0.82, + "content": "x _ { t }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 86, + 626, + 100, + 637 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 105, + 624, + 123, + 638 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 625, + 133, + 636 + ], + "score": 0.87, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 624, + 384, + 638 + ], + "score": 1.0, + "content": ", we further introduce the Pareto suboptimality gap (PSG) [30]", + "type": "text" + }, + { + "bbox": [ + 384, + 625, + 437, + 637 + ], + "score": 0.91, + "content": "\\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 624, + 506, + 638 + ], + "score": 1.0, + "content": ". Then the multi-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 636, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 86, + 639, + 100, + 650 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 104, + 636, + 242, + 654 + ], + "score": 1.0, + "content": "objective regret can be defined as", + "type": "text" + }, + { + "bbox": [ + 243, + 636, + 358, + 650 + ], + "score": 0.92, + "content": "\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 636, + 507, + 654 + ], + "score": 1.0, + "content": ". Now we can formulate the static or", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 86, + 651, + 99, + 660 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 649, + 313, + 662 + ], + "score": 1.0, + "content": "the dynamic variant by specifying the comparator set", + "type": "text" + }, + { + "bbox": [ + 314, + 650, + 324, + 660 + ], + "score": 0.87, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 649, + 484, + 662 + ], + "score": 1.0, + "content": "at each round. 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In Appendix B we show that they are equivalent.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 87, + 72, + 506, + 117 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 86, + 73, + 506, + 119 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 122, + 463, + 142 + ], + "lines": [ + { + "bbox": [ + 147, + 122, + 463, + 142 + ], + "spans": [ + { + "bbox": [ + 147, + 122, + 463, + 142 + ], + "score": 0.89, + "content": "\\Delta ( x ; K ^ { * } , H ) = \\operatorname* { i n f } _ { \\epsilon \\geq 0 } \\epsilon , \\quad \\mathrm { s . t . } \\forall x ^ { \\prime \\prime } \\in K ^ { * } , \\exists i \\in \\{ 1 , . . . , m \\} , h ^ { i } ( x ) - \\epsilon < h ^ { i } ( x ^ { \\prime \\prime } ) .", + "type": "interline_equation", + "image_path": "d7fcf54e6ec56f6b8f193f886213dc0261a83213dacd482a99751aa8661113cb.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 147, + 122, + 463, + 142 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 153, + 506, + 187 + ], + "lines": [ + { + "bbox": [ + 86, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 86, + 155, + 99, + 165 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "Clearly, PSG is a distance-based discrepancy metric that motivated from a purely geometric viewpoint.", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 86, + 167, + 100, + 176 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 104, + 165, + 229, + 177 + ], + "score": 1.0, + "content": "In practice, the comparator set", + "type": "text" + }, + { + "bbox": [ + 229, + 165, + 243, + 175 + ], + "score": 0.88, + "content": "\\kappa ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 165, + 367, + 177 + ], + "score": 1.0, + "content": "is often set to be the Pareto set", + "type": "text" + }, + { + "bbox": [ + 367, + 165, + 398, + 176 + ], + "score": 0.9, + "content": "{ \\mathcal { P } } _ { \\kappa } ( H )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 165, + 476, + 177 + ], + "score": 1.0, + "content": "[30]. 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This is a min-norm solver which finds the weights in the simplex that yields the minimum", + "type": "text" + }, + { + "bbox": [ + 492, + 317, + 504, + 327 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 85, + 329, + 100, + 339 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "norm of the composite gradient. Thus MGDA is also called the min-norm method. Existing works", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 337, + 507, + 351 + ], + "spans": [ + { + "bbox": [ + 86, + 340, + 100, + 349 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 105, + 337, + 507, + 351 + ], + "score": 1.0, + "content": "[7, 29] have shown that MGDA is guaranteed to decrease all the objectives simultaneously until it", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 349, + 486, + 361 + ], + "spans": [ + { + "bbox": [ + 86, + 351, + 100, + 360 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 349, + 384, + 361 + ], + "score": 1.0, + "content": "reaches a Pareto optimal decision (under the convex setting where all", + "type": "text" + }, + { + "bbox": [ + 384, + 349, + 394, + 359 + ], + "score": 0.87, + "content": "h ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 349, + 486, + 361 + ], + "score": 1.0, + "content": "are convex functions).", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + } + ], + "index": 17, + "bbox_fs": [ + 85, + 300, + 508, + 361 + ] + }, + { + "type": "title", + "bbox": [ + 97, + 376, + 356, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 374, + 358, + 395 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 358, + 395 + ], + "score": 1.0, + "content": "3 Multi-Objective Online Convex Optimization", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 102, + 401, + 504, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "In this section, we formally formulate the framework of multi-objective optimization in the online", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 413, + 404, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 404, + 425 + ], + "score": 1.0, + "content": "setting, termed Multi-Objective Online Convex Optimization (MO-OCO).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 401, + 505, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 429, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 441 + ], + "score": 1.0, + "content": "Framework overview. We tailor the famous online convex optimization (OCO) framework to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "multi-objective setting, which can be viewed as a repeated game between an online learner and the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 270, + 464 + ], + "score": 1.0, + "content": "adversarial environment. At each round", + "type": "text" + }, + { + "bbox": [ + 270, + 451, + 330, + 463 + ], + "score": 0.93, + "content": "t \\in \\{ 1 , \\ldots , T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 451, + 464, + 464 + ], + "score": 1.0, + "content": ", the learner generates a decision", + "type": "text" + }, + { + "bbox": [ + 464, + 452, + 474, + 462 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "from a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 249, + 474 + ], + "score": 1.0, + "content": "given convex compact decision set", + "type": "text" + }, + { + "bbox": [ + 249, + 462, + 285, + 472 + ], + "score": 0.9, + "content": "\\mathcal { X } \\subset \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 461, + 506, + 474 + ], + "score": 1.0, + "content": ". Then the adversary replies the decision with a vector", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 163, + 486 + ], + "score": 1.0, + "content": "loss function", + "type": "text" + }, + { + "bbox": [ + 163, + 473, + 241, + 484 + ], + "score": 0.93, + "content": "F _ { t } ( \\bar { x } ) : \\mathcal { X } \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 471, + 287, + 486 + ], + "score": 1.0, + "content": ", where its", + "type": "text" + }, + { + "bbox": [ + 287, + 474, + 292, + 483 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 471, + 354, + 486 + ], + "score": 1.0, + "content": "-th component", + "type": "text" + }, + { + "bbox": [ + 354, + 473, + 424, + 484 + ], + "score": 0.93, + "content": "f _ { t } ^ { i } ( x ) \\ \\bar { : } \\ x \\ \\to \\ \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 471, + 487, + 486 + ], + "score": 1.0, + "content": "belongs to the", + "type": "text" + }, + { + "bbox": [ + 488, + 474, + 492, + 483 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "-th", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 279, + 497 + ], + "score": 1.0, + "content": "objective, and the learner suffers the loss", + "type": "text" + }, + { + "bbox": [ + 279, + 484, + 335, + 496 + ], + "score": 0.92, + "content": "F _ { t } ( x _ { t } ) \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 482, + 506, + 497 + ], + "score": 1.0, + "content": ". The goal of the learner is to generate a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 493, + 469, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 196, + 513 + ], + "score": 1.0, + "content": "sequence of decisions", + "type": "text" + }, + { + "bbox": [ + 196, + 496, + 230, + 509 + ], + "score": 0.93, + "content": "\\{ x _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 493, + 339, + 513 + ], + "score": 1.0, + "content": "so that the cumulative loss", + "type": "text" + }, + { + "bbox": [ + 339, + 495, + 393, + 510 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 493, + 469, + 513 + ], + "score": 1.0, + "content": "can be optimized.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 429, + 506, + 513 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 514, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 86, + 508, + 502, + 535 + ], + "spans": [ + { + "bbox": [ + 86, + 518, + 100, + 527 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 101, + 508, + 372, + 535 + ], + "score": 1.0, + "content": "Recall that, in the single-objective setting, the performance metric", + "type": "text" + }, + { + "bbox": [ + 372, + 514, + 502, + 529 + ], + "score": 0.92, + "content": "\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } ( f _ { t } ( x _ { t } ) - f _ { t } ( z _ { t } ) ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 86, + 529, + 100, + 538 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 288, + 540 + ], + "score": 1.0, + "content": "i.e., the regret, compares the actual decisions", + "type": "text" + }, + { + "bbox": [ + 299, + 527, + 392, + 540 + ], + "score": 1.0, + "content": "with some comparator", + "type": "text" + }, + { + "bbox": [ + 392, + 528, + 422, + 538 + ], + "score": 0.89, + "content": "z _ { t } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 527, + 479, + 540 + ], + "score": 1.0, + "content": "at each round", + "type": "text" + }, + { + "bbox": [ + 480, + 529, + 484, + 537 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 527, + 506, + 540 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 86, + 540, + 100, + 550 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 105, + 537, + 184, + 551 + ], + "score": 1.0, + "content": "the static regret, all", + "type": "text" + }, + { + "bbox": [ + 184, + 540, + 194, + 549 + ], + "score": 0.84, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 537, + 381, + 551 + ], + "score": 1.0, + "content": "are identically set as the fixed optimal decision", + "type": "text" + }, + { + "bbox": [ + 381, + 539, + 393, + 548 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "w.r.t. all losses in hindsight,", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 549, + 507, + 568 + ], + "spans": [ + { + "bbox": [ + 86, + 552, + 100, + 563 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 104, + 549, + 124, + 568 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 124, + 549, + 268, + 564 + ], + "score": 0.89, + "content": "\\begin{array} { r } { z _ { t } \\equiv x ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } f _ { t } ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 549, + 390, + 568 + ], + "score": 1.0, + "content": ". For the dynamic regret, each", + "type": "text" + }, + { + "bbox": [ + 390, + 552, + 400, + 562 + ], + "score": 0.82, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 549, + 507, + 568 + ], + "score": 1.0, + "content": "is selected as the optimal", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 562, + 472, + 576 + ], + "spans": [ + { + "bbox": [ + 86, + 564, + 100, + 573 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 105, + 562, + 142, + 576 + ], + "score": 1.0, + "content": "decision", + "type": "text" + }, + { + "bbox": [ + 142, + 563, + 153, + 574 + ], + "score": 0.89, + "content": "\\boldsymbol { x } _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 562, + 267, + 576 + ], + "score": 1.0, + "content": "w.r.t. the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 267, + 563, + 277, + 573 + ], + "score": 0.87, + "content": "f _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 562, + 350, + 576 + ], + "score": 1.0, + "content": "at that round, i.e.,", + "type": "text" + }, + { + "bbox": [ + 351, + 562, + 469, + 574 + ], + "score": 0.9, + "content": "z _ { t } = x _ { t } ^ { * } \\in \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } f _ { t } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 562, + 472, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 572, + 509, + 601 + ], + "spans": [ + { + "bbox": [ + 86, + 582, + 100, + 593 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 101, + 572, + 324, + 601 + ], + "score": 1.0, + "content": "In analogy, we can define the multi-objective regret as", + "type": "text" + }, + { + "bbox": [ + 325, + 579, + 399, + 594 + ], + "score": 0.93, + "content": "\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 572, + 450, + 601 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 450, + 581, + 463, + 592 + ], + "score": 0.88, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 572, + 509, + 601 + ], + "score": 1.0, + "content": "compares", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 86, + 593, + 99, + 604 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 323, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ". However, in general, no single decision can", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 86, + 605, + 99, + 615 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 603, + 403, + 617 + ], + "score": 1.0, + "content": "optimize all the objectives at the same time. Hence, it is natural to compare", + "type": "text" + }, + { + "bbox": [ + 403, + 605, + 414, + 614 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "with a group of Pareto", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 613, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 86, + 615, + 100, + 627 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 105, + 613, + 312, + 628 + ], + "score": 1.0, + "content": "optimal decisions, which constitute a comparator set", + "type": "text" + }, + { + "bbox": [ + 312, + 614, + 344, + 625 + ], + "score": 0.91, + "content": "\\mathcal { C } _ { t } \\subset \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 613, + 493, + 628 + ], + "score": 1.0, + "content": ". To measure the discrepancy between", + "type": "text" + }, + { + "bbox": [ + 493, + 616, + 504, + 625 + ], + "score": 0.82, + "content": "x _ { t }", + "type": "inline_equation" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 624, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 86, + 626, + 100, + 637 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 105, + 624, + 123, + 638 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 625, + 133, + 636 + ], + "score": 0.87, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 624, + 384, + 638 + ], + "score": 1.0, + "content": ", we further introduce the Pareto suboptimality gap (PSG) [30]", + "type": "text" + }, + { + "bbox": [ + 384, + 625, + 437, + 637 + ], + "score": 0.91, + "content": "\\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 624, + 506, + 638 + ], + "score": 1.0, + "content": ". Then the multi-", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 636, + 507, + 654 + ], + "spans": [ + { + "bbox": [ + 86, + 639, + 100, + 650 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 104, + 636, + 242, + 654 + ], + "score": 1.0, + "content": "objective regret can be defined as", + "type": "text" + }, + { + "bbox": [ + 243, + 636, + 358, + 650 + ], + "score": 0.92, + "content": "\\begin{array} { r } { R ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { C } _ { t } , F _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 636, + 507, + 654 + ], + "score": 1.0, + "content": ". Now we can formulate the static or", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 86, + 651, + 99, + 660 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 649, + 313, + 662 + ], + "score": 1.0, + "content": "the dynamic variant by specifying the comparator set", + "type": "text" + }, + { + "bbox": [ + 314, + 650, + 324, + 660 + ], + "score": 0.87, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 649, + 484, + 662 + ], + "score": 1.0, + "content": "at each round. Specifically, by setting all", + "type": "text" + }, + { + "bbox": [ + 484, + 649, + 494, + 660 + ], + "score": 0.88, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 659, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 663, + 100, + 675 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 104, + 659, + 173, + 678 + ], + "score": 1.0, + "content": "be the Pareto set", + "type": "text" + }, + { + "bbox": [ + 174, + 662, + 188, + 672 + ], + "score": 0.87, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 659, + 277, + 678 + ], + "score": 1.0, + "content": "of the cumulative loss", + "type": "text" + }, + { + "bbox": [ + 278, + 660, + 314, + 675 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 659, + 507, + 678 + ], + "score": 1.0, + "content": ", we formulate the multi-objective static regret", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 669, + 509, + 708 + ], + "spans": [ + { + "bbox": [ + 86, + 677, + 100, + 701 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 106, + 673, + 243, + 689 + ], + "score": 0.92, + "content": "\\begin{array} { r } { R _ { \\mathrm { M O S } } ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { X } ^ { \\ast } , F _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 690, + 135, + 700 + ], + "score": 0.86, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 669, + 313, + 708 + ], + "score": 1.0, + "content": ". By setting each bjective dynamic", + "type": "text" + }, + { + "bbox": [ + 313, + 677, + 323, + 687 + ], + "score": 0.86, + "content": "\\mathcal { C } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 669, + 343, + 708 + ], + "score": 1.0, + "content": "to bgret", + "type": "text" + }, + { + "bbox": [ + 344, + 688, + 480, + 702 + ], + "score": 0.92, + "content": "\\begin{array} { r } { R _ { \\mathrm { M O D } } ( T ) = \\sum _ { t = 1 } ^ { T } \\Delta ( x _ { t } ; \\mathcal { X } _ { t } ^ { \\ast } , F _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 676, + 419, + 687 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { t } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 669, + 509, + 708 + ], + "score": 1.0, + "content": "neous.", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "180", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "Recall that PSG is a zero-order metric motivated in a purely geometric sense, namely, its calculation", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 81, + 503, + 97 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 99, + 95 + ], + "score": 1.0, + "content": "181", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 81, + 431, + 97 + ], + "score": 1.0, + "content": "needs to solve a constrained optimization problem with an unknown boundary", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 431, + 83, + 503, + 96 + ], + "score": 0.91, + "content": "f _ { t } ^ { i } ( x ^ { \\prime \\prime } ) , \\forall x ^ { \\prime \\prime } \\in { \\mathcal { C } } _ { t }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "182", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "Hence, it is not straightforward to design a first-order algorithm to optimize PSG, not to mention", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "183", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "the regret analysis. To motivate algorithm design and analysis, we investigate the two variants in", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "184", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "more detail. We begin with the dynamic variant, since we find that it has an equivalent form, which is", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 85, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "185", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "intuitive and has a strong implication on the design of effective online multiple gradient algorithms.", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + } + ], + "index": 32, + "bbox_fs": [ + 86, + 508, + 507, + 576 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 579, + 506, + 702 + ], + "lines": [], + "index": 39, + "bbox_fs": [ + 86, + 572, + 509, + 708 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 86, + 72, + 505, + 84 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 99, + 84 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 72, + 505, + 84 + ], + "score": 1.0, + "content": "Recall that PSG is a zero-order metric motivated in a purely geometric sense, namely, its calculation", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 81, + 503, + 97 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 99, + 95 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 104, + 81, + 431, + 97 + ], + "score": 1.0, + "content": "needs to solve a constrained optimization problem with an unknown boundary", + "type": "text" + }, + { + "bbox": [ + 431, + 83, + 503, + 96 + ], + "score": 0.91, + "content": "f _ { t } ^ { i } ( x ^ { \\prime \\prime } ) , \\forall x ^ { \\prime \\prime } \\in { \\mathcal { C } } _ { t }", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "Hence, it is not straightforward to design a first-order algorithm to optimize PSG, not to mention", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "the regret analysis. To motivate algorithm design and analysis, we investigate the two variants in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "more detail. We begin with the dynamic variant, since we find that it has an equivalent form, which is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 85, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 85, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "intuitive and has a strong implication on the design of effective online multiple gradient algorithms.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 97, + 143, + 504, + 177 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 478, + 158 + ], + "score": 1.0, + "content": "An equivalent form of the dynamic regret. Surprisingly, the multi-objective dynamic regret", + "type": "text" + }, + { + "bbox": [ + 478, + 144, + 504, + 155 + ], + "score": 0.88, + "content": "R _ { \\mathrm { M O D } }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "can be transformed into an unconstrained max-min form. The derivation utilizes Pareto optimality of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 448, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 120, + 177 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { t } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 165, + 448, + 178 + ], + "score": 1.0, + "content": "and is highly non-trivial, which is deferred to the appendix due to the space limit.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 90, + 179, + 433, + 191 + ], + "lines": [ + { + "bbox": [ + 86, + 177, + 434, + 194 + ], + "spans": [ + { + "bbox": [ + 86, + 177, + 434, + 194 + ], + "score": 1.0, + "content": "189 Proposition 3.1. The multi-objective dynamic regret has an equivalent form, i.e.,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 194, + 425, + 226 + ], + "lines": [ + { + "bbox": [ + 185, + 194, + 425, + 226 + ], + "spans": [ + { + "bbox": [ + 185, + 194, + 425, + 226 + ], + "score": 0.93, + "content": "R _ { \\mathrm { M O D } } ( T ) = \\operatorname* { s u p } _ { \\stackrel { x _ { t } ^ { * } \\in \\mathcal { X } _ { t } ^ { * } , \\ } { 1 \\leq t \\leq T } } \\operatorname* { i n f } _ { \\stackrel { x \\in S _ { m } } { 1 \\leq t \\leq T } } \\sum _ { t = 1 } ^ { T } \\lambda _ { t } ^ { * } { ^ { \\top } ( F _ { t } ( x _ { t } ) - F _ { t } ( x _ { t } ^ { * } ) ) } .", + "type": "interline_equation", + "image_path": "a54de984c645f34484feb27cf9ca2cb5bed88897700e2a277d12ed36c921d08d.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 194, + 425, + 210.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 185, + 210.0, + 425, + 226.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 235, + 506, + 302 + ], + "lines": [ + { + "bbox": [ + 86, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 236, + 100, + 246 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 104, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "Remark. (i) The above form can be understood as a variant of the standard dynamic regret regarding", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 245, + 508, + 262 + ], + "spans": [ + { + "bbox": [ + 85, + 249, + 99, + 259 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 107, + 246, + 158, + 260 + ], + "score": 0.91, + "content": "\\{ \\lambda _ { t } ^ { * } ^ { \\top } F _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 245, + 199, + 262 + ], + "score": 1.0, + "content": ", whereas", + "type": "text" + }, + { + "bbox": [ + 199, + 248, + 210, + 259 + ], + "score": 0.89, + "content": "\\lambda _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 245, + 508, + 262 + ], + "score": 1.0, + "content": "are unknown to the learner. This provides an intuition that we can gen-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 256, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 85, + 260, + 100, + 271 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 104, + 256, + 163, + 274 + ], + "score": 1.0, + "content": "erate weights", + "type": "text" + }, + { + "bbox": [ + 163, + 259, + 199, + 270 + ], + "score": 0.92, + "content": "\\lambda _ { t } \\in \\boldsymbol { S } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 256, + 313, + 274 + ], + "score": 1.0, + "content": "at each round and optimize", + "type": "text" + }, + { + "bbox": [ + 313, + 258, + 356, + 271 + ], + "score": 0.93, + "content": "\\{ \\lambda _ { t } F _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 256, + 507, + 274 + ], + "score": 1.0, + "content": "via single-objective techniques. For", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 85, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 85, + 271, + 100, + 281 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "first-order algorithms, it is equivalent to selecting a convex combination of individual gradients and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 85, + 279, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 85, + 282, + 100, + 292 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 105, + 279, + 493, + 294 + ], + "score": 1.0, + "content": "then applying the composite gradient to model update. Undoubtedly, how to generate the weights", + "type": "text" + }, + { + "bbox": [ + 493, + 281, + 504, + 291 + ], + "score": 0.87, + "content": "\\lambda _ { t }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 292, + 439, + 303 + ], + "spans": [ + { + "bbox": [ + 85, + 293, + 101, + 303 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 106, + 292, + 439, + 303 + ], + "score": 1.0, + "content": "needs some careful designs, which will be explicated later in the algorithm section.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 100, + 303, + 504, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 151, + 316 + ], + "score": 1.0, + "content": "(ii) When", + "type": "text" + }, + { + "bbox": [ + 151, + 303, + 186, + 312 + ], + "score": 0.88, + "content": "m \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 300, + 230, + 316 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 231, + 302, + 280, + 314 + ], + "score": 0.94, + "content": "S _ { m } ~ = ~ \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 300, + 302, + 316 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 302, + 410, + 315 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { X } _ { t } ^ { * } ~ = ~ \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 300, + 450, + 316 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + }, + { + "bbox": [ + 450, + 302, + 505, + 314 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O D } } ( T ) ~ =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 313, + 487, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 237, + 329 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 313, + 451, + 331 + ], + "score": 1.0, + "content": ", which is exactly the single-objective dynamic regret", + "type": "text" + }, + { + "bbox": [ + 451, + 316, + 482, + 328 + ], + "score": 0.92, + "content": "R _ { D } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 313, + 487, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 86, + 332, + 506, + 439 + ], + "lines": [ + { + "bbox": [ + 86, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 86, + 334, + 99, + 344 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 332, + 361, + 344 + ], + "score": 1.0, + "content": "An alternative form of the static regret. Unfortunately, for", + "type": "text" + }, + { + "bbox": [ + 362, + 333, + 385, + 343 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 332, + 505, + 344 + ], + "score": 1.0, + "content": ", the above equivalence form", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 86, + 345, + 99, + 354 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 106, + 343, + 252, + 356 + ], + "score": 1.0, + "content": "does not exist. Here is the reason. In", + "type": "text" + }, + { + "bbox": [ + 252, + 344, + 276, + 354 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 343, + 355, + 356 + ], + "score": 1.0, + "content": ", the comparator set", + "type": "text" + }, + { + "bbox": [ + 355, + 344, + 369, + 353 + ], + "score": 0.88, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "is the Pareto set of the cumulative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 349, + 509, + 374 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 367 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 101, + 349, + 124, + 374 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 354, + 162, + 369 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 349, + 302, + 374 + ], + "score": 1.0, + "content": "rather than the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 302, + 356, + 313, + 367 + ], + "score": 0.87, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 349, + 444, + 374 + ], + "score": 1.0, + "content": ". Hence, at some specific round", + "type": "text" + }, + { + "bbox": [ + 444, + 357, + 450, + 366 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 349, + 509, + 374 + ], + "score": 1.0, + "content": ", the decision", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 85, + 369, + 100, + 379 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 106, + 369, + 117, + 378 + ], + "score": 0.81, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 366, + 264, + 379 + ], + "score": 1.0, + "content": "may Pareto dominate all points in", + "type": "text" + }, + { + "bbox": [ + 279, + 366, + 381, + 379 + ], + "score": 1.0, + "content": "w.r.t. the instantaneous", + "type": "text" + }, + { + "bbox": [ + 382, + 367, + 393, + 378 + ], + "score": 0.87, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 366, + 506, + 379 + ], + "score": 1.0, + "content": ", and we would expect the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 85, + 379, + 100, + 390 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 135, + 391 + ], + "score": 1.0, + "content": "metric", + "type": "text" + }, + { + "bbox": [ + 135, + 379, + 148, + 389 + ], + "score": 0.89, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "to be negative. However, PSG (or other commonly used metrics such as Hypervolume)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 388, + 511, + 423 + ], + "spans": [ + { + "bbox": [ + 85, + 388, + 106, + 423 + ], + "score": 1.0, + "content": "203 204", + "type": "text" + }, + { + "bbox": [ + 135, + 388, + 174, + 423 + ], + "score": 1.0, + "content": "yields no, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 389, + 330, + 400 + ], + "score": 0.88, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 388, + 407, + 423 + ], + "score": 1.0, + "content": "th , w", + "type": "text" + }, + { + "bbox": [ + 408, + 389, + 422, + 400 + ], + "score": 0.88, + "content": "R _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 388, + 511, + 423 + ], + "score": 1.0, + "content": ". For example, whenh can be much looser", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 400, + 397, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 135, + 412 + ], + "score": 0.88, + "content": "m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 400, + 397, + 414 + ], + "score": 0.9, + "content": "\\begin{array} { r } { R _ { \\operatorname { M O S } } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } \\operatorname* { m a x } \\{ F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) , 0 \\} } \\end{array}", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 409, + 509, + 434 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 99, + 427 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 102, + 409, + 192, + 434 + ], + "score": 1.0, + "content": "than the static regret", + "type": "text" + }, + { + "bbox": [ + 192, + 414, + 375, + 428 + ], + "score": 0.89, + "content": "\\begin{array} { r } { R _ { S } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 409, + 470, + 434 + ], + "score": 1.0, + "content": ". Hence the analysis of", + "type": "text" + }, + { + "bbox": [ + 470, + 416, + 495, + 427 + ], + "score": 0.89, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 409, + 509, + 434 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 426, + 502, + 440 + ], + "spans": [ + { + "bbox": [ + 86, + 428, + 99, + 438 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 502, + 440 + ], + "score": 1.0, + "content": "intrinsically complex if we use existing discrepancy metrics that always yield non-negative values.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 86, + 442, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 86, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 86, + 445, + 99, + 454 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "Enlightened by Proposition 3.1, we can formulate the static regret in a different way, i.e., by modifying", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 85, + 455, + 100, + 466 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 435, + 466 + ], + "score": 1.0, + "content": "the equivalent form of dynamic regret. Recall that in Proposition 3.1, at each round", + "type": "text" + }, + { + "bbox": [ + 435, + 455, + 440, + 464 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 453, + 506, + 466 + ], + "score": 1.0, + "content": ", the comparator", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 86, + 466, + 99, + 476 + ], + "score": 1.0, + "content": "209", + "type": "text" + }, + { + "bbox": [ + 106, + 465, + 118, + 477 + ], + "score": 0.88, + "content": "\\boldsymbol { x } _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 464, + 240, + 478 + ], + "score": 1.0, + "content": "is selected from the Pareto set", + "type": "text" + }, + { + "bbox": [ + 241, + 465, + 255, + 477 + ], + "score": 0.9, + "content": "\\mathcal { X } _ { t } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 464, + 356, + 478 + ], + "score": 1.0, + "content": "of the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 357, + 465, + 367, + 476 + ], + "score": 0.88, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 464, + 436, + 478 + ], + "score": 1.0, + "content": ", and the weights", + "type": "text" + }, + { + "bbox": [ + 437, + 465, + 448, + 477 + ], + "score": 0.89, + "content": "\\lambda _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "are generated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 85, + 477, + 100, + 488 + ], + "score": 1.0, + "content": "210", + "type": "text" + }, + { + "bbox": [ + 105, + 475, + 128, + 488 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 476, + 142, + 487 + ], + "score": 0.88, + "content": "S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 475, + 390, + 488 + ], + "score": 1.0, + "content": ". To formulate the static variant, we can use a fixed comparator", + "type": "text" + }, + { + "bbox": [ + 390, + 476, + 401, + 486 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 475, + 479, + 488 + ], + "score": 1.0, + "content": "from the Pareto set", + "type": "text" + }, + { + "bbox": [ + 479, + 476, + 493, + 486 + ], + "score": 0.88, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 486, + 492, + 499 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 99, + 498 + ], + "score": 1.0, + "content": "211", + "type": "text" + }, + { + "bbox": [ + 106, + 487, + 185, + 499 + ], + "score": 1.0, + "content": "the cumulative loss", + "type": "text" + }, + { + "bbox": [ + 185, + 486, + 212, + 499 + ], + "score": 0.92, + "content": "\\sum _ { t } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 487, + 286, + 499 + ], + "score": 1.0, + "content": "and fixed weights", + "type": "text" + }, + { + "bbox": [ + 286, + 487, + 324, + 498 + ], + "score": 0.92, + "content": "\\lambda ^ { * } \\in S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 487, + 492, + 499 + ], + "score": 1.0, + "content": "at all rounds. Now the static variant takes", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 502, + 438, + 527 + ], + "lines": [ + { + "bbox": [ + 172, + 502, + 438, + 527 + ], + "spans": [ + { + "bbox": [ + 172, + 502, + 438, + 527 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O S } } ( T ) : = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\operatorname* { i n f } _ { \\lambda ^ { * } \\in \\mathcal { S } _ { m } } { \\lambda ^ { * } } ^ { \\top } ( \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \\sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) ) .", + "type": "interline_equation", + "image_path": "759e8230542b99bdaacb7436137bb1f0c245c8706ac39c7fa62d608bdfe0a42e.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 172, + 502, + 438, + 527 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 90, + 531, + 504, + 556 + ], + "lines": [ + { + "bbox": [ + 86, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 86, + 531, + 160, + 544 + ], + "score": 1.0, + "content": "212 Remark. (i)", + "type": "text" + }, + { + "bbox": [ + 160, + 531, + 199, + 543 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O S } } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "has a clear physical meaning that optimizing it will impose the cumulative", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 83, + 537, + 432, + 563 + ], + "spans": [ + { + "bbox": [ + 83, + 537, + 124, + 563 + ], + "score": 1.0, + "content": "loss 213", + "type": "text" + }, + { + "bbox": [ + 125, + 542, + 178, + 558 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 537, + 278, + 563 + ], + "score": 1.0, + "content": "to reach the Pareto front", + "type": "text" + }, + { + "bbox": [ + 279, + 545, + 291, + 555 + ], + "score": 0.88, + "content": "{ \\mathcal { P } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 537, + 432, + 563 + ], + "score": 1.0, + "content": ". See more details in Appendix C.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 94, + 558, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 89, + 554, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 89, + 554, + 149, + 574 + ], + "score": 1.0, + "content": "(ii) When 214", + "type": "text" + }, + { + "bbox": [ + 149, + 559, + 179, + 569 + ], + "score": 0.85, + "content": "m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 554, + 184, + 574 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 184, + 558, + 229, + 570 + ], + "score": 0.9, + "content": "S _ { m } = \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 554, + 249, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 249, + 559, + 263, + 569 + ], + "score": 0.87, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 554, + 311, + 574 + ], + "score": 1.0, + "content": "reduces to", + "type": "text" + }, + { + "bbox": [ + 311, + 556, + 412, + 571 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } F _ { t } ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 554, + 453, + 574 + ], + "score": 1.0, + "content": ". Therein", + "type": "text" + }, + { + "bbox": [ + 453, + 558, + 506, + 570 + ], + "score": 0.91, + "content": "R _ { \\mathrm { M O S } } ( T ) =", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 93, + 563, + 502, + 590 + ], + "spans": [ + { + "bbox": [ + 93, + 563, + 106, + 590 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 106, + 570, + 270, + 585 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } F _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } F _ { t } ( x ^ { * } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 563, + 473, + 590 + ], + "score": 1.0, + "content": "x∈X t=1 , which reduces to the single-objective static regret", + "type": "text" + }, + { + "bbox": [ + 473, + 572, + 502, + 584 + ], + "score": 0.92, + "content": "R _ { S } ( T )", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 90, + 598, + 290, + 613 + ], + "lines": [ + { + "bbox": [ + 84, + 596, + 291, + 616 + ], + "spans": [ + { + "bbox": [ + 84, + 596, + 291, + 616 + ], + "score": 1.0, + "content": "216 4 Online Mirror Multiple Descent", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 505, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "In this section, we present the Online Mirror Multiple Descent (OMMD) algorithm, the protocol of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 294, + 646 + ], + "score": 1.0, + "content": "which is given in Algorithm 1. At each round", + "type": "text" + }, + { + "bbox": [ + 295, + 635, + 300, + 644 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 634, + 505, + 646 + ], + "score": 1.0, + "content": ", the learner first computes the gradient of the loss", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "regarding each objective, then determines the composite weights of all these gradients, and finally", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 656, + 365, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 365, + 669 + ], + "score": 1.0, + "content": "applies the composite gradient to the online mirror descent step.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 104, + 679, + 324, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 325, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 325, + 694 + ], + "score": 1.0, + "content": "4.1 Vanilla Min-Norm May Incur Linear Regrets", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 90, + 699, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 88, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 88, + 701, + 101, + 712 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 104, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "The core module of OMMD is the composition of multiple gradients. For simplicity, we represent", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 86, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 86, + 709, + 197, + 724 + ], + "score": 1.0, + "content": "the gradients at round 223", + "type": "text" + }, + { + "bbox": [ + 197, + 712, + 202, + 721 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 709, + 272, + 724 + ], + "score": 1.0, + "content": "in a matrix form", + "type": "text" + }, + { + "bbox": [ + 272, + 710, + 462, + 723 + ], + "score": 0.91, + "content": "\\nabla F _ { t } ( x _ { t } ) = [ \\nabla \\mathsf { \\bar { f } } _ { t } ^ { 1 } ( \\bar { x } _ { t } ) , \\ldots , \\nabla f _ { t } ^ { m } ( x _ { t } ) ] \\in \\bar { \\mathbb { R } } ^ { \\mathsf { \\bar { n } } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 709, + 506, + 724 + ], + "score": 1.0, + "content": ". Then the", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 86, + 72, + 505, + 139 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 85, + 72, + 505, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 97, + 143, + 504, + 177 + ], + "lines": [ + { + "bbox": [ + 104, + 141, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 478, + 158 + ], + "score": 1.0, + "content": "An equivalent form of the dynamic regret. Surprisingly, the multi-objective dynamic regret", + "type": "text" + }, + { + "bbox": [ + 478, + 144, + 504, + 155 + ], + "score": 0.88, + "content": "R _ { \\mathrm { M O D } }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "can be transformed into an unconstrained max-min form. The derivation utilizes Pareto optimality of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 448, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 120, + 177 + ], + "score": 0.89, + "content": "\\mathcal { X } _ { t } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 165, + 448, + 178 + ], + "score": 1.0, + "content": "and is highly non-trivial, which is deferred to the appendix due to the space limit.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 141, + 506, + 178 + ] + }, + { + "type": "text", + "bbox": [ + 90, + 179, + 433, + 191 + ], + "lines": [ + { + "bbox": [ + 86, + 177, + 434, + 194 + ], + "spans": [ + { + "bbox": [ + 86, + 177, + 434, + 194 + ], + "score": 1.0, + "content": "189 Proposition 3.1. The multi-objective dynamic regret has an equivalent form, i.e.,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 86, + 177, + 434, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 194, + 425, + 226 + ], + "lines": [ + { + "bbox": [ + 185, + 194, + 425, + 226 + ], + "spans": [ + { + "bbox": [ + 185, + 194, + 425, + 226 + ], + "score": 0.93, + "content": "R _ { \\mathrm { M O D } } ( T ) = \\operatorname* { s u p } _ { \\stackrel { x _ { t } ^ { * } \\in \\mathcal { X } _ { t } ^ { * } , \\ } { 1 \\leq t \\leq T } } \\operatorname* { i n f } _ { \\stackrel { x \\in S _ { m } } { 1 \\leq t \\leq T } } \\sum _ { t = 1 } ^ { T } \\lambda _ { t } ^ { * } { ^ { \\top } ( F _ { t } ( x _ { t } ) - F _ { t } ( x _ { t } ^ { * } ) ) } .", + "type": "interline_equation", + "image_path": "a54de984c645f34484feb27cf9ca2cb5bed88897700e2a277d12ed36c921d08d.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 194, + 425, + 210.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 185, + 210.0, + 425, + 226.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 235, + 506, + 302 + ], + "lines": [ + { + "bbox": [ + 86, + 233, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 236, + 100, + 246 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 104, + 233, + 506, + 248 + ], + "score": 1.0, + "content": "Remark. (i) The above form can be understood as a variant of the standard dynamic regret regarding", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 245, + 508, + 262 + ], + "spans": [ + { + "bbox": [ + 85, + 249, + 99, + 259 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 107, + 246, + 158, + 260 + ], + "score": 0.91, + "content": "\\{ \\lambda _ { t } ^ { * } ^ { \\top } F _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 245, + 199, + 262 + ], + "score": 1.0, + "content": ", whereas", + "type": "text" + }, + { + "bbox": [ + 199, + 248, + 210, + 259 + ], + "score": 0.89, + "content": "\\lambda _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 245, + 508, + 262 + ], + "score": 1.0, + "content": "are unknown to the learner. This provides an intuition that we can gen-", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 256, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 85, + 260, + 100, + 271 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 104, + 256, + 163, + 274 + ], + "score": 1.0, + "content": "erate weights", + "type": "text" + }, + { + "bbox": [ + 163, + 259, + 199, + 270 + ], + "score": 0.92, + "content": "\\lambda _ { t } \\in \\boldsymbol { S } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 256, + 313, + 274 + ], + "score": 1.0, + "content": "at each round and optimize", + "type": "text" + }, + { + "bbox": [ + 313, + 258, + 356, + 271 + ], + "score": 0.93, + "content": "\\{ \\lambda _ { t } F _ { t } \\} _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 256, + 507, + 274 + ], + "score": 1.0, + "content": "via single-objective techniques. For", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 85, + 271, + 100, + 281 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "first-order algorithms, it is equivalent to selecting a convex combination of individual gradients and", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 279, + 504, + 294 + ], + "spans": [ + { + "bbox": [ + 85, + 282, + 100, + 292 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 105, + 279, + 493, + 294 + ], + "score": 1.0, + "content": "then applying the composite gradient to model update. Undoubtedly, how to generate the weights", + "type": "text" + }, + { + "bbox": [ + 493, + 281, + 504, + 291 + ], + "score": 0.87, + "content": "\\lambda _ { t }", + "type": "inline_equation" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 292, + 439, + 303 + ], + "spans": [ + { + "bbox": [ + 85, + 293, + 101, + 303 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 106, + 292, + 439, + 303 + ], + "score": 1.0, + "content": "needs some careful designs, which will be explicated later in the algorithm section.", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 14.5, + "bbox_fs": [ + 85, + 233, + 508, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 303, + 504, + 328 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 151, + 316 + ], + "score": 1.0, + "content": "(ii) When", + "type": "text" + }, + { + "bbox": [ + 151, + 303, + 186, + 312 + ], + "score": 0.88, + "content": "m \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 300, + 230, + 316 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 231, + 302, + 280, + 314 + ], + "score": 0.94, + "content": "S _ { m } ~ = ~ \\{ 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 300, + 302, + 316 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 302, + 410, + 315 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { X } _ { t } ^ { * } ~ = ~ \\arg \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 300, + 450, + 316 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + }, + { + "bbox": [ + 450, + 302, + 505, + 314 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O D } } ( T ) ~ =", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 313, + 487, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 237, + 329 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - \\operatorname* { m i n } _ { x \\in \\mathcal { X } } F _ { t } ( x ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 313, + 451, + 331 + ], + "score": 1.0, + "content": ", which is exactly the single-objective dynamic regret", + "type": "text" + }, + { + "bbox": [ + 451, + 316, + 482, + 328 + ], + "score": 0.92, + "content": "R _ { D } ( T )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 313, + 487, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 300, + 505, + 331 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 332, + 506, + 439 + ], + "lines": [ + { + "bbox": [ + 86, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 86, + 334, + 99, + 344 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 332, + 361, + 344 + ], + "score": 1.0, + "content": "An alternative form of the static regret. Unfortunately, for", + "type": "text" + }, + { + "bbox": [ + 362, + 333, + 385, + 343 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 332, + 505, + 344 + ], + "score": 1.0, + "content": ", the above equivalence form", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 86, + 345, + 99, + 354 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 106, + 343, + 252, + 356 + ], + "score": 1.0, + "content": "does not exist. Here is the reason. In", + "type": "text" + }, + { + "bbox": [ + 252, + 344, + 276, + 354 + ], + "score": 0.9, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 343, + 355, + 356 + ], + "score": 1.0, + "content": ", the comparator set", + "type": "text" + }, + { + "bbox": [ + 355, + 344, + 369, + 353 + ], + "score": 0.88, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "is the Pareto set of the cumulative", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 349, + 509, + 374 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 367 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 101, + 349, + 124, + 374 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 354, + 162, + 369 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 349, + 302, + 374 + ], + "score": 1.0, + "content": "rather than the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 302, + 356, + 313, + 367 + ], + "score": 0.87, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 349, + 444, + 374 + ], + "score": 1.0, + "content": ". Hence, at some specific round", + "type": "text" + }, + { + "bbox": [ + 444, + 357, + 450, + 366 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 349, + 509, + 374 + ], + "score": 1.0, + "content": ", the decision", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 85, + 369, + 100, + 379 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 106, + 369, + 117, + 378 + ], + "score": 0.81, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 366, + 264, + 379 + ], + "score": 1.0, + "content": "may Pareto dominate all points in", + "type": "text" + }, + { + "bbox": [ + 279, + 366, + 381, + 379 + ], + "score": 1.0, + "content": "w.r.t. the instantaneous", + "type": "text" + }, + { + "bbox": [ + 382, + 367, + 393, + 378 + ], + "score": 0.87, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 366, + 506, + 379 + ], + "score": 1.0, + "content": ", and we would expect the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 85, + 379, + 100, + 390 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 105, + 377, + 135, + 391 + ], + "score": 1.0, + "content": "metric", + "type": "text" + }, + { + "bbox": [ + 135, + 379, + 148, + 389 + ], + "score": 0.89, + "content": "\\Delta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "to be negative. However, PSG (or other commonly used metrics such as Hypervolume)", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 388, + 511, + 423 + ], + "spans": [ + { + "bbox": [ + 85, + 388, + 106, + 423 + ], + "score": 1.0, + "content": "203 204", + "type": "text" + }, + { + "bbox": [ + 135, + 388, + 174, + 423 + ], + "score": 1.0, + "content": "yields no, we have", + "type": "text" + }, + { + "bbox": [ + 306, + 389, + 330, + 400 + ], + "score": 0.88, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 388, + 407, + 423 + ], + "score": 1.0, + "content": "th , w", + "type": "text" + }, + { + "bbox": [ + 408, + 389, + 422, + 400 + ], + "score": 0.88, + "content": "R _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 388, + 511, + 423 + ], + "score": 1.0, + "content": ". For example, whenh can be much looser", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 106, + 400, + 397, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 135, + 412 + ], + "score": 0.88, + "content": "m = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 400, + 397, + 414 + ], + "score": 0.9, + "content": "\\begin{array} { r } { R _ { \\operatorname { M O S } } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } \\operatorname* { m a x } \\{ F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) , 0 \\} } \\end{array}", + "type": "inline_equation" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 409, + 509, + 434 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 99, + 427 + ], + "score": 1.0, + "content": "205", + "type": "text" + }, + { + "bbox": [ + 102, + 409, + 192, + 434 + ], + "score": 1.0, + "content": "than the static regret", + "type": "text" + }, + { + "bbox": [ + 192, + 414, + 375, + 428 + ], + "score": 0.89, + "content": "\\begin{array} { r } { R _ { S } ( T ) = \\operatorname* { s u p } _ { x ^ { * } \\in \\mathcal { X } ^ { * } } \\sum _ { t = 1 } ^ { T } ( F _ { t } ( x _ { t } ) - F _ { t } ( x ^ { * } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 409, + 470, + 434 + ], + "score": 1.0, + "content": ". Hence the analysis of", + "type": "text" + }, + { + "bbox": [ + 470, + 416, + 495, + 427 + ], + "score": 0.89, + "content": "R _ { \\mathrm { M O S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 409, + 509, + 434 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 426, + 502, + 440 + ], + "spans": [ + { + "bbox": [ + 86, + 428, + 99, + 438 + ], + "score": 1.0, + "content": "206", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 502, + 440 + ], + "score": 1.0, + "content": "intrinsically complex if we use existing discrepancy metrics that always yield non-negative values.", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 442, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 86, + 445, + 99, + 454 + ], + "score": 1.0, + "content": "207", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 505, + 456 + ], + "score": 1.0, + "content": "Enlightened by Proposition 3.1, we can formulate the static regret in a different way, i.e., by modifying", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 85, + 455, + 100, + 466 + ], + "score": 1.0, + "content": "208", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 435, + 466 + ], + "score": 1.0, + "content": "the equivalent form of dynamic regret. 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+ }, + { + "bbox": [ + 336, + 266, + 505, + 280 + ], + "score": 1.0, + "content": "is the composite weights. As illustrated in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 86, + 280, + 99, + 289 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "Preliminary, the min-norm method in MGDA [7, 29] is a classic method to determine the composite", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 85, + 290, + 100, + 301 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "weights in the offline setting, which results in a common descent direction that can descend all the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 298, + 454, + 313 + ], + "spans": [ + { + "bbox": [ + 86, + 301, + 99, + 311 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 454, + 313 + ], + "score": 1.0, + "content": "losses simultaneously. Thus, it is tempting to consider applying it to the online setting.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 85, + 315, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 86, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 86, + 318, + 99, + 327 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "However, directly applying the min-norm method to the online setting is not workable, which may", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 85, + 329, + 100, + 339 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "even incur linear regrets of the resulting algorithms. The rationale is as follows. In the vanilla", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 86, + 340, + 99, + 349 + ], + "score": 1.0, + "content": "230", + "type": "text" + }, + { + "bbox": [ + 105, + 338, + 276, + 351 + ], + "score": 1.0, + "content": "min-norm method, the composite weights", + "type": "text" + }, + { + "bbox": [ + 276, + 338, + 287, + 349 + ], + "score": 0.89, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 338, + 443, + 351 + ], + "score": 1.0, + "content": "are determined solely by the gradients", + "type": "text" + }, + { + "bbox": [ + 443, + 338, + 479, + 350 + ], + "score": 0.93, + "content": "\\nabla F _ { t } ( x _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "at the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 347, + 507, + 362 + ], + "spans": [ + { + "bbox": [ + 85, + 350, + 100, + 361 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 104, + 347, + 165, + 362 + ], + "score": 1.0, + "content": "current round", + "type": "text" + }, + { + "bbox": [ + 165, + 349, + 170, + 359 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 347, + 401, + 362 + ], + "score": 1.0, + "content": ", hence they are very sensitive to the instantaneous loss", + "type": "text" + }, + { + "bbox": [ + 401, + 349, + 412, + 360 + ], + "score": 0.88, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 347, + 507, + 362 + ], + "score": 1.0, + "content": ". In the online setting,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 362, + 100, + 371 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "the losses at each round can be adversarially chosen, and thus the corresponding gradients can be", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 85, + 372, + 100, + 382 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 104, + 369, + 506, + 385 + ], + "score": 1.0, + "content": "adversarial. These adversarial gradients may result in undesired composite weights, which may", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 379, + 507, + 396 + ], + "spans": [ + { + "bbox": [ + 86, + 384, + 99, + 393 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 104, + 379, + 507, + 396 + ], + "score": 1.0, + "content": "further produce a composite gradient that even deteriorates the next prediction. In the following,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 86, + 395, + 100, + 404 + ], + "score": 1.0, + "content": "235", + "type": "text" + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "we provide a problem instance in which min-norm incurs a linear regret. We extend OMD to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 403, + 475, + 416 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 100, + 414 + ], + "score": 1.0, + "content": "236", + "type": "text" + }, + { + "bbox": [ + 105, + 403, + 475, + 416 + ], + "score": 1.0, + "content": "multi-objective setting, where the composite weights are directly yielded by min-norm [11].", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 91, + 419, + 503, + 443 + ], + "lines": [ + { + "bbox": [ + 87, + 418, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 87, + 418, + 447, + 433 + ], + "score": 1.0, + "content": "237 Problem instance. We consider a two-objective problem. The decision domain is", + "type": "text" + }, + { + "bbox": [ + 447, + 419, + 504, + 432 + ], + "score": 0.9, + "content": "\\mathcal { X } = \\{ ( u , v ) ~ |", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 87, + 429, + 384, + 445 + ], + "spans": [ + { + "bbox": [ + 87, + 429, + 106, + 445 + ], + "score": 1.0, + "content": "238", + "type": "text" + }, + { + "bbox": [ + 106, + 430, + 231, + 444 + ], + "score": 0.91, + "content": "\\begin{array} { r } { u + v \\leq \\frac { 1 } { 2 } , v - u \\leq \\frac { 1 } { 2 } , v \\geq 0 \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 429, + 384, + 445 + ], + "score": 1.0, + "content": "and the loss function at each round is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 447, + 436, + 482 + ], + "lines": [ + { + "bbox": [ + 173, + 447, + 436, + 482 + ], + "spans": [ + { + "bbox": [ + 173, + 447, + 436, + 482 + ], + "score": 0.93, + "content": "F _ { t } ( x ) = \\left\\{ \\begin{array} { l l } { ( \\| x - a \\| ^ { 2 } , \\| x - b \\| ^ { 2 } ) , ~ t = 2 k - 1 , } & { ~ k = 1 , 2 , . . . ; } \\\\ { ( \\| x - b \\| ^ { 2 } , \\| x - c \\| ^ { 2 } ) , ~ t = 2 k , } & { ~ k = 1 , 2 , . . . , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "0d5a171f637adb9f2eb0528d59a5870302286e4146b4f9aa99c52bd39766479f.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 173, + 447, + 436, + 458.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 173, + 458.6666666666667, + 436, + 470.33333333333337 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 173, + 470.33333333333337, + 436, + 482.00000000000006 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 485, + 506, + 555 + ], + "lines": [ + { + "bbox": [ + 86, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 488, + 99, + 497 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 485, + 133, + 498 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 486, + 291, + 498 + ], + "score": 0.92, + "content": "a = ( - 2 , - 1 ) , b = ( 0 , 1 ) , c = ( 2 , - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 485, + 506, + 498 + ], + "score": 1.0, + "content": ". For simplicity, we first analyze the case where the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 85, + 498, + 100, + 509 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 106, + 497, + 183, + 509 + ], + "score": 1.0, + "content": "total time horizon", + "type": "text" + }, + { + "bbox": [ + 183, + 498, + 192, + 507 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "is an even number. Then we can compute the Pareto set of the cumulative", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 507, + 508, + 525 + ], + "spans": [ + { + "bbox": [ + 85, + 511, + 99, + 522 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 103, + 507, + 124, + 525 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 508, + 162, + 523 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 507, + 183, + 525 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 184, + 509, + 307, + 523 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { X } ^ { * } = \\{ ( u , 0 ) \\mid - \\frac { 1 } { 2 } \\leq u \\leq \\frac { 1 } { 2 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 507, + 395, + 525 + ], + "score": 1.0, + "content": ", which locates at the", + "type": "text" + }, + { + "bbox": [ + 395, + 512, + 402, + 520 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 507, + 508, + 525 + ], + "score": 1.0, + "content": "-axis. For conciseness of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 85, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 85, + 523, + 100, + 533 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "analysis, we instantiate OMD with L2-regularization, which results in the simple OGD algorithm", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 85, + 533, + 100, + 544 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 247, + 545 + ], + "score": 1.0, + "content": "[24]. We start at an arbitrary point", + "type": "text" + }, + { + "bbox": [ + 248, + 532, + 325, + 544 + ], + "score": 0.93, + "content": "x _ { 1 } = ( u _ { 1 } , v _ { 1 } ) \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 531, + 367, + 545 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 368, + 532, + 397, + 543 + ], + "score": 0.91, + "content": "v _ { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 531, + 461, + 545 + ], + "score": 1.0, + "content": ". At each round", + "type": "text" + }, + { + "bbox": [ + 461, + 533, + 466, + 542 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 531, + 506, + 545 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 85, + 541, + 496, + 555 + ], + "spans": [ + { + "bbox": [ + 85, + 544, + 100, + 554 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 157, + 555 + ], + "score": 1.0, + "content": "the decision", + "type": "text" + }, + { + "bbox": [ + 157, + 543, + 231, + 555 + ], + "score": 0.93, + "content": "x _ { t } = ( u _ { t } , v _ { t } ) \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 541, + 400, + 555 + ], + "score": 1.0, + "content": ", then the gradients of each objective w.r.t.", + "type": "text" + }, + { + "bbox": [ + 401, + 545, + 411, + 554 + ], + "score": 0.87, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 541, + 496, + 555 + ], + "score": 1.0, + "content": "can be calculated as", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 558, + 490, + 588 + ], + "lines": [ + { + "bbox": [ + 118, + 558, + 490, + 588 + ], + "spans": [ + { + "bbox": [ + 118, + 558, + 490, + 588 + ], + "score": 0.93, + "content": "g _ { t } ^ { 1 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } + 4 , ~ 2 v _ { t } + 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k . } \\end{array} \\right. } \\qquad g _ { t } ^ { 2 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } - 4 , } & { 2 v _ { t } + 2 ) , } & { t = 2 k . } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "14cb2aa94f549718bcec12a508e4b0249ddaad2970271250ebf8e7e7304f85c8.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 118, + 558, + 490, + 568.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 118, + 568.0, + 490, + 578.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 118, + 578.0, + 490, + 588.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 83, + 593, + 506, + 616 + ], + "lines": [ + { + "bbox": [ + 86, + 591, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 86, + 595, + 100, + 604 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 104, + 591, + 132, + 607 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 592, + 180, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 0 \\leq v _ { t } \\leq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 591, + 506, + 607 + ], + "score": 1.0, + "content": ", we observe that the second entry of either gradient alternates between positive", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 604, + 429, + 616 + ], + "spans": [ + { + "bbox": [ + 85, + 604, + 336, + 616 + ], + "score": 1.0, + "content": "246 and negative. By using min-norm, the composite weights", + "type": "text" + }, + { + "bbox": [ + 337, + 605, + 347, + 615 + ], + "score": 0.9, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 604, + 429, + 616 + ], + "score": 1.0, + "content": "can be computed as", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 619, + 426, + 650 + ], + "lines": [ + { + "bbox": [ + 183, + 619, + 426, + 650 + ], + "spans": [ + { + "bbox": [ + 183, + 619, + 426, + 650 + ], + "score": 0.91, + "content": "\\lambda _ { t } = \\left\\{ { \\begin{array} { l l } { ( ( 1 - u _ { t } - v _ { t } ) / 4 , } & { ( 3 + u _ { t } + v _ { t } ) / 4 ) , t = 2 k - 1 ; } \\\\ { ( ( 3 - u _ { t } + v _ { t } ) / 4 , } & { ( 1 + u _ { t } - v _ { t } ) / 4 ) , t = 2 k . } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "e7d836d82406529953d1663daa665b08f603af784654f260f81479c5d843ade6.jpg" + } + ] + } + ], + "index": 41.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 619, + 426, + 634.5 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 183, + 634.5, + 426, + 650.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 654, + 505, + 688 + ], + "lines": [ + { + "bbox": [ + 86, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 656, + 100, + 666 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 103, + 653, + 423, + 669 + ], + "score": 1.0, + "content": "We observe that both entries of composite weights alternative between above", + "type": "text" + }, + { + "bbox": [ + 423, + 654, + 430, + 667 + ], + "score": 0.87, + "content": "\\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 653, + 477, + 669 + ], + "score": 1.0, + "content": "and below", + "type": "text" + }, + { + "bbox": [ + 477, + 654, + 484, + 667 + ], + "score": 0.86, + "content": "\\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 653, + 506, + 669 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 107, + 665, + 182, + 677 + ], + "score": 0.91, + "content": "\\| \\lambda _ { t + 1 } - \\lambda _ { t } \\| _ { 1 } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 665, + 232, + 678 + ], + "score": 1.0, + "content": ". Recall that", + "type": "text" + }, + { + "bbox": [ + 232, + 666, + 275, + 677 + ], + "score": 0.92, + "content": "\\| \\lambda _ { t } \\| _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", hence the composite weights at two consecutive rounds", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 86, + 676, + 334, + 689 + ], + "spans": [ + { + "bbox": [ + 86, + 679, + 100, + 687 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 334, + 689 + ], + "score": 1.0, + "content": "change radically. 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As illustrated in", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 86, + 280, + 99, + 289 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 106, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "Preliminary, the min-norm method in MGDA [7, 29] is a classic method to determine the composite", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 85, + 290, + 100, + 301 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "weights in the offline setting, which results in a common descent direction that can descend all the", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 298, + 454, + 313 + ], + "spans": [ + { + "bbox": [ + 86, + 301, + 99, + 311 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 298, + 454, + 313 + ], + "score": 1.0, + "content": "losses simultaneously. Thus, it is tempting to consider applying it to the online setting.", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 86, + 318, + 99, + 327 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "However, directly applying the min-norm method to the online setting is not workable, which may", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 85, + 329, + 100, + 339 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "even incur linear regrets of the resulting algorithms. The rationale is as follows. 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In the online setting,", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 362, + 100, + 371 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 105, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "the losses at each round can be adversarially chosen, and thus the corresponding gradients can be", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 369, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 85, + 372, + 100, + 382 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 104, + 369, + 506, + 385 + ], + "score": 1.0, + "content": "adversarial. These adversarial gradients may result in undesired composite weights, which may", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 379, + 507, + 396 + ], + "spans": [ + { + "bbox": [ + 86, + 384, + 99, + 393 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 104, + 379, + 507, + 396 + ], + "score": 1.0, + "content": "further produce a composite gradient that even deteriorates the next prediction. In the following,", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 86, + 395, + 100, + 404 + ], + "score": 1.0, + "content": "235", + "type": "text" + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "we provide a problem instance in which min-norm incurs a linear regret. We extend OMD to the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 403, + 475, + 416 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 100, + 414 + ], + "score": 1.0, + "content": "236", + "type": "text" + }, + { + "bbox": [ + 105, + 403, + 475, + 416 + ], + "score": 1.0, + "content": "multi-objective setting, where the composite weights are directly yielded by min-norm [11].", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 418, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 87, + 418, + 447, + 433 + ], + "score": 1.0, + "content": "237 Problem instance. We consider a two-objective problem. The decision domain is", + "type": "text" + }, + { + "bbox": [ + 447, + 419, + 504, + 432 + ], + "score": 0.9, + "content": "\\mathcal { X } = \\{ ( u , v ) ~ |", + "type": "inline_equation" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 429, + 384, + 445 + ], + "spans": [ + { + "bbox": [ + 87, + 429, + 106, + 445 + ], + "score": 1.0, + "content": "238", + "type": "text" + }, + { + "bbox": [ + 106, + 430, + 231, + 444 + ], + "score": 0.91, + "content": "\\begin{array} { r } { u + v \\leq \\frac { 1 } { 2 } , v - u \\leq \\frac { 1 } { 2 } , v \\geq 0 \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 429, + 384, + 445 + ], + "score": 1.0, + "content": "and the loss function at each round is", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + } + ], + "index": 13.5, + "bbox_fs": [ + 85, + 266, + 505, + 313 + ] + }, + { + "type": "index", + "bbox": [ + 85, + 315, + 505, + 415 + ], + "lines": [], + "index": 20, + "bbox_fs": [ + 85, + 315, + 507, + 416 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 91, + 419, + 503, + 443 + ], + "lines": [], + "index": 25.5, + "bbox_fs": [ + 87, + 418, + 504, + 445 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 447, + 436, + 482 + ], + "lines": [ + { + "bbox": [ + 173, + 447, + 436, + 482 + ], + "spans": [ + { + "bbox": [ + 173, + 447, + 436, + 482 + ], + "score": 0.93, + "content": "F _ { t } ( x ) = \\left\\{ \\begin{array} { l l } { ( \\| x - a \\| ^ { 2 } , \\| x - b \\| ^ { 2 } ) , ~ t = 2 k - 1 , } & { ~ k = 1 , 2 , . . . ; } \\\\ { ( \\| x - b \\| ^ { 2 } , \\| x - c \\| ^ { 2 } ) , ~ t = 2 k , } & { ~ k = 1 , 2 , . . . , } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "0d5a171f637adb9f2eb0528d59a5870302286e4146b4f9aa99c52bd39766479f.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 173, + 447, + 436, + 458.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 173, + 458.6666666666667, + 436, + 470.33333333333337 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 173, + 470.33333333333337, + 436, + 482.00000000000006 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 485, + 506, + 555 + ], + "lines": [ + { + "bbox": [ + 86, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 488, + 99, + 497 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 485, + 133, + 498 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 486, + 291, + 498 + ], + "score": 0.92, + "content": "a = ( - 2 , - 1 ) , b = ( 0 , 1 ) , c = ( 2 , - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 485, + 506, + 498 + ], + "score": 1.0, + "content": ". For simplicity, we first analyze the case where the", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 85, + 498, + 100, + 509 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 106, + 497, + 183, + 509 + ], + "score": 1.0, + "content": "total time horizon", + "type": "text" + }, + { + "bbox": [ + 183, + 498, + 192, + 507 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "is an even number. Then we can compute the Pareto set of the cumulative", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 507, + 508, + 525 + ], + "spans": [ + { + "bbox": [ + 85, + 511, + 99, + 522 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 103, + 507, + 124, + 525 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 125, + 508, + 162, + 523 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { t = 1 } ^ { T } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 507, + 183, + 525 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + }, + { + "bbox": [ + 184, + 509, + 307, + 523 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mathcal { X } ^ { * } = \\{ ( u , 0 ) \\mid - \\frac { 1 } { 2 } \\leq u \\leq \\frac { 1 } { 2 } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 507, + 395, + 525 + ], + "score": 1.0, + "content": ", which locates at the", + "type": "text" + }, + { + "bbox": [ + 395, + 512, + 402, + 520 + ], + "score": 0.79, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 507, + 508, + 525 + ], + "score": 1.0, + "content": "-axis. For conciseness of", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 85, + 523, + 100, + 533 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "analysis, we instantiate OMD with L2-regularization, which results in the simple OGD algorithm", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 531, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 85, + 533, + 100, + 544 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 247, + 545 + ], + "score": 1.0, + "content": "[24]. We start at an arbitrary point", + "type": "text" + }, + { + "bbox": [ + 248, + 532, + 325, + 544 + ], + "score": 0.93, + "content": "x _ { 1 } = ( u _ { 1 } , v _ { 1 } ) \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 531, + 367, + 545 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 368, + 532, + 397, + 543 + ], + "score": 0.91, + "content": "v _ { 1 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 531, + 461, + 545 + ], + "score": 1.0, + "content": ". At each round", + "type": "text" + }, + { + "bbox": [ + 461, + 533, + 466, + 542 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 531, + 506, + 545 + ], + "score": 1.0, + "content": ", suppose", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 541, + 496, + 555 + ], + "spans": [ + { + "bbox": [ + 85, + 544, + 100, + 554 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 157, + 555 + ], + "score": 1.0, + "content": "the decision", + "type": "text" + }, + { + "bbox": [ + 157, + 543, + 231, + 555 + ], + "score": 0.93, + "content": "x _ { t } = ( u _ { t } , v _ { t } ) \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 541, + 400, + 555 + ], + "score": 1.0, + "content": ", then the gradients of each objective w.r.t.", + "type": "text" + }, + { + "bbox": [ + 401, + 545, + 411, + 554 + ], + "score": 0.87, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 541, + 496, + 555 + ], + "score": 1.0, + "content": "can be calculated as", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 85, + 485, + 508, + 555 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 558, + 490, + 588 + ], + "lines": [ + { + "bbox": [ + 118, + 558, + 490, + 588 + ], + "spans": [ + { + "bbox": [ + 118, + 558, + 490, + 588 + ], + "score": 0.93, + "content": "g _ { t } ^ { 1 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } + 4 , ~ 2 v _ { t } + 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k . } \\end{array} \\right. } \\qquad g _ { t } ^ { 2 } = { \\left\\{ \\begin{array} { l l } { ( 2 u _ { t } , } & { 2 v _ { t } - 2 ) , } & { t = 2 k - 1 ; } \\\\ { ( 2 u _ { t } - 4 , } & { 2 v _ { t } + 2 ) , } & { t = 2 k . } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "14cb2aa94f549718bcec12a508e4b0249ddaad2970271250ebf8e7e7304f85c8.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 118, + 558, + 490, + 568.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 118, + 568.0, + 490, + 578.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 118, + 578.0, + 490, + 588.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "index", + "bbox": [ + 83, + 593, + 506, + 616 + ], + "lines": [ + { + "bbox": [ + 86, + 591, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 86, + 595, + 100, + 604 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 104, + 591, + 132, + 607 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 592, + 180, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { 0 \\leq v _ { t } \\leq \\frac { 1 } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 591, + 506, + 607 + ], + "score": 1.0, + "content": ", we observe that the second entry of either gradient alternates between positive", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 604, + 429, + 616 + ], + "spans": [ + { + "bbox": [ + 85, + 604, + 336, + 616 + ], + "score": 1.0, + "content": "246 and negative. By using min-norm, the composite weights", + "type": "text" + }, + { + "bbox": [ + 337, + 605, + 347, + 615 + ], + "score": 0.9, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 604, + 429, + 616 + ], + "score": 1.0, + "content": "can be computed as", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 39.5, + "bbox_fs": [ + 85, + 591, + 506, + 616 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 619, + 426, + 650 + ], + "lines": [ + { + "bbox": [ + 183, + 619, + 426, + 650 + ], + "spans": [ + { + "bbox": [ + 183, + 619, + 426, + 650 + ], + "score": 0.91, + "content": "\\lambda _ { t } = \\left\\{ { \\begin{array} { l l } { ( ( 1 - u _ { t } - v _ { t } ) / 4 , } & { ( 3 + u _ { t } + v _ { t } ) / 4 ) , t = 2 k - 1 ; } \\\\ { ( ( 3 - u _ { t } + v _ { t } ) / 4 , } & { ( 1 + u _ { t } - v _ { t } ) / 4 ) , t = 2 k . } \\end{array} } \\right.", + "type": "interline_equation", + "image_path": "e7d836d82406529953d1663daa665b08f603af784654f260f81479c5d843ade6.jpg" + } + ] + } + ], + "index": 41.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 619, + 426, + 634.5 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 183, + 634.5, + 426, + 650.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 654, + 505, + 688 + ], + "lines": [ + { + "bbox": [ + 86, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 656, + 100, + 666 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 103, + 653, + 423, + 669 + ], + "score": 1.0, + "content": "We observe that both entries of composite weights alternative between above", + "type": "text" + }, + { + "bbox": [ + 423, + 654, + 430, + 667 + ], + "score": 0.87, + "content": "\\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 653, + 477, + 669 + ], + "score": 1.0, + "content": "and below", + "type": "text" + }, + { + "bbox": [ + 477, + 654, + 484, + 667 + ], + "score": 0.86, + "content": "\\frac { 1 } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 653, + 506, + 669 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 667, + 100, + 677 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 107, + 665, + 182, + 677 + ], + "score": 0.91, + "content": "\\| \\lambda _ { t + 1 } - \\lambda _ { t } \\| _ { 1 } \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 665, + 232, + 678 + ], + "score": 1.0, + "content": ". Recall that", + "type": "text" + }, + { + "bbox": [ + 232, + 666, + 275, + 677 + ], + "score": 0.92, + "content": "\\| \\lambda _ { t } \\| _ { 1 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", hence the composite weights at two consecutive rounds", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 676, + 334, + 689 + ], + "spans": [ + { + "bbox": [ + 86, + 679, + 100, + 687 + ], + "score": 1.0, + "content": "249", + "type": "text" + }, + { + "bbox": [ + 105, + 676, + 334, + 689 + ], + "score": 1.0, + "content": "change radically. The resulting composite gradient takes", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + } + ], + "index": 44, + "bbox_fs": [ + 86, + 653, + 506, + 689 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 692, + 424, + 721 + ], + "lines": [ + { + "bbox": [ + 186, + 692, + 424, + 721 + ], + "spans": [ + { + "bbox": [ + 186, + 692, + 424, + 721 + ], + "score": 0.93, + "content": "g _ { t } ^ { c o m p } = \\left\\{ \\begin{array} { l l } { { ( u _ { t } - v _ { t } + 1 , ~ } } & { { - u _ { t } + v _ { t } - 1 ) , t = 2 k - 1 ; } } \\\\ { { ( - u _ { t } - v _ { t } - 1 , } } & { { - u _ { t } - v _ { t } - 1 ) , t = 2 k . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "4dd34499bce826f3562f71baea025f23e37535bd5d8aa7ab1b6d598418786ef5.jpg" + } + ] + } + ], + "index": 46.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 692, + 424, + 706.5 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 186, + 706.5, + 424, + 721.0 + ], + "spans": [], + "index": 47 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 506, + 139 + ], + "lines": [ + { + "bbox": [ + 85, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "The fluctuating composite weights mix with the positive and negative second entries of gradients,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 85, + 80, + 503, + 97 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 99, + 96 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 103, + 80, + 220, + 97 + ], + "score": 1.0, + "content": "making the second entry of", + "type": "text" + }, + { + "bbox": [ + 221, + 83, + 245, + 96 + ], + "score": 0.91, + "content": "g _ { t } ^ { c \\bar { o } m p }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 80, + 334, + 97 + ], + "score": 1.0, + "content": "always negative, i.e.,", + "type": "text" + }, + { + "bbox": [ + 335, + 84, + 409, + 95 + ], + "score": 0.91, + "content": "- u _ { t } + v _ { t } - 1 < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 80, + 428, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 84, + 503, + 95 + ], + "score": 0.9, + "content": "- u _ { t } - v _ { t } - 1 < 0", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 92, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 105, + 92, + 135, + 109 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + }, + { + "bbox": [ + 135, + 94, + 160, + 106 + ], + "score": 0.91, + "content": "{ \\bf { \\bar { \\it g } } } _ { t } ^ { c o m p }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 92, + 224, + 109 + ], + "score": 1.0, + "content": "actually drives", + "type": "text" + }, + { + "bbox": [ + 225, + 97, + 235, + 105 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 92, + 341, + 109 + ], + "score": 1.0, + "content": "away from the Pareto set", + "type": "text" + }, + { + "bbox": [ + 341, + 95, + 355, + 104 + ], + "score": 0.87, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 92, + 452, + 109 + ], + "score": 1.0, + "content": "that coincides with the", + "type": "text" + }, + { + "bbox": [ + 452, + 96, + 459, + 104 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 92, + 507, + 109 + ], + "score": 1.0, + "content": "-axis. This", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "essentially reversely optimizes the loss, hence increases the regret. In fact, we can prove that it even", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 116, + 457, + 129 + ], + "score": 1.0, + "content": "incurs a linear regret2. Due to the lack of space, we leave the proof of linear regret when", + "type": "text" + }, + { + "bbox": [ + 457, + 117, + 466, + 126 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "is an odd", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 128, + 487, + 139 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 105, + 128, + 487, + 139 + ], + "score": 1.0, + "content": "number in the appendix. The above results of the problem instance are summarized as follows.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 97, + 142, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 155 + ], + "score": 1.0, + "content": "Proposition 4.1. For OMD equipped with vanilla min-norm, there exists a multi-objective online", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 153, + 445, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 445, + 166 + ], + "score": 1.0, + "content": "convex optimization problem, in which the resulting algorithm incurs a linear regret.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 89, + 173, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 87, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 87, + 176, + 99, + 185 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "Remark. Stability is a basic requirement to guarantee meaningful regrets in online learning [25].", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 100, + 197 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 105, + 185, + 363, + 198 + ], + "score": 1.0, + "content": "In the single-objective setting, directly regularizing the iterate", + "type": "text" + }, + { + "bbox": [ + 363, + 186, + 374, + 196 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "(e.g., OMD) is already enough.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 99, + 207 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 195, + 338, + 209 + ], + "score": 1.0, + "content": "However, as shown in the above analysis, only regularizing", + "type": "text" + }, + { + "bbox": [ + 338, + 197, + 349, + 207 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "is not enough to attain sublinear regrets", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 86, + 208, + 99, + 218 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 105, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "in the multi-objective setting, since there is another source of instability, i.e., the composite weights,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 87, + 215, + 507, + 231 + ], + "spans": [ + { + "bbox": [ + 87, + 219, + 99, + 228 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 104, + 215, + 507, + 231 + ], + "score": 1.0, + "content": "that affects the direction of the composite gradient. Therefore, in multi-objective online learning,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 228, + 477, + 241 + ], + "spans": [ + { + "bbox": [ + 86, + 230, + 100, + 240 + ], + "score": 1.0, + "content": "263", + "type": "text" + }, + { + "bbox": [ + 105, + 228, + 477, + 241 + ], + "score": 1.0, + "content": "besides regularizing the iterates, we also need to explicitly regularize the composite weights.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 105, + 252, + 354, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 356, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 356, + 269 + ], + "score": 1.0, + "content": "4.2 Doubly Regularized Online Mirror Multiple Descent", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 104, + 272, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 441, + 286 + ], + "score": 1.0, + "content": "Enlightened by the design of regularization in FTRL [25], we consider the regularizer", + "type": "text" + }, + { + "bbox": [ + 441, + 273, + 475, + 285 + ], + "score": 0.93, + "content": "r ( \\lambda , \\lambda _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 272, + 505, + 286 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 118, + 295 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "is the pre-defined composite weight that may reflect the user preference. This results in a new", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 270, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 270, + 308 + ], + "score": 1.0, + "content": "solver called min-regularized-norm, i.e.,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 311, + 389, + 333 + ], + "lines": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "spans": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "score": 0.93, + "content": "\\lambda _ { t } = \\underset { \\lambda \\in S _ { m } } { \\arg \\operatorname* { m i n } } \\| \\nabla F _ { t } ( x _ { t } ) \\lambda \\| _ { 2 } ^ { 2 } + \\alpha r ( \\lambda , \\lambda _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "52361531472f26bd6b6a2fcda139d4374fd851cd8ab1047f09b6327f3470ce75.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 338, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 86, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 86, + 341, + 99, + 350 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 104, + 338, + 134, + 351 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 341, + 142, + 348 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "is the strength of regularization. Equipping OMD with the new solver, we derive the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 86, + 352, + 99, + 361 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 104, + 349, + 420, + 362 + ], + "score": 1.0, + "content": "proposed online algorithm. Note that beyond the regularization on the iterate", + "type": "text" + }, + { + "bbox": [ + 420, + 351, + 430, + 361 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "that is intrinsic in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 361, + 507, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 363, + 99, + 372 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 106, + 361, + 393, + 373 + ], + "score": 1.0, + "content": "online learning, there is another regularization on the composite weights", + "type": "text" + }, + { + "bbox": [ + 394, + 361, + 405, + 371 + ], + "score": 0.88, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 361, + 507, + 373 + ], + "score": 1.0, + "content": "in min-regularized norm.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 86, + 373, + 99, + 383 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "Both regularizations are fundamental and they together ensure the stability in the multi-objective", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 381, + 459, + 395 + ], + "spans": [ + { + "bbox": [ + 86, + 384, + 99, + 393 + ], + "score": 1.0, + "content": "272", + "type": "text" + }, + { + "bbox": [ + 105, + 381, + 459, + 395 + ], + "score": 1.0, + "content": "online setting. Hence we call the algorithm Doubly Regularized OMMD (DR-OMMD).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 86, + 398, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 86, + 399, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 400, + 100, + 410 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 105, + 399, + 160, + 410 + ], + "score": 1.0, + "content": "In principle,", + "type": "text" + }, + { + "bbox": [ + 160, + 401, + 167, + 409 + ], + "score": 0.7, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 399, + 299, + 410 + ], + "score": 1.0, + "content": "can take various forms such as", + "type": "text" + }, + { + "bbox": [ + 299, + 399, + 312, + 410 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 399, + 341, + 410 + ], + "score": 1.0, + "content": "-norm,", + "type": "text" + }, + { + "bbox": [ + 342, + 399, + 354, + 410 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 399, + 505, + 410 + ], + "score": 1.0, + "content": "-norm and KL divergence etc. Here", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 85, + 411, + 100, + 421 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 104, + 409, + 147, + 421 + ], + "score": 1.0, + "content": "we adopt", + "type": "text" + }, + { + "bbox": [ + 147, + 410, + 159, + 420 + ], + "score": 0.88, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 409, + 397, + 421 + ], + "score": 1.0, + "content": "-norm since it aligns well with the simplex constraint of", + "type": "text" + }, + { + "bbox": [ + 397, + 410, + 405, + 419 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 409, + 506, + 421 + ], + "score": 1.0, + "content": ". Min-regularized-norm", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 423, + 99, + 431 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 105, + 420, + 417, + 433 + ], + "score": 1.0, + "content": "can be computed very efficiently, since it has a closed-form solution when", + "type": "text" + }, + { + "bbox": [ + 417, + 421, + 447, + 430 + ], + "score": 0.87, + "content": "m = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 420, + 506, + 433 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 430, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 86, + 433, + 100, + 443 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 104, + 430, + 236, + 445 + ], + "score": 1.0, + "content": "suppose the gradients at round", + "type": "text" + }, + { + "bbox": [ + 237, + 432, + 243, + 441 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 430, + 260, + 445 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 260, + 431, + 271, + 443 + ], + "score": 0.89, + "content": "g _ { t } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 430, + 291, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 431, + 302, + 443 + ], + "score": 0.89, + "content": "g _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 430, + 325, + 445 + ], + "score": 1.0, + "content": ". Set", + "type": "text" + }, + { + "bbox": [ + 326, + 430, + 485, + 443 + ], + "score": 0.89, + "content": "\\gamma _ { L } = ( g _ { 2 } ^ { \\top } ( g _ { 2 } - g _ { 1 } ) - \\alpha ) / \\Vert g _ { 2 } - g _ { 1 } \\Vert ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 430, + 506, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 86, + 444, + 100, + 455 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 106, + 442, + 263, + 455 + ], + "score": 0.9, + "content": "\\gamma _ { R } = ( g _ { 2 } ^ { \\top } ( g _ { 2 } - g _ { 1 } ) + \\alpha ) / \\Vert g _ { 2 } - g _ { 1 } \\Vert ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 443, + 315, + 456 + ], + "score": 1.0, + "content": ". Given any", + "type": "text" + }, + { + "bbox": [ + 315, + 444, + 415, + 455 + ], + "score": 0.91, + "content": "\\lambda _ { 0 } = ( \\gamma _ { 0 } , 1 - \\gamma _ { 0 } ) \\in S _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 443, + 506, + 456 + ], + "score": 1.0, + "content": ", we can compute the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 454, + 282, + 467 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 183, + 467 + ], + "score": 1.0, + "content": "composite weights", + "type": "text" + }, + { + "bbox": [ + 184, + 455, + 194, + 465 + ], + "score": 0.85, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 454, + 207, + 467 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 207, + 455, + 254, + 466 + ], + "score": 0.91, + "content": "( \\gamma _ { t } , 1 - \\gamma _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 454, + 282, + 467 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 470, + 441, + 485 + ], + "lines": [ + { + "bbox": [ + 168, + 470, + 441, + 485 + ], + "spans": [ + { + "bbox": [ + 168, + 470, + 441, + 485 + ], + "score": 0.87, + "content": "\\gamma _ { t } = \\operatorname* { m a x } \\{ \\operatorname* { m i n } \\{ \\gamma _ { t } ^ { \\prime \\prime } , 1 \\} , 0 \\} , \\quad \\mathrm { w h e r e } \\ \\gamma _ { t } ^ { \\prime \\prime } = \\operatorname* { m a x } \\{ \\operatorname* { m i n } \\{ \\gamma _ { 0 } , \\gamma _ { R } \\} , \\gamma _ { L } \\} .", + "type": "interline_equation", + "image_path": "55b0cd35b705b0b6dea4715bc89b3755504987afbf586b3967e9ff91a46e6c9d.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 168, + 470, + 441, + 485 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 94, + 490, + 504, + 513 + ], + "lines": [ + { + "bbox": [ + 91, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 91, + 490, + 178, + 503 + ], + "score": 1.0, + "content": "79 In addition, when", + "type": "text" + }, + { + "bbox": [ + 178, + 491, + 207, + 501 + ], + "score": 0.89, + "content": "m > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 490, + 308, + 503 + ], + "score": 1.0, + "content": ", since the feasible region", + "type": "text" + }, + { + "bbox": [ + 308, + 491, + 322, + 501 + ], + "score": 0.91, + "content": "S _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "is a simplex, we can introduce a Frank-Wolfe", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 91, + 501, + 493, + 513 + ], + "spans": [ + { + "bbox": [ + 91, + 501, + 493, + 513 + ], + "score": 1.0, + "content": "80 solver [14] to compute the composite weights. See the protocol and more details in Appendix D.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "Compared to vanilla min-norm, the composite weights in min-regularized-norm are not fully deter-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "mined by the adversarial gradients. The resulting relative stability of composite weights make the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 552 + ], + "score": 1.0, + "content": "composite gradients more robust to the adversarial environment. In the following, we give a general", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 397, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 397, + 563 + ], + "score": 1.0, + "content": "analysis and prove that DR-OMMD indeed guarantees sublinear regrets.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 107, + 575, + 167, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 168, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 168, + 588 + ], + "score": 1.0, + "content": "4.3 Analysis", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 104, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "We now analyze the static regret and the dynamic regret of DR-OMMD. Our analysis is based on the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 302, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 302, + 619 + ], + "score": 1.0, + "content": "following commonly used assumptions [13, 11].", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 394, + 634 + ], + "score": 1.0, + "content": "Assumption 4.2 (Bregman divergence). The regularization function", + "type": "text" + }, + { + "bbox": [ + 395, + 622, + 403, + 631 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "is 1-strongly convex. In", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 257, + 645 + ], + "score": 1.0, + "content": "addition, the Bregman divergence is", + "type": "text" + }, + { + "bbox": [ + 257, + 634, + 264, + 644 + ], + "score": 0.83, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 632, + 375, + 645 + ], + "score": 1.0, + "content": "-Lipschitz continuous, i.e.,", + "type": "text" + }, + { + "bbox": [ + 375, + 632, + 506, + 644 + ], + "score": 0.88, + "content": "B _ { R } ( x , z ) - B _ { R } ( \\bar { y } , z ) \\leq \\gamma \\| x -", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 643, + 460, + 655 + ], + "spans": [ + { + "bbox": [ + 107, + 643, + 191, + 655 + ], + "score": 0.9, + "content": "y \\| , \\forall x , y , z \\in \\mathrm { d o m } R", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 643, + 222, + 654 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 222, + 643, + 250, + 654 + ], + "score": 0.63, + "content": "\\mathrm { d o m } R", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 643, + 318, + 654 + ], + "score": 1.0, + "content": "is the domain of", + "type": "text" + }, + { + "bbox": [ + 318, + 644, + 327, + 653 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 643, + 379, + 654 + ], + "score": 1.0, + "content": "and satisfies", + "type": "text" + }, + { + "bbox": [ + 379, + 644, + 455, + 654 + ], + "score": 0.86, + "content": "\\mathcal { X } \\subset \\mathrm { d o m } R \\subset \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 643, + 460, + 654 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 315, + 671 + ], + "score": 1.0, + "content": "Assumption 4.3 (Lipschitz continuity). For each", + "type": "text" + }, + { + "bbox": [ + 315, + 658, + 377, + 670 + ], + "score": 0.91, + "content": "i \\in \\{ 1 , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 657, + 506, + 671 + ], + "score": 1.0, + "content": ", there exists some positive and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 129, + 682 + ], + "score": 1.0, + "content": "finite", + "type": "text" + }, + { + "bbox": [ + 129, + 669, + 138, + 679 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 668, + 194, + 682 + ], + "score": 1.0, + "content": "such that, the", + "type": "text" + }, + { + "bbox": [ + 194, + 670, + 199, + 679 + ], + "score": 0.78, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 668, + 228, + 682 + ], + "score": 1.0, + "content": "-th loss", + "type": "text" + }, + { + "bbox": [ + 228, + 669, + 239, + 681 + ], + "score": 0.87, + "content": "f _ { t } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 668, + 295, + 682 + ], + "score": 1.0, + "content": "at each round", + "type": "text" + }, + { + "bbox": [ + 295, + 669, + 356, + 681 + ], + "score": 0.92, + "content": "t \\in \\{ 1 , \\ldots , T \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 668, + 367, + 682 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 367, + 670, + 376, + 679 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 668, + 484, + 682 + ], + "score": 1.0, + "content": "-Lipschitz continuous w.r.t.", + "type": "text" + }, + { + "bbox": [ + 484, + 669, + 502, + 681 + ], + "score": 0.91, + "content": "\\| \\cdot \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 668, + 506, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 679, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 123, + 693 + ], + "score": 1.0, + "content": "i.e.,", + "type": "text" + }, + { + "bbox": [ + 123, + 680, + 246, + 692 + ], + "score": 0.89, + "content": "| f _ { t } ^ { i } ( x ) - f _ { t } ^ { i } ( x ^ { \\prime } ) | \\leq G \\| x - \\bar { x } ^ { \\prime } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 679, + 506, + 693 + ], + "score": 1.0, + "content": ". Note that in the convex setting, this assumption leads to bounded", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 427, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 165, + 704 + ], + "score": 1.0, + "content": "gradients, i.e.,", + "type": "text" + }, + { + "bbox": [ + 166, + 691, + 233, + 703 + ], + "score": 0.9, + "content": "\\| \\nabla f _ { t } ^ { i } ( x ) \\| _ { * } \\leq G", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 690, + 265, + 704 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 266, + 691, + 423, + 703 + ], + "score": 0.91, + "content": "t \\in \\{ 1 , \\ldots , T \\} , i \\in \\{ 1 , \\ldots , m \\} , x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 690, + 427, + 704 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 711, + 360, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 709, + 361, + 724 + ], + "spans": [ + { + "bbox": [ + 118, + 709, + 361, + 724 + ], + "score": 1.0, + "content": "2More concisely, here the regret is the multi-objective static regret.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 86, + 72, + 506, + 139 + ], + "lines": [ + { + "bbox": [ + 85, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "250", + "type": "text" + }, + { + "bbox": [ + 105, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "The fluctuating composite weights mix with the positive and negative second entries of gradients,", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 80, + 503, + 97 + ], + "spans": [ + { + "bbox": [ + 85, + 85, + 99, + 96 + ], + "score": 1.0, + "content": "251", + "type": "text" + }, + { + "bbox": [ + 103, + 80, + 220, + 97 + ], + "score": 1.0, + "content": "making the second entry of", + "type": "text" + }, + { + "bbox": [ + 221, + 83, + 245, + 96 + ], + "score": 0.91, + "content": "g _ { t } ^ { c \\bar { o } m p }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 80, + 334, + 97 + ], + "score": 1.0, + "content": "always negative, i.e.,", + "type": "text" + }, + { + "bbox": [ + 335, + 84, + 409, + 95 + ], + "score": 0.91, + "content": "- u _ { t } + v _ { t } - 1 < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 80, + 428, + 97 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 84, + 503, + 95 + ], + "score": 0.9, + "content": "- u _ { t } - v _ { t } - 1 < 0", + "type": "inline_equation" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 92, + 507, + 109 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "252", + "type": "text" + }, + { + "bbox": [ + 105, + 92, + 135, + 109 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + }, + { + "bbox": [ + 135, + 94, + 160, + 106 + ], + "score": 0.91, + "content": "{ \\bf { \\bar { \\it g } } } _ { t } ^ { c o m p }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 92, + 224, + 109 + ], + "score": 1.0, + "content": "actually drives", + "type": "text" + }, + { + "bbox": [ + 225, + 97, + 235, + 105 + ], + "score": 0.83, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 92, + 341, + 109 + ], + "score": 1.0, + "content": "away from the Pareto set", + "type": "text" + }, + { + "bbox": [ + 341, + 95, + 355, + 104 + ], + "score": 0.87, + "content": "\\mathcal { X } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 92, + 452, + 109 + ], + "score": 1.0, + "content": "that coincides with the", + "type": "text" + }, + { + "bbox": [ + 452, + 96, + 459, + 104 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 92, + 507, + 109 + ], + "score": 1.0, + "content": "-axis. This", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 86, + 108, + 99, + 117 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "essentially reversely optimizes the loss, hence increases the regret. In fact, we can prove that it even", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "254", + "type": "text" + }, + { + "bbox": [ + 105, + 116, + 457, + 129 + ], + "score": 1.0, + "content": "incurs a linear regret2. Due to the lack of space, we leave the proof of linear regret when", + "type": "text" + }, + { + "bbox": [ + 457, + 117, + 466, + 126 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "is an odd", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 128, + 487, + 139 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 100, + 139 + ], + "score": 1.0, + "content": "255", + "type": "text" + }, + { + "bbox": [ + 105, + 128, + 487, + 139 + ], + "score": 1.0, + "content": "number in the appendix. The above results of the problem instance are summarized as follows.", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + } + ], + "index": 2.5, + "bbox_fs": [ + 85, + 73, + 507, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 97, + 142, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 155 + ], + "score": 1.0, + "content": "Proposition 4.1. For OMD equipped with vanilla min-norm, there exists a multi-objective online", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 153, + 445, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 445, + 166 + ], + "score": 1.0, + "content": "convex optimization problem, in which the resulting algorithm incurs a linear regret.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 106, + 141, + 505, + 166 + ] + }, + { + "type": "index", + "bbox": [ + 89, + 173, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 87, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 87, + 176, + 99, + 185 + ], + "score": 1.0, + "content": "258", + "type": "text" + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "Remark. Stability is a basic requirement to guarantee meaningful regrets in online learning [25].", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 185, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 100, + 197 + ], + "score": 1.0, + "content": "259", + "type": "text" + }, + { + "bbox": [ + 105, + 185, + 363, + 198 + ], + "score": 1.0, + "content": "In the single-objective setting, directly regularizing the iterate", + "type": "text" + }, + { + "bbox": [ + 363, + 186, + 374, + 196 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 185, + 506, + 198 + ], + "score": 1.0, + "content": "(e.g., OMD) is already enough.", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 99, + 207 + ], + "score": 1.0, + "content": "260", + "type": "text" + }, + { + "bbox": [ + 105, + 195, + 338, + 209 + ], + "score": 1.0, + "content": "However, as shown in the above analysis, only regularizing", + "type": "text" + }, + { + "bbox": [ + 338, + 197, + 349, + 207 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "is not enough to attain sublinear regrets", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 86, + 208, + 99, + 218 + ], + "score": 1.0, + "content": "261", + "type": "text" + }, + { + "bbox": [ + 105, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "in the multi-objective setting, since there is another source of instability, i.e., the composite weights,", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 215, + 507, + 231 + ], + "spans": [ + { + "bbox": [ + 87, + 219, + 99, + 228 + ], + "score": 1.0, + "content": "262", + "type": "text" + }, + { + "bbox": [ + 104, + 215, + 507, + 231 + ], + "score": 1.0, + "content": "that affects the direction of the composite gradient. Therefore, in multi-objective online learning,", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 228, + 477, + 241 + ], + "spans": [ + { + "bbox": [ + 86, + 230, + 100, + 240 + ], + "score": 1.0, + "content": "263", + "type": "text" + }, + { + "bbox": [ + 105, + 228, + 477, + 241 + ], + "score": 1.0, + "content": "besides regularizing the iterates, we also need to explicitly regularize the composite weights.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + } + ], + "index": 10.5, + "bbox_fs": [ + 86, + 173, + 507, + 241 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 252, + 354, + 265 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 356, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 356, + 269 + ], + "score": 1.0, + "content": "4.2 Doubly Regularized Online Mirror Multiple Descent", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 104, + 272, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 272, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 441, + 286 + ], + "score": 1.0, + "content": "Enlightened by the design of regularization in FTRL [25], we consider the regularizer", + "type": "text" + }, + { + "bbox": [ + 441, + 273, + 475, + 285 + ], + "score": 0.93, + "content": "r ( \\lambda , \\lambda _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 272, + 505, + 286 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 118, + 295 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "is the pre-defined composite weight that may reflect the user preference. This results in a new", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 270, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 270, + 308 + ], + "score": 1.0, + "content": "solver called min-regularized-norm, i.e.,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 272, + 506, + 308 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 311, + 389, + 333 + ], + "lines": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "spans": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "score": 0.93, + "content": "\\lambda _ { t } = \\underset { \\lambda \\in S _ { m } } { \\arg \\operatorname* { m i n } } \\| \\nabla F _ { t } ( x _ { t } ) \\lambda \\| _ { 2 } ^ { 2 } + \\alpha r ( \\lambda , \\lambda _ { 0 } ) ,", + "type": "interline_equation", + "image_path": "52361531472f26bd6b6a2fcda139d4374fd851cd8ab1047f09b6327f3470ce75.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 221, + 311, + 389, + 333 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 338, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 86, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 86, + 341, + 99, + 350 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 104, + 338, + 134, + 351 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 341, + 142, + 348 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "is the strength of regularization. Equipping OMD with the new solver, we derive the", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 86, + 352, + 99, + 361 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 104, + 349, + 420, + 362 + ], + "score": 1.0, + "content": "proposed online algorithm. 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See the protocol and more details in Appendix D.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + } + ], + "index": 31.5, + "bbox_fs": [ + 91, + 490, + 505, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 517, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 506, + 530 + ], + "score": 1.0, + "content": "Compared to vanilla min-norm, the composite weights in min-regularized-norm are not fully deter-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "mined by the adversarial gradients. 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In the following, we give a general", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 397, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 397, + 563 + ], + "score": 1.0, + "content": "analysis and prove that DR-OMMD indeed guarantees sublinear regrets.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 517, + 506, + 563 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 575, + 167, + 587 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 168, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 168, + 588 + ], + "score": 1.0, + "content": "4.3 Analysis", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 104, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "We now analyze the static regret and the dynamic regret of DR-OMMD. Our analysis is based on the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 605, + 302, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 302, + 619 + ], + "score": 1.0, + "content": "following commonly used assumptions [13, 11].", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 594, + 505, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 394, + 634 + ], + "score": 1.0, + "content": "Assumption 4.2 (Bregman divergence). The regularization function", + "type": "text" + }, + { + "bbox": [ + 395, + 622, + 403, + 631 + ], + "score": 0.83, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "is 1-strongly convex. 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Note that in the convex setting, this assumption leads to bounded", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 427, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 165, + 704 + ], + "score": 1.0, + "content": "gradients, i.e.,", + "type": "text" + }, + { + "bbox": [ + 166, + 691, + 233, + 703 + ], + "score": 0.9, + "content": "\\| \\nabla f _ { t } ^ { i } ( x ) \\| _ { * } \\leq G", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 690, + 265, + 704 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 266, + 691, + 423, + 703 + ], + "score": 0.91, + "content": "t \\in \\{ 1 , \\ldots , T \\} , i \\in \\{ 1 , \\ldots , m \\} , x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 690, + 427, + 704 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 657, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 88, + 72, + 496, + 84 + ], + "lines": [ + { + "bbox": [ + 85, + 72, + 496, + 86 + ], + "spans": [ + { + "bbox": [ + 85, + 72, + 496, + 86 + ], + "score": 1.0, + "content": "295 We first provide the static regret bound. The proof is left to the appendix due to the lack of space.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 85, + 87, + 504, + 111 + ], + "lines": [ + { + "bbox": [ + 86, + 86, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 268, + 102 + ], + "score": 1.0, + "content": "Theorem 4.4. Suppose the diameter of 296", + "type": "text" + }, + { + "bbox": [ + 268, + 89, + 277, + 98 + ], + "score": 0.78, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 86, + 337, + 102 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 338, + 88, + 347, + 98 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 86, + 385, + 102 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 385, + 88, + 396, + 99 + ], + "score": 0.86, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 86, + 464, + 102 + ], + "score": 1.0, + "content": "is bounded, i.e.,", + "type": "text" + }, + { + "bbox": [ + 464, + 87, + 505, + 100 + ], + "score": 0.91, + "content": "| f _ { t } ^ { i } ( x ) | \\leq", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 84, + 98, + 440, + 111 + ], + "spans": [ + { + "bbox": [ + 84, + 98, + 105, + 111 + ], + "score": 1.0, + "content": "297", + "type": "text" + }, + { + "bbox": [ + 106, + 99, + 281, + 111 + ], + "score": 0.92, + "content": "F , \\forall x \\in \\mathcal { X } , t \\in \\{ \\bar { 1 } , \\dots , T \\} , i \\in \\{ 1 , \\dots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 98, + 319, + 111 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 319, + 99, + 355, + 110 + ], + "score": 0.89, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 98, + 440, + 111 + ], + "score": 1.0, + "content": ", DR-OMMD attains", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 116, + 455, + 142 + ], + "lines": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "spans": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O S } } ( T ) \\leq \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 4 F } { \\eta } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .", + "type": "interline_equation", + "image_path": "244348e8a04debb30b6e01f798772fa63f1a766a31f073cc4754407ab786b25e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 152, + 506, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 271, + 165 + ], + "score": 1.0, + "content": "Remark. (i) Linearization with weights", + "type": "text" + }, + { + "bbox": [ + 271, + 153, + 309, + 164 + ], + "score": 0.9, + "content": "\\lambda _ { 0 } \\in \\mathcal { S } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "can be viewed as single-objective optimization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 164, + 176 + ], + "score": 1.0, + "content": "on scalar loss", + "type": "text" + }, + { + "bbox": [ + 165, + 163, + 188, + 176 + ], + "score": 0.92, + "content": "\\lambda _ { 0 } ^ { \\top } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 163, + 267, + 176 + ], + "score": 1.0, + "content": ", whose gradient is", + "type": "text" + }, + { + "bbox": [ + 268, + 164, + 336, + 176 + ], + "score": 0.93, + "content": "g _ { t } = \\nabla F _ { t } ( x _ { t } ) \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 163, + 505, + 176 + ], + "score": 1.0, + "content": ". Hence we can directly borrow the tight", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 102, + 172, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 102, + 172, + 342, + 194 + ], + "score": 1.0, + "content": "bound of OMD (Theorem 6.8 in [27]) and derive a bound", + "type": "text" + }, + { + "bbox": [ + 342, + 174, + 504, + 190 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\sum _ { t = 1 } ^ { T } \\frac { \\eta _ { t } } { 2 } \\| \\nabla F _ { t } ( x _ { t } ) \\lambda _ { 0 } \\| _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 510, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 117, + 214 + ], + "score": 0.85, + "content": "\\lambda _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 190, + 200, + 223 + ], + "score": 1.0, + "content": "r linearization. In co, the bound becomes", + "type": "text" + }, + { + "bbox": [ + 200, + 202, + 466, + 216 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } \\operatorname* { m i n } _ { \\lambda \\in { \\cal S } _ { m } } \\{ \\| \\nabla F _ { t } ( x _ { t } ) \\lambda \\| ^ { 2 } + \\alpha \\| \\lambda - \\lambda _ { 0 } \\| _ { 1 } \\} . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 190, + 320, + 202 + ], + "score": 0.92, + "content": "\\alpha = 4 F / \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 190, + 510, + 223 + ], + "score": 1.0, + "content": "ulation of, which is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "smaller than that of linearization. Note that the lower regret of DR-OMMD compared to linearization", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 226, + 351, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 351, + 239 + ], + "score": 1.0, + "content": "is also empirically verified in our experiments (see Figure 1).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 98, + 239, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 149, + 255 + ], + "score": 1.0, + "content": "(ii) When", + "type": "text" + }, + { + "bbox": [ + 149, + 237, + 231, + 255 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\eta = \\frac { \\hat { \\sqrt { 2 \\gamma D } } } { G \\sqrt { T } } , \\alpha = \\frac { 4 F } { \\eta } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 235, + 349, + 255 + ], + "score": 1.0, + "content": ", the bound is in the order of", + "type": "text" + }, + { + "bbox": [ + 350, + 239, + 381, + 252 + ], + "score": 0.93, + "content": "O ( \\sqrt { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 235, + 506, + 255 + ], + "score": 1.0, + "content": ". It matches the optimal static", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 253, + 412, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 248, + 266 + ], + "score": 1.0, + "content": "single-objective regret bound w.r.t.", + "type": "text" + }, + { + "bbox": [ + 248, + 254, + 257, + 264 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 253, + 412, + 266 + ], + "score": 1.0, + "content": "[11] (see more details in Appendix E).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 100, + 270, + 486, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 485, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 485, + 284 + ], + "score": 1.0, + "content": "Then we turn to the dynamic regret. Our analysis relies on an additional assumption [2, 32, 5].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 100, + 284, + 501, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 316, + 298 + ], + "score": 1.0, + "content": "Assumption 4.5 (Temporal variability). For each", + "type": "text" + }, + { + "bbox": [ + 316, + 284, + 378, + 297 + ], + "score": 0.93, + "content": "i \\in \\{ 1 , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 284, + 504, + 298 + ], + "score": 1.0, + "content": ", there exists some positive and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 348, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 129, + 314 + ], + "score": 1.0, + "content": "finite", + "type": "text" + }, + { + "bbox": [ + 129, + 298, + 143, + 309 + ], + "score": 0.87, + "content": "V _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 295, + 182, + 314 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 183, + 296, + 344, + 311 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\operatorname* { s u p } _ { x \\in \\mathcal { X } } | f _ { t } ^ { i } ( x ) - f _ { t + 1 } ^ { i } ( x ) | \\leq V _ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 295, + 348, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 92, + 314, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 85, + 309, + 509, + 333 + ], + "spans": [ + { + "bbox": [ + 85, + 309, + 282, + 333 + ], + "score": 1.0, + "content": "Theorem 4.6. Assume the step size satisfies 310", + "type": "text" + }, + { + "bbox": [ + 282, + 314, + 349, + 329 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 4 V _ { T } } { G ^ { 2 } T } \\leq \\eta \\leq \\frac { 4 V _ { T } } { G ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 309, + 509, + 333 + ], + "score": 1.0, + "content": ". Then under all the above assumptions,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 88, + 326, + 397, + 339 + ], + "spans": [ + { + "bbox": [ + 88, + 326, + 182, + 339 + ], + "score": 1.0, + "content": "311 for any preference", + "type": "text" + }, + { + "bbox": [ + 182, + 327, + 218, + 338 + ], + "score": 0.91, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 326, + 397, + 339 + ], + "score": 1.0, + "content": ", OMMD with min-regularized-norm attains", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 343, + 476, + 371 + ], + "lines": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "spans": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O D } } ( T ) \\leq \\frac { \\eta G ^ { 2 } T } { 2 } + \\frac { 4 \\gamma D V _ { T } } { \\eta ^ { 2 } G ^ { 2 } } + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 8 F G ^ { 2 } T } { V _ { T } } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .", + "type": "interline_equation", + "image_path": "95e535e5a9e21f4de7055d2b2a383d0d2930960b2c050fd2203447d57c6ea6c9.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 382, + 506, + 409 + ], + "lines": [ + { + "bbox": [ + 79, + 377, + 509, + 402 + ], + "spans": [ + { + "bbox": [ + 79, + 377, + 169, + 402 + ], + "score": 1.0, + "content": "Remark. When 312", + "type": "text" + }, + { + "bbox": [ + 169, + 382, + 294, + 398 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\eta = \\frac { 2 } { G } ( \\frac { \\gamma D V _ { T } } { G T } ) ^ { 1 / 3 } , \\alpha = \\frac { 8 F G ^ { 2 } T } { V _ { T } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 377, + 406, + 402 + ], + "score": 1.0, + "content": ", the bound is in the order of", + "type": "text" + }, + { + "bbox": [ + 406, + 382, + 462, + 397 + ], + "score": 0.94, + "content": "O ( T ^ { 2 / 3 } V _ { T } ^ { 1 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 377, + 509, + 402 + ], + "score": 1.0, + "content": ", matching", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 395, + 502, + 410 + ], + "spans": [ + { + "bbox": [ + 83, + 395, + 502, + 410 + ], + "score": 1.0, + "content": "313 the best attainable single-objective dynamic regret bound [2, 35] (see more details in Appendix E).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 99, + 424, + 191, + 438 + ], + "lines": [ + { + "bbox": [ + 104, + 421, + 193, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 193, + 441 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 99, + 448, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "In this section, we conduct extensive experiments to evaluate the effectiveness of DR-OMMD. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "consider two baselines: (i) linearization performs single-objective online learning on the linearized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 124, + 483 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 124, + 470, + 147, + 483 + ], + "score": 0.93, + "content": "\\lambda _ { 0 } ^ { \\top } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 471, + 203, + 483 + ], + "score": 1.0, + "content": "at each round", + "type": "text" + }, + { + "bbox": [ + 203, + 472, + 208, + 481 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 471, + 284, + 483 + ], + "score": 1.0, + "content": ", where the weights", + "type": "text" + }, + { + "bbox": [ + 285, + 471, + 322, + 482 + ], + "score": 0.92, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "are given beforehand; note that it is equivalent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 327, + 495 + ], + "score": 1.0, + "content": "to computing composite gradients with fixed weights", + "type": "text" + }, + { + "bbox": [ + 328, + 482, + 363, + 493 + ], + "score": 0.95, + "content": "\\lambda _ { t } \\equiv \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 481, + 505, + 495 + ], + "score": 1.0, + "content": ". (ii) min-norm equips OMD with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 493, + 295, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 295, + 505 + ], + "score": 1.0, + "content": "vanilla min-norm [7] for gradient composition.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 101, + 517, + 352, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 353, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 353, + 532 + ], + "score": 1.0, + "content": "5.1 Simulation Experiments: Tracking the Pareto Front", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 105, + 537, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 351, + 552 + ], + "score": 1.0, + "content": "As summarized in Figure 1 (a), the goal is to track two points", + "type": "text" + }, + { + "bbox": [ + 352, + 537, + 376, + 550 + ], + "score": 0.92, + "content": "\\xi _ { t } ^ { 1 } , \\xi _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 536, + 464, + 552 + ], + "score": 1.0, + "content": "cycling along a circle", + "type": "text" + }, + { + "bbox": [ + 464, + 538, + 505, + 550 + ], + "score": 0.9, + "content": "{ \\mathcal { C } } = \\{ \\xi \\in { }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 546, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 107, + 548, + 171, + 561 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { 2 } \\mid \\| \\xi \\| _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 546, + 213, + 563 + ], + "score": 1.0, + "content": ". 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The proof is left to the appendix due to the lack of space.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 85, + 72, + 496, + 86 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 87, + 504, + 111 + ], + "lines": [ + { + "bbox": [ + 86, + 86, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 268, + 102 + ], + "score": 1.0, + "content": "Theorem 4.4. Suppose the diameter of 296", + "type": "text" + }, + { + "bbox": [ + 268, + 89, + 277, + 98 + ], + "score": 0.78, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 86, + 337, + 102 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 338, + 88, + 347, + 98 + ], + "score": 0.77, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 86, + 385, + 102 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 385, + 88, + 396, + 99 + ], + "score": 0.86, + "content": "F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 86, + 464, + 102 + ], + "score": 1.0, + "content": "is bounded, i.e.,", + "type": "text" + }, + { + "bbox": [ + 464, + 87, + 505, + 100 + ], + "score": 0.91, + "content": "| f _ { t } ^ { i } ( x ) | \\leq", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 84, + 98, + 440, + 111 + ], + "spans": [ + { + "bbox": [ + 84, + 98, + 105, + 111 + ], + "score": 1.0, + "content": "297", + "type": "text" + }, + { + "bbox": [ + 106, + 99, + 281, + 111 + ], + "score": 0.92, + "content": "F , \\forall x \\in \\mathcal { X } , t \\in \\{ \\bar { 1 } , \\dots , T \\} , i \\in \\{ 1 , \\dots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 98, + 319, + 111 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 319, + 99, + 355, + 110 + ], + "score": 0.89, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 98, + 440, + 111 + ], + "score": 1.0, + "content": ", DR-OMMD attains", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 84, + 86, + 505, + 111 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 155, + 116, + 455, + 142 + ], + "lines": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "spans": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O S } } ( T ) \\leq \\frac { 1 } { \\eta } B _ { R } ( x ^ { * } , x _ { 1 } ) + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 4 F } { \\eta } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .", + "type": "interline_equation", + "image_path": "244348e8a04debb30b6e01f798772fa63f1a766a31f073cc4754407ab786b25e.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 155, + 116, + 455, + 142 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 152, + 506, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 271, + 165 + ], + "score": 1.0, + "content": "Remark. 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It matches the optimal static", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 253, + 412, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 248, + 266 + ], + "score": 1.0, + "content": "single-objective regret bound w.r.t.", + "type": "text" + }, + { + "bbox": [ + 248, + 254, + 257, + 264 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 253, + 412, + 266 + ], + "score": 1.0, + "content": "[11] (see more details in Appendix E).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 235, + 506, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 270, + 486, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 485, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 485, + 284 + ], + "score": 1.0, + "content": "Then we turn to the dynamic regret. Our analysis relies on an additional assumption [2, 32, 5].", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 268, + 485, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 100, + 284, + 501, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 284, + 504, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 316, + 298 + ], + "score": 1.0, + "content": "Assumption 4.5 (Temporal variability). For each", + "type": "text" + }, + { + "bbox": [ + 316, + 284, + 378, + 297 + ], + "score": 0.93, + "content": "i \\in \\{ 1 , \\ldots , m \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 284, + 504, + 298 + ], + "score": 1.0, + "content": ", there exists some positive and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 348, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 129, + 314 + ], + "score": 1.0, + "content": "finite", + "type": "text" + }, + { + "bbox": [ + 129, + 298, + 143, + 309 + ], + "score": 0.87, + "content": "V _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 295, + 182, + 314 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 183, + 296, + 344, + 311 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { t = 1 } ^ { T - 1 } \\operatorname* { s u p } _ { x \\in \\mathcal { X } } | f _ { t } ^ { i } ( x ) - f _ { t + 1 } ^ { i } ( x ) | \\leq V _ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 295, + 348, + 314 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 284, + 504, + 314 + ] + }, + { + "type": "text", + "bbox": [ + 92, + 314, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 85, + 309, + 509, + 333 + ], + "spans": [ + { + "bbox": [ + 85, + 309, + 282, + 333 + ], + "score": 1.0, + "content": "Theorem 4.6. Assume the step size satisfies 310", + "type": "text" + }, + { + "bbox": [ + 282, + 314, + 349, + 329 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\frac { 4 V _ { T } } { G ^ { 2 } T } \\leq \\eta \\leq \\frac { 4 V _ { T } } { G ^ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 309, + 509, + 333 + ], + "score": 1.0, + "content": ". Then under all the above assumptions,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 88, + 326, + 397, + 339 + ], + "spans": [ + { + "bbox": [ + 88, + 326, + 182, + 339 + ], + "score": 1.0, + "content": "311 for any preference", + "type": "text" + }, + { + "bbox": [ + 182, + 327, + 218, + 338 + ], + "score": 0.91, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 326, + 397, + 339 + ], + "score": 1.0, + "content": ", OMMD with min-regularized-norm attains", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 85, + 309, + 509, + 339 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 343, + 476, + 371 + ], + "lines": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "spans": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "score": 0.92, + "content": "R _ { \\mathrm { M O D } } ( T ) \\leq \\frac { \\eta G ^ { 2 } T } { 2 } + \\frac { 4 \\gamma D V _ { T } } { \\eta ^ { 2 } G ^ { 2 } } + \\frac { \\eta } { 2 } \\sum _ { t = 1 } ^ { T } ( \\Vert \\nabla F _ { t } ( x _ { t } ) \\lambda _ { t } \\Vert _ { 2 } ^ { 2 } + \\frac { 8 F G ^ { 2 } T } { V _ { T } } \\Vert \\lambda _ { t } - \\lambda _ { 0 } \\Vert _ { 1 } ) .", + "type": "interline_equation", + "image_path": "95e535e5a9e21f4de7055d2b2a383d0d2930960b2c050fd2203447d57c6ea6c9.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 135, + 343, + 476, + 371 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 382, + 506, + 409 + ], + "lines": [ + { + "bbox": [ + 79, + 377, + 509, + 402 + ], + "spans": [ + { + "bbox": [ + 79, + 377, + 169, + 402 + ], + "score": 1.0, + "content": "Remark. When 312", + "type": "text" + }, + { + "bbox": [ + 169, + 382, + 294, + 398 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\eta = \\frac { 2 } { G } ( \\frac { \\gamma D V _ { T } } { G T } ) ^ { 1 / 3 } , \\alpha = \\frac { 8 F G ^ { 2 } T } { V _ { T } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 377, + 406, + 402 + ], + "score": 1.0, + "content": ", the bound is in the order of", + "type": "text" + }, + { + "bbox": [ + 406, + 382, + 462, + 397 + ], + "score": 0.94, + "content": "O ( T ^ { 2 / 3 } V _ { T } ^ { 1 / 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 377, + 509, + 402 + ], + "score": 1.0, + "content": ", matching", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 395, + 502, + 410 + ], + "spans": [ + { + "bbox": [ + 83, + 395, + 502, + 410 + ], + "score": 1.0, + "content": "313 the best attainable single-objective dynamic regret bound [2, 35] (see more details in Appendix E).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 79, + 377, + 509, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 99, + 424, + 191, + 438 + ], + "lines": [ + { + "bbox": [ + 104, + 421, + 193, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 193, + 441 + ], + "score": 1.0, + "content": "5 Experiments", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 99, + 448, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "In this section, we conduct extensive experiments to evaluate the effectiveness of DR-OMMD. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "consider two baselines: (i) linearization performs single-objective online learning on the linearized", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 124, + 483 + ], + "score": 1.0, + "content": "loss", + "type": "text" + }, + { + "bbox": [ + 124, + 470, + 147, + 483 + ], + "score": 0.93, + "content": "\\lambda _ { 0 } ^ { \\top } F _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 471, + 203, + 483 + ], + "score": 1.0, + "content": "at each round", + "type": "text" + }, + { + "bbox": [ + 203, + 472, + 208, + 481 + ], + "score": 0.76, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 471, + 284, + 483 + ], + "score": 1.0, + "content": ", where the weights", + "type": "text" + }, + { + "bbox": [ + 285, + 471, + 322, + 482 + ], + "score": 0.92, + "content": "\\lambda _ { 0 } \\in { S _ { m } }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "are given beforehand; note that it is equivalent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 481, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 327, + 495 + ], + "score": 1.0, + "content": "to computing composite gradients with fixed weights", + "type": "text" + }, + { + "bbox": [ + 328, + 482, + 363, + 493 + ], + "score": 0.95, + "content": "\\lambda _ { t } \\equiv \\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 481, + 505, + 495 + ], + "score": 1.0, + "content": ". 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(i) protein is a bioinformatics dataset for protein", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 85, + 427, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 85, + 429, + 100, + 439 + ], + "score": 1.0, + "content": "354", + "type": "text" + }, + { + "bbox": [ + 104, + 427, + 505, + 441 + ], + "score": 1.0, + "content": "type classification [31], which has 17 thousand instances with 357 features. 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Other results are deferred to the appendix due to the lack of space. The results show that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 575, + 342, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 342, + 590 + ], + "score": 1.0, + "content": "DR-OMMD consistently outperforms fixed regularization.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5 + }, + { + "type": "title", + "bbox": [ + 105, + 604, + 187, + 617 + ], + "lines": [ + { + "bbox": [ + 104, + 601, + 190, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 190, + 619 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "In this paper, we give a systematic study of multi-objective optimization in the online setting. We", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "first formulate the framework of Multi-Objective Online Convex Optimization. Then we devise", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "the Doubly Regularized Online Mirror Multiple Descent algorithm, which has a special design for", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "score": 1.0, + "content": "gradient composition in online learning, namely min-regularized-norm. 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For fixed regularization, the strength", + "type": "text" + }, + { + "bbox": [ + 330, + 522, + 399, + 534 + ], + "score": 0.93, + "content": "\\sigma = ( 1 - \\lambda _ { 0 } ^ { 1 } ) / \\lambda _ { 0 } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "is determined by the some", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 152, + 547 + ], + "score": 1.0, + "content": "preference", + "type": "text" + }, + { + "bbox": [ + 152, + 532, + 196, + 545 + ], + "score": 0.94, + "content": "\\lambda _ { 0 } ^ { 1 } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 532, + 393, + 547 + ], + "score": 1.0, + "content": ", which is essentially linearization with weights", + "type": "text" + }, + { + "bbox": [ + 393, + 532, + 468, + 545 + ], + "score": 0.93, + "content": "\\overset { \\vartriangle } { \\lambda _ { 0 } } = ( \\lambda _ { 0 } ^ { 1 } , 1 - \\lambda _ { 0 } ^ { \\bar { 1 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 532, + 506, + 547 + ], + "score": 1.0, + "content": ". We run", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 286, + 557 + ], + "score": 1.0, + "content": "both algorithms with varying initial weights", + "type": "text" + }, + { + "bbox": [ + 286, + 543, + 363, + 556 + ], + "score": 0.93, + "content": "\\lambda _ { 0 } ^ { 1 } \\in \\{ 0 , 0 . 1 , . . . , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 543, + 506, + 557 + ], + "score": 1.0, + "content": ". In Figure 2, we plot (a) their final", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 234, + 568 + ], + "score": 1.0, + "content": "performance w.r.t. the choice of", + "type": "text" + }, + { + "bbox": [ + 234, + 555, + 245, + 565 + ], + "score": 0.88, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 554, + 417, + 568 + ], + "score": 1.0, + "content": "and (b) their learning curves with desirable", + "type": "text" + }, + { + "bbox": [ + 417, + 555, + 429, + 565 + ], + "score": 0.87, + "content": "\\lambda _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "(e.g., (0.1, 0.9) on", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "protein). Other results are deferred to the appendix due to the lack of space. The results show that", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 575, + 342, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 342, + 590 + ], + "score": 1.0, + "content": "DR-OMMD consistently outperforms fixed regularization.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 500, + 506, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 604, + 187, + 617 + ], + "lines": [ + { + "bbox": [ + 104, + 601, + 190, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 190, + 619 + ], + "score": 1.0, + "content": "6 Conclusions", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 51 + }, + { + "type": "text", + "bbox": [ + 106, + 629, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "score": 1.0, + "content": "In this paper, we give a systematic study of multi-objective optimization in the online setting. We", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "first formulate the framework of Multi-Objective Online Convex Optimization. Then we devise", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "the Doubly Regularized Online Mirror Multiple Descent algorithm, which has a special design for", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "score": 1.0, + "content": "gradient composition in online learning, namely min-regularized-norm. We provide non-trivial regret", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 672, + 475, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 475, + 686 + ], + "score": 1.0, + "content": "bounds for DR-OMMD and conduct extensive experiments to demonstrate its effectiveness.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54, + "bbox_fs": [ + 105, + 628, + 506, + 686 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 689, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "Limitations. As the first step of studying multiple gradient algorithm in online learning, we conduct", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "our analysis in the convex setting. 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