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+# Robustifying $\ell _ { \infty }$ Adversarial Training to the Union of Perturbation Models
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Classical adversarial training (AT) frameworks are designed to achieve high ad  
+2 versarial accuracy against a single attack type, typically $\ell _ { \infty }$ norm-bounded per  
+3 turbations. Recent extensions in AT have focused on defending against the union   
+4 of multiple perturbation models but this benefit is obtained at the expense of a   
+5 significant (up to $1 0 \times$ ) increase in training complexity over single-attack $\ell _ { \infty }$ AT.   
+6 In this work, we expand the capabilities of widely popular single-attack $\ell _ { \infty }$ AT   
+7 frameworks to provide robustness to the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations while   
+8 preserving their training efficiency. Our technique, referred to as Shaped Noise   
+9 Augmented Processing (SNAP), exploits a well-established byproduct of single  
+10 attack AT frameworks – the reduction in the curvature of the decision boundary of   
+11 networks. SNAP prepends a given deep net with a shaped noise augmentation layer   
+12 whose distribution is learned along with network parameters using any standard   
+13 single-attack AT. As a result, SNAP enhances adversarial accuracy of ResNet-18   
+14 on CIFAR-10 against the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations by $1 \dot { 4 } \%$ -to- $2 0 \%$ for   
+15 four state-of-the-art (SOTA) single-attack $\ell _ { \infty }$ AT frameworks, and, for the first   
+16 time, establishes a benchmark for ResNet-50 and ResNet-101 on ImageNet.
+
+# 17 1 Introduction
+
+18 Today adversarial training (AT) provides state-of-the-art (SOTA) empirical defense against adver  
+19 sarial perturbations. For this, adversarial perturbations are used during training to optimize a robust   
+20 loss function [20, 41, 30, 35]. Early AT frameworks [20, 41] were $7 \times$ -to- $1 0 \times$ more computationally   
+21 demanding than vanilla training. More recent works [30, 35, 40] have significantly reduced the   
+22 computational demands of AT via single-step attacks and superconvergence.   
+23 However, today’s AT frameworks predominantly focus on a single-attack, i.e., they seek robustness   
+24 to a single perturbation, typically $\ell _ { \infty }$ -bounded [30, 35, 37, 41, 43, 40, 39, 26, 9, 34, 42, 10, 11, 14].   
+25 This results in low performance against other perturbations such as $\ell _ { 2 } , \ell _ { 1 }$ , or the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ .   
+26 Indeed, as showemploying only $\ell _ { \infty }$ Fig. 1, four state-of-the-art (SOTA) single-attack AT fra-bounded perturbations achieve low adversarial accuracy ${ \mathcal { A } } _ { \mathrm { a d v } } ^ { ( U ) }$ rksof $\approx 1 5 \%$ mar-to- $2 0 \%$   
+28 against the union of perturbations. Recent extensions in AT [21, 32, 18] do seek higher   
+29 A(U) but only at the expense of $6 \times$ -to- $. 1 0 \times$ increase in the total training time (blue markers in   
+30 Fig. 1). The large training time of these AT frameworks has inhibited their application to large-scale   
+31 datasets such as ImageNet, e.g., Maini et al. [21], Tramèr & Boneh [32] show results for MNIST and   
+32 CIFAR-10 only, while Laidlaw et al. [18] only additionally show $6 4 \times 6 4$ ImageNet-100 results.   
+33 The high training time for AT frameworks arises from two sources: (i) the need to employ larger   
+34 networks, e.g., MSD [21] with ResNet-18 achieves higher $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ than PAT [18] with ResNet-50 (see   
+35 Fig. 1); and (ii) the need to incorporate multiple perturbations during each attack step and a higher   
+36 overall number of attack steps, e.g., 50 in MSD [21], 20 in AVG [32]. Obviously one can always   
+37 reduce the number of attack steps in MSD/AVG to proportionally reduce training time. Doing so   
+38 results in training time and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ to rapidly approach the training complexity and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \bar { U } ) }$ of standard AT   
+39 frameworks, e.g., a 5-step MSD and 2-step AVG is equivalent in training time and accuracy to PGD   
+40 and TRADES, respectively. Notwithstanding the expensive nature of 50-step multi-attack training,   
+41 today MSD [21] achieves a SOTA $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ of $47 \%$ with ResNet-18 on CIFAR-10.   
+42 This poses a question: can we approach the high   
+43 robustness of multiple-attack AT such as 50-step   
+44 MSD against the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations   
+45 while maintaining the low training time of fast single  
+46 attack AT frameworks such as FreeAdv [30] and Fas  
+47 tAdv [35]?   
+48 In our quest to answer this question we find that noise   
+49 augmentation using adequately shaped noise within   
+50 standard single-attack AT frameworks employing $\ell _ { \infty }$ -   
+51 bounded perturbations significantly improves robust  
+52 ness against the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations.   
+53 The improvement appears to be a consequence of a   
+54 well-established byproduct of AT frameworks – the   
+55 reduction in the curvature of the decision boundary of   
+56 networks trained using single-attack AT [6, 23]. We   
+57 confirm this connection by quantifying the impact   
+58 of single-attack AT on the geometric orientations of   
+59 different perturbations.   
+60 Based on this insight, we propose Shaped Noise   
+61 Augmented Processing (SNAP) – a method to en  
+62 hance robustness against the union of perturbation types by augmenting single-attack AT frameworks.   
+63 SNAP prepends a deep net with a shaped noise (SN) augmentation layer (see Fig. 4) whose dis  
+64 tribution parameter $\Sigma$ is learned with that of the network $\mathbf { \eta } ^ { ( \theta ) }$ within any standard single-attack AT   
+65 framework. SNAP improves the robustness of four SOTA $\ell _ { \infty }$ -AT frameworks against the union of   
+66 $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations by $15 \%$ -to- $20 \%$ on CIFAR-10 (red markers in Fig. 1) with only a modest   
+67 $( \sim 1 0 \% )$ increase in training time. This expands the capabilities of widely popular single-attack $\ell _ { \infty }$   
+68 AT frameworks to providing robustness to the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations without sacrificing   
+69 training efficiency. We validate SNAP’s benefits via thorough comparisons with nine SOTA adver  
+70 sarial training and randomized smoothing frameworks across different operating regimes on both   
+71 CIFAR-10 and ImageNet.
+
+![](images/8ee097816275189000c70f1f5fd6a48cadaa34dbb3058651c5d7bd4b3780565e.jpg)  
+Figure 1: Adv uracy $( \mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } )$ $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ wall-clock total training time on CIFAR10 with different AT frameworks on single NVIDIA TESLA P100 GPU. $\epsilon =$ (0.031, 0.5, 12) for $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations, respectively. SNAP enhances robustness with a small increase in training time. All frameworks except PAT employ ResNet-18.
+
+72 Onenetw73 ome of our woeNet that achieve $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } = 3 2 \% ( 3 5 \% )$ e for the first time R against the union of $( \ell _ { \infty } ( \epsilon = 2 / 2 5 5 ) , \ell _ { 2 } ( \epsilon =$ $\ell _ { 1 } ( \epsilon = 7 2 . 0 )$ ) perturbations. Our code and trained models will be shared publicly on GitHub.
+
+# 2 Related Work
+
+76 We categorize works on adversarial vulnerability of DNNs as follows:
+
+77 Low-complexity adversarial training: The high computational needs of AT frameworks has spurred   
+78 significant efforts in reducing their complexity [40, 30, 35, 43]. FreeAdv [30] updates weights while   
+79 accumulating multiple attack iterations. FastAdv [35] employs appropriate use of single-step attacks,   
+80 while Zheng et al. [43] leverage inter-epoch similarity between adversarial perturbations. However,   
+81 these fast AT methods seek robustness against a single perturbation type, e.g., $\ell _ { \infty }$ norm-bounded   
+82 perturbations. In contrast, SNAP expands the capabilities of these AT frameworks by enhancing   
+83 robustness to the union of three perturbation types $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ , while preserving their efficiency.
+
+Robustness against union of perturbation models: The focus on the robustness against the union of multiple perturbation types is relatively new. Kang et al. [16] studied transferability between different perturbation types, while Jordan et al. [15] considered combination attacks with low perceptual distortion. Stutz et al. [31] proposed a modification in AT to detect images with different models of perturbations via confidence thresholding, but they don’t attempt to classify perturbed images correctly. For accurate classification in the presence of different perturbation models, Tramèr &
+
+90 Boneh [32] studied empirical and theoretical trade-offs involved in including multiple perturbation   
+91 types simultaneously during training. Maini et al. [21] further built upon this work to propose the   
+92 multi steepest descent (MSD) AT framework which chooses one among the three perturbation models   
+93 $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ in each attack iteration during training, achieving SOTA adversarial accuracy on CIFAR  
+94 10 against the union of the $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbation models, albeit at a high $( 1 0 \times )$ training time. In   
+95 contrast, SNAP provides high robustness against the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbation models using   
+96 established single-attack $\ell _ { \infty }$ AT frameworks. This enables to showcase the benefits of our approach   
+97 on large-scale datasets such as ImageNet.   
+98 Recently, Laidlaw et al. [18] developed a novel AT framework (PAT) with low perceptual distortion   
+99 attacks to demonstrate impressive generalization to unseen attacks. In contrast, we focus on extending   
+100 the capabilities of widely popular $\ell _ { \infty }$ -AT frameworks to providing robustness against the union of   
+101 $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations, while preserving their training efficiency.   
+102 Noise augmentation: Multiple recent works have investigated the role of randomization in enhancing   
+103 adversarial robustness [12, 24, 8, 25] with theoretical guarantees. Another prominent line of work   
+104 in this category is randomized smoothing [5, 29, 19, 38], where random noise is used as a tool to   
+105 compute certification bounds. Rusak et al. [28] also explored the role of noise augmentation for   
+106 improving the robustness against common-corruptions [13]. In contrast, in SNAP, noise augmentation   
+107 is used as a means to enable widely popular $\ell _ { \infty }$ -AT frameworks to efficiently achieve high robustness   
+108 against the union of multiple norm-bounded perturbations. As is the characteristic of AT works, our   
+109 results are primarily empirical in nature. Hence, we follow recent guidelines [33, 21] to evaluate the   
+110 accuracy against the strongest possible adversaries. We do explicitly compare $\ell _ { \infty } { - } \mathrm { A T } { + } \mathrm { S N A P }$ with   
+111 randomized smoothing approaches in the Appendix.
+
+# 112 3 Subspace Analysis of Adversarial Perturbations
+
+113 In this section, we employ subspace methods to compre  
+114 hend the distinction between $\ell _ { \infty }$ , $\ell _ { 2 }$ and $\ell _ { 1 }$ perturbations.   
+115 For each input $\pmb { x } _ { i } \in \mathbb { R } ^ { D }$ in dataset $X$ , consider adversar  
+116 ial perturbations $\alpha _ { i }$ , $\beta _ { i }$ , and $\gamma _ { i }$ bounded within $\ell _ { \infty }$ , $\ell _ { 2 }$ ,   
+117 and $\ell _ { 1 }$ norms, respectively.   
+118 We begin with a hypothesis (see Fig. 2): The perturbations   
+119 $\alpha , \beta ,$ , and $\gamma$ corresponding to input x have directions that   
+120 differ significantly if the curvature of the decision bound  
+121 ary is high in the neighborhood of $_ { \textbf { \em x } }$ . Conversely, if the   
+122 curvature of the decision boundary is low, the perturba  
+123 tions $_ \alpha$ , $\beta$ , and $\gamma$ tend to point in similar directions.   
+124 Since, prior works [6, 23] have found that single-attack   
+125 AT reduces the curvature of the decision boundary, we test   
+126 our hypothesis by studying the following two networks
+
+![](images/0e677add87485345fde4d1a80ac940c129edf1c919a91163701690bfb2c81a7a.jpg)  
+!: ℓ! norm bounded; $\beta \colon \ell _ { 2 }$ norm bounded; %: $\ell _ { 1 }$ norm bounded   
+Figure 2: Illustration of the role of decision boundary curvature on the distinction between different types of perturbations $_ { \pmb { \alpha } }$ , $\beta$ and $\gamma$ of the given input $_ { \textbf { \em x } }$ .
+
+on CIFAR-10 data: a non-robust ResNet18 $f _ { \theta } ^ { \mathrm { v a n } }$ trained using vanilla training, and a robust ResNet18 $f _ { \theta } ^ { \mathrm { r o b } }$ trained using the TRADES [41] AT framework employing $\ell _ { \infty }$ perturbations.
+
+129 We compute perturbations $\alpha _ { i } , \beta _ { i }$ , and $\gamma _ { i }$ for each $\pmb { x } _ { i } \in X$ for both networks, i.e., $\kappa \in \{ \mathrm { v a n } , \mathrm { r o b } \}$ . We   
+130 compute the singular vector basis ${ \mathcal { P } } ^ { \kappa }$ for the set of $\ell _ { 2 }$ bounded perturbations $\Delta ^ { \kappa } = \{ \beta _ { 1 } ^ { \kappa } , \ldots , \beta _ { | X | } ^ { \kappa } \}$   
+131 The normalized mean squared projections of the three types of perturbation vectors on the singular   
+132 vector basis ${ \mathcal { P } } ^ { \kappa }$ of vanilla trained ResNet-18 $( \mathcal { P } ^ { \mathrm { v a n } } )$ )(Fig. 3(a)) and TRADES trained ResNet-18   
+133 $( \mathcal { P } ^ { \mathrm { r o b } } )$ )(Fig. 3(b)) shows a clear contrast.   
+134 The perturbations of a vanilla trained network roll-off gradually to occupy a larger subspace as   
+135 indicated in Fig. 3(a). Specifically, the projections of $_ { \pmb { \alpha } }$ and $\gamma$ occupy almost all 3000 directions in   
+136 the basis $\mathcal { P } ^ { \mathrm { v a n } }$ since their mean squared projections are within $\sim 1 0 \%$ of the maximum value $m _ { \mathrm { m a x } }$   
+137 This shows that the dominant singular vectors of $\beta$ are not well-aligned with $_ { \pmb { \alpha } }$ and $\gamma$ in a vanilla   
+138 trained network. With TRADES AT (Fig. 3(b)), however, all three types of perturbations are squeezed   
+139 into a much smaller subspace spanning only the top 250 singular vectors in the perturbation basis   
+140 ${ \mathcal { P } } ^ { \mathrm { r o b } }$ . Outside these 250 dimensions, the mean squared projections fall to $< 1 0 \%$ of their maximum   
+141 value.   
+142 In summary, the results in Fig. 3 validate the hypothesis that single-attack AT increases the average   
+143 alignment of different perturbation types due to the reduction in the decision boundary curvature. In   
+44 Sec. 4, we exploit this behavior of single-attack $\ell _ { \infty }$ AT to improve its robustness against the union of   
+45 multiple perturbation models via SNAP.
+
+![](images/fbe4288bbbab0d5c0fab3b22cc11e5381333f4db5d9aa210c49425d1122424a9.jpg)  
+Figure 3: Normalized mean squared projections of three perturbation types on the singular vector basis ${ \mathcal { P } } ^ { \kappa }$ of $\ell _ { 2 }$ perturbations of ResNet18 on CIFAR-10 after: (a) vanilla training $\kappa \equiv \operatorname { v a n } )$ , and (b) TRADES training $\kappa \equiv \mathrm { r o b }$ ). The singular vectors $\mathbf { \Delta } _ { \mathbf { \mathcal { p } } _ { i } ^ { \kappa } }$ comprising $\mathcal { P } ^ { \kappa } = \{ p _ { 1 } ^ { \kappa } , \ldots , p _ { D } ^ { \kappa } \}$ are ordered in descending order of their singular values.
+
+# 146 4 Shaped Noise Augmented Processing (SNAP)
+
+We show that single-attack AT can be enhanced to address multiple perturbations by introducing noise to appropriately wiggle the $\ell _ { \infty }$ -bounded perturbations (Fig. 4(a)). However, to do so, the noise distribution needs to be chosen and shaped appropriately to minimize its impact on natural accuracy and robustness to $\ell _ { \infty }$ -bounded perturbations.
+
+153 We experiment with both $\ell _ { \infty }$ and $\ell _ { 2 }$ perturbations in single  
+154 attack AT frameworks and find $\ell _ { \infty } { \cdot } \mathbf { A } \mathrm { T }$ to be suitable for   
+155 our proposed shaped noise augmentation (see Sec. 5.2.1 for   
+156 details). Hence, in this section, we describe SNAP for single  
+157 attack AT frameworks employing $\ell _ { \infty }$ perturbations.
+
+# 4.1 SNAPnet
+
+A deep net $f _ { \theta } ( \pmb { x } ) _ { \mathbf { \lambda } } \colon \mathbb { R } ^ { D }  \{ 0 , 1 \} ^ { C }$ parametrized by $\theta$ maps the input $\pmb { x } \in \mathbb { R } ^ { D }$ to a one-hot vector ${ \pmb y } \in \{ 0 , 1 \} ^ { \tilde { C } }$ over $C$ classes.
+
+162 We construct a SNAP-based deep net (SNAPnet) $f _ { \boldsymbol { \theta } , \Sigma } ^ { \mathrm { S N } } ( \boldsymbol { x } )$ by   
+163 introducing an additive shaped noise (SN) layer (Fig. 4(b)),   
+164 where the noise distribution parameter $\Sigma$ is learned during   
+165 training. Formally,
+
+![](images/a3164964467f30c335a054e979640698625cdd6cce794036dbdb04ad576db1af.jpg)  
+Figure 4: SNAP: (a) intuition underlying SNAP (not an exact depiction), and (b) SNAPnet $f _ { \boldsymbol { \theta } , \Sigma } ^ { \mathrm { S N } } ( \boldsymbol { x } )$ constructed from a given deep net $f _ { \boldsymbol { \theta } } ( \pmb { x } )$ by prepending a shaped noise (SN) augmentation layer which perturbs the primary input $_ { \textbf { \em x } }$ with noise $\mathbf { n }$ whose distribution parameter $\Sigma$ is learned during AT along with the base network parameter $\theta$ .
+
+$$
+{ \pmb y } = f _ { \theta , \Sigma } ^ { \mathrm { S N } } ( { \pmb x } ) = f _ { \theta } \big ( { \pmb x } + { \pmb n } \big ) = f _ { \theta } \big ( { \pmb x } + V \Sigma { \pmb n } _ { 0 } \big ) ,
+$$
+
+166 where $\mathbf { n } _ { 0 } \sim \mathcal { L } ( 0 , \mathbf { I } _ { D \times D } )$ is a zero-mean isotropic Laplace   
+167 noise vector, $\Sigma = \mathrm { D i a g } [ \sigma _ { 1 } , \dots , \sigma _ { D } ]$ is a distribution param  
+168 eter denoting its per-dimension standard deviation, ${ \bf { I } } _ { D \times D }$   
+169 denotes the $D \times D$ identity matrix, and $V = [ \pmb { v } _ { 1 } , \dots , \pmb { v } _ { D } ]$ denotes a basis in $\mathbb { R } ^ { D }$ . We also studied   
+170 Gaussian and Uniform distributed ${ \bf n } _ { 0 }$ , but empirically find the Laplace distribution to yield better   
+171 results (Sec. 5.2.1). We use $V = \mathbf { I } _ { D \times D }$ for all our experiments in the main text and study other   
+172 options for $V$ in the Appendix.
+
+173 The final classification decision $d$ is computed via
+
+$$
+d = \arg \operatorname* { m a x } _ { c } { \left[ \mathbb { E } _ { \mathbf { n } } [ \pmb { y } ] \right] } _ { c } ,
+$$
+
+174 where $[ \pmb { a } ] _ { c }$ denotes the $c$ -th element of vector $^ { a }$ . Note, the shaped noise perturbs the input $_ { \textbf { \em x } }$ with a   
+175 noise source $\mathbf { n } = V \Sigma \mathbf { n } _ { 0 }$ (Eq. (1)). The distribution parameter $\Sigma$ is learned in the presence of any   
+176 standard AT method [20, 41, 30] used for learning deep net parameters $\theta$ as described next.
+
+# Algorithm 1 Training SNAPnet
+
+Input: training set $X$ ; basis $V = [ \pmb { v } _ { 1 } , \dots , \pmb { v } _ { D } ]$ ; total noise power $P _ { \mathrm { n o i s e } }$ ; minibatch size $r$ ; baseline training method BASE; noise variance update frequency $U _ { f }$ ; Total number of epochs $T$
+
+Initialize: noise variances $\Sigma _ { 0 } = \mathrm { { D i a g } } [ \sigma _ { 1 , 0 } , . . . , \sigma _ { D , 0 } ]$ .   
+Output: robust network $f _ { \boldsymbol { \theta } , \Sigma } ^ { \mathrm { S N } }$ , noise variances $\Sigma _ { T } = \operatorname { \bar { D i a g } } [ \sigma _ { 1 , T } ^ { 2 } , \dots , \sigma _ { D , T } ^ { 2 } ]$ .   
+1: for epoch $t = 1 \dots T$ do   
+2: for mini-batch $B = \{ { \pmb x } _ { 1 } , \ldots , { \pmb x } _ { r } \}$ do θ ← BASE\`∞ f SNθ,Σt {xi}ri=1
, θ . BASE() Training   
+3: end for   
+4: if $t$ mod $U _ { f } = 0$ then . SNAP Distribution Update once every $U _ { f }$ epochs   
+5: for mini-batch $B = \{ { \pmb x } _ { 1 } , \ldots , { \pmb x } _ { r } \}$ do   
+6: $\begin{array} { r l r } & { } & { \{ \pmb { x } _ { i } ^ { \mathrm { a d v } } \} _ { i = 1 } ^ { r }  \mathrm { P G D } _ { \ell _ { 2 } } ^ { ( K ) } \bigg ( f _ { \theta , \Sigma _ { t } } ^ { \mathrm { S N } } \big ( \{ \pmb { x } _ { i } \} _ { i = 1 } ^ { r } \big ) \bigg ) ; \quad \eta _ { i } = \pmb { x } _ { i } ^ { \mathrm { a d v } } - \pmb { x } _ { i } \forall i \in \{ 1 , \dots , r \} } \\ & { } & { \gamma _ { j }  \gamma _ { j } + \sum _ { i = 1 } ^ { r } \big ( \langle \pmb { v } _ { j } , \eta _ { i } \rangle \big ) ^ { 2 } \quad \forall j \in \{ 1 , \dots , D \} \qquad \mathrm { ~ \mathbb { b } \ c c u m u l a t e \ p r o j e c t i o n s } ; } \\ & { } & { \frac { \mathrm { n d \ d \Pi \ f o r } } { j , t + 1 } = P _ { \mathrm { n o i s e } } \frac { \sqrt { \gamma _ { j } } } { \sum _ { k = 1 } ^ { D } \sqrt { \gamma _ { k } } } \quad \forall j \in \{ 1 , \dots , D \} \quad \mathrm { ~ \mathbb { b } \ c n o r m a l i z e \ a c c u m u l a t e d \ p r o j e c t i o n s } ; } \end{array}$   
+7: See Eq. (3)   
+8: 9: eσ See Eq. (3)   
+10: else   
+11: $\Sigma _ { t + 1 }  \Sigma _ { t }$   
+12: end if   
+13: end for
+
+# 177 4.2 Training SNAPnet
+
+178 Algorithm 1 summarizes the procedure for training SNAPnet $f _ { \boldsymbol { \theta } , \Sigma } ^ { \mathrm { S N } } ( \boldsymbol { x } )$ . In each epoch, an arbitrary   
+179 AT method BASE() (line 2) updates network parameters $\theta$ with input perturbed by noise $\mathbf { n }$ . Here   
+180 BASE() can be any established AT framework [20, 41, 30, 35] employing $\ell _ { \infty }$ perturbation.   
+181 The SNAP parameter $\Sigma$ is updated once every $U _ { f } = 1 0$ epochs via a SNAP distribution update (lines   
+182 4-10). In this update, the per-dimension noise variance $\sigma _ { j } ^ { 2 }$ is updated proportional to the root mean   
+183 squared projection of the adversarial perturbations $\eta$ on the basis $V$ given a total noise constraint   
+184 $\begin{array} { r } { \bar { \sum _ { j = 1 } ^ { D } } \sigma _ { j } ^ { 2 } = P _ { \mathrm { n o i s e } } } \end{array}$ , where $P _ { \mathrm { n o i s e } }$ denotes the total noise power. Formally,
+
+$$
+\sigma _ { j } ^ { 2 } \propto \sqrt { \mathbb { E } _ { \pmb { x } \in X } \left( \langle \pmb { \eta } , \pmb { v } _ { j } \rangle ^ { 2 } \right) } \quad \mathrm { s . t . } \quad \sum _ { j = 1 } ^ { D } \sigma _ { j } ^ { 2 } = P _ { \mathrm { n o i s e } } ,
+$$
+
+185 where $\eta$ is the $\ell _ { 2 }$ norm-bounded PGD adversarial perturbation for the given input $\boldsymbol { x } \in \boldsymbol { X }$ (line 6).   
+186 Note that these $\ell _ { 2 }$ perturbations are employed only for noise shaping and are distinct from the $\ell _ { \infty }$   
+187 perturbations employed by BASE() AT (line 2). Also, $\ell _ { \infty }$ perturbations cannot be used here since   
+188 their projections are constant $\forall j$ when $V = \mathbf { I } _ { D \times D }$ , whereas employing $\ell _ { 1 }$ perturbations leads to poor   
+189 shaping due to high sparsity.   
+190 Thus, in SNAP, the average squared $\ell _ { 2 }$ norm of the noise vector $\mathbf { n }$ is held constant at $P _ { \mathrm { n o i s e } }$ while   
+191 adapting the noise variances in the individual dimensions so as to align the noise vectors with the   
+192 adversarial perturbations on average. Intuitively, the decision boundary is pushed aggressively in   
+193 those directions.
+
+# 194 4.3 Remarks
+
+195 Note that the SNAP distribution update is distinct from BASE() AT. Hence, SNAP doesn’t require any   
+196 hyperparameter tuning in BASE(). For fairness to baselines we keep all hyperparameters identical   
+197 when introducing SNAP in all our experiments. However, SNAP introduces a new hyperparameter   
+198 $P _ { \mathrm { n o i s e } }$ , which permits to trade adversarial robustness $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ for natural accuracy ${ \mathcal { A } } _ { \mathrm { n a t } }$ . This trade-off is   
+199 explored in Sec. 5.2.2.   
+200 The computational overhead of SNAP is small $( \sim 1 0 \%$ ) since the SNAP Distribution Update occurs   
+201 once in 10 epochs using just $20 \%$ of the training data to update the noise standard deviations $\sigma _ { j }$ . We   
+202 provide more details about the SNAP Distribution Update in the Appendix.
+
+<table><tr><td>Method</td><td>Anat</td><td>A ∈=0.03</td><td>A ∈=0.5</td><td>A ∈= 12</td><td>A</td></tr><tr><td colspan="6">PGD AT withloo perturbations</td></tr><tr><td>PGD +SNAP[G] +SNAP[U]</td><td>84.6 80.7 85.1</td><td>48.8 45.7 42.7</td><td>62.3 66.9 66.7</td><td>15.0 34.6 28.6</td><td>15.0 31.9 26.6</td></tr><tr><td colspan="6">+SNAP[L] 83.0 44.8 68.6 40.1 PGD AT withl2 perturbations</td></tr><tr><td>PGD +SNAP[G] +SNAP[U]</td><td>89.3 83.0</td><td>28.8 35.0</td><td>67.3 65.8</td><td>31.8 39.9</td><td>25.1 30.2</td></tr></table>
+
+Table 1: ResNet-18 CIFAR-10 results showing the impact of SNAP augmentation of PGD [20] AT framework with $\ell _ { \infty }$ (top) and $\ell _ { 2 }$ (bottom) perturbations where [G], [U], and [L], denote shaped Gaussian, Uniform, and Laplace noise.
+
+<table><tr><td>Method</td><td>Anat</td><td>A ∈= 0.03</td><td>A ∈=0.5</td><td>A e=12</td><td>A</td></tr><tr><td colspan="6">High Complexity ATwith loperturbations</td></tr><tr><td>PGD +SNAP</td><td>84.6 83.0</td><td>48.8 44.8</td><td>62.3 68.6</td><td>15.0 40.1</td><td>15.0 35.6</td></tr><tr><td>TRADES +SNAP</td><td>82.1 80.9</td><td>50.2 45.2</td><td>59.6 66.9</td><td>19.8 46.6</td><td>19.7 41.2</td></tr><tr><td colspan="6">Low Complexity AT with looperturbations</td></tr><tr><td>FreeAdv +SNAP</td><td>81.7 83.5</td><td>46.1 39.7</td><td>59 66.2</td><td>15.0 34.3</td><td>15.0 29.6</td></tr><tr><td>FastAdv +SNAP</td><td>85.7 84.2</td><td>46.2 40.4</td><td>60.0 67.9</td><td>13.2 36.6</td><td>13.2 30.8</td></tr></table>
+
+Table 2: ResNet-18 CIFAR-10 results showing the impact of SNAP augmentation of established $\ell _ { \infty }$ -AT frameworks. The computational overhead of SNAP is limited to $\sim 1 0 \%$ .
+
+# 03 5 Experimental Results
+
+# 5.1 Setup
+
+Following experimental settings of prior work [41, 30, 21], we employ a ResNet-18 network for CIFAR-10 experiments and both ResNet-50 and ResNet-101 networks for ImageNet experiments. Accuracy on clreferred to via $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ t , $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) }$ s refe, and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ o wit, for $\ell _ { \infty }$ ${ \mathcal { A } } _ { \mathrm { n a t } }$ $\ell _ { 2 }$ and a, and $\ell _ { 1 }$ uracy on adversarially perturbed test data isnorm bounded perturbations, respectively. Accuracy against the union of all three perturbations is denoted by $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ .
+
+For a fair robustness comparison, our evaluation setup closely follows the setup of Maini et al. [21] for CIFAR-10 data: (1) choose norm bounds $\epsilon = ( 0 . 0 3 1 , 0 . 5 , 1 2 . 0 )$ for $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations, respectively; (2) scale norm bounds for images to lie between $[ 0 , 1 ]$ ; (3) choose the PGD attack configuration to be $I O O$ iterations with $I O$ random restarts for all perturbation types1; and (4) estimate $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ as the fraction of test data that is simultaneously resistant to all three perturbation models.
+
+Following the guidelines of Tramer et al. [33], we carefully design adaptive PGD attacks that target the full defense – SN layer – since SNAPnet is end-to-end differentiable. Specifically, we backpropagate to primary input $_ { \textbf { \em x } }$ through the SN layer (see Fig. 4). Thus, the final shaped noise distribution is exposed to the adversary. We also account for the expectation $\mathbb { E } _ { \mathbf { n } } [ \cdot ]$ in Eq. (2) by explicitly averaging deep net logits over $N _ { 0 } ( = 8 )$ noise samples before computing the gradient, which eliminates any gradient obfuscation, and is known to be the strongest attack against noise augmented models [29]. In the Appendix we also show robustness stress tests and evaluate more attacks.
+
+On CIFAR-10 data, we compare with the following seven key SOTA AT frameworks: PGD [20], TRADES [41], FreeAdv [30], FastAdv [35], AVG [32], MSD [21], PAT [18]. We also compare with two randomized smoothing frameworks [5, 29] in the Appendix. Thanks to their GitHub code releases, we first successfully reproduce their results with a ResNet-18 network in our environment. In the case of PAT [18], we evaluate and compare with their pretrained ResNet-50 model on CIFAR-10. We compare all training times on a single NVIDIA P100 GPU. On ImageNet data, we primarily compare to FreeAdv [30]. We train ResNet-50 and its SNAPnet version with FreeAdv on a Google Cloud server with four NVIDIA P100 GPUs to compare their accuracy and training times. We will release our pretrained models and code on GitHub.
+
+# 5.2 Ablation Studies
+
+# 5.2.1 Impact of Noise Distribution and Model of BASE() AT Perturbations
+
+line 2 in Alg. 1) on In this subsection, we first study the impact of employing $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ . For each choice, we further experiment with three $\ell _ { \infty }$ vs. $\ell _ { 2 }$ perturbations in BASE AT() (see ributions for the $\ell _ { 1 }$
+
+237 BASE AT() since Maini et al. [21] showed that employing $\ell _ { 1 }$ single-attack AT achieves very low   
+238 robustness to all attacks. We choose PGD [20] AT as BASE AT() for this ablation study. For a fair   
+239 comparison across the noise distributions, we fix $P _ { \mathrm { n o i s e } } = 1 6 0$ , enforcing all noise vectors to have the   
+240 same average $\ell _ { 2 }$ norm. For each distribution, the noise is shaped per the procedure summarized in   
+241 Alg. 1.
+
+As observed in Table 1, $\ell _ { \infty }$ -PGD AT achieves much lower $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ than $\ell _ { 2 }$ -PGD AT, an observation also reported by Maini et al. [21]. With SNAP, however, we find that there is an interaction between the perturbation model in PGD AT and the noise distribution in SNAP. For instance, SNAP[U] enhances $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ by $11 \%$ with $\ell _ { \infty }$ -PGD AT while not achieving any improvement with $\ell _ { 2 }$ -PGD AT. In fact, SNAP appears to be particularly suitable for $\ell _ { \infty }$ -AT, since it always improves $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ by $11 \%$ -to- $2 0 . 6 \%$
+
+Finally, of the three noise distributions, we find the Laplace distribution to be distinctly superior, achieving the highest A(U)adv ( $3 5 . 6 \%$ and $3 0 . 8 \%$ ) due to a significant improvement in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ for both $\ell _ { \infty }$ and $\ell _ { 2 }$ PGD AT, respectively. The superiority of the Laplace distribution in achieving high $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ stems from its heavier tail compared to the Gaussian and Uniform distributions with the same variance. Shaped Laplace noise generates the highest fraction of extreme values in a given noise sample. Hence, it is more effective in improving accuracy against $\ell _ { 1 }$ -bounded attacks, which are the strongest when perturbing few pixels by a large magnitude [21, 32]. We discuss this further in the Appendix. Henceforth, unless otherwise mentioned, we choose Laplace noise for SNAP and $\ell _ { \infty }$ perturbations for BASE() AT as the default setting since it achieves the highest ${ \mathcal { A } } _ { \mathrm { a d v } } ^ { ( U ) }$ .
+
+# 5.2.2 Impact of $P _ { \mathrm { n o i s e } }$
+
+Next, we explore the impact of the SNAP hyperparameter $P _ { \mathrm { n o i s e } }$ , which constrains the average squared $\ell _ { 2 }$ norm of the noise vector n. It enables to trade between adversarial and natural accuracy.
+
+Fig. 5 shows that, as $P _ { \mathrm { n o i s e } }$ increases, $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ improves from $31 \%$ to $47 \%$ , accompanied by a graceful $( 5 \% )$ drop in ${ \mathcal { A } } _ { \mathrm { n a t } }$ and a small drop of $2 \%$ in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ that stabilizes to $\approx 4 5 \%$ . These results show: (1) SNAP preserves the impact of $\ell _ { \infty }$ perturbations which is not surprising since PGD AT [20] explicitly includes those, and (2) $P _ { \mathrm { n o i s e } }$ provides an explicit knob to control the $\boldsymbol { A } _ { \mathrm { n a t } }$ vs. ${ \mathcal { A } } _ { \mathrm { a d v } }$ trade-off. Henceforth, we choose $P _ { \mathrm { n o i s e } }$ values that incur $< 1 . 5 \%$ drop in ${ \mathcal { A } } _ { \mathrm { n a t } }$ for all $\mathrm { S N A P { + } A T }$ experiments.
+
+![](images/e64006dc9658f7ac2e31b75327a197074e59c75a52f4e4cc991275b7cd824aa2.jpg)  
+Figure 5: ResNet-18 CIFAR-10 results: ad-(a) versarial racy $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ , $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ , an ural ${ \mathcal { A } } _ { \mathrm { n a t } }$ vs. total noise power $P _ { \mathrm { n o i s e } }$ $\mathrm { P G D + S N A P }$ .
+
+# 5.2.3 SNAP augmented SOTA AT Frameworks
+
+Table 2 shows the effectiveness of SNAP for four SOTA AT frameworks: high complexity frameworks, such as PGD [20], TRADES [41], and low complexity frameworks such as FreeAdv [30], FastAdv [35]. All are trained against $\ell _ { \infty }$ attacks with $\epsilon = 0 . 0 3 1$ . As expected, while they achieve high $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ , their $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) }$ and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ are lower.
+
+For high-complexity AT, SNAP enhances $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) }$ and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ by $\sim 6 \%$ and $\sim 2 5 \%$ , respectively, while incurring only a drop of $\sim 5 \%$ in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ adv adv . Thus overall, SNAP improves robustness $( \mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } )$ by $\sim 2 0 \%$ against the union of the three perturbation models. Note that this robustness improvement comes robustness at only a $\sim 1 \%$ $( \mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } )$ drop in are also significant ${ \mathcal { A } } _ { \mathrm { n a t } }$ (see Table 2). For low-complexity ATs, SNAP improvements in union $( \sim 1 5 \% )$ ). Again, presence of SNAP improves $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) }$ and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ This time the drop in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ is $\sim 7 \%$ . We believe this is due to the fact that thes frameworks employ weaker single-step attacks during training. Note that in the case of FreeAdv $^ +$ observe a $\sim 2 \%$ increase in ${ \mathcal { A } } _ { \mathrm { n a t } }$ , a trend we also observe in the ImageNet experiments described later.
+
+Table 3: CIFAR-10 results for comparing adversarial accuracy A(U)adv vs. training time (on single NVIDIA P100 GPU) for different AT frameworks and the improvements by introducing proposed SNAP technique. All frameworks except PAT [18] (which employs ResNet-50) employ ResNet-18.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>LR schedule</td><td rowspan=1 colspan=1>Epochs</td><td rowspan=1 colspan=1>Anat</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Total time(minutes)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetA:Total Time</td><td rowspan=1 colspan=1>SetA:Total Time≥</td><td rowspan=1 colspan=1>12 Hrs</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=5 colspan=1>AVG 50Step[32]AVG 20 Step [32]AVG 10 Step [32]PAT[18]MSD 50 Step [21]MSD 30 Step [21]</td><td rowspan=1 colspan=1>cyclic</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>84.8</td><td rowspan=1 colspan=1>40.4</td><td rowspan=5 colspan=1>4217183495613641693978</td></tr><tr><td rowspan=4 colspan=1>cycliccyclicstepcycliccyclic</td><td rowspan=2 colspan=1>5050</td><td rowspan=2 colspan=1>85.686.7</td><td rowspan=1 colspan=1>40.438.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=3 colspan=1>501005050</td><td></td></tr><tr><td rowspan=2 colspan=1>86.782.481.782.4</td><td></td></tr><tr><td rowspan=1 colspan=1>38.936.647.044.9</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetB:8Hrs</td><td rowspan=1 colspan=3>Set B:8 Hrs&lt;Total Time&lt;12 Hrs</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>AVG 5 Step [32]MSD 20 Step [21]TRADES [41]TRADES+SNAP</td><td rowspan=1 colspan=1>cycliccyclicstepstep</td><td rowspan=1 colspan=1>5050100100</td><td rowspan=1 colspan=1>87.883.082.080.9</td><td rowspan=1 colspan=1>33.737.319.741.2</td><td rowspan=1 colspan=1>489690516566</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetC:5Hr</td><td rowspan=1 colspan=1><TotalTi</td><td rowspan=1 colspan=1>ne<8Hi</td><td rowspan=1 colspan=1>S</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>MSD 10 Step [21]PGD [20]PGD+SNAP</td><td rowspan=1 colspan=1>cyclicstepstep</td><td rowspan=1 colspan=1>50100100</td><td rowspan=1 colspan=1>83.684.683.0</td><td rowspan=1 colspan=1>33.315.035.6</td><td rowspan=1 colspan=1>342354403</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetD:2Hr</td><td rowspan=1 colspan=1><TotalTi</td><td rowspan=1 colspan=1>ne<5Hi</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>AVG 2 Step [32]MSD 5 Step [21]PGD [20]TRADES [41]PGD+SNAPTRADES+SNAP</td><td rowspan=1 colspan=1>cycliccycliccycliccycliccycliccyclic</td><td rowspan=1 colspan=1>505050505050</td><td rowspan=1 colspan=1>88.484.082.880.082.378.8</td><td rowspan=1 colspan=1>22.012.615.721.433.540.8</td><td rowspan=1 colspan=1>232185177258199280</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetE:</td><td rowspan=1 colspan=1>TotalTime<</td><td rowspan=1 colspan=1>2Hrs</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>FreeAdv [30]FastAdv [35]FreeAdv+SNAPFastAdv+SNAP</td><td rowspan=2 colspan=1>stepcyclicstepcyclic</td><td rowspan=1 colspan=1>200</td><td rowspan=1 colspan=1>81.7</td><td rowspan=1 colspan=1>15.0</td><td rowspan=2 colspan=1>66478869</td></tr><tr><td rowspan=1 colspan=1>5020050</td><td rowspan=1 colspan=1>85.783.584.2</td><td rowspan=1 colspan=1>13.229.630.8</td></tr></table>
+
+<table><tr><td>Training</td><td>Anat (%)</td><td>A e= 2/255</td><td>A ∈ = 2.0</td><td>A e = 72.0</td><td>A</td><td>Total time (minutes)</td></tr><tr><td colspan="7">ResNet-50</td></tr><tr><td>FreeAdv [30]</td><td>61.7</td><td>47.8</td><td>19.9</td><td>14.8</td><td>12.6</td><td>3590</td></tr><tr><td>FreeAdv+SNAP</td><td>66.8</td><td>46.1</td><td>37.8</td><td>37.4</td><td>32.4</td><td>3756</td></tr><tr><td colspan="7">ResNet-101</td></tr><tr><td>FreeAdv [30]</td><td>65.4</td><td>51.8</td><td>22.8</td><td>18.8</td><td>16.1</td><td>5678</td></tr><tr><td>FreeAdv+SNAP</td><td>69.7</td><td>50.3</td><td>41.1</td><td>40.2</td><td>35.4</td><td>5904</td></tr></table>
+
+Table 4: ImageNet results: Iso-hyperparameter introduction of SNAP yields $\sim 2 0 \%$ improvement in adversarial accuracy $( \mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } )$ with modest impact on training time for ResNet-50 and ResNet-101.
+
+# 286 5.3 Robustness vs. Training Complexity
+
+Next we quantify adversarial robustness vs. training time trade-offs. Table 3 shows that SNAP augmentation of single-attack AT frame rks achieves the highest $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ , when training time is $\mathbf { E }$
+
+For instance, TRADES $^ +$ SNAP achieves a $4 \%$ higher $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } ( = 4 1 \% )$ than MSD-20 with 2 hours lower training time (Set B in Table 3). Similarly, PGD $^ +$ SNAP achieves a $2 \%$ higher $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ than MSD-10 while having a similar training time (Set C). Note that both PGD and TRADES here use 100 training epochs with standard step learning rate (LR) schedule, while MSD frameworks employ a cyclic learning rate schedule to achieve superconvergence in 50 epochs.
+
+295 In Set D, following Maini et al. [21], we employ a cyclic learning rate schedule for PGD, TRADES,   
+296 as well as for $\mathrm { P G D + S N A P }$ and TRADES $^ +$ SNAP to achieve convergence in 50 epochs. Improvements   
+297 in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ for PGD $^ +$ SNAP and TRADES $^ +$ SNAP are similar to those in Sets $\mathbf { B }$ and C. Most notably,
+
+298 $\mathrm { P G D + S N A P }$ with cyclic learning rate achieves $\sim 2 0 \%$ and $1 1 . 5 \%$ high $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ than MSD-5 and 299 AVG-2, respectively, while having aTable 2 with training times. FastAdv300 training time  and FreeAdv $\sim 3$ hours). Set AP achieve $\mathbf { E }$ auigh a from, while $+ { \mathrm { S N A P } }$ $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) } \sim 3 0 \%$ 301 $1 8 \%$ erving thigher $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ ining efficiency of both Fthan MSD-5, while being $\sim 2 . 7 \times$ nd FreeAdv. Notably, FastAdv+SNAP achievesmore efficient to train.
+
+# 5.4 ImageNet Results
+
+Thanks to SNAP’s low computational overhead combined with FreeAdv’s fast training time, we are for the first time able to report adversarial accuracy of ResNet-50 and ResNet-101 against the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ attacks on ImageNet.
+
+We closely follow the evaluation setup of Shafahi et al. [30]. Specifically, we use 100 step PGD attack, one of the strongest adversaries considered by Shafahi et al. [30], and evaluate on the entire test set. We first reproduce FreeAdv [30] results using the same hyperparameters and then introduce SNAP. All hyperparameter details are specified in the Appendix.
+
+In order to clearly demonstrate the contrast between robustness to different perturbation models, we evaluate with FreeAdv achi $\epsilon \overset { \cdot } { = } ( 2 / 2 5 5 , 2 . 0 , 7 2 . 0 )$ $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ attacks, respectivet-50, but a lower n inand adv , and consequently, a low $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) } = 4 7 . 8 \%$ $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ o f $1 2 . 6 \%$ against the union of the perturbations. In contrast, $A _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) } = 2 0 \%$ $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) } =$ FreeAdv+SNAP improves $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 2 } ) }$ and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { 1 } ) }$ by $1 7 \%$ and $2 2 \%$ , respectively, accompanied by a $5 \%$ improvement in ${ \mathcal { A } } _ { \mathrm { n a t } }$ and a small $2 \%$ loss in $\mathcal { A } _ { \mathrm { a d v } } ^ { ( \ell _ { \infty } ) }$ . This results in an overall robustness improvement of $\mathrm { \bar { 2 0 \% } }$ against the union of the perturbation models, setting a first benchmark for ResNet-50 on ImageNet. Upon increasing the network to ResNet-101, both natural and adversarial accuracies improve by $\approx 4 \%$ for FreeAdv, a trend also observed by Shafahi et al. [30]. SNAP further improves FreeAdv’s results for $\boldsymbol { A } _ { \mathrm { n a t } }$ and $\mathcal { A } _ { \mathrm { a d v } } ^ { ( U ) }$ by $4 . 3 \%$ and $1 9 . 3 \%$ .
+
+# 6 Discussion
+
+Given the wide popularity of $\ell _ { \infty } { \cdot } \mathbf { A } \mathbf { T } .$ , in this paper, we propose SNAP as an augmentation that generalizes the effectiveness of $\ell _ { \infty }$ -AT to the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations. SNAP’s strength is its simplicity and efficiency. Consequently, this work sets a first benchmark for ResNet-50 and ResNet101 networks which are resilient to the union of $( \ell _ { \infty } , \ell _ { 2 } , \ell _ { 1 } )$ perturbations on ImageNet. Note that norm-bounded perturbations include a large class of attacks, e.g., gradient-based [20, 27, 32, 21, 4, 22], decision-based [3] and black-box [1] attacks.
+
+More work is needed to extend the proposed SNAP technique to attacks beyond norm-bounded additive perturbations, e.g., functional [17, 36], rotation [7], texture [2], etc. We provide preliminary evaluations in this direction in the Appendix. It is important to note that SNAP is meant to be an efficient technique for improving $\ell _ { \infty }$ -AT, and not a new defense. Indeed defending against a large variety of attacks simultaneously remains an open problem, with encouraging results from recent efforts [21, 18].
+
+Another limitation of our approach is that its benefits are demonstrated empirically. It is an inevitable consequence of a lack of any theoretical guarantees for underlying AT frameworks. An interesting direction of future work is to explore whether any theoretical guarantees can be derived for anisotropic shaped noise distributions in SNAP by building upon the recent developments in randomized smoothing [29, 38]. This could be a potential avenue for bridging the gap between certification bounds and empirical adversarial accuracy.
+
+Finally, we believe that any effort on improving adversarial robustness of deep nets has net positive societal impact. However, recent past in this field has shown that any improvements in defense techniques also lead to more effective threat models. While such a cat-and-mouse game is of great intellectual value in the academic setting, it does have an unintentional negative societal consequence of equipping malicious outside actors with a broad set of tools. This further underscores the wellrecognized need for provable defenses.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] An URL to our code as well as the pretrained models is provided in the Appendix.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] All training hyperparameters are mentioned in the Appendix. Our technique does introduce a new hyperparameter, whose impact is discussed in Sec. 5.2.2.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We do run a subset of experiments multiple times to obtain error bars (see Appendix). In doing so we confirm that our technique is effective across random initializations. However, some of the training runs in our work are too expensive to run multiple times.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Yes, we explicitly mention the type of GPUs used and the training times in Section 5.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We appropriately cite the relevant papers while using their code to reproduce/extend their results.   
+(b) Did you mention the license of the assets? [Yes] Our own codes & models, as well as, all the other codes that we use are available freely in public domain. We do mention so explicitly in Section 5.1   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We do share our own code and pretrained model as a part of the supplemental material   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
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+        "type": "text",
+        "text": "Robustifying $\\ell _ { \\infty }$ Adversarial Training to the Union of Perturbation Models ",
+        "text_level": 1,
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 Classical adversarial training (AT) frameworks are designed to achieve high ad  \n2 versarial accuracy against a single attack type, typically $\\ell _ { \\infty }$ norm-bounded per  \n3 turbations. Recent extensions in AT have focused on defending against the union   \n4 of multiple perturbation models but this benefit is obtained at the expense of a   \n5 significant (up to $1 0 \\times$ ) increase in training complexity over single-attack $\\ell _ { \\infty }$ AT.   \n6 In this work, we expand the capabilities of widely popular single-attack $\\ell _ { \\infty }$ AT   \n7 frameworks to provide robustness to the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations while   \n8 preserving their training efficiency. Our technique, referred to as Shaped Noise   \n9 Augmented Processing (SNAP), exploits a well-established byproduct of single  \n10 attack AT frameworks – the reduction in the curvature of the decision boundary of   \n11 networks. SNAP prepends a given deep net with a shaped noise augmentation layer   \n12 whose distribution is learned along with network parameters using any standard   \n13 single-attack AT. As a result, SNAP enhances adversarial accuracy of ResNet-18   \n14 on CIFAR-10 against the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations by $1 \\dot { 4 } \\%$ -to- $2 0 \\%$ for   \n15 four state-of-the-art (SOTA) single-attack $\\ell _ { \\infty }$ AT frameworks, and, for the first   \n16 time, establishes a benchmark for ResNet-50 and ResNet-101 on ImageNet. ",
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+        "text": "17 1 Introduction ",
+        "text_level": 1,
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+        "text": "18 Today adversarial training (AT) provides state-of-the-art (SOTA) empirical defense against adver  \n19 sarial perturbations. For this, adversarial perturbations are used during training to optimize a robust   \n20 loss function [20, 41, 30, 35]. Early AT frameworks [20, 41] were $7 \\times$ -to- $1 0 \\times$ more computationally   \n21 demanding than vanilla training. More recent works [30, 35, 40] have significantly reduced the   \n22 computational demands of AT via single-step attacks and superconvergence.   \n23 However, today’s AT frameworks predominantly focus on a single-attack, i.e., they seek robustness   \n24 to a single perturbation, typically $\\ell _ { \\infty }$ -bounded [30, 35, 37, 41, 43, 40, 39, 26, 9, 34, 42, 10, 11, 14].   \n25 This results in low performance against other perturbations such as $\\ell _ { 2 } , \\ell _ { 1 }$ , or the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ .   \n26 Indeed, as showemploying only $\\ell _ { \\infty }$ Fig. 1, four state-of-the-art (SOTA) single-attack AT fra-bounded perturbations achieve low adversarial accuracy ${ \\mathcal { A } } _ { \\mathrm { a d v } } ^ { ( U ) }$ rksof $\\approx 1 5 \\%$ mar-to- $2 0 \\%$   \n28 against the union of perturbations. Recent extensions in AT [21, 32, 18] do seek higher   \n29 A(U) but only at the expense of $6 \\times$ -to- $. 1 0 \\times$ increase in the total training time (blue markers in   \n30 Fig. 1). The large training time of these AT frameworks has inhibited their application to large-scale   \n31 datasets such as ImageNet, e.g., Maini et al. [21], Tramèr & Boneh [32] show results for MNIST and   \n32 CIFAR-10 only, while Laidlaw et al. [18] only additionally show $6 4 \\times 6 4$ ImageNet-100 results.   \n33 The high training time for AT frameworks arises from two sources: (i) the need to employ larger   \n34 networks, e.g., MSD [21] with ResNet-18 achieves higher $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ than PAT [18] with ResNet-50 (see   \n35 Fig. 1); and (ii) the need to incorporate multiple perturbations during each attack step and a higher   \n36 overall number of attack steps, e.g., 50 in MSD [21], 20 in AVG [32]. Obviously one can always   \n37 reduce the number of attack steps in MSD/AVG to proportionally reduce training time. Doing so   \n38 results in training time and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ to rapidly approach the training complexity and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\bar { U } ) }$ of standard AT   \n39 frameworks, e.g., a 5-step MSD and 2-step AVG is equivalent in training time and accuracy to PGD   \n40 and TRADES, respectively. Notwithstanding the expensive nature of 50-step multi-attack training,   \n41 today MSD [21] achieves a SOTA $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ of $47 \\%$ with ResNet-18 on CIFAR-10.   \n42 This poses a question: can we approach the high   \n43 robustness of multiple-attack AT such as 50-step   \n44 MSD against the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations   \n45 while maintaining the low training time of fast single  \n46 attack AT frameworks such as FreeAdv [30] and Fas  \n47 tAdv [35]?   \n48 In our quest to answer this question we find that noise   \n49 augmentation using adequately shaped noise within   \n50 standard single-attack AT frameworks employing $\\ell _ { \\infty }$ -   \n51 bounded perturbations significantly improves robust  \n52 ness against the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations.   \n53 The improvement appears to be a consequence of a   \n54 well-established byproduct of AT frameworks – the   \n55 reduction in the curvature of the decision boundary of   \n56 networks trained using single-attack AT [6, 23]. We   \n57 confirm this connection by quantifying the impact   \n58 of single-attack AT on the geometric orientations of   \n59 different perturbations.   \n60 Based on this insight, we propose Shaped Noise   \n61 Augmented Processing (SNAP) – a method to en  \n62 hance robustness against the union of perturbation types by augmenting single-attack AT frameworks.   \n63 SNAP prepends a deep net with a shaped noise (SN) augmentation layer (see Fig. 4) whose dis  \n64 tribution parameter $\\Sigma$ is learned with that of the network $\\mathbf { \\eta } ^ { ( \\theta ) }$ within any standard single-attack AT   \n65 framework. SNAP improves the robustness of four SOTA $\\ell _ { \\infty }$ -AT frameworks against the union of   \n66 $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations by $15 \\%$ -to- $20 \\%$ on CIFAR-10 (red markers in Fig. 1) with only a modest   \n67 $( \\sim 1 0 \\% )$ increase in training time. This expands the capabilities of widely popular single-attack $\\ell _ { \\infty }$   \n68 AT frameworks to providing robustness to the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations without sacrificing   \n69 training efficiency. We validate SNAP’s benefits via thorough comparisons with nine SOTA adver  \n70 sarial training and randomized smoothing frameworks across different operating regimes on both   \n71 CIFAR-10 and ImageNet. ",
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+        "image_caption": [
+            "Figure 1: Adv uracy $( \\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } )$ $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ wall-clock total training time on CIFAR10 with different AT frameworks on single NVIDIA TESLA P100 GPU. $\\epsilon =$ (0.031, 0.5, 12) for $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations, respectively. SNAP enhances robustness with a small increase in training time. All frameworks except PAT employ ResNet-18. "
+        ],
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+        "type": "text",
+        "text": "72 Onenetw73 ome of our woeNet that achieve $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } = 3 2 \\% ( 3 5 \\% )$ e for the first time R against the union of $( \\ell _ { \\infty } ( \\epsilon = 2 / 2 5 5 ) , \\ell _ { 2 } ( \\epsilon =$ $\\ell _ { 1 } ( \\epsilon = 7 2 . 0 )$ ) perturbations. Our code and trained models will be shared publicly on GitHub. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    },
+    {
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+        "text": "76 We categorize works on adversarial vulnerability of DNNs as follows: ",
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+        "text": "77 Low-complexity adversarial training: The high computational needs of AT frameworks has spurred   \n78 significant efforts in reducing their complexity [40, 30, 35, 43]. FreeAdv [30] updates weights while   \n79 accumulating multiple attack iterations. FastAdv [35] employs appropriate use of single-step attacks,   \n80 while Zheng et al. [43] leverage inter-epoch similarity between adversarial perturbations. However,   \n81 these fast AT methods seek robustness against a single perturbation type, e.g., $\\ell _ { \\infty }$ norm-bounded   \n82 perturbations. In contrast, SNAP expands the capabilities of these AT frameworks by enhancing   \n83 robustness to the union of three perturbation types $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ , while preserving their efficiency. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Robustness against union of perturbation models: The focus on the robustness against the union of multiple perturbation types is relatively new. Kang et al. [16] studied transferability between different perturbation types, while Jordan et al. [15] considered combination attacks with low perceptual distortion. Stutz et al. [31] proposed a modification in AT to detect images with different models of perturbations via confidence thresholding, but they don’t attempt to classify perturbed images correctly. For accurate classification in the presence of different perturbation models, Tramèr & ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "90 Boneh [32] studied empirical and theoretical trade-offs involved in including multiple perturbation   \n91 types simultaneously during training. Maini et al. [21] further built upon this work to propose the   \n92 multi steepest descent (MSD) AT framework which chooses one among the three perturbation models   \n93 $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ in each attack iteration during training, achieving SOTA adversarial accuracy on CIFAR  \n94 10 against the union of the $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbation models, albeit at a high $( 1 0 \\times )$ training time. In   \n95 contrast, SNAP provides high robustness against the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbation models using   \n96 established single-attack $\\ell _ { \\infty }$ AT frameworks. This enables to showcase the benefits of our approach   \n97 on large-scale datasets such as ImageNet.   \n98 Recently, Laidlaw et al. [18] developed a novel AT framework (PAT) with low perceptual distortion   \n99 attacks to demonstrate impressive generalization to unseen attacks. In contrast, we focus on extending   \n100 the capabilities of widely popular $\\ell _ { \\infty }$ -AT frameworks to providing robustness against the union of   \n101 $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations, while preserving their training efficiency.   \n102 Noise augmentation: Multiple recent works have investigated the role of randomization in enhancing   \n103 adversarial robustness [12, 24, 8, 25] with theoretical guarantees. Another prominent line of work   \n104 in this category is randomized smoothing [5, 29, 19, 38], where random noise is used as a tool to   \n105 compute certification bounds. Rusak et al. [28] also explored the role of noise augmentation for   \n106 improving the robustness against common-corruptions [13]. In contrast, in SNAP, noise augmentation   \n107 is used as a means to enable widely popular $\\ell _ { \\infty }$ -AT frameworks to efficiently achieve high robustness   \n108 against the union of multiple norm-bounded perturbations. As is the characteristic of AT works, our   \n109 results are primarily empirical in nature. Hence, we follow recent guidelines [33, 21] to evaluate the   \n110 accuracy against the strongest possible adversaries. We do explicitly compare $\\ell _ { \\infty } { - } \\mathrm { A T } { + } \\mathrm { S N A P }$ with   \n111 randomized smoothing approaches in the Appendix. ",
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+    },
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+    {
+        "type": "text",
+        "text": "112 3 Subspace Analysis of Adversarial Perturbations ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "113 In this section, we employ subspace methods to compre  \n114 hend the distinction between $\\ell _ { \\infty }$ , $\\ell _ { 2 }$ and $\\ell _ { 1 }$ perturbations.   \n115 For each input $\\pmb { x } _ { i } \\in \\mathbb { R } ^ { D }$ in dataset $X$ , consider adversar  \n116 ial perturbations $\\alpha _ { i }$ , $\\beta _ { i }$ , and $\\gamma _ { i }$ bounded within $\\ell _ { \\infty }$ , $\\ell _ { 2 }$ ,   \n117 and $\\ell _ { 1 }$ norms, respectively.   \n118 We begin with a hypothesis (see Fig. 2): The perturbations   \n119 $\\alpha , \\beta ,$ , and $\\gamma$ corresponding to input x have directions that   \n120 differ significantly if the curvature of the decision bound  \n121 ary is high in the neighborhood of $_ { \\textbf { \\em x } }$ . Conversely, if the   \n122 curvature of the decision boundary is low, the perturba  \n123 tions $_ \\alpha$ , $\\beta$ , and $\\gamma$ tend to point in similar directions.   \n124 Since, prior works [6, 23] have found that single-attack   \n125 AT reduces the curvature of the decision boundary, we test   \n126 our hypothesis by studying the following two networks ",
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+        "image_caption": [
+            "!: ℓ! norm bounded; $\\beta \\colon \\ell _ { 2 }$ norm bounded; %: $\\ell _ { 1 }$ norm bounded ",
+            "Figure 2: Illustration of the role of decision boundary curvature on the distinction between different types of perturbations $_ { \\pmb { \\alpha } }$ , $\\beta$ and $\\gamma$ of the given input $_ { \\textbf { \\em x } }$ . "
+        ],
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+        "type": "text",
+        "text": "on CIFAR-10 data: a non-robust ResNet18 $f _ { \\theta } ^ { \\mathrm { v a n } }$ trained using vanilla training, and a robust ResNet18 $f _ { \\theta } ^ { \\mathrm { r o b } }$ trained using the TRADES [41] AT framework employing $\\ell _ { \\infty }$ perturbations. ",
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+        "type": "text",
+        "text": "129 We compute perturbations $\\alpha _ { i } , \\beta _ { i }$ , and $\\gamma _ { i }$ for each $\\pmb { x } _ { i } \\in X$ for both networks, i.e., $\\kappa \\in \\{ \\mathrm { v a n } , \\mathrm { r o b } \\}$ . We   \n130 compute the singular vector basis ${ \\mathcal { P } } ^ { \\kappa }$ for the set of $\\ell _ { 2 }$ bounded perturbations $\\Delta ^ { \\kappa } = \\{ \\beta _ { 1 } ^ { \\kappa } , \\ldots , \\beta _ { | X | } ^ { \\kappa } \\}$   \n131 The normalized mean squared projections of the three types of perturbation vectors on the singular   \n132 vector basis ${ \\mathcal { P } } ^ { \\kappa }$ of vanilla trained ResNet-18 $( \\mathcal { P } ^ { \\mathrm { v a n } } )$ )(Fig. 3(a)) and TRADES trained ResNet-18   \n133 $( \\mathcal { P } ^ { \\mathrm { r o b } } )$ )(Fig. 3(b)) shows a clear contrast.   \n134 The perturbations of a vanilla trained network roll-off gradually to occupy a larger subspace as   \n135 indicated in Fig. 3(a). Specifically, the projections of $_ { \\pmb { \\alpha } }$ and $\\gamma$ occupy almost all 3000 directions in   \n136 the basis $\\mathcal { P } ^ { \\mathrm { v a n } }$ since their mean squared projections are within $\\sim 1 0 \\%$ of the maximum value $m _ { \\mathrm { m a x } }$   \n137 This shows that the dominant singular vectors of $\\beta$ are not well-aligned with $_ { \\pmb { \\alpha } }$ and $\\gamma$ in a vanilla   \n138 trained network. With TRADES AT (Fig. 3(b)), however, all three types of perturbations are squeezed   \n139 into a much smaller subspace spanning only the top 250 singular vectors in the perturbation basis   \n140 ${ \\mathcal { P } } ^ { \\mathrm { r o b } }$ . Outside these 250 dimensions, the mean squared projections fall to $< 1 0 \\%$ of their maximum   \n141 value.   \n142 In summary, the results in Fig. 3 validate the hypothesis that single-attack AT increases the average   \n143 alignment of different perturbation types due to the reduction in the decision boundary curvature. In   \n44 Sec. 4, we exploit this behavior of single-attack $\\ell _ { \\infty }$ AT to improve its robustness against the union of   \n45 multiple perturbation models via SNAP. ",
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+        "img_path": "images/fbe4288bbbab0d5c0fab3b22cc11e5381333f4db5d9aa210c49425d1122424a9.jpg",
+        "image_caption": [
+            "Figure 3: Normalized mean squared projections of three perturbation types on the singular vector basis ${ \\mathcal { P } } ^ { \\kappa }$ of $\\ell _ { 2 }$ perturbations of ResNet18 on CIFAR-10 after: (a) vanilla training $\\kappa \\equiv \\operatorname { v a n } )$ , and (b) TRADES training $\\kappa \\equiv \\mathrm { r o b }$ ). The singular vectors $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { p } } _ { i } ^ { \\kappa } }$ comprising $\\mathcal { P } ^ { \\kappa } = \\{ p _ { 1 } ^ { \\kappa } , \\ldots , p _ { D } ^ { \\kappa } \\}$ are ordered in descending order of their singular values. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "146 4 Shaped Noise Augmented Processing (SNAP) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We show that single-attack AT can be enhanced to address multiple perturbations by introducing noise to appropriately wiggle the $\\ell _ { \\infty }$ -bounded perturbations (Fig. 4(a)). However, to do so, the noise distribution needs to be chosen and shaped appropriately to minimize its impact on natural accuracy and robustness to $\\ell _ { \\infty }$ -bounded perturbations. ",
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+        "text": "153 We experiment with both $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ perturbations in single  \n154 attack AT frameworks and find $\\ell _ { \\infty } { \\cdot } \\mathbf { A } \\mathrm { T }$ to be suitable for   \n155 our proposed shaped noise augmentation (see Sec. 5.2.1 for   \n156 details). Hence, in this section, we describe SNAP for single  \n157 attack AT frameworks employing $\\ell _ { \\infty }$ perturbations. ",
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+        "type": "text",
+        "text": "4.1 SNAPnet ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A deep net $f _ { \\theta } ( \\pmb { x } ) _ { \\mathbf { \\lambda } } \\colon \\mathbb { R } ^ { D }  \\{ 0 , 1 \\} ^ { C }$ parametrized by $\\theta$ maps the input $\\pmb { x } \\in \\mathbb { R } ^ { D }$ to a one-hot vector ${ \\pmb y } \\in \\{ 0 , 1 \\} ^ { \\tilde { C } }$ over $C$ classes. ",
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+        "text": "162 We construct a SNAP-based deep net (SNAPnet) $f _ { \\boldsymbol { \\theta } , \\Sigma } ^ { \\mathrm { S N } } ( \\boldsymbol { x } )$ by   \n163 introducing an additive shaped noise (SN) layer (Fig. 4(b)),   \n164 where the noise distribution parameter $\\Sigma$ is learned during   \n165 training. Formally, ",
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+        "img_path": "images/a3164964467f30c335a054e979640698625cdd6cce794036dbdb04ad576db1af.jpg",
+        "image_caption": [
+            "Figure 4: SNAP: (a) intuition underlying SNAP (not an exact depiction), and (b) SNAPnet $f _ { \\boldsymbol { \\theta } , \\Sigma } ^ { \\mathrm { S N } } ( \\boldsymbol { x } )$ constructed from a given deep net $f _ { \\boldsymbol { \\theta } } ( \\pmb { x } )$ by prepending a shaped noise (SN) augmentation layer which perturbs the primary input $_ { \\textbf { \\em x } }$ with noise $\\mathbf { n }$ whose distribution parameter $\\Sigma$ is learned during AT along with the base network parameter $\\theta$ . "
+        ],
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+        "type": "equation",
+        "img_path": "images/458bf583769dff042b5c1d6467079fc5e5375e0d0cfc0af3010fe98fbf8c5a48.jpg",
+        "text": "$$\n{ \\pmb y } = f _ { \\theta , \\Sigma } ^ { \\mathrm { S N } } ( { \\pmb x } ) = f _ { \\theta } \\big ( { \\pmb x } + { \\pmb n } \\big ) = f _ { \\theta } \\big ( { \\pmb x } + V \\Sigma { \\pmb n } _ { 0 } \\big ) ,\n$$",
+        "text_format": "latex",
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+        "text": "166 where $\\mathbf { n } _ { 0 } \\sim \\mathcal { L } ( 0 , \\mathbf { I } _ { D \\times D } )$ is a zero-mean isotropic Laplace   \n167 noise vector, $\\Sigma = \\mathrm { D i a g } [ \\sigma _ { 1 } , \\dots , \\sigma _ { D } ]$ is a distribution param  \n168 eter denoting its per-dimension standard deviation, ${ \\bf { I } } _ { D \\times D }$   \n169 denotes the $D \\times D$ identity matrix, and $V = [ \\pmb { v } _ { 1 } , \\dots , \\pmb { v } _ { D } ]$ denotes a basis in $\\mathbb { R } ^ { D }$ . We also studied   \n170 Gaussian and Uniform distributed ${ \\bf n } _ { 0 }$ , but empirically find the Laplace distribution to yield better   \n171 results (Sec. 5.2.1). We use $V = \\mathbf { I } _ { D \\times D }$ for all our experiments in the main text and study other   \n172 options for $V$ in the Appendix. ",
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+        "type": "text",
+        "text": "173 The final classification decision $d$ is computed via ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f939c2af8b4066d839c822e0799d3805fdbe1c328e8070aa8acfaab1cd94052d.jpg",
+        "text": "$$\nd = \\arg \\operatorname* { m a x } _ { c } { \\left[ \\mathbb { E } _ { \\mathbf { n } } [ \\pmb { y } ] \\right] } _ { c } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "174 where $[ \\pmb { a } ] _ { c }$ denotes the $c$ -th element of vector $^ { a }$ . Note, the shaped noise perturbs the input $_ { \\textbf { \\em x } }$ with a   \n175 noise source $\\mathbf { n } = V \\Sigma \\mathbf { n } _ { 0 }$ (Eq. (1)). The distribution parameter $\\Sigma$ is learned in the presence of any   \n176 standard AT method [20, 41, 30] used for learning deep net parameters $\\theta$ as described next. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Training SNAPnet ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Input: training set $X$ ; basis $V = [ \\pmb { v } _ { 1 } , \\dots , \\pmb { v } _ { D } ]$ ; total noise power $P _ { \\mathrm { n o i s e } }$ ; minibatch size $r$ ; baseline training method BASE; noise variance update frequency $U _ { f }$ ; Total number of epochs $T$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Initialize: noise variances $\\Sigma _ { 0 } = \\mathrm { { D i a g } } [ \\sigma _ { 1 , 0 } , . . . , \\sigma _ { D , 0 } ]$ .   \nOutput: robust network $f _ { \\boldsymbol { \\theta } , \\Sigma } ^ { \\mathrm { S N } }$ , noise variances $\\Sigma _ { T } = \\operatorname { \\bar { D i a g } } [ \\sigma _ { 1 , T } ^ { 2 } , \\dots , \\sigma _ { D , T } ^ { 2 } ]$ .   \n1: for epoch $t = 1 \\dots T$ do   \n2: for mini-batch $B = \\{ { \\pmb x } _ { 1 } , \\ldots , { \\pmb x } _ { r } \\}$ do θ ← BASE\\`∞ \u0012f SNθ,Σt \u0000{xi}ri=1\u0001, θ\u0013 . BASE() Training   \n3: end for   \n4: if $t$ mod $U _ { f } = 0$ then . SNAP Distribution Update once every $U _ { f }$ epochs   \n5: for mini-batch $B = \\{ { \\pmb x } _ { 1 } , \\ldots , { \\pmb x } _ { r } \\}$ do   \n6: $\\begin{array} { r l r } & { } & { \\{ \\pmb { x } _ { i } ^ { \\mathrm { a d v } } \\} _ { i = 1 } ^ { r }  \\mathrm { P G D } _ { \\ell _ { 2 } } ^ { ( K ) } \\bigg ( f _ { \\theta , \\Sigma _ { t } } ^ { \\mathrm { S N } } \\big ( \\{ \\pmb { x } _ { i } \\} _ { i = 1 } ^ { r } \\big ) \\bigg ) ; \\quad \\eta _ { i } = \\pmb { x } _ { i } ^ { \\mathrm { a d v } } - \\pmb { x } _ { i } \\forall i \\in \\{ 1 , \\dots , r \\} } \\\\ & { } & { \\gamma _ { j }  \\gamma _ { j } + \\sum _ { i = 1 } ^ { r } \\big ( \\langle \\pmb { v } _ { j } , \\eta _ { i } \\rangle \\big ) ^ { 2 } \\quad \\forall j \\in \\{ 1 , \\dots , D \\} \\qquad \\mathrm { ~ \\mathbb { b } \\ c c u m u l a t e \\ p r o j e c t i o n s } ; } \\\\ & { } & { \\frac { \\mathrm { n d \\ d \\Pi \\ f o r } } { j , t + 1 } = P _ { \\mathrm { n o i s e } } \\frac { \\sqrt { \\gamma _ { j } } } { \\sum _ { k = 1 } ^ { D } \\sqrt { \\gamma _ { k } } } \\quad \\forall j \\in \\{ 1 , \\dots , D \\} \\quad \\mathrm { ~ \\mathbb { b } \\ c n o r m a l i z e \\ a c c u m u l a t e d \\ p r o j e c t i o n s } ; } \\end{array}$   \n7: See Eq. (3)   \n8: 9: eσ See Eq. (3)   \n10: else   \n11: $\\Sigma _ { t + 1 }  \\Sigma _ { t }$   \n12: end if   \n13: end for ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "177 4.2 Training SNAPnet ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "178 Algorithm 1 summarizes the procedure for training SNAPnet $f _ { \\boldsymbol { \\theta } , \\Sigma } ^ { \\mathrm { S N } } ( \\boldsymbol { x } )$ . In each epoch, an arbitrary   \n179 AT method BASE() (line 2) updates network parameters $\\theta$ with input perturbed by noise $\\mathbf { n }$ . Here   \n180 BASE() can be any established AT framework [20, 41, 30, 35] employing $\\ell _ { \\infty }$ perturbation.   \n181 The SNAP parameter $\\Sigma$ is updated once every $U _ { f } = 1 0$ epochs via a SNAP distribution update (lines   \n182 4-10). In this update, the per-dimension noise variance $\\sigma _ { j } ^ { 2 }$ is updated proportional to the root mean   \n183 squared projection of the adversarial perturbations $\\eta$ on the basis $V$ given a total noise constraint   \n184 $\\begin{array} { r } { \\bar { \\sum _ { j = 1 } ^ { D } } \\sigma _ { j } ^ { 2 } = P _ { \\mathrm { n o i s e } } } \\end{array}$ , where $P _ { \\mathrm { n o i s e } }$ denotes the total noise power. Formally, ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/41985f5aa95aea4049d3e2c57bc168e9c40b3f4bada4cc9c4681474693afe37f.jpg",
+        "text": "$$\n\\sigma _ { j } ^ { 2 } \\propto \\sqrt { \\mathbb { E } _ { \\pmb { x } \\in X } \\left( \\langle \\pmb { \\eta } , \\pmb { v } _ { j } \\rangle ^ { 2 } \\right) } \\quad \\mathrm { s . t . } \\quad \\sum _ { j = 1 } ^ { D } \\sigma _ { j } ^ { 2 } = P _ { \\mathrm { n o i s e } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "185 where $\\eta$ is the $\\ell _ { 2 }$ norm-bounded PGD adversarial perturbation for the given input $\\boldsymbol { x } \\in \\boldsymbol { X }$ (line 6).   \n186 Note that these $\\ell _ { 2 }$ perturbations are employed only for noise shaping and are distinct from the $\\ell _ { \\infty }$   \n187 perturbations employed by BASE() AT (line 2). Also, $\\ell _ { \\infty }$ perturbations cannot be used here since   \n188 their projections are constant $\\forall j$ when $V = \\mathbf { I } _ { D \\times D }$ , whereas employing $\\ell _ { 1 }$ perturbations leads to poor   \n189 shaping due to high sparsity.   \n190 Thus, in SNAP, the average squared $\\ell _ { 2 }$ norm of the noise vector $\\mathbf { n }$ is held constant at $P _ { \\mathrm { n o i s e } }$ while   \n191 adapting the noise variances in the individual dimensions so as to align the noise vectors with the   \n192 adversarial perturbations on average. Intuitively, the decision boundary is pushed aggressively in   \n193 those directions. ",
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+    {
+        "type": "text",
+        "text": "194 4.3 Remarks ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "195 Note that the SNAP distribution update is distinct from BASE() AT. Hence, SNAP doesn’t require any   \n196 hyperparameter tuning in BASE(). For fairness to baselines we keep all hyperparameters identical   \n197 when introducing SNAP in all our experiments. However, SNAP introduces a new hyperparameter   \n198 $P _ { \\mathrm { n o i s e } }$ , which permits to trade adversarial robustness $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ for natural accuracy ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ . This trade-off is   \n199 explored in Sec. 5.2.2.   \n200 The computational overhead of SNAP is small $( \\sim 1 0 \\%$ ) since the SNAP Distribution Update occurs   \n201 once in 10 epochs using just $20 \\%$ of the training data to update the noise standard deviations $\\sigma _ { j }$ . We   \n202 provide more details about the SNAP Distribution Update in the Appendix. ",
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+    {
+        "type": "table",
+        "img_path": "images/e3430ea99b3d0f1dd243ef304808b269a1d236de29859655fed9bee6f65177fd.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: ResNet-18 CIFAR-10 results showing the impact of SNAP augmentation of PGD [20] AT framework with $\\ell _ { \\infty }$ (top) and $\\ell _ { 2 }$ (bottom) perturbations where [G], [U], and [L], denote shaped Gaussian, Uniform, and Laplace noise. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Anat</td><td>A ∈=0.03</td><td>A ∈=0.5</td><td>A ∈= 12</td><td>A</td></tr><tr><td colspan=\"6\">PGD AT withloo perturbations</td></tr><tr><td>PGD +SNAP[G] +SNAP[U]</td><td>84.6 80.7 85.1</td><td>48.8 45.7 42.7</td><td>62.3 66.9 66.7</td><td>15.0 34.6 28.6</td><td>15.0 31.9 26.6</td></tr><tr><td colspan=\"6\">+SNAP[L] 83.0 44.8 68.6 40.1 PGD AT withl2 perturbations</td></tr><tr><td>PGD +SNAP[G] +SNAP[U]</td><td>89.3 83.0</td><td>28.8 35.0</td><td>67.3 65.8</td><td>31.8 39.9</td><td>25.1 30.2</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: ResNet-18 CIFAR-10 results showing the impact of SNAP augmentation of established $\\ell _ { \\infty }$ -AT frameworks. The computational overhead of SNAP is limited to $\\sim 1 0 \\%$ . "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Anat</td><td>A ∈= 0.03</td><td>A ∈=0.5</td><td>A e=12</td><td>A</td></tr><tr><td colspan=\"6\">High Complexity ATwith loperturbations</td></tr><tr><td>PGD +SNAP</td><td>84.6 83.0</td><td>48.8 44.8</td><td>62.3 68.6</td><td>15.0 40.1</td><td>15.0 35.6</td></tr><tr><td>TRADES +SNAP</td><td>82.1 80.9</td><td>50.2 45.2</td><td>59.6 66.9</td><td>19.8 46.6</td><td>19.7 41.2</td></tr><tr><td colspan=\"6\">Low Complexity AT with looperturbations</td></tr><tr><td>FreeAdv +SNAP</td><td>81.7 83.5</td><td>46.1 39.7</td><td>59 66.2</td><td>15.0 34.3</td><td>15.0 29.6</td></tr><tr><td>FastAdv +SNAP</td><td>85.7 84.2</td><td>46.2 40.4</td><td>60.0 67.9</td><td>13.2 36.6</td><td>13.2 30.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "03 5 Experimental Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 Setup",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following experimental settings of prior work [41, 30, 21], we employ a ResNet-18 network for CIFAR-10 experiments and both ResNet-50 and ResNet-101 networks for ImageNet experiments. Accuracy on clreferred to via $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ t , $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) }$ s refe, and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ o wit, for $\\ell _ { \\infty }$ ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ $\\ell _ { 2 }$ and a, and $\\ell _ { 1 }$ uracy on adversarially perturbed test data isnorm bounded perturbations, respectively. Accuracy against the union of all three perturbations is denoted by $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ . ",
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+    {
+        "type": "text",
+        "text": "For a fair robustness comparison, our evaluation setup closely follows the setup of Maini et al. [21] for CIFAR-10 data: (1) choose norm bounds $\\epsilon = ( 0 . 0 3 1 , 0 . 5 , 1 2 . 0 )$ for $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations, respectively; (2) scale norm bounds for images to lie between $[ 0 , 1 ]$ ; (3) choose the PGD attack configuration to be $I O O$ iterations with $I O$ random restarts for all perturbation types1; and (4) estimate $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ as the fraction of test data that is simultaneously resistant to all three perturbation models. ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Following the guidelines of Tramer et al. [33], we carefully design adaptive PGD attacks that target the full defense – SN layer – since SNAPnet is end-to-end differentiable. Specifically, we backpropagate to primary input $_ { \\textbf { \\em x } }$ through the SN layer (see Fig. 4). Thus, the final shaped noise distribution is exposed to the adversary. We also account for the expectation $\\mathbb { E } _ { \\mathbf { n } } [ \\cdot ]$ in Eq. (2) by explicitly averaging deep net logits over $N _ { 0 } ( = 8 )$ noise samples before computing the gradient, which eliminates any gradient obfuscation, and is known to be the strongest attack against noise augmented models [29]. In the Appendix we also show robustness stress tests and evaluate more attacks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On CIFAR-10 data, we compare with the following seven key SOTA AT frameworks: PGD [20], TRADES [41], FreeAdv [30], FastAdv [35], AVG [32], MSD [21], PAT [18]. We also compare with two randomized smoothing frameworks [5, 29] in the Appendix. Thanks to their GitHub code releases, we first successfully reproduce their results with a ResNet-18 network in our environment. In the case of PAT [18], we evaluate and compare with their pretrained ResNet-50 model on CIFAR-10. We compare all training times on a single NVIDIA P100 GPU. On ImageNet data, we primarily compare to FreeAdv [30]. We train ResNet-50 and its SNAPnet version with FreeAdv on a Google Cloud server with four NVIDIA P100 GPUs to compare their accuracy and training times. We will release our pretrained models and code on GitHub. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2.1 Impact of Noise Distribution and Model of BASE() AT Perturbations ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "line 2 in Alg. 1) on In this subsection, we first study the impact of employing $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ . For each choice, we further experiment with three $\\ell _ { \\infty }$ vs. $\\ell _ { 2 }$ perturbations in BASE AT() (see ributions for the $\\ell _ { 1 }$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "237 BASE AT() since Maini et al. [21] showed that employing $\\ell _ { 1 }$ single-attack AT achieves very low   \n238 robustness to all attacks. We choose PGD [20] AT as BASE AT() for this ablation study. For a fair   \n239 comparison across the noise distributions, we fix $P _ { \\mathrm { n o i s e } } = 1 6 0$ , enforcing all noise vectors to have the   \n240 same average $\\ell _ { 2 }$ norm. For each distribution, the noise is shaped per the procedure summarized in   \n241 Alg. 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As observed in Table 1, $\\ell _ { \\infty }$ -PGD AT achieves much lower $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ than $\\ell _ { 2 }$ -PGD AT, an observation also reported by Maini et al. [21]. With SNAP, however, we find that there is an interaction between the perturbation model in PGD AT and the noise distribution in SNAP. For instance, SNAP[U] enhances $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ by $11 \\%$ with $\\ell _ { \\infty }$ -PGD AT while not achieving any improvement with $\\ell _ { 2 }$ -PGD AT. In fact, SNAP appears to be particularly suitable for $\\ell _ { \\infty }$ -AT, since it always improves $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ by $11 \\%$ -to- $2 0 . 6 \\%$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, of the three noise distributions, we find the Laplace distribution to be distinctly superior, achieving the highest A(U)adv ( $3 5 . 6 \\%$ and $3 0 . 8 \\%$ ) due to a significant improvement in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ for both $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ PGD AT, respectively. The superiority of the Laplace distribution in achieving high $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ stems from its heavier tail compared to the Gaussian and Uniform distributions with the same variance. Shaped Laplace noise generates the highest fraction of extreme values in a given noise sample. Hence, it is more effective in improving accuracy against $\\ell _ { 1 }$ -bounded attacks, which are the strongest when perturbing few pixels by a large magnitude [21, 32]. We discuss this further in the Appendix. Henceforth, unless otherwise mentioned, we choose Laplace noise for SNAP and $\\ell _ { \\infty }$ perturbations for BASE() AT as the default setting since it achieves the highest ${ \\mathcal { A } } _ { \\mathrm { a d v } } ^ { ( U ) }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2.2 Impact of $P _ { \\mathrm { n o i s e } }$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            424,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we explore the impact of the SNAP hyperparameter $P _ { \\mathrm { n o i s e } }$ , which constrains the average squared $\\ell _ { 2 }$ norm of the noise vector n. It enables to trade between adversarial and natural accuracy. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 5 shows that, as $P _ { \\mathrm { n o i s e } }$ increases, $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ improves from $31 \\%$ to $47 \\%$ , accompanied by a graceful $( 5 \\% )$ drop in ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ and a small drop of $2 \\%$ in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ that stabilizes to $\\approx 4 5 \\%$ . These results show: (1) SNAP preserves the impact of $\\ell _ { \\infty }$ perturbations which is not surprising since PGD AT [20] explicitly includes those, and (2) $P _ { \\mathrm { n o i s e } }$ provides an explicit knob to control the $\\boldsymbol { A } _ { \\mathrm { n a t } }$ vs. ${ \\mathcal { A } } _ { \\mathrm { a d v } }$ trade-off. Henceforth, we choose $P _ { \\mathrm { n o i s e } }$ values that incur $< 1 . 5 \\%$ drop in ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ for all $\\mathrm { S N A P { + } A T }$ experiments. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e64006dc9658f7ac2e31b75327a197074e59c75a52f4e4cc991275b7cd824aa2.jpg",
+        "image_caption": [
+            "Figure 5: ResNet-18 CIFAR-10 results: ad-(a) versarial racy $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ , $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ , an ural ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ vs. total noise power $P _ { \\mathrm { n o i s e } }$ $\\mathrm { P G D + S N A P }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.2.3 SNAP augmented SOTA AT Frameworks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the effectiveness of SNAP for four SOTA AT frameworks: high complexity frameworks, such as PGD [20], TRADES [41], and low complexity frameworks such as FreeAdv [30], FastAdv [35]. All are trained against $\\ell _ { \\infty }$ attacks with $\\epsilon = 0 . 0 3 1$ . As expected, while they achieve high $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ , their $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) }$ and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ are lower. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For high-complexity AT, SNAP enhances $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) }$ and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ by $\\sim 6 \\%$ and $\\sim 2 5 \\%$ , respectively, while incurring only a drop of $\\sim 5 \\%$ in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ adv adv . Thus overall, SNAP improves robustness $( \\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } )$ by $\\sim 2 0 \\%$ against the union of the three perturbation models. Note that this robustness improvement comes robustness at only a $\\sim 1 \\%$ $( \\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } )$ drop in are also significant ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ (see Table 2). For low-complexity ATs, SNAP improvements in union $( \\sim 1 5 \\% )$ ). Again, presence of SNAP improves $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) }$ and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ This time the drop in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ is $\\sim 7 \\%$ . We believe this is due to the fact that thes frameworks employ weaker single-step attacks during training. Note that in the case of FreeAdv $^ +$ observe a $\\sim 2 \\%$ increase in ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ , a trend we also observe in the ImageNet experiments described later. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/22ea25bf97acdd1eb3d7369b23ab30fc4d52ed481c7f3ede45cf629e5088d860.jpg",
+        "table_caption": [
+            "Table 3: CIFAR-10 results for comparing adversarial accuracy A(U)adv vs. training time (on single NVIDIA P100 GPU) for different AT frameworks and the improvements by introducing proposed SNAP technique. All frameworks except PAT [18] (which employs ResNet-50) employ ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>LR schedule</td><td rowspan=1 colspan=1>Epochs</td><td rowspan=1 colspan=1>Anat</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Total time(minutes)</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetA:Total Time</td><td rowspan=1 colspan=1>SetA:Total Time≥</td><td rowspan=1 colspan=1>12 Hrs</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=5 colspan=1>AVG 50Step[32]AVG 20 Step [32]AVG 10 Step [32]PAT[18]MSD 50 Step [21]MSD 30 Step [21]</td><td rowspan=1 colspan=1>cyclic</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>84.8</td><td rowspan=1 colspan=1>40.4</td><td rowspan=5 colspan=1>4217183495613641693978</td></tr><tr><td rowspan=4 colspan=1>cycliccyclicstepcycliccyclic</td><td rowspan=2 colspan=1>5050</td><td rowspan=2 colspan=1>85.686.7</td><td rowspan=1 colspan=1>40.438.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=3 colspan=1>501005050</td><td></td></tr><tr><td rowspan=2 colspan=1>86.782.481.782.4</td><td></td></tr><tr><td rowspan=1 colspan=1>38.936.647.044.9</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetB:8Hrs</td><td rowspan=1 colspan=3>Set B:8 Hrs&lt;Total Time&lt;12 Hrs</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>AVG 5 Step [32]MSD 20 Step [21]TRADES [41]TRADES+SNAP</td><td rowspan=1 colspan=1>cycliccyclicstepstep</td><td rowspan=1 colspan=1>5050100100</td><td rowspan=1 colspan=1>87.883.082.080.9</td><td rowspan=1 colspan=1>33.737.319.741.2</td><td rowspan=1 colspan=1>489690516566</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetC:5Hr</td><td rowspan=1 colspan=1><TotalTi</td><td rowspan=1 colspan=1>ne<8Hi</td><td rowspan=1 colspan=1>S</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>MSD 10 Step [21]PGD [20]PGD+SNAP</td><td rowspan=1 colspan=1>cyclicstepstep</td><td rowspan=1 colspan=1>50100100</td><td rowspan=1 colspan=1>83.684.683.0</td><td rowspan=1 colspan=1>33.315.035.6</td><td rowspan=1 colspan=1>342354403</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetD:2Hr</td><td rowspan=1 colspan=1><TotalTi</td><td rowspan=1 colspan=1>ne<5Hi</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>AVG 2 Step [32]MSD 5 Step [21]PGD [20]TRADES [41]PGD+SNAPTRADES+SNAP</td><td rowspan=1 colspan=1>cycliccycliccycliccycliccycliccyclic</td><td rowspan=1 colspan=1>505050505050</td><td rowspan=1 colspan=1>88.484.082.880.082.378.8</td><td rowspan=1 colspan=1>22.012.615.721.433.540.8</td><td rowspan=1 colspan=1>232185177258199280</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SetE:</td><td rowspan=1 colspan=1>TotalTime<</td><td rowspan=1 colspan=1>2Hrs</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>FreeAdv [30]FastAdv [35]FreeAdv+SNAPFastAdv+SNAP</td><td rowspan=2 colspan=1>stepcyclicstepcyclic</td><td rowspan=1 colspan=1>200</td><td rowspan=1 colspan=1>81.7</td><td rowspan=1 colspan=1>15.0</td><td rowspan=2 colspan=1>66478869</td></tr><tr><td rowspan=1 colspan=1>5020050</td><td rowspan=1 colspan=1>85.783.584.2</td><td rowspan=1 colspan=1>13.229.630.8</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/343a8e5083461ec98ba1f6c0fe96984a4ae0cb09beafab653ffdef8ed02e3af2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Training</td><td>Anat (%)</td><td>A e= 2/255</td><td>A ∈ = 2.0</td><td>A e = 72.0</td><td>A</td><td>Total time (minutes)</td></tr><tr><td colspan=\"7\">ResNet-50</td></tr><tr><td>FreeAdv [30]</td><td>61.7</td><td>47.8</td><td>19.9</td><td>14.8</td><td>12.6</td><td>3590</td></tr><tr><td>FreeAdv+SNAP</td><td>66.8</td><td>46.1</td><td>37.8</td><td>37.4</td><td>32.4</td><td>3756</td></tr><tr><td colspan=\"7\">ResNet-101</td></tr><tr><td>FreeAdv [30]</td><td>65.4</td><td>51.8</td><td>22.8</td><td>18.8</td><td>16.1</td><td>5678</td></tr><tr><td>FreeAdv+SNAP</td><td>69.7</td><td>50.3</td><td>41.1</td><td>40.2</td><td>35.4</td><td>5904</td></tr></table>",
+        "bbox": [
+            225,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4: ImageNet results: Iso-hyperparameter introduction of SNAP yields $\\sim 2 0 \\%$ improvement in adversarial accuracy $( \\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } )$ with modest impact on training time for ResNet-50 and ResNet-101. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "286 5.3 Robustness vs. Training Complexity ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            704,
+            462,
+            719
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Next we quantify adversarial robustness vs. training time trade-offs. Table 3 shows that SNAP augmentation of single-attack AT frame rks achieves the highest $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ , when training time is $\\mathbf { E }$ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For instance, TRADES $^ +$ SNAP achieves a $4 \\%$ higher $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } ( = 4 1 \\% )$ than MSD-20 with 2 hours lower training time (Set B in Table 3). Similarly, PGD $^ +$ SNAP achieves a $2 \\%$ higher $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ than MSD-10 while having a similar training time (Set C). Note that both PGD and TRADES here use 100 training epochs with standard step learning rate (LR) schedule, while MSD frameworks employ a cyclic learning rate schedule to achieve superconvergence in 50 epochs. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "295 In Set D, following Maini et al. [21], we employ a cyclic learning rate schedule for PGD, TRADES,   \n296 as well as for $\\mathrm { P G D + S N A P }$ and TRADES $^ +$ SNAP to achieve convergence in 50 epochs. Improvements   \n297 in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ for PGD $^ +$ SNAP and TRADES $^ +$ SNAP are similar to those in Sets $\\mathbf { B }$ and C. Most notably, ",
+        "bbox": [
+            145,
+            866,
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+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "298 $\\mathrm { P G D + S N A P }$ with cyclic learning rate achieves $\\sim 2 0 \\%$ and $1 1 . 5 \\%$ high $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ than MSD-5 and 299 AVG-2, respectively, while having aTable 2 with training times. FastAdv300 training time  and FreeAdv $\\sim 3$ hours). Set AP achieve $\\mathbf { E }$ auigh a from, while $+ { \\mathrm { S N A P } }$ $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) } \\sim 3 0 \\%$ 301 $1 8 \\%$ erving thigher $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ ining efficiency of both Fthan MSD-5, while being $\\sim 2 . 7 \\times$ nd FreeAdv. Notably, FastAdv+SNAP achievesmore efficient to train. ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4 ImageNet Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            184,
+            334,
+            199
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Thanks to SNAP’s low computational overhead combined with FreeAdv’s fast training time, we are for the first time able to report adversarial accuracy of ResNet-50 and ResNet-101 against the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ attacks on ImageNet. ",
+        "bbox": [
+            173,
+            210,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We closely follow the evaluation setup of Shafahi et al. [30]. Specifically, we use 100 step PGD attack, one of the strongest adversaries considered by Shafahi et al. [30], and evaluate on the entire test set. We first reproduce FreeAdv [30] results using the same hyperparameters and then introduce SNAP. All hyperparameter details are specified in the Appendix. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In order to clearly demonstrate the contrast between robustness to different perturbation models, we evaluate with FreeAdv achi $\\epsilon \\overset { \\cdot } { = } ( 2 / 2 5 5 , 2 . 0 , 7 2 . 0 )$ $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ attacks, respectivet-50, but a lower n inand adv , and consequently, a low $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) } = 4 7 . 8 \\%$ $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ o f $1 2 . 6 \\%$ against the union of the perturbations. In contrast, $A _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) } = 2 0 \\%$ $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) } =$ FreeAdv+SNAP improves $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 2 } ) }$ and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { 1 } ) }$ by $1 7 \\%$ and $2 2 \\%$ , respectively, accompanied by a $5 \\%$ improvement in ${ \\mathcal { A } } _ { \\mathrm { n a t } }$ and a small $2 \\%$ loss in $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( \\ell _ { \\infty } ) }$ . This results in an overall robustness improvement of $\\mathrm { \\bar { 2 0 \\% } }$ against the union of the perturbation models, setting a first benchmark for ResNet-50 on ImageNet. Upon increasing the network to ResNet-101, both natural and adversarial accuracies improve by $\\approx 4 \\%$ for FreeAdv, a trend also observed by Shafahi et al. [30]. SNAP further improves FreeAdv’s results for $\\boldsymbol { A } _ { \\mathrm { n a t } }$ and $\\mathcal { A } _ { \\mathrm { a d v } } ^ { ( U ) }$ by $4 . 3 \\%$ and $1 9 . 3 \\%$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            491,
+            294,
+            507
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Given the wide popularity of $\\ell _ { \\infty } { \\cdot } \\mathbf { A } \\mathbf { T } .$ , in this paper, we propose SNAP as an augmentation that generalizes the effectiveness of $\\ell _ { \\infty }$ -AT to the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations. SNAP’s strength is its simplicity and efficiency. Consequently, this work sets a first benchmark for ResNet-50 and ResNet101 networks which are resilient to the union of $( \\ell _ { \\infty } , \\ell _ { 2 } , \\ell _ { 1 } )$ perturbations on ImageNet. Note that norm-bounded perturbations include a large class of attacks, e.g., gradient-based [20, 27, 32, 21, 4, 22], decision-based [3] and black-box [1] attacks. ",
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+        "text": "More work is needed to extend the proposed SNAP technique to attacks beyond norm-bounded additive perturbations, e.g., functional [17, 36], rotation [7], texture [2], etc. We provide preliminary evaluations in this direction in the Appendix. It is important to note that SNAP is meant to be an efficient technique for improving $\\ell _ { \\infty }$ -AT, and not a new defense. Indeed defending against a large variety of attacks simultaneously remains an open problem, with encouraging results from recent efforts [21, 18]. ",
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+        "text": "Another limitation of our approach is that its benefits are demonstrated empirically. It is an inevitable consequence of a lack of any theoretical guarantees for underlying AT frameworks. An interesting direction of future work is to explore whether any theoretical guarantees can be derived for anisotropic shaped noise distributions in SNAP by building upon the recent developments in randomized smoothing [29, 38]. This could be a potential avenue for bridging the gap between certification bounds and empirical adversarial accuracy. ",
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+        "text": "Finally, we believe that any effort on improving adversarial robustness of deep nets has net positive societal impact. However, recent past in this field has shown that any improvements in defense techniques also lead to more effective threat models. While such a cat-and-mouse game is of great intellectual value in the academic setting, it does have an unintentional negative societal consequence of equipping malicious outside actors with a broad set of tools. This further underscores the wellrecognized need for provable defenses. ",
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+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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A simple way to make neural networks robust against diverse image corruptions. In European Conference on Computer Vision, pp. 53–69. Springer, 2020.   \n[29] Salman, H., Li, J., Razenshteyn, I., Zhang, P., Zhang, H., Bubeck, S., and Yang, G. Provably robust deep learning via adversarially trained smoothed classifiers. In Advances in Neural Information Processing Systems, pp. 11289–11300, 2019.   \n[30] Shafahi, A., Najibi, M., Ghiasi, A., Xu, Z., Dickerson, J., Studer, C., Davis, L. S., Taylor, G., and Goldstein, T. Adversarial training for free! Advances in Neural Information Processing Systems (NeurIPS), 2019.   \n[31] Stutz, D., Hein, M., and Schiele, B. Confidence-calibrated adversarial training: Generalizing to unseen attacks. In International Conference on Machine Learning, pp. 9155–9166. PMLR, 2020.   \n[32] Tramèr, F. and Boneh, D. Adversarial training and robustness for multiple perturbations. In Advances in Neural Information Processing Systems, pp. 5858–5868, 2019.   \n[33] Tramer, F., Carlini, N., Brendel, W., and Madry, A. On adaptive attacks to adversarial example defenses. arXiv preprint arXiv:2002.08347, 2020.   \n[34] Vivek, B. and Babu, R. V. Single-step adversarial training with dropout scheduling. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 947–956. IEEE, 2020.   \n[35] Wong, E., Rice, L., and Kolter, J. Z. Fast is better than free: Revisiting adversarial training. In International Conference on Machine Learning (ICLR), 2020.   \n[36] Xiao, C., Zhu, J.-Y., Li, B., He, W., Liu, M., and Song, D. Spatially transformed adversarial examples. In International Conference on Learning Representations, 2018.   \n[37] Xie, C. and Yuille, A. Intriguing properties of adversarial training at scale. In International Conference on Learning Representations, 2020.   \n[38] Yang, G., Duan, T., Hu, E., Salman, H., Razenshteyn, I., and Li, J. Randomized smoothing of all shapes and sizes. International Conference on Machine Learning (ICML), 2020.   \n[39] Yang, Y.-Y., Rashtchian, C., Zhang, H., Salakhutdinov, R., and Chaudhuri, K. A closer look at accuracy vs. robustness. Advances in Neural Information Processing Systems, 33, 2020.   \n[40] Zhang, D., Zhang, T., Lu, Y., Zhu, Z., and Dong, B. You only propagate once: Accelerating adversarial training via maximal principle. arXiv preprint arXiv:1905.00877, 2019.   \n[41] Zhang, H., Yu, Y., Jiao, J., Xing, E., El Ghaoui, L., and Jordan, M. Theoretically principled trade-off between robustness and accuracy. In International Conference on Machine Learning (ICML), 2019.   \n[42] Zhang, J., Xu, X., Han, B., Niu, G., Cui, L., Sugiyama, M., and Kankanhalli, M. Attacks which do not kill training make adversarial learning stronger. In International Conference on Machine Learning, pp. 11278–11287. PMLR, 2020.   \n[43] Zheng, H., Zhang, Z., Gu, J., Lee, H., and Prakash, A. Efficient adversarial training with transferable adversarial examples. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1181–1190, 2020. ",
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parse/train/B1M8JF9xx/B1M8JF9xx.md

@@ -0,0 +1,324 @@
+# ON THE QUANTITATIVE ANALYSIS OF DECODERBASED GENERATIVE MODELS
+
+Yuri Burda   
+OpenAI   
+yburda@openai.com
+
+Yuhuai Wu Department of Computer Science University of Toronto ywu@cs.toronto.edu
+
+Ruslan Salakhutdinov School of Computer Science Carnegie Mellon University rsalakhu@cs.cmu.edu
+
+Roger Grosse Department of Computer Science University of Toronto rgrosse@cs.toronto.edu
+
+# ABSTRACT
+
+The past several years have seen remarkable progress in generative models which produce convincing samples of images and other modalities. A shared component of many powerful generative models is a decoder network, a parametric deep neural net that defines a generative distribution. Examples include variational autoencoders, generative adversarial networks, and generative moment matching networks. Unfortunately, it can be difficult to quantify the performance of these models because of the intractability of log-likelihood estimation, and inspecting samples can be misleading. We propose to use Annealed Importance Sampling for evaluating log-likelihoods for decoder-based models and validate its accuracy using bidirectional Monte Carlo. The evaluation code is provided at https:// github.com/tonywu95/eval_gen. Using this technique, we analyze the performance of decoder-based models, the effectiveness of existing log-likelihood estimators, the degree of overfitting, and the degree to which these models miss important modes of the data distribution.
+
+# 1 INTRODUCTION
+
+In recent years, deep generative models have dramatically pushed forward the state-of-the-art in generative modelling by generating convincing samples of images (Radford et al., 2016), achieving state-of-the-art semi-supervised learning results (Salimans et al., 2016), and enabling automatic image manipulation (Zhu et al., 2016). Many of the most successful approaches are defined in terms of a process which samples latent variables from a simple fixed distribution (such as Gaussian or uniform) and then applies a learned deterministic mapping which we will refer to as a decoder network. Important examples include variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014), generative adversarial networks (GANs) (Goodfellow et al., 2014), generative moment matching networks (GMMNs) (Li & Swersky, 2015; Dziugaite et al., 2015), and nonlinear independent components estimation (Dinh et al., 2014). We refer to this set of models collectively as decoder-based models, also known as density networks (MacKay & Gibbs, 1998).
+
+While many decoder-based models are able to produce convincing samples (Denton et al., 2015; Radford et al., 2016), rigorous evaluation remains a challenge. Comparing models by inspecting samples is labor-intensive, and potentially misleading (Theis et al., 2016). While alternative quantitative criteria have been proposed (Bounliphone et al., 2016; Im et al., 2016; Salimans et al., 2016), log-likelihood of held-out test data remains one of the most important measures of a generative model’s performance. Unfortunately, unless the decoder is designed to be reversible (Dinh et al., 2014; 2016), log-likelihood estimation in decoder-based models is typically intractable. In the case of VAE-based models, a learned encoder network gives a tractable lower bound, but for GANs and GMMNs it is not obvious how even to compute a good lower bound. Even when lower bounds are available, their accuracy may be hard to determine. Because of the difficulty of log-likelihood evaluation, it is hard to answer basic questions such as whether the networks are simply memorizing training examples, or whether they are missing important modes of the data distribution.
+
+![](images/abdb08b1a142504877599e275d9dd7235f7c6fed6aafaaa4479232bccf4e6298.jpg)  
+Figure 1: (a) samples from a GAN with 10 latent dimensions, (b) and (c) samples from a GAN with 50 latent dimensions at different epochs of training. While it is difficult to visually discern differences between these three models, their log-likelihood (LLD) values span almost 300 nats.
+
+The most widely used estimator of log-likelihood for GANs and GMMNs is the Kernel Density Estimator (KDE) (Parzen, 1962). It estimates the likelihood under an approximation to the model’s distribution obtained by simulating from the model and convolving the set of samples with a kernel (typically Gaussian). Unfortunately, KDE is notoriously inaccurate for estimating likelihood in high dimensions, because it is hard to tile a high-dimensional manifold with spherical Gaussians (Theis et al., 2016).
+
+In this paper, we propose to use annealed importance sampling (AIS; (Neal, 2001)) to estimate log-likelihoods of decoder-based generative models and to obtain approximate posterior samples. Importantly, we validate this approach using Bidirectional Monte Carlo (BDMC) (Grosse et al., 2015), which provably bounds the log-likelihood estimation error and the KL divergence from the true posterior distribution for data simulated from a model. For most models we consider, we find that AIS is two orders of magnitude more accurate than KDE, and is accurate enough to perform fine-grained comparisons between generative models. In the case of VAEs, we show that AIS can be further sped up by using the recognition network to determine the initial distribution; this yields an estimator which is fast enough to be run repeatedly during training.
+
+Using the proposed method, we analyze several scientific questions central to understanding decoderbased generative models. First, we measure the accuracy of KDE and of the importance weighting bound which is commonly used to evaluate VAEs. We find that the KDE error is larger than the (quite significant) log-likelihood differences between different models, and that KDE can lead to misleading conclusions. The importance weighted bound, while reasonably accurate, can also yield misleading results in some cases.
+
+Second, we compare the log-likelihoods of VAEs, GANs, and GMMNs, and find that VAEs achieve log-likelihoods several hundred nats higher than the other models (even though KDE considers all three models to have roughly the same log-likelihood). Third, we analyze the degree of overfitting in VAEs, GANs, and GMMNs. Contrary to a commonly proposed hypothesis, we find that GANs and GMMNs are not simply memorizing their training data; in fact, their log-likelihood gaps between training and test data are much smaller relative to comparably-sized VAEs. Finally, by visualizing (approximate) posterior samples obtained from AIS, we observe that GANs miss important modes of the data distribution, even ones which are represented in the training data.
+
+We emphasize that none of the above phenomena can be measured using KDE or the importance weighted bound, or by inspecting samples. (See Fig. 1 for an example where it is tricky to compare models based on samples.) While log-likelihood is by no means a perfect measure, we find that the ability to accurately estimate log-likelihoods of decoder-based models yields crucial insight into their behavior and suggests directions for improving them.
+
+# 2 BACKGROUND
+
+# 2.1 DECODER-BASED GENERATIVE MODELS
+
+In generative modelling, a decoder network is often used to define a generative distribution by transforming samples from some simple distribution (e.g. normal) to the data manifold. In this
+
+paper, we consider three kinds of decoder-based generative models: Variational Autoencoder (VAE) (Kingma & Welling, 2014), Generative Adversarial Network (GAN) (Goodfellow et al., 2014), and Generative Moment Matching Network (GMMN) (Li & Swersky, 2015; Dziugaite et al., 2015).
+
+# 2.1.1 VARIATIONAL AUTOENCODER
+
+A variational autoencoder (VAE) (Kingma & Welling, 2014) is a probabilistic directed graphical model. It is defined by a joint distribution over a set of latent random variables $z$ and the observed variables $x$ $: p ( x , z ) = p ( x | z ) p ( z )$ . The prior over the latent random variables, $p ( z )$ , is usually chosen to be a standard Gaussian distribution. The data likelihood $p ( x | z )$ is usually a Gaussian or Bernoulli distribution whose parameters depend on $z$ through a deep neural network, known as the decoder network. It also uses an approximate inference model called an encoder or recognition network, that serves as a variational approximation $q ( z | x )$ to the posterior $p ( z | x )$ . The decoder network and the encoder networks are jointly trained to maximize the evidence lower bound (ELBO):
+
+$$
+\log p ( x ) \geq \mathbb { E } _ { q ( z | x ) } [ \log p ( x | z ) ] - K L ( q ( z | x ) | | p ( z ) )
+$$
+
+In addition, the reparametrization trick is used to reduce the variance of the gradient estimate.
+
+# 2.1.2 GENERATIVE ADVERSARIAL NETWORK (GAN)
+
+A generative adversarial network (GAN) (Goodfellow et al., 2014) is a generative model trained by a game between a decoder network and a discriminator network. It defines the generative model by sampling the latent variable $z$ from some simple prior distribution $p ( z )$ (e.g., Gaussian) followed through the decoder network. The discriminator network $D ( \cdot )$ outputs a probability of a given sample coming from the data distribution. Its task is to distinguish samples from the generator distribution from real data. The decoder network, on the other hand, tries to produce samples as realistic as possible, in order to fool the discriminator into accepting its outputs as being real. The competition between the two networks results in the following minimax problem:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } \mathbb { E } _ { x \sim p _ { d a t a } } [ \log D ( x ) ] + \mathbb { E } _ { z \sim p ( z ) } [ \log ( 1 - D ( G ( z ) ) ]
+$$
+
+Unlike VAE, the objective is not explicitly related to the log-likelihood of the data. Moreover, the generative distribution is a deterministic mapping, i.e., $\bar { p } ( x | z )$ is a Dirac delta distribution, parametrized by the deterministic decoder. This can make data likelihood ill-defined, as the probability density of any particular point $x$ can be either infinite, or exactly zero.
+
+# 2.1.3 GENERATIVE MOMENT MATCHING NETWORK (GMMN)
+
+Generative moment matching networks (GMMNs) (Li & Swersky, 2015; Dziugaite et al., 2015) adopt maximum mean discrepancy (MMD) as the training objective, a moment matching criterion where kernel mean embedding techniques are used to avoid unnecessary assumptions of the distributions. It has the same issue as GAN in that the log-likelihood is undefined.
+
+# 2.2 ANNEALED IMPORTANCE SAMPLING
+
+We are interested in estimating the probability $\begin{array} { r } { p ( x ) = \int p ( z ) p ( x | z ) \mathrm { d } z } \end{array}$ a model assigns to an observation $x$ . This is equivalent to computing the normalizing constant of the unnormalized distribution $f ( z ) ~ = ~ p ( z , { \bar { x } } )$ . One naïve approach is likelihood weighting, where one samples $\{ z ^ { ( k ) } \} _ { k = 1 } ^ { K } \sim p ( z )$ and averages the conditional likelihoods $p ( x | z ^ { ( k ) } )$ . This is justified by the following identity:
+
+$$
+p ( x ) = \int \frac { p ( x , z ) } { p ( z ) } p ( z ) \mathrm { d } z = \mathbb { E } _ { z \sim p ( z ) } [ p ( x | z ) ]
+$$
+
+Likelihood weighting can be viewed as simple importance sampling, where the proposal distribution is the prior $p ( z )$ and the target distribution is the posterior $p ( z | x )$ . Unfortunately, importance sampling works well only when the proposal distribution is a good match for the target distribution. For the models considered in this paper, the (very broad) prior can be drastically different than the (highly concentrated) posterior, leading to inaccurate estimates of the likelihood.
+
+Annealed importance sampling (AIS; Neal, 2001) is a Monte Carlo algorithm commonly used to estimate (ratios of) normalizing constants. Roughly speaking, it computes a sequence of importance sampling based estimates, each of which is stable because it involves two distributions which are very similar. In particular, suppose one is interested in estimating the normalizing constant $\begin{array} { r } { \mathcal { Z } = \int f ( \boldsymbol { z } ) \mathrm { d } \dot { \boldsymbol { z } } } \end{array}$ of an unnormalized distribution $f ( z )$ . (In the likelihood estimation setting, $f ( z ) = p ( z , x )$ and ${ \mathcal { Z } } = p ( x )$ .) One must specify a sequence of distributions $q _ { 1 } , . . . , q _ { T }$ , where $q _ { t } = f _ { t } / Z _ { t }$ , and $f _ { T } = f$ is the target distribution. It is required that one can obtain one or more exact samples from the initial distribution $q _ { 1 }$ . One must also specify a sequence of reversible MCMC transition operators $\mathcal { T } _ { 1 } , . . . , \mathcal { T } _ { T }$ , where $\mathcal { T } _ { t }$ leaves $q _ { t }$ invariant.
+
+AIS produces a (nonnegative) unbiased estimate of the ratio $\mathcal { Z } _ { T } / \mathcal { Z } _ { 1 }$ as follows: first, we sample a random initial state $z _ { 1 } \sim q _ { 1 }$ and set the initial weight $w _ { 1 } = 1$ . For every stage $t \geq 2$ we update the weight $w$ and sample the state $z _ { t }$ according to
+
+$$
+w _ { t }  w _ { t - 1 } \frac { f _ { t } ( z _ { t - 1 } ) } { f _ { t - 1 } ( z _ { t - 1 } ) } \qquad z _ { t } \sim \mathcal { T } _ { t } ( z | z _ { t - 1 } )
+$$
+
+As demonstrated by Neal (2001), this procedure produces a nonnegative weight $w _ { T }$ such that $\mathbb { E } [ w _ { T } ] = \mathcal { Z } _ { T } / \mathcal { Z } _ { 1 }$ . Typically, btains the un $\mathcal { Z } _ { 1 }$ is known,ed estimate le independent AIS weights. In the likelihood estimation $\{ w _ { T } ^ { ( K ) } \} _ { k = 1 } ^ { K }$ $\begin{array} { r } { \hat { \mathcal { Z } } _ { T } = \mathcal { Z } _ { 1 } \frac { 1 } { K } \sum _ { k = 1 } ^ { K } w _ { T } ^ { ( K ) } } \end{array}$   
+setting, $\mathcal { Z } _ { 1 } = 1$ and , so we denote this estimator as ${ \hat { p } } ( x )$ .
+
+Typically, the unnormalized intermediate distributions are simply defined to be geometric averages $\dot { f _ { t } ( z ) } = \dot { \sp { \prime } } f _ { 1 } ( z ) \sp { 1 - \beta _ { t } } f _ { T } ( z ) \sp { \beta _ { t } }$ , where the $\beta _ { t }$ are monotonically increasing parameters with $\beta _ { 1 } = 0$ and $\beta _ { T } = 1$ . For $f _ { 1 } ( z ) = p ( z )$ and $f _ { T } ( z ) = p ( z , x )$ , this gives
+
+$$
+f _ { t } ( z ) = p ( z ) p ( x | z ) ^ { \beta _ { t } } .
+$$
+
+As shown by Neal (2001), under certain regularity conditions, the variance of $\hat { \mathcal { Z } } _ { T }$ tends to zero as the number of intermediate distributions is increased. AIS is very effective in practice, and has been used to estimate normalizing constants of complex high-dimensional distributions (Salakhutdinov $\&$ Murray, 2008).
+
+# 2.3 BIDIRECTIONAL MONTE CARLO
+
+AIS provides a nonnegative unbiased estimate ${ \hat { p } } ( x )$ of $p ( x )$ . However, it is often more practical to estimate $p ( x )$ in the log space, i.e. $\log p ( x )$ , because of underflow problem of dealing with many products of probability measure. In general, we note that logarithm of a nonnegative unbiased estimate is a stochastic lower bound of the log estimand (Grosse et al., 2015). In particular, $\log { \hat { p } } ( x )$ is a stochastic lower bound on $\log p ( x )$ , satisfying $\mathbb { E } [ \log { \hat { p } ( x ) } ] \leq \log p ( x )$ and $\operatorname* { P r } ( \log { \hat { p } } ( x ) > \log p ( x ) +$ $b ) < e ^ { - b }$ .
+
+Grosse et al. (2015) pointed out that if AIS is run in reverse starting from an exact posterior sample, it yields an unbiased estimate of $1 / p ( x )$ , which (by the above argument) can be seen as a stochastic upper bound on $\log p ( x )$ . The combination of lower and upper bounds from forward and reverse AIS is known as bidirectional Monte Carlo (BDMC). In many cases, the combination of bounds can pinpoint the true value quite precisely. While posterior sampling is just as hard as log-likelihood estimation (Jerrum et al., 1986), in the case of log-likelihood estimation for simulated data, one has available a single exact posterior sample: the parameters and/or latent variables which generated the data. Because this trick is only applicable to simulated data, BDMC is most useful for measuring the accuracy of a log-likelihood estimator on simulated data.
+
+Grosse et al. (2016) observed that BDMC can also be used to validate posterior inference algorithms, as the gap between upper and lower bounds is itself a bound on the KL divergence of approximate samples from the true posterior distribution.
+
+# 3 METHODOLOGY
+
+For a given generative distribution $p ( x , z ) = p ( z ) p ( x | z )$ , our task is to measure the log-likelihood of test examples $\log p ( x _ { t e s t } )$ . We first discuss how we define the generative distribution for decoderbased networks. For VAE, the generative distribution is defined in the standard way, where $p ( z )$ is a standard normal distribution and $p ( x | z )$ is a normal distribution parametrized by mean $\mu _ { \boldsymbol { \theta } } ( z )$ and $\sigma _ { \theta } ( z )$ , predicted by the generator given the latent code. However, the observation distribution for GANs and GMMNs is typically taken to be a delta function, so that the model’s distribution covers only a submanifold of the space of observables. In order for the likelihood to be well-defined, we follow the same assumption made when evaluating using Kernel Density Estimator (Parzen, 1962): we assume a Gaussian observation model with a fixed variance hyperparameter $\sigma ^ { 2 }$ . We will refer to the distribution defined by this Gaussian observation model as $p _ { \sigma }$ .
+
+Observe that the KDE estimate is given by
+
+$$
+\hat { p } _ { \sigma } ( x ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } p _ { \sigma } ( x | z ^ { ( k ) } ) ,
+$$
+
+wher e {z(k)}K are samples from the prior $p ( z )$ . This is equivalent to likelihood weighting for the distribution $p _ { \sigma }$ , which is an instance of simple importance sampling (SIS). Because SIS is an unbiased estimator of the likelihood, $\log \hat { p } _ { \sigma } ( x )$ is a stochastic lower bound on $\log p _ { \sigma } ( x )$ (Grosse et al., 2015). Unfortunately, SIS can result in very poor estimates when the evidence has low prior probability (i.e. the posterior is very dissimilar to the prior). This suggests that AIS might be able to yield much more accurate log-likelihood estimates under $p _ { \sigma }$ . We note that KDE can be viewed as a special case of AIS where the number of intermediate distributions is set to 0.
+
+We now describe specifically how we carry out evaluation using AIS. In most of our experiments, we choose the initial distribution of AIS to be $p ( z )$ , the same prior distribution used in training decoderbased models. If the model provides an encoder network (e.g., VAE), we can take the approximated distribution predicted by the encoder $q ( z | x )$ as the initial distribution of the AIS chain. For continuous data, we define the unnormalized density of target distribution to be the joint generative distribution with the Gaussian noise model, $p _ { \sigma } ( x , z ) = p _ { \sigma } ( x | z ) p ( z )$ . For the small subset of experiments done on the binary data, we define the observation model to be a Bernoulli model with mean predicted by the decoder. Our intermediate distributions are geometric averages of the prior and posterior, as in Eqn. 5. Since all of our experiments are done using continuous latent space, we use Hamiltonian Monte Carlo (Neal, 2010) as the transition operator for sampling latent samples along annealing. The evaluation code is provided at https://github.com/tonywu95/eval_gen.
+
+# 4 RELATED WORK
+
+AIS is known to be a powerful technique of estimating the partition function of the model. One influential example was the use of AIS to evaluate deep belief networks (Salakhutdinov & Murray, 2008). Although we used the same technique, the problem we consider is completely different. First of all, the model they consider is undirected graphical models, whereas decoder-based models are directed graphical models. Secondly, their model has a well-defined probabilistic density function in terms of energy function, whereas we need to consider different probabilistic model for one in which the the likelihood is ill-defined. In addition, we validate our estimates using BDMC.
+
+Theis et al. (2016) give an in-depth analysis of issues that might come up in evaluating generative models. They also point out that a model that completely fails at modelling the proportion of modes of the distribution might still achieve a high likelihood score. Salimans et al. (2016) propose an image-quality measure which they find to be highly correlated with human visual judgement. They propose to feed the samples $x$ of the model to the “inception” model to obtain a conditional label distribution $p ( y | x )$ , and evaluate the score defined by exp $\mathbb { E } _ { x } \mathrm { K L } ( p ( y | x ) | | p ( y ) )$ , which is motivated by having a low entropy of $p ( y | x )$ but a large entropy of $p ( y )$ . However, the measure is largely based on visual quality of the sample, and we argue that the visual quality can be a misleading way to evaluate a model.
+
+# 5 EXPERIMENTS
+
+# 5.1 DATASETS
+
+All of our experiments were performed on the MNIST dataset of images of handwritten digits (LeCun et al., 1998). For consistency with prior work on evaluating decoder-based models, most of our experiments used the continuous inputs. We dequantized the data following Uria et al. (2013), by adding a uniform noise of $\scriptstyle { \frac { 1 } { 2 5 6 } }$ to the data and rescaling it to be in $[ 0 , 1 ] ^ { D }$ after dequantization. We use the standard split of MNIST into 60,000 training and 10,000 test examples, and used 50,000 images from the training set for training, and remaining 10,000 images for validation. In addition, some of our experiments used the binarized MNIST dataset with a Bernoulli observation model (Salakhutdinov & Murray, 2008).
+
+# 5.2 MODELS
+
+For most of our experiments, we considered two decoder architectures: a small one with 10 latent dimensions, and a larger one with 50 latent dimensions. We use standard Normal distribution as prior for training all of our models. All layers were fully connected, and the number of units in each layer was 10–64–256–256-1024–784 for the smaller architecture and 50–1024–1024–1024–784 for the larger one. We trained both architectures using the VAE, GAN, and GMMN objectives, resulting in six networks which we refer to as VAE-10, VAE-50, etc. In general, the larger architecture performed substantially better on both the training and test sets, but we analyze the smaller architecture as well because it better highlights some of the differences between the training criteria. Additional architectural details are given in Appendix A.1.
+
+In order to enable a direct comparison between training criteria, all models used a spherical Gaussian observation model with fixed variance. This is consistent with previous protocols for evaluating GANs and GMMNs. However, we note that this observation model is a nontrivial constraint on the VAEs, which could instead be trained with a more flexible diagonal Gaussian observation model where the variances depend on the latent state. Such observation models can easily achieve much higher log-likelihood scores, for instance by noticing that boundary pixels are always close to 0. (E.g., we trained a VAE with the more general observation model which achieved a log-likelihood of at least 2200 nats on continuous MNIST.) Therefore, the log-likelihood values we report should not be compared directly against networks which have a more flexible observation model.
+
+# 5.3 VALIDATION OF LOG-LIKELIHOOD ESTIMATES
+
+Before we analyze the performance of the trained networks, we must first determine the accuracy of the log-likelihood estimators. In this section, we validate the accuracy of our AIS-based estimates using BDMC. We then analyze the error in the KDE and IWAE estimates and highlight some cases where these measures miss important phenomena.
+
+# 5.3.1 VALIDATION OF AIS
+
+We used AIS to estimate log-likelihoods for all models under consideration. Except where otherwise specified, all AIS estimates were obtained using 16 independent chains, 10,000 intermediate distributions of the form in Eqn. 5, and a transition operator consisting of one proposed HMC trajectory with 10 leapfrog steps.1 Following Ranzato et al. (2010), the HMC stepsize was tuned to achieve an acceptance rate of 0.65 (as recommended by Neal (2010)).
+
+For all six models, we evaluated the accuracy of this estimation procedure using BDMC on data sampled from the model’s distribution on 1000 simulated examples. The gap between the loglikelihood estimates produced by forward AIS (which gives a lower bound) and reverse AIS (which gives an upper bound) bounds the error of the AIS estimates on simulated data. We refer to this gap as the BDMC gap. For five of the six networks under consideration, we found the BDMC gap to be less than 1 nat. For the remaining model (GAN-50), the gap was about 10 nats. Both gaps are much smaller than our measured log-likelihood differences between models. If these gaps are representative of the true error in the estimates on the real data, then this indicates AIS is accurate enough to make fine-grained comparisons between models and to benchmark other log-likelihood estimators. (The BDMC gap is not guaranteed to hold for the real data, although Grosse et al. (2016) found the behavior of AIS to match closely between real and simulated data.)
+
+# 5.3.2 HOW ACCURATE IS KERNEL DENSITY ESTIMATION?
+
+Kernel density estimation (KDE) (Parzen, 1962) is widely used to evaluate decoder-based models (Goodfellow et al., 2014; Li & Swersky, 2015), and a variant was proposed in the setting of evaluating Boltzmann machines (Bengio et al., 2013). Papers reporting KDE estimates often caution that the
+
+![](images/1147904ce6942a6a038f58b1f5ced578a74bce4a984505b6a275f0164f1a6300.jpg)
+
+Figure 2: (a) Log-likelihood of GAN-50, under different choices of variance parameter. (b) Log-likelihood of GMMN-10 on 100 simulated examples evaluated by AIS and KDE vs. the corresponding running time. We show the BDMC gap converges to almost zero as we increase the running time. (c) Log-likelihood of IWAE on 10,000 test examples evaluated by AIS and IWAE bound vs. running time. (a), (b) are results on continuous MNIST, and (c) is on binarized MNIST. Note that AIS/AIS $^ +$ encoder dominates the other estimate in both estimation accuracy and running time   
+Table 1: AIS vs. IWAE bound on 10,000 test examples of binarized MNIST. “# dist” denotes the number of intermediate distributions used for evalution. We find AIS estimate is consistently 1 nat higher than IWAE bound; AIS+encoder can achieve about the same estimate as AIS, but with 1 order of magnitude less number of intermediate distributions.   
+
+<table><tr><td rowspan=2 colspan=2>(Nats)IWAE</td><td rowspan=1 colspan=1>AIS</td><td rowspan=1 colspan=1>AIS+encoder</td><td rowspan=1 colspan=1>IWAE bound</td><td rowspan=1 colspan=1># dist AIS</td><td rowspan=1 colspan=1># dist AIS+encoder</td><td rowspan=1 colspan=1># samples</td></tr><tr><td rowspan=1 colspan=1>IWAE</td><td rowspan=1 colspan=1>-85.679-85.619</td><td rowspan=1 colspan=1>-85.754-85.621</td><td rowspan=1 colspan=1>-86.902-86.464</td><td rowspan=1 colspan=1>100010000</td><td rowspan=1 colspan=1>1001000</td><td rowspan=1 colspan=1>10000100000</td></tr></table>
+
+KDE is not meant to be applied in high-dimensional spaces and that the results might therefore be inaccurate. Nevertheless, KDE remains the standard protocol for evaluating decoder-based models. We analyzed the accuracy of the KDE estimates by comparing against AIS. Both estimates are stochastic lower bounds on the true log-likelihood (see Section 3), so larger values are guaranteed (with high probability) to be more accurate.
+
+For each estimator, we varied one parameter influencing the computational budget; for AIS, this was the number of intermediate distributions (chosen from $\{ 1 0 0 , 5 \bar { 0 } 0 , 1 0 0 0 , 2 0 0 0 , \bar { 1 } 0 0 0 0 \} )$ , and for KDE, it was the number of samples (chosen from $\{ 1 0 0 0 0 , 1 0 0 0 0 0 , 5 0 0 0 0 0 , 1 0 0 0 0 0 0 , 2 0 0 0 0 0 \} )$ . Using GMMN-10 for illustration, we plot both log-likelihood estimates 100 simulated examples as a function of evaluation time in Fig. 2(b). We also plot the upper bound of likelihood given by running AIS in reverse direction. We see that the BDMC gap approaches to zero, validating the accuracy of AIS. We also see that the AIS estimator achieves much more accurate estimates during similar evaluation time. Furthermore, the KDE estimates appear to level off, suggesting one cannot obtain accurate results even using orders of magnitude more samples.
+
+The KDE estimation error also impacts the estimate of the observation noise $\sigma$ , since a large value of $\sigma$ is needed for the samples to cover the full distribution. We compared the log-likelihoods estimated by AIS and KDE with varying choices of $\sigma$ on 100 training and validation examples of MNIST. We used 1 million simulated samples for KDE evaluation, which takes almost the same time as running AIS estimation. In Fig. 2(a), we show the log-likelihood of GAN-50 estimated by KDE and AIS as a function of $\sigma$ . Because the accuracy of KDE declines sharply for small $\sigma$ values, it creates a strong bias towards large $\sigma$ .
+
+# 5.3.3 HOW ACCURATE IS THE IWAE BOUND?
+
+In principle, one could estimate VAE likelihoods using the VAE objective function (which is a lower bound on the true log-likelihood). However, it is more common to use importance weighting, where the proposal distribution is computed by the recognition network. This is provably more accurate than the VAE bound (Burda et al., 2016). Because the importance weighted estimate corresponds to the objective function used by the Importance Weighted Autoencoder (IWAE) (Burda et al., 2016), we will refer to it as the IWAE bound.
+
+On continuous MNIST, the IWAE bound underestimated the true log-likelihoods by at least 33.2 nats on the training set and 187.4 nats on the test set. While this is considerably more accurate than KDE, the error is still significant. Interestingly, this result also suggests that the recognition network overfits the training data.
+
+<table><tr><td>(Nats)</td><td>AIS Test</td><td>AISTrain</td><td>BDMC gap</td><td>KDE Test</td><td>IWAE Test</td></tr><tr><td>VAE-50</td><td>991.435±6.477</td><td>1298.830±0.863</td><td>1.540</td><td>351.213</td><td>826.325</td></tr><tr><td>GAN-50</td><td>627.297±8.813</td><td>648.283±21.115</td><td>10.045</td><td>300.331</td><td>/</td></tr><tr><td>GMMN-50</td><td>593.472±8.591</td><td>607.272±1.451</td><td>1.146</td><td>277.193</td><td>/</td></tr><tr><td>VAE-10</td><td>705.375±7.411</td><td>791.029±0.810</td><td>0.832</td><td>408.659</td><td>486.466</td></tr><tr><td>GAN-10</td><td>328.772±5.538</td><td>346.640±4.260</td><td>0.934</td><td>259.673</td><td>/</td></tr><tr><td>GMMN-10</td><td>346.679±5.860</td><td>358.943±6.485</td><td>0.605</td><td>262.73</td><td>/</td></tr></table>
+
+Table 2: Model comparisons on 1000 test and training examples of continuous MNIST. Confidence intervals reflect the variability from the choice of training or test examples (which appears to be the dominant source of error for the AIS values). AIS, KDE, and IWAE are all stochastic lower bounds on the log-likelihood.
+
+Since VAE and IWAE results have customarily been reported on binarized MNIST, we additionally trained an IWAE in this setting. The training details are given in Appendix A.2. To show the practicality of our method, we evaluated the IWAE on the full 10000 test using AIS and IWAE bound, with different choices of intermediate distribution and number of simulated samples, shown in Table 1. We also evaluate AIS with the initial distribution defined by encoders of VAEs, denoted as AIS+encoder. We find that the IWAE bound underestimates the true value by at least 1 nat, which is a large difference by the standards of binarized MNIST. (E.g., it represents about half of the gap between a state-of-the-art permutation-invariant model (Tran et al., 2016) and one which exploits structure (van den Oord et al., 2016).) The AIS and IWAE estimates are compared in terms of evaluation time in Fig. 2 (c).
+
+# 5.4 SCIENTIFIC FINDINGS
+
+Having validated the accuracy of AIS, we now use it to analyze the effectiveness of various training criteria. We also highlight phenomena which would not be observable using existing log-likelihood estimators or by inspecting samples. For all experiments in this section, we used 10,000 intermediate distributions for AIS, 1 million simulated samples for KDE, and 200,000 importance samples for the IWAE bound. (These settings resulted in similar computation time for all three estimators.)
+
+# 5.4.1 MODEL LIKELIHOOD COMPARISON
+
+We evaluated the trained models using AIS and KDE on 1000 test examples of MNIST; results are shown in Table 2. We find that for all three training criteria, the larger architectures consistently outperformed the smaller ones. We also find that for both the 10- and 50-dimensional architectures, the VAEs achieved substantially higher log-likelihoods than GANs or GMMNs. It is not surprising that the VAEs achieved higher likelihood, because they were trained using a likelihood-based objective while the GANs and GMMNs were not. However, it is interesting that the difference in log-likelihoods was so large; in the rest of this section, we attempt to analyze what exactly is causing this large difference.
+
+We note that the KDE errors were of the same order of magnitude as the differences between models, indicating that it cannot be used reliably to compare log-likelihoods. Furthermore, KDE did not identify the correct ordering of models; for instance, it estimated a lower log-likelihood for VAE50 than for VAE-10, even though its true log-likelihood was almost 300 nats higher. KDE also underestimated by an order of magnitude the log-likelihood improvements that resulted from using the larger architectures. (E.g., it estimated a 15 nat difference between GMMN-10 and GMMN-50, even though the true difference was 247 nats as estimated by AIS.)
+
+These differences are also hard to observe simply by looking at samples; for instance, we were unable to visually distinguish the quality of samples for GAN-10 and GAN-50 (see Fig. 1), even though their log-likelihoods differed by almost 300 nats on both the training and test sets.
+
+# 5.4.2 MEASURING THE DEGREE OF OVERFITTING
+
+One question that arises in evaluation of decoder-based generative models is whether they memorize parts of the training dataset. One cannot test this by looking only at model samples. The commonly reported nearest-neighbors from the training set can be misleading (Theis et al., 2016), and interpolation in the latent space between different samples can be visually appealing, but does not provide a quantitative measure of the degree of generalization.
+
+![](images/1d90aedac8cf9ebef7190dc2d2bf71283ad6a43386db9195176d7f1f571d15ab.jpg)  
+Figure 3: Training curves for (a) GAN-50, (b) VAE-50, and (c) GMMN-10, as measured by AIS, KDE, and (if applicable) the IWAE lower bound. All estimates shown here are lower bounds. In (c), the gap between training and validation log-likelihoods is not fairly small (see Table 2).
+
+To analyze the degree of overfitting, Fig. 3 shows training curves for three networks as measured by AIS, KDE, and the IWAE bound. We observe that GAN-50’s training and test log-likelihoods are nearly identical throughout training, disconfirming the hypothesis that it was memorizing training data. Both GAN-50 and GMMN-50 overfit less than VAE-50.
+
+We also observed two phenomena which could not be measured using existing techniques. First, in the case of VAE-50, the IWAE lower bound starts to decline after 200 epochs, while the AIS estimates hold steady, suggesting it is the recognition network rather than the generative network which is overfitting most. Second, the GMMN-50 training and validation error continue to improve at 10,000 epochs, even though KDE erroneously indicates that performance has leveled off.
+
+# 5.4.3 HOW APPROPRIATE IS THE OBSERVATION MODEL?
+
+Appendix B addresses the questions of whether the spherical Gaussian observation model is a good fit and whether the log-likelihood differences could be an artifact of the observation model. We find that all of the models can be substantially improved by accounting for non-Gaussianity, but that this effect is insufficient to explain the gap between the VAEs and the other models.
+
+# 5.4.4 ARE THE NETWORKS MISSING MODES?
+
+It was previously observed that one of the potential failure modes of Boltzmann machines is to fail to generate one or more modes of a distribution or to drastically misallocate probability mass between modes (Salakhutdinov & Murray, 2008). Here we analyze this for decoder-based models.
+
+First, we ask a coarse-grained version of this question: do the networks allocate probability mass correctly between the 10 digit classes, and if not, can this explain the difference in log-likelihood scores? In Fig. 1, we see that GAN-50’s distribution of digit classes was heavily skewed: out of 100 samples, it generated 37 images of 1’s, but only a single 2. This appears to be a large effect, but it does not explain the magnitude of the log-likelihood difference from VAEs. In particular, if the allocation of digit classes were off by a factor of 10, this effect by itself could cost at most l $\mathrm { 9 g 1 0 \approx 2 . 3 }$ nats of log-likelihood. Since VAE-50 outperformed GAN-50 by 364 nats, this effect cannot explain the difference.
+
+However, MNIST has many factors of variability beyond simply the 10 digit classes. In order to determine whether any of the models missed more fine-grained modes, we visualized posterior samples for each model conditioned on training and test images. In particular, for each image $x$ under consideration, we used AIS to approximately sample $z$ from the posterior distribution $p ( \bar { z } | x )$ , and then ran the decoder on $z$ . While these samples are approximate, Grosse et al. (2016) point out that the BDMC gap also bounds the KL divergence of approximate samples from the true posterior. With the exception of GAN-50, our BDMC gaps were on the order of 1 nat, suggesting our approximate posterior samples are fairly representative. The results are shown in Fig. 4. Further posterior visualizations for digit class 2 (the most difficult for the models we considered) are shown in Appendix C.
+
+Both VAEs’ posterior samples match the observations almost perfectly. (We observed a few poorly reconstructed examples on the test set, but not on the training set.) The GANs and GMMNs fail to reconstruct some of the examples on both the training and validation sets, suggesting that they failed to learn some modes of the distribution.
+
+![](images/278f1d9bc0c8cf3988cf3078c5b37ece1ef0b552aa500512c17b223cfaf6956e.jpg)  
+Figure 4: (a) and (b) show visualization of posterior samples of 10 training/validation examples. (c) shows visualization of posterior samples of 10 training examples of digit “2". Each column of 10 digits comes from true data and the six models. The order of visualization is: True data, GAN-10, VAE-10, GMMN-10, GAN-50, VAE-50, GMMN-50.
+
+# ACKNOWLEDGMENTS
+
+We like to thank Yujia Li for providing his original GMMN model and codebase, and thank Jimmy Ba for advice on training GANs. Ruslan Salakhutdinov is supported in part by Disney and ONR grant N000141310721. We also thank the developers of Lasagne (Battenberg et al., 2014) and Theano (Al-Rfou et al., 2016).
+
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+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 2672–2680. Curran Associates, Inc., 2014. URL http://papers.nips. cc/paper/5423-generative-adversarial-nets.pdf.
+
+Roger Grosse, Siddharth Ancha, and Daniel M. Roy. Measuring the reliability of MCMC inference with bidirectional Monte Carlo. In NIPS, 2016.
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+Roger B. Grosse, Zoubin Ghahramani, and Ryan P. Adams. Sandwiching the marginal likelihood using bidirectional monte carlo. arXiv preprint arXiv:1511.02543, 2015.
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+Daniel Jiwoong Im, Chris Dongjoo Kim, Hui Jiang, and Roland Memisevic. Generating images with recurrent adversarial networks. arXiv preprint arXiv:1602.05110, 2016.
+
+Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43:169–188, 1986.
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+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In ICLR, 2014.
+
+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, November 1998.
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+Yujia Li and Kevin Swersky. Generative moment matching networks. In In ICML 32, 2015.
+
+D. J. C. MacKay and M. N. Gibbs. Density networks. In J. W. Kay and D. M. Titterington (eds.), Statistics and Neural Networks, pp. 129–146. O.U.P., 1998.
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+Radford M. Neal. Annealed importance sampling. Statistics and Computing, 11(2):125–139, April 2001. ISSN 0960-3174. doi: 10.1023/A:1008923215028. URL http://dx.doi.org/10. 1023/A:1008923215028.
+
+Radford M. Neal. MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo, 54:113–162, 2010.
+
+Emanuel Parzen. On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33(3):pp. 1065–1076, 1962. ISSN 00034851. URL http://www.jstor. org/stable/2237880.
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+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016.
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+Marc’Aurelio Ranzato, Alex Krizhevsky, and Geoffrey E Hinton. Factored 3-way restricted Boltzmann machines for modeling natural images. In International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 621–628, 2010.
+
+Danilo J. Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Tony Jebara and Eric P. Xing (eds.), Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 1278–1286. JMLR Workshop and Conference Proceedings, 2014. URL http://jmlr.org/proceedings/ papers/v32/rezende14.pdf.
+
+Ruslan Salakhutdinov and Iain Murray. On the quantitative analysis of Deep Belief Networks. In Andrew McCallum and Sam Roweis (eds.), Proceedings of the 25th Annual International Conference on Machine Learning (ICML 2008), pp. 872–879. Omnipress, 2008.
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+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In NIPS, 2016.
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+Lucas Theis, Aäron van den Oord, and Matthias Bethge. A note on the evaluation of generative models. In ICLR, 2016.
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+Dustin Tran, Rajesh Ranganath, and David M. Blei. The variational Gaussian process. In ICLR, 2016.
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+Benigno Uria, Iain Murray, and Hugo Larochelle. RNADE: The real-valued neural autoregressive density-estimator. In Advances in Neural Information Processing Systems 26, pp. 2175–2183. 2013. URL http://www.benignouria.com/en/research/papers/Uria2013.pdf.
+
+Aäron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In ICML, 2016.
+
+Martin J. Wainwright and Eero P. Simoncelli. Scale mixtures of Gaussians and the statistics of natural images. In NIPS, 1999.
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+Jun-Yan Zhu, Philipp Krähenbühl, Eli Shechtman, and Alexei A. Efros. Generative visual manipulation on the natural image manifold. In Proceedings of European Conference on Computer Vision (ECCV), 2016.
+
+# A NETWORK ARCHITECTURES/TRAINING
+
+# A.1 MODELS ON CONTINUOUS MNIST
+
+The decoders have all fully connected layers, and the number of units in each layer was 10–64–256– 256-1024–784 for the smaller architecture and 50–1024–1024–1024–784 for the larger one. Other architecture details are summarized as follows.
+
+• For GAN-10, we used a discriminator with the architecture 784-512-256-1, where each layer used dropout with parameter 0.5. For GAN-50, we used a discriminator with architecture 784-4096-4096-4096-4096-1. All hidden layers used dropout with parameter 0.8. All hidden layers in both networks used the tanh activation function, and the output layers used the logistic function. The larger model uses an encoder of an architecture 784-1024-1024-1024-100. We add dropout layer between each hidden layer, with a dropout rate of 0.2. The smaller model uses an encoder of an architecture 784-256-64-20. Generator’s hidden layers use tanh activation function, and the output layer uses sigmoid unit. Encoder’s hidden layers use tanh activation function, and the output layer uses linear activation.   
+• GMMN: The hidden layers use ReLU activation function, and the output layer uses sigomid unit.
+
+For training GAN/VAE, we use our own implementation. We use Adam for optimization, and perform grid search of learning rate from $\{ 0 . 0 0 1 , 0 . 0 0 0 1 , 0 . 0 0 0 0 1 \}$ . For training GMMN, we take the implementation from https://github.com/yujiali/gmmn.git. Following the implementation, we use SGD with momentum for optimization, and perform grid search of learning rate from $\lbrace 0 . 1 , 0 . 5 , 1 , 2 \rbrace$ , with momentum 0.9.
+
+# A.2 MODELS ON BINARIZED MNIST
+
+Its decoder has the architecture 50-200-200-784 with all tanh hidden layers and sigmoid output layer, and its encoder is symmetric in architecture, with linear output layer. We take the implementation at https://github.com/yburda/iwae.git for training the IWAE model.The IWAE bound was computed with 50 samples during training. We keep all the hyperparameter choices the same as in the implementation.
+
+B HOW PROBLEMATIC IS THE GAUSSIAN OBSERVATION MODEL?   
+Table 3: Optimal variance vs. Fixed variance   
+
+<table><tr><td>(Nats)</td><td colspan="3">Train</td><td colspan="3">Valid</td></tr><tr><td></td><td>Optimal</td><td>Fixed</td><td>Improvement</td><td>Optimal</td><td>Fixed</td><td>Improvement</td></tr><tr><td>GAN-50</td><td>711.405</td><td>620.498</td><td>90.907</td><td>702.699</td><td>623.492</td><td>79.207</td></tr><tr><td>GMMN-50</td><td>655.807</td><td>571.803</td><td>84.004</td><td>661.652</td><td>594.612</td><td>67.040</td></tr><tr><td>GAN-10</td><td>376.788</td><td>318.948</td><td>57.840</td><td>368.585</td><td>316.614</td><td>51.971</td></tr><tr><td>GMMN-10</td><td>393.976</td><td>345.177</td><td>48.799</td><td>371.325</td><td>332.360</td><td>38.965</td></tr></table>
+
+In this section, we consider whether the difference in log-likelihood between models could be an artifact of the Gaussian noise model (which we know to be a poor fit). In principle, the Gaussian noise assumption could be unfair to the GANs and GMMNs, because the VAE training uses the correct observation model, while the GAN and GMMN objectives do not have any particular observation model built in.
+
+To determine the size of this effect, we evaluated the models under a different regime where, instead of choosing a fixed value of the observation noise $\sigma$ on a validation set, $\sigma$ was tuned independently for each example.2 This is not a proper generative model, but it can be viewed as an upper bound on the log-likelihood that would be achievable with a heavy-tailed and radially symmetric noise model.3 Results are shown in Table 3. We see that adapting $\sigma$ for each example results in a log-likelihood improvement between 30 and 100 nats for all of the networks. In general, the examples which show the largest performance jump are images of 1’s (which prefer smaller $\sigma$ ) and 2’s (which prefer larger $\sigma _ { . }$ ). This is a significant effect, and suggests that one could significantly improve the log-likelihood scores by picking a better observation model. However, this effect is smaller in magnitude than the differences between VAE and GAN/GMMN log-likelihoods, so it fails to explain the likelihood difference.
+
+# C POSTERIOR VISUALIZATION OF DIGIT “2"
+
+According to the log-likelihood evaluation, we find digit “2" is the hardest digit for modelling. In this section we investigate the quality of modelling $" 2 "$ of each model. We randomly sampled a fixed set of 100 samples of digit “2" from training data and compare whether model capture this mode. We show the plots of “2" for GAN-10, GAN-50, VAE-10 and true data in the following figures for illustration. We see that GAN-10 fails at capturing many instances of digit “2" in the training data! We see instead of generating $" 2 "$ , it tries to generate digit “1", “7" “9", “4", “8" from reconstruction. GAN-50 does much better, its reconstruction are all digit “2" and there is only some style difference from the true data. VAE-10 totally dominates this competition, where it perfectly reconstructs all the samples of digit “2". We emphasize if directly sampling from each model, samples look visually indistinguishable (see Fig. 1), but we can clearly see differences in posterior samples.
+
+![](images/88c8505b1c7a133c4cd7e4e0394f445680a5f92fc9b45f6da2b5c004c592c196.jpg)  
+Figure 5: Posterior samples of digit “2" for GAN-10.
+
+![](images/fa33a858f8d884b0f849fcbc7b769618396fd345f19cf5ffa8a53b83c11284f2.jpg)  
+Figure 6: Posterior samples of digit “2" for GAN-50.
+
+![](images/c441ffa6076e604d1acaba867f49898ce2211f0fad5cb5d3a733afee18cc6aef.jpg)  
+Figure 7: Posterior samples of digit “2" for VAE-10.
+
+![](images/1de2e6c0013fbeaec9a869eeb2d58370872a4aef70a5dcec5670c18cacf35204.jpg)  
+Figure 8: 100 digit “2" from training data.

parse/train/B1M8JF9xx/B1M8JF9xx_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ON THE QUANTITATIVE ANALYSIS OF DECODERBASED GENERATIVE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Yuri Burda   \nOpenAI   \nyburda@openai.com ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yuhuai Wu Department of Computer Science University of Toronto ywu@cs.toronto.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ruslan Salakhutdinov School of Computer Science Carnegie Mellon University rsalakhu@cs.cmu.edu ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Roger Grosse Department of Computer Science University of Toronto rgrosse@cs.toronto.edu ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            340,
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+            354
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The past several years have seen remarkable progress in generative models which produce convincing samples of images and other modalities. A shared component of many powerful generative models is a decoder network, a parametric deep neural net that defines a generative distribution. Examples include variational autoencoders, generative adversarial networks, and generative moment matching networks. Unfortunately, it can be difficult to quantify the performance of these models because of the intractability of log-likelihood estimation, and inspecting samples can be misleading. We propose to use Annealed Importance Sampling for evaluating log-likelihoods for decoder-based models and validate its accuracy using bidirectional Monte Carlo. The evaluation code is provided at https:// github.com/tonywu95/eval_gen. Using this technique, we analyze the performance of decoder-based models, the effectiveness of existing log-likelihood estimators, the degree of overfitting, and the degree to which these models miss important modes of the data distribution. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            595,
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+            611
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, deep generative models have dramatically pushed forward the state-of-the-art in generative modelling by generating convincing samples of images (Radford et al., 2016), achieving state-of-the-art semi-supervised learning results (Salimans et al., 2016), and enabling automatic image manipulation (Zhu et al., 2016). Many of the most successful approaches are defined in terms of a process which samples latent variables from a simple fixed distribution (such as Gaussian or uniform) and then applies a learned deterministic mapping which we will refer to as a decoder network. Important examples include variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014), generative adversarial networks (GANs) (Goodfellow et al., 2014), generative moment matching networks (GMMNs) (Li & Swersky, 2015; Dziugaite et al., 2015), and nonlinear independent components estimation (Dinh et al., 2014). We refer to this set of models collectively as decoder-based models, also known as density networks (MacKay & Gibbs, 1998). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While many decoder-based models are able to produce convincing samples (Denton et al., 2015; Radford et al., 2016), rigorous evaluation remains a challenge. Comparing models by inspecting samples is labor-intensive, and potentially misleading (Theis et al., 2016). While alternative quantitative criteria have been proposed (Bounliphone et al., 2016; Im et al., 2016; Salimans et al., 2016), log-likelihood of held-out test data remains one of the most important measures of a generative model’s performance. Unfortunately, unless the decoder is designed to be reversible (Dinh et al., 2014; 2016), log-likelihood estimation in decoder-based models is typically intractable. In the case of VAE-based models, a learned encoder network gives a tractable lower bound, but for GANs and GMMNs it is not obvious how even to compute a good lower bound. Even when lower bounds are available, their accuracy may be hard to determine. Because of the difficulty of log-likelihood evaluation, it is hard to answer basic questions such as whether the networks are simply memorizing training examples, or whether they are missing important modes of the data distribution. ",
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+    {
+        "type": "image",
+        "img_path": "images/abdb08b1a142504877599e275d9dd7235f7c6fed6aafaaa4479232bccf4e6298.jpg",
+        "image_caption": [
+            "Figure 1: (a) samples from a GAN with 10 latent dimensions, (b) and (c) samples from a GAN with 50 latent dimensions at different epochs of training. While it is difficult to visually discern differences between these three models, their log-likelihood (LLD) values span almost 300 nats. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The most widely used estimator of log-likelihood for GANs and GMMNs is the Kernel Density Estimator (KDE) (Parzen, 1962). It estimates the likelihood under an approximation to the model’s distribution obtained by simulating from the model and convolving the set of samples with a kernel (typically Gaussian). Unfortunately, KDE is notoriously inaccurate for estimating likelihood in high dimensions, because it is hard to tile a high-dimensional manifold with spherical Gaussians (Theis et al., 2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose to use annealed importance sampling (AIS; (Neal, 2001)) to estimate log-likelihoods of decoder-based generative models and to obtain approximate posterior samples. Importantly, we validate this approach using Bidirectional Monte Carlo (BDMC) (Grosse et al., 2015), which provably bounds the log-likelihood estimation error and the KL divergence from the true posterior distribution for data simulated from a model. For most models we consider, we find that AIS is two orders of magnitude more accurate than KDE, and is accurate enough to perform fine-grained comparisons between generative models. In the case of VAEs, we show that AIS can be further sped up by using the recognition network to determine the initial distribution; this yields an estimator which is fast enough to be run repeatedly during training. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Using the proposed method, we analyze several scientific questions central to understanding decoderbased generative models. First, we measure the accuracy of KDE and of the importance weighting bound which is commonly used to evaluate VAEs. We find that the KDE error is larger than the (quite significant) log-likelihood differences between different models, and that KDE can lead to misleading conclusions. The importance weighted bound, while reasonably accurate, can also yield misleading results in some cases. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Second, we compare the log-likelihoods of VAEs, GANs, and GMMNs, and find that VAEs achieve log-likelihoods several hundred nats higher than the other models (even though KDE considers all three models to have roughly the same log-likelihood). Third, we analyze the degree of overfitting in VAEs, GANs, and GMMNs. Contrary to a commonly proposed hypothesis, we find that GANs and GMMNs are not simply memorizing their training data; in fact, their log-likelihood gaps between training and test data are much smaller relative to comparably-sized VAEs. Finally, by visualizing (approximate) posterior samples obtained from AIS, we observe that GANs miss important modes of the data distribution, even ones which are represented in the training data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We emphasize that none of the above phenomena can be measured using KDE or the importance weighted bound, or by inspecting samples. (See Fig. 1 for an example where it is tricky to compare models based on samples.) While log-likelihood is by no means a perfect measure, we find that the ability to accurately estimate log-likelihoods of decoder-based models yields crucial insight into their behavior and suggests directions for improving them. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "2.1 DECODER-BASED GENERATIVE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In generative modelling, a decoder network is often used to define a generative distribution by transforming samples from some simple distribution (e.g. normal) to the data manifold. In this ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "paper, we consider three kinds of decoder-based generative models: Variational Autoencoder (VAE) (Kingma & Welling, 2014), Generative Adversarial Network (GAN) (Goodfellow et al., 2014), and Generative Moment Matching Network (GMMN) (Li & Swersky, 2015; Dziugaite et al., 2015). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1.1 VARIATIONAL AUTOENCODER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A variational autoencoder (VAE) (Kingma & Welling, 2014) is a probabilistic directed graphical model. It is defined by a joint distribution over a set of latent random variables $z$ and the observed variables $x$ $: p ( x , z ) = p ( x | z ) p ( z )$ . The prior over the latent random variables, $p ( z )$ , is usually chosen to be a standard Gaussian distribution. The data likelihood $p ( x | z )$ is usually a Gaussian or Bernoulli distribution whose parameters depend on $z$ through a deep neural network, known as the decoder network. It also uses an approximate inference model called an encoder or recognition network, that serves as a variational approximation $q ( z | x )$ to the posterior $p ( z | x )$ . The decoder network and the encoder networks are jointly trained to maximize the evidence lower bound (ELBO): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad69188252821eeae81e7092212f0c22a948a83ab7b9dffe0c3fd4ccbbbdea21.jpg",
+        "text": "$$\n\\log p ( x ) \\geq \\mathbb { E } _ { q ( z | x ) } [ \\log p ( x | z ) ] - K L ( q ( z | x ) | | p ( z ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition, the reparametrization trick is used to reduce the variance of the gradient estimate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1.2 GENERATIVE ADVERSARIAL NETWORK (GAN) ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+    },
+    {
+        "type": "text",
+        "text": "A generative adversarial network (GAN) (Goodfellow et al., 2014) is a generative model trained by a game between a decoder network and a discriminator network. It defines the generative model by sampling the latent variable $z$ from some simple prior distribution $p ( z )$ (e.g., Gaussian) followed through the decoder network. The discriminator network $D ( \\cdot )$ outputs a probability of a given sample coming from the data distribution. Its task is to distinguish samples from the generator distribution from real data. The decoder network, on the other hand, tries to produce samples as realistic as possible, in order to fool the discriminator into accepting its outputs as being real. The competition between the two networks results in the following minimax problem: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/21d86ccafdf6b9d93ba7147673667ba2e0a3e5827d364c14191e89e629216e19.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { G } \\operatorname* { m a x } _ { D } \\mathbb { E } _ { x \\sim p _ { d a t a } } [ \\log D ( x ) ] + \\mathbb { E } _ { z \\sim p ( z ) } [ \\log ( 1 - D ( G ( z ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unlike VAE, the objective is not explicitly related to the log-likelihood of the data. Moreover, the generative distribution is a deterministic mapping, i.e., $\\bar { p } ( x | z )$ is a Dirac delta distribution, parametrized by the deterministic decoder. This can make data likelihood ill-defined, as the probability density of any particular point $x$ can be either infinite, or exactly zero. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1.3 GENERATIVE MOMENT MATCHING NETWORK (GMMN) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Generative moment matching networks (GMMNs) (Li & Swersky, 2015; Dziugaite et al., 2015) adopt maximum mean discrepancy (MMD) as the training objective, a moment matching criterion where kernel mean embedding techniques are used to avoid unnecessary assumptions of the distributions. It has the same issue as GAN in that the log-likelihood is undefined. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2.2 ANNEALED IMPORTANCE SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We are interested in estimating the probability $\\begin{array} { r } { p ( x ) = \\int p ( z ) p ( x | z ) \\mathrm { d } z } \\end{array}$ a model assigns to an observation $x$ . This is equivalent to computing the normalizing constant of the unnormalized distribution $f ( z ) ~ = ~ p ( z , { \\bar { x } } )$ . One naïve approach is likelihood weighting, where one samples $\\{ z ^ { ( k ) } \\} _ { k = 1 } ^ { K } \\sim p ( z )$ and averages the conditional likelihoods $p ( x | z ^ { ( k ) } )$ . This is justified by the following identity: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c126581e543ea475c0109189b3ab2d66f46ea1f27952e511513c1868226d43b7.jpg",
+        "text": "$$\np ( x ) = \\int \\frac { p ( x , z ) } { p ( z ) } p ( z ) \\mathrm { d } z = \\mathbb { E } _ { z \\sim p ( z ) } [ p ( x | z ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Likelihood weighting can be viewed as simple importance sampling, where the proposal distribution is the prior $p ( z )$ and the target distribution is the posterior $p ( z | x )$ . Unfortunately, importance sampling works well only when the proposal distribution is a good match for the target distribution. For the models considered in this paper, the (very broad) prior can be drastically different than the (highly concentrated) posterior, leading to inaccurate estimates of the likelihood. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Annealed importance sampling (AIS; Neal, 2001) is a Monte Carlo algorithm commonly used to estimate (ratios of) normalizing constants. Roughly speaking, it computes a sequence of importance sampling based estimates, each of which is stable because it involves two distributions which are very similar. In particular, suppose one is interested in estimating the normalizing constant $\\begin{array} { r } { \\mathcal { Z } = \\int f ( \\boldsymbol { z } ) \\mathrm { d } \\dot { \\boldsymbol { z } } } \\end{array}$ of an unnormalized distribution $f ( z )$ . (In the likelihood estimation setting, $f ( z ) = p ( z , x )$ and ${ \\mathcal { Z } } = p ( x )$ .) One must specify a sequence of distributions $q _ { 1 } , . . . , q _ { T }$ , where $q _ { t } = f _ { t } / Z _ { t }$ , and $f _ { T } = f$ is the target distribution. It is required that one can obtain one or more exact samples from the initial distribution $q _ { 1 }$ . One must also specify a sequence of reversible MCMC transition operators $\\mathcal { T } _ { 1 } , . . . , \\mathcal { T } _ { T }$ , where $\\mathcal { T } _ { t }$ leaves $q _ { t }$ invariant. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "AIS produces a (nonnegative) unbiased estimate of the ratio $\\mathcal { Z } _ { T } / \\mathcal { Z } _ { 1 }$ as follows: first, we sample a random initial state $z _ { 1 } \\sim q _ { 1 }$ and set the initial weight $w _ { 1 } = 1$ . For every stage $t \\geq 2$ we update the weight $w$ and sample the state $z _ { t }$ according to ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f48af92756204d4e075dea4ea95c890a9339c504573fd6343dc389eb3632202e.jpg",
+        "text": "$$\nw _ { t }  w _ { t - 1 } \\frac { f _ { t } ( z _ { t - 1 } ) } { f _ { t - 1 } ( z _ { t - 1 } ) } \\qquad z _ { t } \\sim \\mathcal { T } _ { t } ( z | z _ { t - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As demonstrated by Neal (2001), this procedure produces a nonnegative weight $w _ { T }$ such that $\\mathbb { E } [ w _ { T } ] = \\mathcal { Z } _ { T } / \\mathcal { Z } _ { 1 }$ . Typically, btains the un $\\mathcal { Z } _ { 1 }$ is known,ed estimate le independent AIS weights. In the likelihood estimation $\\{ w _ { T } ^ { ( K ) } \\} _ { k = 1 } ^ { K }$ $\\begin{array} { r } { \\hat { \\mathcal { Z } } _ { T } = \\mathcal { Z } _ { 1 } \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } w _ { T } ^ { ( K ) } } \\end{array}$   \nsetting, $\\mathcal { Z } _ { 1 } = 1$ and , so we denote this estimator as ${ \\hat { p } } ( x )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Typically, the unnormalized intermediate distributions are simply defined to be geometric averages $\\dot { f _ { t } ( z ) } = \\dot { \\sp { \\prime } } f _ { 1 } ( z ) \\sp { 1 - \\beta _ { t } } f _ { T } ( z ) \\sp { \\beta _ { t } }$ , where the $\\beta _ { t }$ are monotonically increasing parameters with $\\beta _ { 1 } = 0$ and $\\beta _ { T } = 1$ . For $f _ { 1 } ( z ) = p ( z )$ and $f _ { T } ( z ) = p ( z , x )$ , this gives ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e77948b2890a68041a9dbc6305754e5eca68281a75c8586defda2e39df30cc9.jpg",
+        "text": "$$\nf _ { t } ( z ) = p ( z ) p ( x | z ) ^ { \\beta _ { t } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown by Neal (2001), under certain regularity conditions, the variance of $\\hat { \\mathcal { Z } } _ { T }$ tends to zero as the number of intermediate distributions is increased. AIS is very effective in practice, and has been used to estimate normalizing constants of complex high-dimensional distributions (Salakhutdinov $\\&$ Murray, 2008). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 BIDIRECTIONAL MONTE CARLO ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "AIS provides a nonnegative unbiased estimate ${ \\hat { p } } ( x )$ of $p ( x )$ . However, it is often more practical to estimate $p ( x )$ in the log space, i.e. $\\log p ( x )$ , because of underflow problem of dealing with many products of probability measure. In general, we note that logarithm of a nonnegative unbiased estimate is a stochastic lower bound of the log estimand (Grosse et al., 2015). In particular, $\\log { \\hat { p } } ( x )$ is a stochastic lower bound on $\\log p ( x )$ , satisfying $\\mathbb { E } [ \\log { \\hat { p } ( x ) } ] \\leq \\log p ( x )$ and $\\operatorname* { P r } ( \\log { \\hat { p } } ( x ) > \\log p ( x ) +$ $b ) < e ^ { - b }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Grosse et al. (2015) pointed out that if AIS is run in reverse starting from an exact posterior sample, it yields an unbiased estimate of $1 / p ( x )$ , which (by the above argument) can be seen as a stochastic upper bound on $\\log p ( x )$ . The combination of lower and upper bounds from forward and reverse AIS is known as bidirectional Monte Carlo (BDMC). In many cases, the combination of bounds can pinpoint the true value quite precisely. While posterior sampling is just as hard as log-likelihood estimation (Jerrum et al., 1986), in the case of log-likelihood estimation for simulated data, one has available a single exact posterior sample: the parameters and/or latent variables which generated the data. Because this trick is only applicable to simulated data, BDMC is most useful for measuring the accuracy of a log-likelihood estimator on simulated data. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Grosse et al. (2016) observed that BDMC can also be used to validate posterior inference algorithms, as the gap between upper and lower bounds is itself a bound on the KL divergence of approximate samples from the true posterior distribution. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a given generative distribution $p ( x , z ) = p ( z ) p ( x | z )$ , our task is to measure the log-likelihood of test examples $\\log p ( x _ { t e s t } )$ . We first discuss how we define the generative distribution for decoderbased networks. For VAE, the generative distribution is defined in the standard way, where $p ( z )$ is a standard normal distribution and $p ( x | z )$ is a normal distribution parametrized by mean $\\mu _ { \\boldsymbol { \\theta } } ( z )$ and $\\sigma _ { \\theta } ( z )$ , predicted by the generator given the latent code. However, the observation distribution for GANs and GMMNs is typically taken to be a delta function, so that the model’s distribution covers only a submanifold of the space of observables. In order for the likelihood to be well-defined, we follow the same assumption made when evaluating using Kernel Density Estimator (Parzen, 1962): we assume a Gaussian observation model with a fixed variance hyperparameter $\\sigma ^ { 2 }$ . We will refer to the distribution defined by this Gaussian observation model as $p _ { \\sigma }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Observe that the KDE estimate is given by ",
+        "bbox": [
+            176,
+            166,
+            452,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8569563b808c6570b433f85b36ca95825e6e118b2f42230014d4011e611ceb90.jpg",
+        "text": "$$\n\\hat { p } _ { \\sigma } ( x ) = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } p _ { \\sigma } ( x | z ^ { ( k ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "wher e {z(k)}K are samples from the prior $p ( z )$ . This is equivalent to likelihood weighting for the distribution $p _ { \\sigma }$ , which is an instance of simple importance sampling (SIS). Because SIS is an unbiased estimator of the likelihood, $\\log \\hat { p } _ { \\sigma } ( x )$ is a stochastic lower bound on $\\log p _ { \\sigma } ( x )$ (Grosse et al., 2015). Unfortunately, SIS can result in very poor estimates when the evidence has low prior probability (i.e. the posterior is very dissimilar to the prior). This suggests that AIS might be able to yield much more accurate log-likelihood estimates under $p _ { \\sigma }$ . We note that KDE can be viewed as a special case of AIS where the number of intermediate distributions is set to 0. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now describe specifically how we carry out evaluation using AIS. In most of our experiments, we choose the initial distribution of AIS to be $p ( z )$ , the same prior distribution used in training decoderbased models. If the model provides an encoder network (e.g., VAE), we can take the approximated distribution predicted by the encoder $q ( z | x )$ as the initial distribution of the AIS chain. For continuous data, we define the unnormalized density of target distribution to be the joint generative distribution with the Gaussian noise model, $p _ { \\sigma } ( x , z ) = p _ { \\sigma } ( x | z ) p ( z )$ . For the small subset of experiments done on the binary data, we define the observation model to be a Bernoulli model with mean predicted by the decoder. Our intermediate distributions are geometric averages of the prior and posterior, as in Eqn. 5. Since all of our experiments are done using continuous latent space, we use Hamiltonian Monte Carlo (Neal, 2010) as the transition operator for sampling latent samples along annealing. The evaluation code is provided at https://github.com/tonywu95/eval_gen. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            518,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "AIS is known to be a powerful technique of estimating the partition function of the model. One influential example was the use of AIS to evaluate deep belief networks (Salakhutdinov & Murray, 2008). Although we used the same technique, the problem we consider is completely different. First of all, the model they consider is undirected graphical models, whereas decoder-based models are directed graphical models. Secondly, their model has a well-defined probabilistic density function in terms of energy function, whereas we need to consider different probabilistic model for one in which the the likelihood is ill-defined. In addition, we validate our estimates using BDMC. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theis et al. (2016) give an in-depth analysis of issues that might come up in evaluating generative models. They also point out that a model that completely fails at modelling the proportion of modes of the distribution might still achieve a high likelihood score. Salimans et al. (2016) propose an image-quality measure which they find to be highly correlated with human visual judgement. They propose to feed the samples $x$ of the model to the “inception” model to obtain a conditional label distribution $p ( y | x )$ , and evaluate the score defined by exp $\\mathbb { E } _ { x } \\mathrm { K L } ( p ( y | x ) | | p ( y ) )$ , which is motivated by having a low entropy of $p ( y | x )$ but a large entropy of $p ( y )$ . However, the measure is largely based on visual quality of the sample, and we argue that the visual quality can be a misleading way to evaluate a model. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "5.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "All of our experiments were performed on the MNIST dataset of images of handwritten digits (LeCun et al., 1998). For consistency with prior work on evaluating decoder-based models, most of our experiments used the continuous inputs. We dequantized the data following Uria et al. (2013), by adding a uniform noise of $\\scriptstyle { \\frac { 1 } { 2 5 6 } }$ to the data and rescaling it to be in $[ 0 , 1 ] ^ { D }$ after dequantization. We use the standard split of MNIST into 60,000 training and 10,000 test examples, and used 50,000 images from the training set for training, and remaining 10,000 images for validation. In addition, some of our experiments used the binarized MNIST dataset with a Bernoulli observation model (Salakhutdinov & Murray, 2008). ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For most of our experiments, we considered two decoder architectures: a small one with 10 latent dimensions, and a larger one with 50 latent dimensions. We use standard Normal distribution as prior for training all of our models. All layers were fully connected, and the number of units in each layer was 10–64–256–256-1024–784 for the smaller architecture and 50–1024–1024–1024–784 for the larger one. We trained both architectures using the VAE, GAN, and GMMN objectives, resulting in six networks which we refer to as VAE-10, VAE-50, etc. In general, the larger architecture performed substantially better on both the training and test sets, but we analyze the smaller architecture as well because it better highlights some of the differences between the training criteria. Additional architectural details are given in Appendix A.1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to enable a direct comparison between training criteria, all models used a spherical Gaussian observation model with fixed variance. This is consistent with previous protocols for evaluating GANs and GMMNs. However, we note that this observation model is a nontrivial constraint on the VAEs, which could instead be trained with a more flexible diagonal Gaussian observation model where the variances depend on the latent state. Such observation models can easily achieve much higher log-likelihood scores, for instance by noticing that boundary pixels are always close to 0. (E.g., we trained a VAE with the more general observation model which achieved a log-likelihood of at least 2200 nats on continuous MNIST.) Therefore, the log-likelihood values we report should not be compared directly against networks which have a more flexible observation model. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3 VALIDATION OF LOG-LIKELIHOOD ESTIMATES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Before we analyze the performance of the trained networks, we must first determine the accuracy of the log-likelihood estimators. In this section, we validate the accuracy of our AIS-based estimates using BDMC. We then analyze the error in the KDE and IWAE estimates and highlight some cases where these measures miss important phenomena. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3.1 VALIDATION OF AIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We used AIS to estimate log-likelihoods for all models under consideration. Except where otherwise specified, all AIS estimates were obtained using 16 independent chains, 10,000 intermediate distributions of the form in Eqn. 5, and a transition operator consisting of one proposed HMC trajectory with 10 leapfrog steps.1 Following Ranzato et al. (2010), the HMC stepsize was tuned to achieve an acceptance rate of 0.65 (as recommended by Neal (2010)). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For all six models, we evaluated the accuracy of this estimation procedure using BDMC on data sampled from the model’s distribution on 1000 simulated examples. The gap between the loglikelihood estimates produced by forward AIS (which gives a lower bound) and reverse AIS (which gives an upper bound) bounds the error of the AIS estimates on simulated data. We refer to this gap as the BDMC gap. For five of the six networks under consideration, we found the BDMC gap to be less than 1 nat. For the remaining model (GAN-50), the gap was about 10 nats. Both gaps are much smaller than our measured log-likelihood differences between models. If these gaps are representative of the true error in the estimates on the real data, then this indicates AIS is accurate enough to make fine-grained comparisons between models and to benchmark other log-likelihood estimators. (The BDMC gap is not guaranteed to hold for the real data, although Grosse et al. (2016) found the behavior of AIS to match closely between real and simulated data.) ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3.2 HOW ACCURATE IS KERNEL DENSITY ESTIMATION? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Kernel density estimation (KDE) (Parzen, 1962) is widely used to evaluate decoder-based models (Goodfellow et al., 2014; Li & Swersky, 2015), and a variant was proposed in the setting of evaluating Boltzmann machines (Bengio et al., 2013). Papers reporting KDE estimates often caution that the ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1147904ce6942a6a038f58b1f5ced578a74bce4a984505b6a275f0164f1a6300.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0a7c1e09e382dbfcfc0cfa29ed0d45801d1efe7cdb7e7a256278059a4b23f8d8.jpg",
+        "table_caption": [
+            "Figure 2: (a) Log-likelihood of GAN-50, under different choices of variance parameter. (b) Log-likelihood of GMMN-10 on 100 simulated examples evaluated by AIS and KDE vs. the corresponding running time. We show the BDMC gap converges to almost zero as we increase the running time. (c) Log-likelihood of IWAE on 10,000 test examples evaluated by AIS and IWAE bound vs. running time. (a), (b) are results on continuous MNIST, and (c) is on binarized MNIST. Note that AIS/AIS $^ +$ encoder dominates the other estimate in both estimation accuracy and running time ",
+            "Table 1: AIS vs. IWAE bound on 10,000 test examples of binarized MNIST. “# dist” denotes the number of intermediate distributions used for evalution. We find AIS estimate is consistently 1 nat higher than IWAE bound; AIS+encoder can achieve about the same estimate as AIS, but with 1 order of magnitude less number of intermediate distributions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=2>(Nats)IWAE</td><td rowspan=1 colspan=1>AIS</td><td rowspan=1 colspan=1>AIS+encoder</td><td rowspan=1 colspan=1>IWAE bound</td><td rowspan=1 colspan=1># dist AIS</td><td rowspan=1 colspan=1># dist AIS+encoder</td><td rowspan=1 colspan=1># samples</td></tr><tr><td rowspan=1 colspan=1>IWAE</td><td rowspan=1 colspan=1>-85.679-85.619</td><td rowspan=1 colspan=1>-85.754-85.621</td><td rowspan=1 colspan=1>-86.902-86.464</td><td rowspan=1 colspan=1>100010000</td><td rowspan=1 colspan=1>1001000</td><td rowspan=1 colspan=1>10000100000</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "KDE is not meant to be applied in high-dimensional spaces and that the results might therefore be inaccurate. Nevertheless, KDE remains the standard protocol for evaluating decoder-based models. We analyzed the accuracy of the KDE estimates by comparing against AIS. Both estimates are stochastic lower bounds on the true log-likelihood (see Section 3), so larger values are guaranteed (with high probability) to be more accurate. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For each estimator, we varied one parameter influencing the computational budget; for AIS, this was the number of intermediate distributions (chosen from $\\{ 1 0 0 , 5 \\bar { 0 } 0 , 1 0 0 0 , 2 0 0 0 , \\bar { 1 } 0 0 0 0 \\} )$ , and for KDE, it was the number of samples (chosen from $\\{ 1 0 0 0 0 , 1 0 0 0 0 0 , 5 0 0 0 0 0 , 1 0 0 0 0 0 0 , 2 0 0 0 0 0 \\} )$ . Using GMMN-10 for illustration, we plot both log-likelihood estimates 100 simulated examples as a function of evaluation time in Fig. 2(b). We also plot the upper bound of likelihood given by running AIS in reverse direction. We see that the BDMC gap approaches to zero, validating the accuracy of AIS. We also see that the AIS estimator achieves much more accurate estimates during similar evaluation time. Furthermore, the KDE estimates appear to level off, suggesting one cannot obtain accurate results even using orders of magnitude more samples. ",
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+    },
+    {
+        "type": "text",
+        "text": "The KDE estimation error also impacts the estimate of the observation noise $\\sigma$ , since a large value of $\\sigma$ is needed for the samples to cover the full distribution. We compared the log-likelihoods estimated by AIS and KDE with varying choices of $\\sigma$ on 100 training and validation examples of MNIST. We used 1 million simulated samples for KDE evaluation, which takes almost the same time as running AIS estimation. In Fig. 2(a), we show the log-likelihood of GAN-50 estimated by KDE and AIS as a function of $\\sigma$ . Because the accuracy of KDE declines sharply for small $\\sigma$ values, it creates a strong bias towards large $\\sigma$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3.3 HOW ACCURATE IS THE IWAE BOUND? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In principle, one could estimate VAE likelihoods using the VAE objective function (which is a lower bound on the true log-likelihood). However, it is more common to use importance weighting, where the proposal distribution is computed by the recognition network. This is provably more accurate than the VAE bound (Burda et al., 2016). Because the importance weighted estimate corresponds to the objective function used by the Importance Weighted Autoencoder (IWAE) (Burda et al., 2016), we will refer to it as the IWAE bound. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "On continuous MNIST, the IWAE bound underestimated the true log-likelihoods by at least 33.2 nats on the training set and 187.4 nats on the test set. While this is considerably more accurate than KDE, the error is still significant. Interestingly, this result also suggests that the recognition network overfits the training data. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4db7595a8dc6adfefe5335164e11cb074bc2763fa6adbb96598d718fa78f4f6e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>(Nats)</td><td>AIS Test</td><td>AISTrain</td><td>BDMC gap</td><td>KDE Test</td><td>IWAE Test</td></tr><tr><td>VAE-50</td><td>991.435±6.477</td><td>1298.830±0.863</td><td>1.540</td><td>351.213</td><td>826.325</td></tr><tr><td>GAN-50</td><td>627.297±8.813</td><td>648.283±21.115</td><td>10.045</td><td>300.331</td><td>/</td></tr><tr><td>GMMN-50</td><td>593.472±8.591</td><td>607.272±1.451</td><td>1.146</td><td>277.193</td><td>/</td></tr><tr><td>VAE-10</td><td>705.375±7.411</td><td>791.029±0.810</td><td>0.832</td><td>408.659</td><td>486.466</td></tr><tr><td>GAN-10</td><td>328.772±5.538</td><td>346.640±4.260</td><td>0.934</td><td>259.673</td><td>/</td></tr><tr><td>GMMN-10</td><td>346.679±5.860</td><td>358.943±6.485</td><td>0.605</td><td>262.73</td><td>/</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Model comparisons on 1000 test and training examples of continuous MNIST. Confidence intervals reflect the variability from the choice of training or test examples (which appears to be the dominant source of error for the AIS values). AIS, KDE, and IWAE are all stochastic lower bounds on the log-likelihood. ",
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+    {
+        "type": "text",
+        "text": "Since VAE and IWAE results have customarily been reported on binarized MNIST, we additionally trained an IWAE in this setting. The training details are given in Appendix A.2. To show the practicality of our method, we evaluated the IWAE on the full 10000 test using AIS and IWAE bound, with different choices of intermediate distribution and number of simulated samples, shown in Table 1. We also evaluate AIS with the initial distribution defined by encoders of VAEs, denoted as AIS+encoder. We find that the IWAE bound underestimates the true value by at least 1 nat, which is a large difference by the standards of binarized MNIST. (E.g., it represents about half of the gap between a state-of-the-art permutation-invariant model (Tran et al., 2016) and one which exploits structure (van den Oord et al., 2016).) The AIS and IWAE estimates are compared in terms of evaluation time in Fig. 2 (c). ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4 SCIENTIFIC FINDINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having validated the accuracy of AIS, we now use it to analyze the effectiveness of various training criteria. We also highlight phenomena which would not be observable using existing log-likelihood estimators or by inspecting samples. For all experiments in this section, we used 10,000 intermediate distributions for AIS, 1 million simulated samples for KDE, and 200,000 importance samples for the IWAE bound. (These settings resulted in similar computation time for all three estimators.) ",
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+    {
+        "type": "text",
+        "text": "5.4.1 MODEL LIKELIHOOD COMPARISON ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluated the trained models using AIS and KDE on 1000 test examples of MNIST; results are shown in Table 2. We find that for all three training criteria, the larger architectures consistently outperformed the smaller ones. We also find that for both the 10- and 50-dimensional architectures, the VAEs achieved substantially higher log-likelihoods than GANs or GMMNs. It is not surprising that the VAEs achieved higher likelihood, because they were trained using a likelihood-based objective while the GANs and GMMNs were not. However, it is interesting that the difference in log-likelihoods was so large; in the rest of this section, we attempt to analyze what exactly is causing this large difference. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We note that the KDE errors were of the same order of magnitude as the differences between models, indicating that it cannot be used reliably to compare log-likelihoods. Furthermore, KDE did not identify the correct ordering of models; for instance, it estimated a lower log-likelihood for VAE50 than for VAE-10, even though its true log-likelihood was almost 300 nats higher. KDE also underestimated by an order of magnitude the log-likelihood improvements that resulted from using the larger architectures. (E.g., it estimated a 15 nat difference between GMMN-10 and GMMN-50, even though the true difference was 247 nats as estimated by AIS.) ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "These differences are also hard to observe simply by looking at samples; for instance, we were unable to visually distinguish the quality of samples for GAN-10 and GAN-50 (see Fig. 1), even though their log-likelihoods differed by almost 300 nats on both the training and test sets. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.4.2 MEASURING THE DEGREE OF OVERFITTING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One question that arises in evaluation of decoder-based generative models is whether they memorize parts of the training dataset. One cannot test this by looking only at model samples. The commonly reported nearest-neighbors from the training set can be misleading (Theis et al., 2016), and interpolation in the latent space between different samples can be visually appealing, but does not provide a quantitative measure of the degree of generalization. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d90aedac8cf9ebef7190dc2d2bf71283ad6a43386db9195176d7f1f571d15ab.jpg",
+        "image_caption": [
+            "Figure 3: Training curves for (a) GAN-50, (b) VAE-50, and (c) GMMN-10, as measured by AIS, KDE, and (if applicable) the IWAE lower bound. All estimates shown here are lower bounds. In (c), the gap between training and validation log-likelihoods is not fairly small (see Table 2). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To analyze the degree of overfitting, Fig. 3 shows training curves for three networks as measured by AIS, KDE, and the IWAE bound. We observe that GAN-50’s training and test log-likelihoods are nearly identical throughout training, disconfirming the hypothesis that it was memorizing training data. Both GAN-50 and GMMN-50 overfit less than VAE-50. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also observed two phenomena which could not be measured using existing techniques. First, in the case of VAE-50, the IWAE lower bound starts to decline after 200 epochs, while the AIS estimates hold steady, suggesting it is the recognition network rather than the generative network which is overfitting most. Second, the GMMN-50 training and validation error continue to improve at 10,000 epochs, even though KDE erroneously indicates that performance has leveled off. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4.3 HOW APPROPRIATE IS THE OBSERVATION MODEL?",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Appendix B addresses the questions of whether the spherical Gaussian observation model is a good fit and whether the log-likelihood differences could be an artifact of the observation model. We find that all of the models can be substantially improved by accounting for non-Gaussianity, but that this effect is insufficient to explain the gap between the VAEs and the other models. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4.4 ARE THE NETWORKS MISSING MODES? ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It was previously observed that one of the potential failure modes of Boltzmann machines is to fail to generate one or more modes of a distribution or to drastically misallocate probability mass between modes (Salakhutdinov & Murray, 2008). Here we analyze this for decoder-based models. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "First, we ask a coarse-grained version of this question: do the networks allocate probability mass correctly between the 10 digit classes, and if not, can this explain the difference in log-likelihood scores? In Fig. 1, we see that GAN-50’s distribution of digit classes was heavily skewed: out of 100 samples, it generated 37 images of 1’s, but only a single 2. This appears to be a large effect, but it does not explain the magnitude of the log-likelihood difference from VAEs. In particular, if the allocation of digit classes were off by a factor of 10, this effect by itself could cost at most l $\\mathrm { 9 g 1 0 \\approx 2 . 3 }$ nats of log-likelihood. Since VAE-50 outperformed GAN-50 by 364 nats, this effect cannot explain the difference. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "However, MNIST has many factors of variability beyond simply the 10 digit classes. In order to determine whether any of the models missed more fine-grained modes, we visualized posterior samples for each model conditioned on training and test images. In particular, for each image $x$ under consideration, we used AIS to approximately sample $z$ from the posterior distribution $p ( \\bar { z } | x )$ , and then ran the decoder on $z$ . While these samples are approximate, Grosse et al. (2016) point out that the BDMC gap also bounds the KL divergence of approximate samples from the true posterior. With the exception of GAN-50, our BDMC gaps were on the order of 1 nat, suggesting our approximate posterior samples are fairly representative. The results are shown in Fig. 4. Further posterior visualizations for digit class 2 (the most difficult for the models we considered) are shown in Appendix C. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Both VAEs’ posterior samples match the observations almost perfectly. (We observed a few poorly reconstructed examples on the test set, but not on the training set.) The GANs and GMMNs fail to reconstruct some of the examples on both the training and validation sets, suggesting that they failed to learn some modes of the distribution. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/278f1d9bc0c8cf3988cf3078c5b37ece1ef0b552aa500512c17b223cfaf6956e.jpg",
+        "image_caption": [
+            "Figure 4: (a) and (b) show visualization of posterior samples of 10 training/validation examples. (c) shows visualization of posterior samples of 10 training examples of digit “2\". Each column of 10 digits comes from true data and the six models. The order of visualization is: True data, GAN-10, VAE-10, GMMN-10, GAN-50, VAE-50, GMMN-50. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            362,
+            326,
+            376
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We like to thank Yujia Li for providing his original GMMN model and codebase, and thank Jimmy Ba for advice on training GANs. Ruslan Salakhutdinov is supported in part by Disney and ONR grant N000141310721. We also thank the developers of Lasagne (Battenberg et al., 2014) and Theano (Al-Rfou et al., 2016). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "Jun-Yan Zhu, Philipp Krähenbühl, Eli Shechtman, and Alexei A. Efros. Generative visual manipulation on the natural image manifold. In Proceedings of European Conference on Computer Vision (ECCV), 2016. ",
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A NETWORK ARCHITECTURES/TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            212,
+            535,
+            228
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 MODELS ON CONTINUOUS MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            242,
+            460,
+            257
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The decoders have all fully connected layers, and the number of units in each layer was 10–64–256– 256-1024–784 for the smaller architecture and 50–1024–1024–1024–784 for the larger one. Other architecture details are summarized as follows. ",
+        "bbox": [
+            176,
+            268,
+            826,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• For GAN-10, we used a discriminator with the architecture 784-512-256-1, where each layer used dropout with parameter 0.5. For GAN-50, we used a discriminator with architecture 784-4096-4096-4096-4096-1. All hidden layers used dropout with parameter 0.8. All hidden layers in both networks used the tanh activation function, and the output layers used the logistic function. The larger model uses an encoder of an architecture 784-1024-1024-1024-100. We add dropout layer between each hidden layer, with a dropout rate of 0.2. The smaller model uses an encoder of an architecture 784-256-64-20. Generator’s hidden layers use tanh activation function, and the output layer uses sigmoid unit. Encoder’s hidden layers use tanh activation function, and the output layer uses linear activation.   \n• GMMN: The hidden layers use ReLU activation function, and the output layer uses sigomid unit. ",
+        "bbox": [
+            215,
+            323,
+            825,
+            501
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For training GAN/VAE, we use our own implementation. We use Adam for optimization, and perform grid search of learning rate from $\\{ 0 . 0 0 1 , 0 . 0 0 0 1 , 0 . 0 0 0 0 1 \\}$ . For training GMMN, we take the implementation from https://github.com/yujiali/gmmn.git. Following the implementation, we use SGD with momentum for optimization, and perform grid search of learning rate from $\\lbrace 0 . 1 , 0 . 5 , 1 , 2 \\rbrace$ , with momentum 0.9. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            583
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 MODELS ON BINARIZED MNIST ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            599,
+            442,
+            614
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Its decoder has the architecture 50-200-200-784 with all tanh hidden layers and sigmoid output layer, and its encoder is symmetric in architecture, with linear output layer. We take the implementation at https://github.com/yburda/iwae.git for training the IWAE model.The IWAE bound was computed with 50 samples during training. We keep all the hyperparameter choices the same as in the implementation. ",
+        "bbox": [
+            173,
+            626,
+            825,
+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/5a4612741b69346253739f5b98081177cc091aed359cb0cb8a4f9f741660130f.jpg",
+        "table_caption": [
+            "B HOW PROBLEMATIC IS THE GAUSSIAN OBSERVATION MODEL? ",
+            "Table 3: Optimal variance vs. Fixed variance "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>(Nats)</td><td colspan=\"3\">Train</td><td colspan=\"3\">Valid</td></tr><tr><td></td><td>Optimal</td><td>Fixed</td><td>Improvement</td><td>Optimal</td><td>Fixed</td><td>Improvement</td></tr><tr><td>GAN-50</td><td>711.405</td><td>620.498</td><td>90.907</td><td>702.699</td><td>623.492</td><td>79.207</td></tr><tr><td>GMMN-50</td><td>655.807</td><td>571.803</td><td>84.004</td><td>661.652</td><td>594.612</td><td>67.040</td></tr><tr><td>GAN-10</td><td>376.788</td><td>318.948</td><td>57.840</td><td>368.585</td><td>316.614</td><td>51.971</td></tr><tr><td>GMMN-10</td><td>393.976</td><td>345.177</td><td>48.799</td><td>371.325</td><td>332.360</td><td>38.965</td></tr></table>",
+        "bbox": [
+            227,
+            752,
+            766,
+            842
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider whether the difference in log-likelihood between models could be an artifact of the Gaussian noise model (which we know to be a poor fit). In principle, the Gaussian noise assumption could be unfair to the GANs and GMMNs, because the VAE training uses the correct observation model, while the GAN and GMMN objectives do not have any particular observation model built in. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To determine the size of this effect, we evaluated the models under a different regime where, instead of choosing a fixed value of the observation noise $\\sigma$ on a validation set, $\\sigma$ was tuned independently for each example.2 This is not a proper generative model, but it can be viewed as an upper bound on the log-likelihood that would be achievable with a heavy-tailed and radially symmetric noise model.3 Results are shown in Table 3. We see that adapting $\\sigma$ for each example results in a log-likelihood improvement between 30 and 100 nats for all of the networks. In general, the examples which show the largest performance jump are images of 1’s (which prefer smaller $\\sigma$ ) and 2’s (which prefer larger $\\sigma _ { . }$ ). This is a significant effect, and suggests that one could significantly improve the log-likelihood scores by picking a better observation model. However, this effect is smaller in magnitude than the differences between VAE and GAN/GMMN log-likelihoods, so it fails to explain the likelihood difference. ",
+        "bbox": [
+            174,
+            140,
+            825,
+            291
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C POSTERIOR VISUALIZATION OF DIGIT “2\" ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            311,
+            553,
+            327
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "According to the log-likelihood evaluation, we find digit “2\" is the hardest digit for modelling. In this section we investigate the quality of modelling $\" 2 \"$ of each model. We randomly sampled a fixed set of 100 samples of digit “2\" from training data and compare whether model capture this mode. We show the plots of “2\" for GAN-10, GAN-50, VAE-10 and true data in the following figures for illustration. We see that GAN-10 fails at capturing many instances of digit “2\" in the training data! We see instead of generating $\" 2 \"$ , it tries to generate digit “1\", “7\" “9\", “4\", “8\" from reconstruction. GAN-50 does much better, its reconstruction are all digit “2\" and there is only some style difference from the true data. VAE-10 totally dominates this competition, where it perfectly reconstructs all the samples of digit “2\". We emphasize if directly sampling from each model, samples look visually indistinguishable (see Fig. 1), but we can clearly see differences in posterior samples. ",
+        "bbox": [
+            174,
+            343,
+            825,
+            483
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/88c8505b1c7a133c4cd7e4e0394f445680a5f92fc9b45f6da2b5c004c592c196.jpg",
+        "image_caption": [
+            "Figure 5: Posterior samples of digit “2\" for GAN-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            261,
+            803,
+            727
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/fa33a858f8d884b0f849fcbc7b769618396fd345f19cf5ffa8a53b83c11284f2.jpg",
+        "image_caption": [
+            "Figure 6: Posterior samples of digit “2\" for GAN-50. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            276,
+            810,
+            731
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c441ffa6076e604d1acaba867f49898ce2211f0fad5cb5d3a733afee18cc6aef.jpg",
+        "image_caption": [
+            "Figure 7: Posterior samples of digit “2\" for VAE-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            270,
+            797,
+            727
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1de2e6c0013fbeaec9a869eeb2d58370872a4aef70a5dcec5670c18cacf35204.jpg",
+        "image_caption": [
+            "Figure 8: 100 digit “2\" from training data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            276,
+            790,
+            727
+        ],
+        "page_idx": 16
+    }
+]

parse/train/BJfOXnActQ/BJfOXnActQ.md

@@ -0,0 +1,284 @@
+# LEARNING TO LEARNWITH CONDITIONAL CLASS DEPENDENCIES
+
+Xiang Jiang?†, Mohammad Havaei?, Farshid Varno?†, Gabriel Chartrand?,   
+Nicolas Chapados?, Stan Matwin†   
+?Imagia Inc., †Dalhousie University   
+{xiang.jiang,mohammad,farshid.varno,gabriel,nic}@imagia.com, stan@cs.dal.ca
+
+# ABSTRACT
+
+Neural networks can learn to extract statistical properties from data, but they seldom make use of structured information from the label space to help representation learning. For example “cat” and “dog” are closer than “cat” and “truck”. Although some label structure can implicitly be obtained when training on huge amounts of data, in a few-shot learning context where little data is available, making explicit use of the label structure can inform the model to reshape the representation space to reflect a global sense of class dependencies. We propose a meta-learning framework, Conditional class-Aware Meta-Learning (CAML), that conditionally transforms feature representations based on a metric space that is trained to capture inter-class dependencies. This enables a conditional modulation of the feature representations of the base-learner to impose regularities informed by the label space. Experiments show that the conditional transformation in CAML leads to more disentangled representations and achieves competitive results on the miniImageNet benchmark.
+
+# 1 INTRODUCTION
+
+In machine learning, the objective of classification is to train a model to categorize inputs into various classes. We usually assume a categorical distribution over the label space, and thus effectively ignore dependencies among them. However, class structure does exist in real world and is also present in most datasets. Although class structure can be implicitly obtained as a by-product during learning, it is not commonly exploited in an explicit manner to develop better learning systems. The use of label structure might not be of prime importance when having access to huge amounts of data, such the full ImageNet dataset. However, in the case of few-shot learning where little data is available, meta-information such as dependencies in the label space can be crucial.
+
+In recent years, few-shot learning—learning from few examples across many tasks—has received considerable attention (Ravi & Larochelle, 2016; Snell et al., 2017; Finn et al., 2017; Vinyals et al., 2016). In particular, the concept of meta-learning has been shown to provide effective tools for few-shot learning tasks. In contrast to common transfer learning methods that aim to fine-tune a pre-trained model, meta-learning systems are trained by being exposed to a large number of tasks and evaluated in their ability to learn new tasks effectively. In meta-training, learning happens at two levels: a meta-learner that learns across many tasks, and a base-learner that optimizes for each task. Model-Agnostic Meta-Learning (MAML) is a gradient-based meta-learning algorithm that provides a mechanism for rapid adaptation by optimizing only for the initial parameters of the base-learner (Finn et al., 2017).
+
+Our motivation stems from a core challenge in gradient-based meta-learning, wherein the quality of gradient information is key to fast generalization: it is known that gradient-based optimization fails to converge adequately when trained from only a few examples (Ravi & Larochelle, 2016), hampering the effectiveness of gradient-based meta-learning techniques. We hypothesize that under such circumstances, introducing a metric space trained to encode regularities of the label structure can impose global class dependencies on the model. This class structure can then provide a high-level view of the input examples, in turn leading to learning more disentangled representations.
+
+We propose a meta-learning framework taking advantage of this class structure information, which is available in a number of applications. The Conditional class-Aware Meta-Learning (CAML) model is tasked with producing activations in a manner similar to a standard neural network, but with the additional flexibility to shift and scale those activations conditioned on some auxiliary meta-information. While there are no restrictions on the nature of the conditioning factor, in this work we model class dependencies by means of a metric space. We aim to learn a function mapping inputs to a metric space where semantic distances between instances follow an Euclidean geometry—classes that are semantically close lie in close proximity in an $\ell ^ { p }$ sense. The goal of the conditional class-aware transformation is to make explicit use of the label structure to inform the model to reshape the representation landscape in a manner that incorporates a global sense of class structure.
+
+The contributions of this work are threefold: (i) We provide a meta-learning framework that makes use of structured class information in the form of a metric space to modulate representations in few-shot learning tasks; (ii) We introduce class-aware grouping to improve the statistical strength of few-shot learning tasks; (iii) We show experimentally that our proposed algorithm learns more disentangled representation and achieves competitive results on the miniImageNet benchmark.
+
+# 2 BACKGROUND
+
+We start by describing the meta-learning formulation proposed by Vinyals et al. (2016) and Ravi & Larochelle (2016), and review MAML (Finn et al., 2017), of which CAML is an instance.
+
+# 2.1 META-LEARNING PROBLEM FORMULATION
+
+The goal of meta-learning is to learn from a distribution of tasks. The learning happens on two levels: (i) a meta-level model, or meta-learner, that learns across many tasks, and (ii) a base-level model, or base-learner, that operates within each specific task. Meta-learning happens in task space, where each task can be treated as one meta-example. In the meta-learning formulation, we define a collection of regular tasks as meta-sets $\mathcal { D }$ , and each task $\mathcal { D } \in \mathcal { D }$ has its own $\mathcal { D } ^ { \mathrm { t r a i n } }$ and $\mathcal { D } ^ { \mathrm { t e s t } }$ split. $\mathcal { D } ^ { \mathrm { t r a i n } }$ is often denoted as the “support set” and $\mathcal { D } ^ { \mathrm { t e s t } }$ the “query set”. The resulting meta-learner objective is to choose parameters $\theta$ that minimize the expected loss $\mathcal { L } ( \cdot ; \theta )$ across all tasks in $\mathcal { D }$ ,
+
+$$
+\begin{array} { r } { \theta ^ { * } = \mathrm { a r g m i n } _ { \theta } \mathbb { E } _ { \mathcal { D } \sim \mathcal { D } } [ \mathcal { L } ( \mathcal { D } ; \theta ) ] . } \end{array}
+$$
+
+At the meta-level, the meta-sets $\mathcal { D }$ can be further split into disjoint meta-training set $\mathcal { D } _ { \mathrm { m e t a - t r a i n } }$ , meta-validation set $\mathcal { D } _ { \mathrm { m e t a - v a l i d } }$ and meta-test set $\mathcal { D } _ { \mathrm { m e t a - t e s t } }$ . The meta-learner is trained on $\mathcal { D } _ { \mathrm { m e t a - t r a i n } }$ , validated on $\mathcal { D } _ { \mathrm { m e t a - v a l i d } }$ and finally evaluated on $\mathcal { D } _ { \mathrm { m e t a - t e s t } }$ .
+
+# 2.2 MODEL-AGNOSTIC META-LEARNING
+
+Model-Agnostic Meta-Learning (Finn et al., 2017) is a meta-learning algorithm that aims to learn representations that encourage fast adaptation across different tasks. The meta-learner and base-learner share the same network structure, and the parameters learned by the meta-learner are used to initialize the base-learner on a new task.
+
+To optimize the meta-learner, we first sample a set of tasks $\{ \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , . . . , \mathcal { D } _ { S } \}$ from the meta-training set $\mathcal { D } _ { \mathrm { m e t a - t r a i n } }$ . For a meta-learner parameterized by $\theta$ , we compute its adapted parameters $\theta _ { i }$ for each sampled task $\mathcal { D } _ { i }$ . The adapted parameters $\theta _ { i }$ are task-specific and tell us the effectiveness of $\theta$ as to whether it can achieve generalization through one or a few additional gradient steps. The objective of the meta-learner is to optimize the representation $\theta$ such that it leads to good task-specific adaptations $\theta _ { i }$ with only a few gradient steps. The meta-learner performs slow learning at the meta-level across many tasks to support fast learning on new tasks. At meta-test time, we initialize the base-learner with the meta-learned representation $\theta ^ { * }$ followed by gradient-based fine-tuning.
+
+# 3 METHOD
+
+# 3.1 CONDITIONAL CLASS-AWARE META-LEARNING
+
+As shown in Figure 1, the proposed Conditional class-Aware Meta-Learning (CAML) is composed of four components: an embedding function $f _ { \phi }$ that maps inputs to a metric space, a base-learner $f _ { \theta }$ that learns each individual task, an adaptation function $f _ { c }$ that conditionally modulates the representations of the base-learner, and a meta-learner that learns across different tasks. Figure 1 depicts a toy illustration of the task inference procedure where examples from three classes are mapped onto a metric space using $f _ { \phi }$ , which are further used to modulate the base-learner $f _ { \theta }$ through a conditional transformation function $f _ { c }$ .
+
+![](images/03b9b29c15eeaae6bc3b0296cbffabc73ecb65988e8285e024bd8cb326348430.jpg)  
+Figure 1: Overview of Conditional class-Aware Meta-Learning. Inputs to the model are mapped onto an embedding space using $f _ { \phi }$ which are then used to modulate the base-learner $f _ { \theta }$ through a conditional transformation $f _ { c }$ . We use MAML (not shown) to meta-learn $f _ { c } , f _ { \theta }$ , and a metric loss to pre-train $f _ { \phi }$
+
+The main contribution of this paper is to incorporate metric-based conditional transformations $( f _ { c } )$ into the meta-learning framework at the instance level. A notable feature of the proposed method is that the model has a global sense of the label space through the embedding function $f _ { \phi }$ by mapping examples onto the semantically meaningful metric space. The embeddings on the metric space inform the base-learner $f _ { \theta }$ about the label structure which in turn helps disentangle representations from different classes. This structured information can also provide a global view of the input examples to improve gradient-based meta-learning.
+
+In a simplistic form, our proposed model makes predictions using
+
+$$
+\hat { y } = f _ { \theta } \Big ( x ; f _ { c } \big ( f _ { \phi } ( x ) \big ) \Big ) ,
+$$
+
+where the base-learner $f _ { \theta }$ is conditioned on the embedding space $f _ { \phi } \left( x \right)$ through the conditional transformation $f _ { c }$ . This is in contrast with a regular base-learner where $\hat { y } = f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ . In our framework, we use MAML to meta-learn $f _ { c }$ and $f _ { \theta }$ . The metric space is pre-trained using distance-based loss function.
+
+# 3.2 METRIC SPACE AS CONDITIONAL INFORMATION
+
+We encode information of the label structure through $f _ { \phi }$ in the form of an $M$ -dimensional metric space, where each input example is reduced to a point in the metric space. The goal of the metric learning step is to optimize parameter $\phi$ such that distances between examples in the metric space are semantically meaningful. Given the parameters of the metric space $\phi$ , which is represented by a convolutional network, we calculate a centroid $\mathbf { c } _ { t }$ for each class $t$ ,
+
+$$
+\mathbf { c } _ { t } = \frac { 1 } { K } \sum _ { ( \mathbf { x } _ { i } , y _ { i } ) \in \mathcal { D } ^ { \mathrm { t r a i n } } } \mathbb { 1 } _ { \{ y _ { i } = t \} } f _ { \phi } ( \mathbf { x } _ { i } ) ,
+$$
+
+where $K$ denotes the number of examples for class $t$ , $\mathbb { 1 } _ { \{ y _ { i } = t \} }$ denotes an indicator function of $y _ { i }$ which takes value 1 when $y _ { i } = t$ and 0 otherwise. The centroid $\mathbf { c } _ { t }$ is the sample mean among all instances from the same class which is treated as a prototype representation of the class $t$ . The mapping function $f _ { \phi }$ is optimized to minimize the negative log-probability defined in Eq. (1) by minimizing the Euclidean distance $d$ between an example and its corresponding class centroid $\mathbf { c } _ { t }$ while maximizing its Euclidean distance to other class centroids $\mathbf { c } _ { t ^ { \prime } }$ :
+
+$$
+\underset { \phi } { \mathrm { a r g m i n } } \mathbb { E } \Bigg [ d ( f _ { \phi } ( \mathbf { x } _ { i } ) , \mathbf { c } _ { t } ) ) + \log \sum _ { t ^ { \prime } } \mathrm { e x p } ( - d ( f _ { \phi } ( \mathbf { x } _ { i } ) , \mathbf { c } _ { t ^ { \prime } } ) ) \Bigg ] .
+$$
+
+In relation to prototypical networks (Snell et al., 2017), we use the same loss function for metric learning. However, these frameworks differ in the test mode: we are not interested in example-centroid distances for label assignment, but rather in the projection $f _ { \phi } ( \mathbf { x } _ { i } )$ from the input space to the metric space that encapsulates inferred class regularities given the input example $\mathbf { x } _ { i }$ . In relation to other pre-training methods that use the meta-train classes to train a 64-way classifier, our use of the metric space imposes distance-based constraints to learn embeddings that follow semantically meaningful distance measures.
+
+![](images/cdcf20b42bc1172191b67df2011db6bbcd6f5b0981538cbc0305f308566a3f6b.jpg)  
+Figure 2: Conditionally transformed convolutional block. The convolutional feature maps are conditionally scaled and shifted based on the input image’s representation in the pre-trained metric space.
+
+We empirically find it difficult to optimize both the metric space and base-learner end-to-end. The metric space is pre-trained on the meta-train data and it is not updated during meta-learning. This also ensures the metric space is trained on a large number of classes to capture the global class dependencies.
+
+# 3.3 CONDITIONALLY TRANSFORMED CONVOLUTIONAL BLOCK
+
+We now turn to describing the conditionally transformed convolutional block, shown in Figure 2, which uses the metric space described in Section 3.2 to inform the base-learner about the label structure of a task. The conditional transformation $f _ { c }$ receives embeddings from the metric space and produces transformation operations to modulate convolutional representations of the base-learner $f _ { \theta }$ .
+
+Our conditional transformation has close relation to Batch Normalization (BN) (Ioffe & Szegedy, 2015) that normalizes the input to every layer of a neural network. In order to conditionally modulate feature representations, we use Conditional Batch Normalization (CBN) (Dumoulin et al., 2017) to predict scale and shift operators from conditional input $\mathbf { s } _ { i }$ :
+
+$$
+\begin{array} { r } { \hat { \gamma } _ { c } = f _ { c , \gamma } ( \mathbf { s } _ { i } ) , \qquad \hat { \beta } _ { c } = f _ { c , \beta } ( \mathbf { s } _ { i } ) , } \end{array}
+$$
+
+where $f _ { c , \gamma }$ and $f _ { c , \beta }$ can be any differentiable function. This gives our model the flexibility to shift or scale the intermediate representations based on some source information in $\mathbf { s } _ { i }$ . Since examples belonging to the same class are conceptually close, we exploit this inherent relationship in the metric space to modulate the feature maps at the example level in a way that encodes the label structure.
+
+Once we obtained the embedding function $f _ { \phi }$ , we use two auxiliary networks, learned end-to-end together with the meta-learner, to predict the shift and scale factors of the convolutional feature map:
+
+$$
+\hat { \gamma } _ { i , c } = f _ { c , \gamma } ( f _ { \phi } ( \mathbf { x } _ { i } ) ) , \hat { \beta } _ { i , c } = f _ { c , \beta } ( f _ { \phi } ( \mathbf { x } _ { i } ) ) .
+$$
+
+Having computed $\hat { \gamma } _ { i , c }$ and $\hat { \beta } _ { i , c }$ , Conditional Batch Normalization (CBN) is applied as follows:
+
+$$
+\mathrm { C B N } ( \mathbf { R } _ { i , c } | \hat { \gamma } _ { i , c } , \hat { \beta } _ { i , c } ) = \hat { \gamma } _ { i , c } \frac { { \mathbf { R } _ { i , c } - \mathbb { E } [ \mathbf { R } _ { c } ] } } { \sqrt { \mathrm { V a r } [ \mathbf { R } _ { c } ] + \epsilon } } + \hat { \beta } _ { i , c } ,
+$$
+
+where $\mathbf { R } _ { i , c }$ refers to the $c ^ { t h }$ feature map from the $i ^ { t h }$ example, $\epsilon$ is a small constant, $\beta _ { c }$ and $\gamma _ { c }$ are learnable parameters shared within a task. $\mathbb { E } [ \mathbf { R } _ { c } ]$ and $\mathrm { V a r } [ \mathbf { R } _ { c } ]$ are batch mean and variance of $\mathbf { R } _ { c }$ .
+
+It is worthwhile to note the effect of conditional transformation. The conditional bias transformation with $\hat { \beta } _ { i , c }$ is analogous to concatenation-based conditioning where the conditional information is concatenated to the feature maps (Dumoulin et al., 2018). The conditional scaling factor provides multiplicative interactions between the metric space and the feature maps to aggregate information.
+
+Furthermore, the goal of the conditionally transformed convolutional block is to simultaneously capture the two views of a classification task: a global view that is aware of the relationships among all classes, and a local views of the current N-way K-shot classification task. The metric space, or the global view, is pre-trained in a way that is independent of the current N-way K-shot task; while the base-learner, or the local view, attempts to develop representations for the current classification task. Although the metric space is never trained on the meta-test classes, we expect the learned metric space to generalize from the meta-train tasks to the meta-test tasks.
+
+We further describe parameter sharing for CBN learning in Section 3.3.1, and class-aware grouping in Section 3.3.2 which provides more statistical strength for more effective few-shot learning.
+
+![](images/6827a286c9f904ceb91c65069c40b2fe7aaed0f0dba2dee08f13df56a310a3e8.jpg)  
+Figure 3: CBN shared architecture
+
+![](images/c7846b989a4ca1056287b41bd4a58573b2bbce0af7614f0fd64281fae6ae7405.jpg)  
+Figure 4: Normalization methods. ‘C’ denotes channels, $\mathrm { ^ { 6 } H }$ , W’ spatial dimensions and $\mathbf { \Delta } ^ { \mathsf {  } } \mathbf { N } ^ { \mathsf { \prime } }$ examples.
+
+# 3.3.1 MULTITASK LEARNING OF CBN
+
+Although one can predict $\hat { \gamma } _ { c }$ and $\hat { \boldsymbol \beta } _ { c }$ using two separate functions, we find it beneficial to use shared parameters as shown in Figure 3. The shared representations are more efficient at producing conditional transformations which also provide a strong inductive bias to help learning (Caruana, 1997).
+
+# 3.3.2 CLASS-AWARE GROUPING
+
+We propose class-aware grouping, as shown in Figure 4 (b), to further exploit properties of metric space. The motivation stems from a lack of statistical strength when learning from only a few examples. As an example, in $N .$ -way 1-shot learning, the model is required to find the most meaningful way to distinguish different classes. However, gradient-based optimization may lead to the discovery of irrelevant features which coincide with the class labels, such as background colors.
+
+We address this problem by class-aware grouping that is guided by our metric space. This is related to “transduction”, which is a standard technique in MAML-based methods. Transduction as discussed in Nichol et al. (2018), makes use of the channel mean $\mathbb { E } [ \mathbf { R } _ { c } ]$ and variance $\mathrm { V a r } [ \mathbf { R } _ { c } ]$ , defined in Eq. (4), of query examples when evaluating a base-learner. In contrast to standard transduction methods that calculate mean and variance over all examples of the current batch, we introduce class-aware grouping that clusters examples into different groups and use group-based mean and variance to normalize different channels. The grouping is determined by distance measures in the metric space where examples are grouped together based on their nearest centroid $\mathbf { } c _ { t }$ defined in Section 3.2. Class-aware grouping is integrated into CBN as:
+
+$$
+\mathrm { C B N } ( \mathbf { R } _ { i , c } | \hat { \gamma } _ { i , c } , \hat { \beta } _ { i , c } ) = \hat { \gamma } _ { i , c } \frac { \mathbf { R } _ { i , c } - \mathbb { E } [ \mathbf { R } _ { i , c } \cdot \mathbb { 1 } _ { \{ x _ { i } \in t \} } ] } { \sqrt { \mathrm { V a r } [ \mathbf { R } _ { i , c } \cdot \mathbb { 1 } _ { \{ x _ { i } \in t \} } ] + \epsilon } } + \hat { \beta } _ { i , c } ,
+$$
+
+where $\mathbb { 1 } _ { \{ x _ { i } \in t \} }$ indicates if an example $x _ { i }$ belongs to cluster $t$ , and $\mathbb { E } [ \mathbf { R } _ { i , c } { \cdot } \mathbb { 1 } _ { \{ x _ { i } \in t \} } ]$ represents the average of channel $\mathbf { R } _ { c }$ among examples clustered at $c _ { t }$ . This is depicted in Figure 4 where the channel mean and variance are calculated for every group.This approach informs the base-learner about what to expect from the query examples at the class level through channel mean and variance, which provides more explicit guidance to the meta-learning procedure.
+
+# 3.4 TRAINING DETAILS
+
+The base-learner $( f _ { \theta } )$ is composed of 4 layers of $3 \times 3$ convolutions with a $4 \times 4$ skip connections from the input to the final convolutional layer. The use of skip connections is to improve the gradient flow as MAML unfolds the inner loop into one computational graph. The use of skip connections is empirically important to the proposed model. Each convolutional layer has 30 channels and is followed by CBN, ReLU and $2 \times 2$ max-pooling operations. The output of the final convolution is flattened and fed to a 1-layer dense classifier. For learning the metric space $( f _ { \phi } )$ , we use the same residual network (ResNet-12) as Oreshkin et al. (2018). The metric space is pre-trained on the same meta-training dataset for 30,000 episodes and not updated while learning the base-learner. The meta-learner is trained for 50,000 episodes. We empirically observe that training the metric space and meta-learner end-to-end is overly complex and prone to over-fitting. For CBN functions $( f _ { c } )$ , we use 3 dense layers with 30 hidden units each. Every layer is followed by a ReLU except for the last layer where no activation is used. For the meta-learner, we use MAML with 1 gradient step for 1-shot learning and 5 gradient steps for 5-shot learning. We use the Adam (Kingma & Ba, 2014) optimizer and clip the L2 norm of gradients with an upper bound of 5. Similar to MAML, we use transduction where the statistics of the current batch is used for $\mathbb { E } ( . )$ and $\mathrm { V a r } ( . )$ in Eq. (4) for both training and testing.
+
+# 4 RELATED WORK
+
+# 4.1 META-LEARNING
+
+Meta-learning or “learning-to-learn” (Schmidhuber, 1987; Bengio et al., 1992; Mitchell & Thrun, 1993; Vilalta & Drissi, 2002) has been studied as a means to acquire meta-knowledge across many tasks. In recent years, meta-learning has become an important approach for few-shot learning. A number of approaches aim to learn universal learning procedure approximators by supplying training examples to the meta-learner that outputs predictions on testing examples (Hochreiter et al., 2001; Vinyals et al., 2016; Santoro et al., 2016; Mishra et al., 2017). Other approaches learn to generate model parameters conditioned on training examples (Gomez & Schmidhuber, 2005; Munkhdalai & Yu, 2017; Ha et al., 2016; Gidaris & Komodakis, 2018), or learning optimization algorithms across different tasks (Ravi & Larochelle, 2016; Andrychowicz et al., 2016; Li & Malik, 2017).
+
+# 4.1.1 GRADIENT-BASED META-LEARNING
+
+Our work is more inline with gradient-based meta-learning that aims to learn representations that encourage fast adaptation on new tasks. These methods are based on model-agnostic meta-learning (MAML) introduced by Finn et al. (2017). While the original MAML requires second-order gradients in meta-optimization, REPTILE (Nichol et al., 2018) only uses first-order gradient information. Furthermore, Latent Embedding Optimization (LEO) (Rusu et al., 2018) is proposed to perform gradient-based optimization on a lowdimensional latent space instead of the original high-dimensional parameter space. We emphasize that all those methods do not make explicit use of structured label information, which is a main novelty in this paper.
+
+# 4.1.2 METRIC-BASED META-LEARNING
+
+Our work also relates closely to metric-based meta-learning that learns a metric space across different tasks. Siamese networks (Koch et al., 2015) learn a similarity measure between inputs using a shared network architecture that outputs high probability when paired examples are from the same class. Matching networks (Vinyals et al., 2016) use full context embeddings to encode examples to the metric space and use attention as a similarity measure for predictions. Prototypical networks (Snell et al., 2017) compute a centroid, or prototype, for every class that are later used for distance-based queries of new examples. Task dependent adaptive metric (TADAM) (Oreshkin et al., 2018) uses metric scaling based on tasks representations to learn a task-dependent metric space.
+
+A notable difference between the metric-based methods and our approach is that, the metric space in our model is not aimed for distance-based classification. Rather, we use the metric space to represent class structure which facilitates the gradient-based meta learning towards better generalization. Another difference between our method and TADAM is that, TADAM scales the metric space at the task level where all examples within a task are scaled in the same manner. In contrast, our method provides instance-based conditioning that makes use of the precise representation of each example. Put another way, TADAM modulates the inference on a metric space from a task perspective, while CAML uses example-level representation to modulate the representation at the content level.
+
+# 4.2 CONDITIONAL TRANSFORMATION
+
+In style transfer, conditional instance normalization is proposed by Dumoulin et al. (2017) that transforms the content image conditioned on the domain of the style image. In visual question answering, De Vries et al. (2017) have shown that it is beneficial to modulate early visual signals of a pre-trained residual network by language in the form of conditional batch normalization. It was further shown that feature-wise linear modulation (Perez et al., 2017; Dumoulin et al., 2018) can efficiently select meaningful representations for visual reasoning.
+
+![](images/56fc28b20335c5d617cdd3dc2d850d31a4b4169fd8b5dbe3e0d18e91a83340b0.jpg)  
+Figure 5: t-SNE visualization of the learned metric space colored by category.
+
+The notion that is common to all these methods is the use of an additional input source, e.g., style or language, to conditionally transform intermediate representations of a network. In few-shot learning, Zhou et al. (2018) suggested that it is easier to operate in the concept space in the form of a lower dimensional representation. This is compatible with our proposed approach that uses the metric space as concept-level representation to modulate intermediate features of the base-learner.
+
+# 5 EXPERIMENTS
+
+We use miniImageNet to evaluate the proposed Conditional class-Aware Meta-Learning algorithm. miniImageNet (Vinyals et al., 2016) is composed of $8 4 \times 8 4$ colored images from 100 classes, with 600 examples in each class. We adopt the class split by Ravi & Larochelle (2016) that uses 64 classes for training, 16 for validation, and 20 for test. For $N .$ -way $K$ -shot training, we randomly sample $N$ classes from the meta-train classes each containing $K$ examples for training and 20 examples for testing. At meta-testing time, we randomly sample 600 $N$ -way $K$ -shot tasks from the test classes.
+
+# 5.1 RESULTS
+
+The results presented in Table 1 show that our proposed algorithm has comparable performance on the state-of-the-art miniImageNet 5-way 1-shot classification task, and competitive results on the 5-way 5-shot task. Unlike LEO (Rusu et al., 2018) that applies meta-learning on pre-trained representations, our meta-learner is able to effectively operate on the high-dimensional parameter space. Our method also does not require co-training compared with TADAM (Oreshkin et al., 2018).
+
+Figure 5 shows the t-SNE plot of the learned metric space for both meta-train and meta-validation classes. As seen in Figure 4b, examples from the meta-validation set form clusters consistent with their class membership, even though the metric space is not trained on these classes. For example, “mierkat”, “tundrarum” and “podenco” are all animals and they are clustered close together.
+
+The first main baseline we report is MAML. CAML improves upon MAML by about $10 \%$ on both 1-shot and 5-shot tasks. This means incorporating class dependencies in the form of a metric space can greatly facilitate gradient-based meta-learning. We also compare with MAML using our base-learner architecture equipped with skip connections from the input to the last convolutional layer. MAML trained with our base-learner’s architecture yields similar performance as the original MAML, suggesting the improvement is resulted from the proposed CAML framework, rather than changes in the base-learner’s architecture.
+
+Table 1: miniImageNet classification accuracy with $9 5 \%$ confidence intervals.   
+
+<table><tr><td>Model</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>Meta-Learner LSTM (Ravi &amp; Larochelle, 2016)</td><td>43.44% ± 0.77%</td><td>60.60% ± 0.71%</td></tr><tr><td>Matching Networks (Vinyals et al.,2016)</td><td>46.6%</td><td>60.0%</td></tr><tr><td>Prototypical Network with Soft k-Means (Ren et al., 2018)</td><td>50.41% ± 0.31%</td><td>69.88%± 0.20%</td></tr><tr><td>MetaNet (Munkhdalai &amp; Yu, 2017)</td><td>49.21% ± 0.96%</td><td></td></tr><tr><td>TCML (Mishra et al., 2018)</td><td>55.71% ± 0.99%</td><td>68.88% ± 0.92%</td></tr><tr><td>adaResNet (Munkhdalai etal., 2018)</td><td>56.88% ± 0.62%</td><td>71.94 ± 0.57%</td></tr><tr><td>Cosine Classifier (Gidaris &amp; Komodakis, 2018)</td><td>56.20% ± 0.86%</td><td>73.00% ± 0.64%</td></tr><tr><td>TADAM (Oreshkin et al., 2018)</td><td>58.5%</td><td>76.7%</td></tr><tr><td>LEO (Rusu et al., 2018)</td><td>61.76% ± 0.08%</td><td>77.59% ± 0.12%</td></tr><tr><td>MAML (Finn et al., 2017)</td><td>48.7% ±1.84%</td><td>63.11% ± 0.92%</td></tr><tr><td>MAML on our architecture</td><td>48.26% ±1.04%</td><td>64.25% ± 0.78%</td></tr><tr><td>Prototypical Network (Snell et al., 2017)</td><td>49.42% ± 0.78%</td><td>68.2% ± 0.66%</td></tr><tr><td>Prototypical Network on our metric space</td><td>55.96% ± 0.91%</td><td>71.64% ± 0.70%</td></tr><tr><td>CAML (with multitask learning alone)</td><td>52.56% ± 0.83%</td><td>71.35% ±1.13%</td></tr><tr><td>CAML (with class-aware grouping alone)</td><td>55.28% ±0.90%</td><td>71.14% ± 0.81%</td></tr><tr><td>CAML (full model)</td><td>59.23% ± 0.99%</td><td>72.35% ± 0.71%</td></tr></table>
+
+The confidence intervals are constructed by sampling 600 evaluation tasks from the meta-test classes.
+
+![](images/024b362258691f6916775cedf52202f77f2793ffd2486cb7713693298a9ebeed.jpg)  
+Figure 6: PCA visualization of feature maps from the last convolutional layer colored by category.
+
+The second baseline we use is prototypical network. We measure the classification ability of our metric space using prototypical network as a classifier, shown in Table 1 (Prototypical Network in our metric space). These results suggest that making predictions on the metric space alone is inferior to CAML.This can be explained by CAML’s ability to fast-adapt representations even when the metric space does not provide good separations. We also find that CAML has larger improvements in 1-shot tasks than 5-shot ones. This is because, in 1-shot learning, metric-based methods estimate class representations from a single example, making it difficult to provide a robust class estimation.
+
+# 5.2 THE EFFECT OF CONDITIONAL TRANSFORMATION
+
+We compare activations before and after the conditional transformation to better understand how conditional transformation modulates the feature representations. Figure 6 shows the PCA projections of the last convolutional layer in the base-learner. We observe in Figure 5a that, before conditional transformation, examples from three classes (“parallel bars”, “tile roof” and “reel”) are mixed together. In Figure 5b, after the conditional transformation is applied, one of the previously cluttered classes (“tile roof”) become separated from the rest classes. This confirms that metric space can alleviate the difficulty in few-shot learning by means of conditional transformations.
+
+We undertake ablation studies to show the impact of multitask learning and class-aware grouping. Empirical results in Table 1 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is not affected by those techniques. This is owing to a lack of statistical strength in 1-shot learning, which requires more explicit guidance in the training procedure. This means exploiting metric-based channel mean and variance can provide valuable information to improve meta-learning. More detailed ablation studies are included in Appendix A.
+
+# 6 CONCLUSION
+
+In this work, we propose Conditional class-Aware Meta-Learning (CAML) that incorporates class information by means of an embedding space to conditionally modulate representations of the base-learner. By conditionally transforming the intermediate representations of the base-learner, our goal is to reshape the representation with a global sense of class structure. Experiments reveal that the proposed conditional transformation can modulate the convolutional feature maps towards a more disentangled representation. We also introduce class-aware grouping to address a lack of statistical strength in few-shot learning. The proposed approach obtains competitive results with the current state-of-the-art performance on 5-way 1-shot and 5-shot miniImageNet benchmark.
+
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+
+# A ADDITIONAL ABLATION STUDIES
+
+# A.1 THE IMPACT OF MULTITASK LEARNING AND CLASS-AWARE GROUPING
+
+To better understand the role of different components in the proposed conditional transformation, we undertake ablation studies to provide further insights into CBN. We study the impact of multitask learning detailed in Section 3.3.1 and class-aware grouping described in Section 3.3.2.
+
+Table 2: Ablation study on the impact of multitask learning and class-aware grouping miniImageNet.   
+
+<table><tr><td>CBN</td><td>multitask</td><td>class-aware grouping</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>×</td><td>×</td><td>×</td><td>48.26% ± 1.04%</td><td>64.25% ± 0.78%</td></tr><tr><td>√</td><td>×</td><td>×</td><td>52.06% ± 1.12%</td><td>69.84% ± 1.28%</td></tr><tr><td>√</td><td>√</td><td>×</td><td>52.56% ± 0.83%</td><td>71.35% ± 1.13%</td></tr><tr><td>√</td><td>×</td><td>√</td><td>55.28% ± 0.90%</td><td>71.14% ± 0.81%</td></tr><tr><td>√</td><td>√</td><td>√</td><td>59.23% ± 0.99%</td><td>72.35% ± 0.71%</td></tr></table>
+
+Table 3: Ablation study on the impact of conditional transformation operators for miniImageNet.   
+
+<table><tr><td>Model</td><td>BN</td><td>CBN with βc alone （</td><td> CBN with c alone</td><td>CBN</td></tr><tr><td>5-way 1-shot48.7% ± 1.84%</td><td></td><td>56.04% ± 0.99%</td><td>57.83% ± 1.04%</td><td>59.23% ± 0.99%</td></tr></table>
+
+Empirical results from Table 2 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is less sensitive those techniques. This is owing to a lack of sufficient training examples in 1-shot learning tasks, which requires more explicit guidance in the training procedure. We further note that, in 1-shot learning, using class-aware grouping alone can improve CBN’s performance by $3 \%$ . This means exploiting metric-based channel mean and variance can provide valuable information for gradient-based meta-learning.
+
+# A.2 THE IMPACT OF SCALE AND SHIFT TRANSFORMATIONS
+
+For CBN parameters, we observe that more than half of the predicted $\hat { \boldsymbol \beta } _ { c }$ are negative. This is inline with findings from Perez et al. (2017) that CBN selectively suppresses activations of a feature map when followed by a ReLU. To further examine the impact of the scale and shift operators, we train CBN with each operator alone. Table 3 shows CBN works the best when both $\hat { \gamma } _ { c }$ and $\hat { \boldsymbol \beta } _ { c }$ are used, and $\hat { \gamma } _ { c }$ contributes more than $\hat { \boldsymbol { \beta } } _ { c }$ , owing to its multiplicative interactions between the metric space and convolutional feature representations.

parse/train/BJfOXnActQ/BJfOXnActQ_content_list.json

@@ -0,0 +1,1537 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING TO LEARNWITH CONDITIONAL CLASS DEPENDENCIES",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Xiang Jiang?†, Mohammad Havaei?, Farshid Varno?†, Gabriel Chartrand?,   \nNicolas Chapados?, Stan Matwin†   \n?Imagia Inc., †Dalhousie University   \n{xiang.jiang,mohammad,farshid.varno,gabriel,nic}@imagia.com, stan@cs.dal.ca ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            455,
+            265,
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+            280
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks can learn to extract statistical properties from data, but they seldom make use of structured information from the label space to help representation learning. For example “cat” and “dog” are closer than “cat” and “truck”. Although some label structure can implicitly be obtained when training on huge amounts of data, in a few-shot learning context where little data is available, making explicit use of the label structure can inform the model to reshape the representation space to reflect a global sense of class dependencies. We propose a meta-learning framework, Conditional class-Aware Meta-Learning (CAML), that conditionally transforms feature representations based on a metric space that is trained to capture inter-class dependencies. This enables a conditional modulation of the feature representations of the base-learner to impose regularities informed by the label space. Experiments show that the conditional transformation in CAML leads to more disentangled representations and achieves competitive results on the miniImageNet benchmark. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            508,
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+            523
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In machine learning, the objective of classification is to train a model to categorize inputs into various classes. We usually assume a categorical distribution over the label space, and thus effectively ignore dependencies among them. However, class structure does exist in real world and is also present in most datasets. Although class structure can be implicitly obtained as a by-product during learning, it is not commonly exploited in an explicit manner to develop better learning systems. The use of label structure might not be of prime importance when having access to huge amounts of data, such the full ImageNet dataset. However, in the case of few-shot learning where little data is available, meta-information such as dependencies in the label space can be crucial. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "In recent years, few-shot learning—learning from few examples across many tasks—has received considerable attention (Ravi & Larochelle, 2016; Snell et al., 2017; Finn et al., 2017; Vinyals et al., 2016). In particular, the concept of meta-learning has been shown to provide effective tools for few-shot learning tasks. In contrast to common transfer learning methods that aim to fine-tune a pre-trained model, meta-learning systems are trained by being exposed to a large number of tasks and evaluated in their ability to learn new tasks effectively. In meta-training, learning happens at two levels: a meta-learner that learns across many tasks, and a base-learner that optimizes for each task. Model-Agnostic Meta-Learning (MAML) is a gradient-based meta-learning algorithm that provides a mechanism for rapid adaptation by optimizing only for the initial parameters of the base-learner (Finn et al., 2017). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our motivation stems from a core challenge in gradient-based meta-learning, wherein the quality of gradient information is key to fast generalization: it is known that gradient-based optimization fails to converge adequately when trained from only a few examples (Ravi & Larochelle, 2016), hampering the effectiveness of gradient-based meta-learning techniques. We hypothesize that under such circumstances, introducing a metric space trained to encode regularities of the label structure can impose global class dependencies on the model. This class structure can then provide a high-level view of the input examples, in turn leading to learning more disentangled representations. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a meta-learning framework taking advantage of this class structure information, which is available in a number of applications. The Conditional class-Aware Meta-Learning (CAML) model is tasked with producing activations in a manner similar to a standard neural network, but with the additional flexibility to shift and scale those activations conditioned on some auxiliary meta-information. While there are no restrictions on the nature of the conditioning factor, in this work we model class dependencies by means of a metric space. We aim to learn a function mapping inputs to a metric space where semantic distances between instances follow an Euclidean geometry—classes that are semantically close lie in close proximity in an $\\ell ^ { p }$ sense. The goal of the conditional class-aware transformation is to make explicit use of the label structure to inform the model to reshape the representation landscape in a manner that incorporates a global sense of class structure. ",
+        "bbox": [
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+        "page_idx": 0
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "The contributions of this work are threefold: (i) We provide a meta-learning framework that makes use of structured class information in the form of a metric space to modulate representations in few-shot learning tasks; (ii) We introduce class-aware grouping to improve the statistical strength of few-shot learning tasks; (iii) We show experimentally that our proposed algorithm learns more disentangled representation and achieves competitive results on the miniImageNet benchmark. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            311,
+            320,
+            329
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We start by describing the meta-learning formulation proposed by Vinyals et al. (2016) and Ravi & Larochelle (2016), and review MAML (Finn et al., 2017), of which CAML is an instance. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 META-LEARNING PROBLEM FORMULATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
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+    {
+        "type": "text",
+        "text": "The goal of meta-learning is to learn from a distribution of tasks. The learning happens on two levels: (i) a meta-level model, or meta-learner, that learns across many tasks, and (ii) a base-level model, or base-learner, that operates within each specific task. Meta-learning happens in task space, where each task can be treated as one meta-example. In the meta-learning formulation, we define a collection of regular tasks as meta-sets $\\mathcal { D }$ , and each task $\\mathcal { D } \\in \\mathcal { D }$ has its own $\\mathcal { D } ^ { \\mathrm { t r a i n } }$ and $\\mathcal { D } ^ { \\mathrm { t e s t } }$ split. $\\mathcal { D } ^ { \\mathrm { t r a i n } }$ is often denoted as the “support set” and $\\mathcal { D } ^ { \\mathrm { t e s t } }$ the “query set”. The resulting meta-learner objective is to choose parameters $\\theta$ that minimize the expected loss $\\mathcal { L } ( \\cdot ; \\theta )$ across all tasks in $\\mathcal { D }$ , ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/5686ae010ea4187e2afac77ce46b5b18b54dd9a3ec92a0251e27200f403a4779.jpg",
+        "text": "$$\n\\begin{array} { r } { \\theta ^ { * } = \\mathrm { a r g m i n } _ { \\theta } \\mathbb { E } _ { \\mathcal { D } \\sim \\mathcal { D } } [ \\mathcal { L } ( \\mathcal { D } ; \\theta ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            520,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "At the meta-level, the meta-sets $\\mathcal { D }$ can be further split into disjoint meta-training set $\\mathcal { D } _ { \\mathrm { m e t a - t r a i n } }$ , meta-validation set $\\mathcal { D } _ { \\mathrm { m e t a - v a l i d } }$ and meta-test set $\\mathcal { D } _ { \\mathrm { m e t a - t e s t } }$ . The meta-learner is trained on $\\mathcal { D } _ { \\mathrm { m e t a - t r a i n } }$ , validated on $\\mathcal { D } _ { \\mathrm { m e t a - v a l i d } }$ and finally evaluated on $\\mathcal { D } _ { \\mathrm { m e t a - t e s t } }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 MODEL-AGNOSTIC META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Model-Agnostic Meta-Learning (Finn et al., 2017) is a meta-learning algorithm that aims to learn representations that encourage fast adaptation across different tasks. The meta-learner and base-learner share the same network structure, and the parameters learned by the meta-learner are used to initialize the base-learner on a new task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To optimize the meta-learner, we first sample a set of tasks $\\{ \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } , . . . , \\mathcal { D } _ { S } \\}$ from the meta-training set $\\mathcal { D } _ { \\mathrm { m e t a - t r a i n } }$ . For a meta-learner parameterized by $\\theta$ , we compute its adapted parameters $\\theta _ { i }$ for each sampled task $\\mathcal { D } _ { i }$ . The adapted parameters $\\theta _ { i }$ are task-specific and tell us the effectiveness of $\\theta$ as to whether it can achieve generalization through one or a few additional gradient steps. The objective of the meta-learner is to optimize the representation $\\theta$ such that it leads to good task-specific adaptations $\\theta _ { i }$ with only a few gradient steps. The meta-learner performs slow learning at the meta-level across many tasks to support fast learning on new tasks. At meta-test time, we initialize the base-learner with the meta-learned representation $\\theta ^ { * }$ followed by gradient-based fine-tuning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "3.1 CONDITIONAL CLASS-AWARE META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1, the proposed Conditional class-Aware Meta-Learning (CAML) is composed of four components: an embedding function $f _ { \\phi }$ that maps inputs to a metric space, a base-learner $f _ { \\theta }$ that learns each individual task, an adaptation function $f _ { c }$ that conditionally modulates the representations of the base-learner, and a meta-learner that learns across different tasks. Figure 1 depicts a toy illustration of the task inference procedure where examples from three classes are mapped onto a metric space using $f _ { \\phi }$ , which are further used to modulate the base-learner $f _ { \\theta }$ through a conditional transformation function $f _ { c }$ . ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "image",
+        "img_path": "images/03b9b29c15eeaae6bc3b0296cbffabc73ecb65988e8285e024bd8cb326348430.jpg",
+        "image_caption": [
+            "Figure 1: Overview of Conditional class-Aware Meta-Learning. Inputs to the model are mapped onto an embedding space using $f _ { \\phi }$ which are then used to modulate the base-learner $f _ { \\theta }$ through a conditional transformation $f _ { c }$ . We use MAML (not shown) to meta-learn $f _ { c } , f _ { \\theta }$ , and a metric loss to pre-train $f _ { \\phi }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "The main contribution of this paper is to incorporate metric-based conditional transformations $( f _ { c } )$ into the meta-learning framework at the instance level. A notable feature of the proposed method is that the model has a global sense of the label space through the embedding function $f _ { \\phi }$ by mapping examples onto the semantically meaningful metric space. The embeddings on the metric space inform the base-learner $f _ { \\theta }$ about the label structure which in turn helps disentangle representations from different classes. This structured information can also provide a global view of the input examples to improve gradient-based meta-learning. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In a simplistic form, our proposed model makes predictions using ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cacb4856d5c00323afbb2bdf01533a0a140adb99a65edd18cd48ab1d990630e.jpg",
+        "text": "$$\n\\hat { y } = f _ { \\theta } \\Big ( x ; f _ { c } \\big ( f _ { \\phi } ( x ) \\big ) \\Big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "where the base-learner $f _ { \\theta }$ is conditioned on the embedding space $f _ { \\phi } \\left( x \\right)$ through the conditional transformation $f _ { c }$ . This is in contrast with a regular base-learner where $\\hat { y } = f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } )$ . In our framework, we use MAML to meta-learn $f _ { c }$ and $f _ { \\theta }$ . The metric space is pre-trained using distance-based loss function. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 METRIC SPACE AS CONDITIONAL INFORMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We encode information of the label structure through $f _ { \\phi }$ in the form of an $M$ -dimensional metric space, where each input example is reduced to a point in the metric space. The goal of the metric learning step is to optimize parameter $\\phi$ such that distances between examples in the metric space are semantically meaningful. Given the parameters of the metric space $\\phi$ , which is represented by a convolutional network, we calculate a centroid $\\mathbf { c } _ { t }$ for each class $t$ , ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/626114dce79b1e2506a8d4d0f83f05f9ed69c89a800815c1df41fc1a193d6cac.jpg",
+        "text": "$$\n\\mathbf { c } _ { t } = \\frac { 1 } { K } \\sum _ { ( \\mathbf { x } _ { i } , y _ { i } ) \\in \\mathcal { D } ^ { \\mathrm { t r a i n } } } \\mathbb { 1 } _ { \\{ y _ { i } = t \\} } f _ { \\phi } ( \\mathbf { x } _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            712,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $K$ denotes the number of examples for class $t$ , $\\mathbb { 1 } _ { \\{ y _ { i } = t \\} }$ denotes an indicator function of $y _ { i }$ which takes value 1 when $y _ { i } = t$ and 0 otherwise. The centroid $\\mathbf { c } _ { t }$ is the sample mean among all instances from the same class which is treated as a prototype representation of the class $t$ . The mapping function $f _ { \\phi }$ is optimized to minimize the negative log-probability defined in Eq. (1) by minimizing the Euclidean distance $d$ between an example and its corresponding class centroid $\\mathbf { c } _ { t }$ while maximizing its Euclidean distance to other class centroids $\\mathbf { c } _ { t ^ { \\prime } }$ : ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/e7ff61ab60fd342d551329902053c2bb9189a6b0e295e9475a0939aac69f94c8.jpg",
+        "text": "$$\n\\underset { \\phi } { \\mathrm { a r g m i n } } \\mathbb { E } \\Bigg [ d ( f _ { \\phi } ( \\mathbf { x } _ { i } ) , \\mathbf { c } _ { t } ) ) + \\log \\sum _ { t ^ { \\prime } } \\mathrm { e x p } ( - d ( f _ { \\phi } ( \\mathbf { x } _ { i } ) , \\mathbf { c } _ { t ^ { \\prime } } ) ) \\Bigg ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In relation to prototypical networks (Snell et al., 2017), we use the same loss function for metric learning. However, these frameworks differ in the test mode: we are not interested in example-centroid distances for label assignment, but rather in the projection $f _ { \\phi } ( \\mathbf { x } _ { i } )$ from the input space to the metric space that encapsulates inferred class regularities given the input example $\\mathbf { x } _ { i }$ . In relation to other pre-training methods that use the meta-train classes to train a 64-way classifier, our use of the metric space imposes distance-based constraints to learn embeddings that follow semantically meaningful distance measures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdcf20b42bc1172191b67df2011db6bbcd6f5b0981538cbc0305f308566a3f6b.jpg",
+        "image_caption": [
+            "Figure 2: Conditionally transformed convolutional block. The convolutional feature maps are conditionally scaled and shifted based on the input image’s representation in the pre-trained metric space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            259,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We empirically find it difficult to optimize both the metric space and base-learner end-to-end. The metric space is pre-trained on the meta-train data and it is not updated during meta-learning. This also ensures the metric space is trained on a large number of classes to capture the global class dependencies. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 CONDITIONALLY TRANSFORMED CONVOLUTIONAL BLOCK ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now turn to describing the conditionally transformed convolutional block, shown in Figure 2, which uses the metric space described in Section 3.2 to inform the base-learner about the label structure of a task. The conditional transformation $f _ { c }$ receives embeddings from the metric space and produces transformation operations to modulate convolutional representations of the base-learner $f _ { \\theta }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our conditional transformation has close relation to Batch Normalization (BN) (Ioffe & Szegedy, 2015) that normalizes the input to every layer of a neural network. In order to conditionally modulate feature representations, we use Conditional Batch Normalization (CBN) (Dumoulin et al., 2017) to predict scale and shift operators from conditional input $\\mathbf { s } _ { i }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/69f9854de94fc72bb2acd074f8742a7b4a2601418b47b405348e04e671f7e2d7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\gamma } _ { c } = f _ { c , \\gamma } ( \\mathbf { s } _ { i } ) , \\qquad \\hat { \\beta } _ { c } = f _ { c , \\beta } ( \\mathbf { s } _ { i } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { c , \\gamma }$ and $f _ { c , \\beta }$ can be any differentiable function. This gives our model the flexibility to shift or scale the intermediate representations based on some source information in $\\mathbf { s } _ { i }$ . Since examples belonging to the same class are conceptually close, we exploit this inherent relationship in the metric space to modulate the feature maps at the example level in a way that encodes the label structure. ",
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+    {
+        "type": "text",
+        "text": "Once we obtained the embedding function $f _ { \\phi }$ , we use two auxiliary networks, learned end-to-end together with the meta-learner, to predict the shift and scale factors of the convolutional feature map: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02b5a2f88a13867502b0e273c1b56bcfc4bfb7171c38297387b97eac103085e5.jpg",
+        "text": "$$\n\\hat { \\gamma } _ { i , c } = f _ { c , \\gamma } ( f _ { \\phi } ( \\mathbf { x } _ { i } ) ) , \\hat { \\beta } _ { i , c } = f _ { c , \\beta } ( f _ { \\phi } ( \\mathbf { x } _ { i } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Having computed $\\hat { \\gamma } _ { i , c }$ and $\\hat { \\beta } _ { i , c }$ , Conditional Batch Normalization (CBN) is applied as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a089c02b563fd21c44104698da86c6e7fd7be00c088233a104af822a64ffe11.jpg",
+        "text": "$$\n\\mathrm { C B N } ( \\mathbf { R } _ { i , c } | \\hat { \\gamma } _ { i , c } , \\hat { \\beta } _ { i , c } ) = \\hat { \\gamma } _ { i , c } \\frac { { \\mathbf { R } _ { i , c } - \\mathbb { E } [ \\mathbf { R } _ { c } ] } } { \\sqrt { \\mathrm { V a r } [ \\mathbf { R } _ { c } ] + \\epsilon } } + \\hat { \\beta } _ { i , c } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { R } _ { i , c }$ refers to the $c ^ { t h }$ feature map from the $i ^ { t h }$ example, $\\epsilon$ is a small constant, $\\beta _ { c }$ and $\\gamma _ { c }$ are learnable parameters shared within a task. $\\mathbb { E } [ \\mathbf { R } _ { c } ]$ and $\\mathrm { V a r } [ \\mathbf { R } _ { c } ]$ are batch mean and variance of $\\mathbf { R } _ { c }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It is worthwhile to note the effect of conditional transformation. The conditional bias transformation with $\\hat { \\beta } _ { i , c }$ is analogous to concatenation-based conditioning where the conditional information is concatenated to the feature maps (Dumoulin et al., 2018). The conditional scaling factor provides multiplicative interactions between the metric space and the feature maps to aggregate information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Furthermore, the goal of the conditionally transformed convolutional block is to simultaneously capture the two views of a classification task: a global view that is aware of the relationships among all classes, and a local views of the current N-way K-shot classification task. The metric space, or the global view, is pre-trained in a way that is independent of the current N-way K-shot task; while the base-learner, or the local view, attempts to develop representations for the current classification task. Although the metric space is never trained on the meta-test classes, we expect the learned metric space to generalize from the meta-train tasks to the meta-test tasks. ",
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+        "type": "text",
+        "text": "We further describe parameter sharing for CBN learning in Section 3.3.1, and class-aware grouping in Section 3.3.2 which provides more statistical strength for more effective few-shot learning. ",
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+    {
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+        "img_path": "images/6827a286c9f904ceb91c65069c40b2fe7aaed0f0dba2dee08f13df56a310a3e8.jpg",
+        "image_caption": [
+            "Figure 3: CBN shared architecture "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/c7846b989a4ca1056287b41bd4a58573b2bbce0af7614f0fd64281fae6ae7405.jpg",
+        "image_caption": [
+            "Figure 4: Normalization methods. ‘C’ denotes channels, $\\mathrm { ^ { 6 } H }$ , W’ spatial dimensions and $\\mathbf { \\Delta } ^ { \\mathsf {  } } \\mathbf { N } ^ { \\mathsf { \\prime } }$ examples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.3.1 MULTITASK LEARNING OF CBN ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Although one can predict $\\hat { \\gamma } _ { c }$ and $\\hat { \\boldsymbol \\beta } _ { c }$ using two separate functions, we find it beneficial to use shared parameters as shown in Figure 3. The shared representations are more efficient at producing conditional transformations which also provide a strong inductive bias to help learning (Caruana, 1997). ",
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+    {
+        "type": "text",
+        "text": "3.3.2 CLASS-AWARE GROUPING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose class-aware grouping, as shown in Figure 4 (b), to further exploit properties of metric space. The motivation stems from a lack of statistical strength when learning from only a few examples. As an example, in $N .$ -way 1-shot learning, the model is required to find the most meaningful way to distinguish different classes. However, gradient-based optimization may lead to the discovery of irrelevant features which coincide with the class labels, such as background colors. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We address this problem by class-aware grouping that is guided by our metric space. This is related to “transduction”, which is a standard technique in MAML-based methods. Transduction as discussed in Nichol et al. (2018), makes use of the channel mean $\\mathbb { E } [ \\mathbf { R } _ { c } ]$ and variance $\\mathrm { V a r } [ \\mathbf { R } _ { c } ]$ , defined in Eq. (4), of query examples when evaluating a base-learner. In contrast to standard transduction methods that calculate mean and variance over all examples of the current batch, we introduce class-aware grouping that clusters examples into different groups and use group-based mean and variance to normalize different channels. The grouping is determined by distance measures in the metric space where examples are grouped together based on their nearest centroid $\\mathbf { } c _ { t }$ defined in Section 3.2. Class-aware grouping is integrated into CBN as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b194fc2f54ae42c5dc0e93e46508cc4fae593901707ddc93bd319c32478724f9.jpg",
+        "text": "$$\n\\mathrm { C B N } ( \\mathbf { R } _ { i , c } | \\hat { \\gamma } _ { i , c } , \\hat { \\beta } _ { i , c } ) = \\hat { \\gamma } _ { i , c } \\frac { \\mathbf { R } _ { i , c } - \\mathbb { E } [ \\mathbf { R } _ { i , c } \\cdot \\mathbb { 1 } _ { \\{ x _ { i } \\in t \\} } ] } { \\sqrt { \\mathrm { V a r } [ \\mathbf { R } _ { i , c } \\cdot \\mathbb { 1 } _ { \\{ x _ { i } \\in t \\} } ] + \\epsilon } } + \\hat { \\beta } _ { i , c } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            713,
+            678,
+            750
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { 1 } _ { \\{ x _ { i } \\in t \\} }$ indicates if an example $x _ { i }$ belongs to cluster $t$ , and $\\mathbb { E } [ \\mathbf { R } _ { i , c } { \\cdot } \\mathbb { 1 } _ { \\{ x _ { i } \\in t \\} } ]$ represents the average of channel $\\mathbf { R } _ { c }$ among examples clustered at $c _ { t }$ . This is depicted in Figure 4 where the channel mean and variance are calculated for every group.This approach informs the base-learner about what to expect from the query examples at the class level through channel mean and variance, which provides more explicit guidance to the meta-learning procedure. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The base-learner $( f _ { \\theta } )$ is composed of 4 layers of $3 \\times 3$ convolutions with a $4 \\times 4$ skip connections from the input to the final convolutional layer. The use of skip connections is to improve the gradient flow as MAML unfolds the inner loop into one computational graph. The use of skip connections is empirically important to the proposed model. Each convolutional layer has 30 channels and is followed by CBN, ReLU and $2 \\times 2$ max-pooling operations. The output of the final convolution is flattened and fed to a 1-layer dense classifier. For learning the metric space $( f _ { \\phi } )$ , we use the same residual network (ResNet-12) as Oreshkin et al. (2018). The metric space is pre-trained on the same meta-training dataset for 30,000 episodes and not updated while learning the base-learner. The meta-learner is trained for 50,000 episodes. We empirically observe that training the metric space and meta-learner end-to-end is overly complex and prone to over-fitting. For CBN functions $( f _ { c } )$ , we use 3 dense layers with 30 hidden units each. Every layer is followed by a ReLU except for the last layer where no activation is used. For the meta-learner, we use MAML with 1 gradient step for 1-shot learning and 5 gradient steps for 5-shot learning. We use the Adam (Kingma & Ba, 2014) optimizer and clip the L2 norm of gradients with an upper bound of 5. Similar to MAML, we use transduction where the statistics of the current batch is used for $\\mathbb { E } ( . )$ and $\\mathrm { V a r } ( . )$ in Eq. (4) for both training and testing. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 META-LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Meta-learning or “learning-to-learn” (Schmidhuber, 1987; Bengio et al., 1992; Mitchell & Thrun, 1993; Vilalta & Drissi, 2002) has been studied as a means to acquire meta-knowledge across many tasks. In recent years, meta-learning has become an important approach for few-shot learning. A number of approaches aim to learn universal learning procedure approximators by supplying training examples to the meta-learner that outputs predictions on testing examples (Hochreiter et al., 2001; Vinyals et al., 2016; Santoro et al., 2016; Mishra et al., 2017). Other approaches learn to generate model parameters conditioned on training examples (Gomez & Schmidhuber, 2005; Munkhdalai & Yu, 2017; Ha et al., 2016; Gidaris & Komodakis, 2018), or learning optimization algorithms across different tasks (Ravi & Larochelle, 2016; Andrychowicz et al., 2016; Li & Malik, 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1.1 GRADIENT-BASED META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our work is more inline with gradient-based meta-learning that aims to learn representations that encourage fast adaptation on new tasks. These methods are based on model-agnostic meta-learning (MAML) introduced by Finn et al. (2017). While the original MAML requires second-order gradients in meta-optimization, REPTILE (Nichol et al., 2018) only uses first-order gradient information. Furthermore, Latent Embedding Optimization (LEO) (Rusu et al., 2018) is proposed to perform gradient-based optimization on a lowdimensional latent space instead of the original high-dimensional parameter space. We emphasize that all those methods do not make explicit use of structured label information, which is a main novelty in this paper. ",
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+    {
+        "type": "text",
+        "text": "4.1.2 METRIC-BASED META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our work also relates closely to metric-based meta-learning that learns a metric space across different tasks. Siamese networks (Koch et al., 2015) learn a similarity measure between inputs using a shared network architecture that outputs high probability when paired examples are from the same class. Matching networks (Vinyals et al., 2016) use full context embeddings to encode examples to the metric space and use attention as a similarity measure for predictions. Prototypical networks (Snell et al., 2017) compute a centroid, or prototype, for every class that are later used for distance-based queries of new examples. Task dependent adaptive metric (TADAM) (Oreshkin et al., 2018) uses metric scaling based on tasks representations to learn a task-dependent metric space. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A notable difference between the metric-based methods and our approach is that, the metric space in our model is not aimed for distance-based classification. Rather, we use the metric space to represent class structure which facilitates the gradient-based meta learning towards better generalization. Another difference between our method and TADAM is that, TADAM scales the metric space at the task level where all examples within a task are scaled in the same manner. In contrast, our method provides instance-based conditioning that makes use of the precise representation of each example. Put another way, TADAM modulates the inference on a metric space from a task perspective, while CAML uses example-level representation to modulate the representation at the content level. ",
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+    {
+        "type": "text",
+        "text": "4.2 CONDITIONAL TRANSFORMATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In style transfer, conditional instance normalization is proposed by Dumoulin et al. (2017) that transforms the content image conditioned on the domain of the style image. In visual question answering, De Vries et al. (2017) have shown that it is beneficial to modulate early visual signals of a pre-trained residual network by language in the form of conditional batch normalization. It was further shown that feature-wise linear modulation (Perez et al., 2017; Dumoulin et al., 2018) can efficiently select meaningful representations for visual reasoning. ",
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+    {
+        "type": "image",
+        "img_path": "images/56fc28b20335c5d617cdd3dc2d850d31a4b4169fd8b5dbe3e0d18e91a83340b0.jpg",
+        "image_caption": [
+            "Figure 5: t-SNE visualization of the learned metric space colored by category. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The notion that is common to all these methods is the use of an additional input source, e.g., style or language, to conditionally transform intermediate representations of a network. In few-shot learning, Zhou et al. (2018) suggested that it is easier to operate in the concept space in the form of a lower dimensional representation. This is compatible with our proposed approach that uses the metric space as concept-level representation to modulate intermediate features of the base-learner. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use miniImageNet to evaluate the proposed Conditional class-Aware Meta-Learning algorithm. miniImageNet (Vinyals et al., 2016) is composed of $8 4 \\times 8 4$ colored images from 100 classes, with 600 examples in each class. We adopt the class split by Ravi & Larochelle (2016) that uses 64 classes for training, 16 for validation, and 20 for test. For $N .$ -way $K$ -shot training, we randomly sample $N$ classes from the meta-train classes each containing $K$ examples for training and 20 examples for testing. At meta-testing time, we randomly sample 600 $N$ -way $K$ -shot tasks from the test classes. ",
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+    {
+        "type": "text",
+        "text": "5.1 RESULTS ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results presented in Table 1 show that our proposed algorithm has comparable performance on the state-of-the-art miniImageNet 5-way 1-shot classification task, and competitive results on the 5-way 5-shot task. Unlike LEO (Rusu et al., 2018) that applies meta-learning on pre-trained representations, our meta-learner is able to effectively operate on the high-dimensional parameter space. Our method also does not require co-training compared with TADAM (Oreshkin et al., 2018). ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Figure 5 shows the t-SNE plot of the learned metric space for both meta-train and meta-validation classes. As seen in Figure 4b, examples from the meta-validation set form clusters consistent with their class membership, even though the metric space is not trained on these classes. For example, “mierkat”, “tundrarum” and “podenco” are all animals and they are clustered close together. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The first main baseline we report is MAML. CAML improves upon MAML by about $10 \\%$ on both 1-shot and 5-shot tasks. This means incorporating class dependencies in the form of a metric space can greatly facilitate gradient-based meta-learning. We also compare with MAML using our base-learner architecture equipped with skip connections from the input to the last convolutional layer. MAML trained with our base-learner’s architecture yields similar performance as the original MAML, suggesting the improvement is resulted from the proposed CAML framework, rather than changes in the base-learner’s architecture. ",
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+    {
+        "type": "table",
+        "img_path": "images/b37763941d23fc34e829d96081e2da690b87a8e59b20ca0d577d316e67e97369.jpg",
+        "table_caption": [
+            "Table 1: miniImageNet classification accuracy with $9 5 \\%$ confidence intervals. "
+        ],
+        "table_footnote": [
+            "The confidence intervals are constructed by sampling 600 evaluation tasks from the meta-test classes. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>Meta-Learner LSTM (Ravi &amp; Larochelle, 2016)</td><td>43.44% ± 0.77%</td><td>60.60% ± 0.71%</td></tr><tr><td>Matching Networks (Vinyals et al.,2016)</td><td>46.6%</td><td>60.0%</td></tr><tr><td>Prototypical Network with Soft k-Means (Ren et al., 2018)</td><td>50.41% ± 0.31%</td><td>69.88%± 0.20%</td></tr><tr><td>MetaNet (Munkhdalai &amp; Yu, 2017)</td><td>49.21% ± 0.96%</td><td></td></tr><tr><td>TCML (Mishra et al., 2018)</td><td>55.71% ± 0.99%</td><td>68.88% ± 0.92%</td></tr><tr><td>adaResNet (Munkhdalai etal., 2018)</td><td>56.88% ± 0.62%</td><td>71.94 ± 0.57%</td></tr><tr><td>Cosine Classifier (Gidaris &amp; Komodakis, 2018)</td><td>56.20% ± 0.86%</td><td>73.00% ± 0.64%</td></tr><tr><td>TADAM (Oreshkin et al., 2018)</td><td>58.5%</td><td>76.7%</td></tr><tr><td>LEO (Rusu et al., 2018)</td><td>61.76% ± 0.08%</td><td>77.59% ± 0.12%</td></tr><tr><td>MAML (Finn et al., 2017)</td><td>48.7% ±1.84%</td><td>63.11% ± 0.92%</td></tr><tr><td>MAML on our architecture</td><td>48.26% ±1.04%</td><td>64.25% ± 0.78%</td></tr><tr><td>Prototypical Network (Snell et al., 2017)</td><td>49.42% ± 0.78%</td><td>68.2% ± 0.66%</td></tr><tr><td>Prototypical Network on our metric space</td><td>55.96% ± 0.91%</td><td>71.64% ± 0.70%</td></tr><tr><td>CAML (with multitask learning alone)</td><td>52.56% ± 0.83%</td><td>71.35% ±1.13%</td></tr><tr><td>CAML (with class-aware grouping alone)</td><td>55.28% ±0.90%</td><td>71.14% ± 0.81%</td></tr><tr><td>CAML (full model)</td><td>59.23% ± 0.99%</td><td>72.35% ± 0.71%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/024b362258691f6916775cedf52202f77f2793ffd2486cb7713693298a9ebeed.jpg",
+        "image_caption": [
+            "Figure 6: PCA visualization of feature maps from the last convolutional layer colored by category. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            429,
+            746,
+            628
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The second baseline we use is prototypical network. We measure the classification ability of our metric space using prototypical network as a classifier, shown in Table 1 (Prototypical Network in our metric space). These results suggest that making predictions on the metric space alone is inferior to CAML.This can be explained by CAML’s ability to fast-adapt representations even when the metric space does not provide good separations. We also find that CAML has larger improvements in 1-shot tasks than 5-shot ones. This is because, in 1-shot learning, metric-based methods estimate class representations from a single example, making it difficult to provide a robust class estimation. ",
+        "bbox": [
+            174,
+            684,
+            825,
+            781
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 THE EFFECT OF CONDITIONAL TRANSFORMATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            800,
+            544,
+            814
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare activations before and after the conditional transformation to better understand how conditional transformation modulates the feature representations. Figure 6 shows the PCA projections of the last convolutional layer in the base-learner. We observe in Figure 5a that, before conditional transformation, examples from three classes (“parallel bars”, “tile roof” and “reel”) are mixed together. In Figure 5b, after the conditional transformation is applied, one of the previously cluttered classes (“tile roof”) become separated from the rest classes. This confirms that metric space can alleviate the difficulty in few-shot learning by means of conditional transformations. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            922
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We undertake ablation studies to show the impact of multitask learning and class-aware grouping. Empirical results in Table 1 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is not affected by those techniques. This is owing to a lack of statistical strength in 1-shot learning, which requires more explicit guidance in the training procedure. This means exploiting metric-based channel mean and variance can provide valuable information to improve meta-learning. More detailed ablation studies are included in Appendix A. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            207,
+            313,
+            223
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose Conditional class-Aware Meta-Learning (CAML) that incorporates class information by means of an embedding space to conditionally modulate representations of the base-learner. By conditionally transforming the intermediate representations of the base-learner, our goal is to reshape the representation with a global sense of class structure. Experiments reveal that the proposed conditional transformation can modulate the convolutional feature maps towards a more disentangled representation. We also introduce class-aware grouping to address a lack of statistical strength in few-shot learning. The proposed approach obtains competitive results with the current state-of-the-art performance on 5-way 1-shot and 5-shot miniImageNet benchmark. ",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
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+    {
+        "type": "text",
+        "text": "A ADDITIONAL ABLATION STUDIES ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "A.1 THE IMPACT OF MULTITASK LEARNING AND CLASS-AWARE GROUPING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            856,
+            683,
+            869
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To better understand the role of different components in the proposed conditional transformation, we undertake ablation studies to provide further insights into CBN. We study the impact of multitask learning detailed in Section 3.3.1 and class-aware grouping described in Section 3.3.2. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/7cc9b17cfd3bca9da61b489bdb8100949ca29a60eb6e3ebaeface14fa3b69d56.jpg",
+        "table_caption": [
+            "Table 2: Ablation study on the impact of multitask learning and class-aware grouping miniImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CBN</td><td>multitask</td><td>class-aware grouping</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td>×</td><td>×</td><td>×</td><td>48.26% ± 1.04%</td><td>64.25% ± 0.78%</td></tr><tr><td>√</td><td>×</td><td>×</td><td>52.06% ± 1.12%</td><td>69.84% ± 1.28%</td></tr><tr><td>√</td><td>√</td><td>×</td><td>52.56% ± 0.83%</td><td>71.35% ± 1.13%</td></tr><tr><td>√</td><td>×</td><td>√</td><td>55.28% ± 0.90%</td><td>71.14% ± 0.81%</td></tr><tr><td>√</td><td>√</td><td>√</td><td>59.23% ± 0.99%</td><td>72.35% ± 0.71%</td></tr></table>",
+        "bbox": [
+            230,
+            123,
+            767,
+            237
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/a0a25836fa057e3b937f3c93a7e677a1726c7b560b134ec22ca41e993355cd15.jpg",
+        "table_caption": [
+            "Table 3: Ablation study on the impact of conditional transformation operators for miniImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>BN</td><td>CBN with βc alone （</td><td> CBN with c alone</td><td>CBN</td></tr><tr><td>5-way 1-shot48.7% ± 1.84%</td><td></td><td>56.04% ± 0.99%</td><td>57.83% ± 1.04%</td><td>59.23% ± 0.99%</td></tr></table>",
+        "bbox": [
+            196,
+            287,
+            800,
+            335
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Empirical results from Table 2 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is less sensitive those techniques. This is owing to a lack of sufficient training examples in 1-shot learning tasks, which requires more explicit guidance in the training procedure. We further note that, in 1-shot learning, using class-aware grouping alone can improve CBN’s performance by $3 \\%$ . This means exploiting metric-based channel mean and variance can provide valuable information for gradient-based meta-learning. ",
+        "bbox": [
+            173,
+            366,
+            826,
+            450
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 THE IMPACT OF SCALE AND SHIFT TRANSFORMATIONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            468,
+            581,
+            481
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For CBN parameters, we observe that more than half of the predicted $\\hat { \\boldsymbol \\beta } _ { c }$ are negative. This is inline with findings from Perez et al. (2017) that CBN selectively suppresses activations of a feature map when followed by a ReLU. To further examine the impact of the scale and shift operators, we train CBN with each operator alone. Table 3 shows CBN works the best when both $\\hat { \\gamma } _ { c }$ and $\\hat { \\boldsymbol \\beta } _ { c }$ are used, and $\\hat { \\gamma } _ { c }$ contributes more than $\\hat { \\boldsymbol { \\beta } } _ { c }$ , owing to its multiplicative interactions between the metric space and convolutional feature representations. ",
+        "bbox": [
+            174,
+            484,
+            825,
+            556
+        ],
+        "page_idx": 10
+    }
+]

parse/train/CO87OIEOGU8/CO87OIEOGU8.md

@@ -0,0 +1,229 @@
+# Celebrating Diversity in Shared Multi-Agent Reinforcement Learning
+
+Chenghao Li, Tonghan Wang, Chengjie Wu, Qianchuan Zhao, Jun Yang∗, Chongjie Zhang∗
+
+Tsinghua University {lich18, wangth18, wucj19} $@$ mails.tsinghua.edu.cn, {zhaoqc, yangjun603, chongjie}@tsinghua.edu.cn
+
+# Abstract
+
+Recently, deep multi-agent reinforcement learning (MARL) has shown the promise to solve complex cooperative tasks. Its success is partly because of parameter sharing among agents. However, such sharing may lead agents to behave similarly and limit their coordination capacity. In this paper, we aim to introduce diversity in both optimization and representation of shared multi-agent reinforcement learning. Specifically, we propose an information-theoretical regularization to maximize the mutual information between agents’ identities and their trajectories, encouraging extensive exploration and diverse individualized behaviors. In representation, we incorporate agent-specific modules in the shared neural network architecture, which are regularized by L1-norm to promote learning sharing among agents while keeping necessary diversity. Empirical results show that our method achieves state-of-the-art performance on Google Research Football and super hard StarCraft II micromanagement tasks†.
+
+# 1 Introduction
+
+Cooperative multi-agent reinforcement learning (MARL) has drawn increasing interest in recent years, which provides a promise for solving many real-world challenging problems, such as sensor networks [1], traffic management [2], and coordination of robot swarms [3]. However, learning effective policies for such complex multi-agent systems remains challenging. One central problem is that the joint action-observation space grows exponentially with the number of agents, which imposes high demand on the scalability of learning algorithms.
+
+To address this scalability challenge, policy decentralization with shared parameters (PDSP) is widely used, where agents share their neural network weights. Parameter sharing significantly improves learning efficiency because it dramatically reduces the total number of policy parameters, while experiences and gradients of one agent can be used to train others. Enjoying these advantages, many advanced deep MARL approaches adopt the PDSP paradigm, including value-based methods [4–8], policy gradients [9–13] and communication learning algorithms [14, 15]. These approaches achieve state-of-the-art performance on tasks such as StarCraft II micromanagement [16].
+
+While parameter sharing has been proven to accelerate training [17], its drawbacks are also apparent in complex tasks. These tasks typically require substantial exploration and diversified strategies among agents. When parameters are shared, agents tend to acquire homogeneous behaviors because they typically adopt similar actions under similar observations, preventing efficient exploration and the emergence of sophisticated cooperative policies. This tendency becomes particularly problematic for many challenging multi-agent coordination tasks, hindering deep MARL from broader applications. For example, the unsatisfactory performance of state-of-the-art MARL algorithms on Google Research Football (Fig. 1, and [18]) highlights an urgent demand for diverse behaviors.
+
+Notably, sacrificing the merits of parameter sharing for diversity is also unfavorable. Like humans, sharing necessary experience or understanding of tasks can broadly accelerate cooperation learning. Without parameter sharing, agents search in a much larger parameter space, which may be wasteful because they do not need to behave differently all the time. Therefore, the question is how to adaptively trade-off diversity and sharing. In this paper, we solve this dilemma by proposing several structural and learning novelties.
+
+To encourage diversity, we propose a novel information-theoretical objective to maximize the mutual information between agents’ identi
+
+![](images/4e2aaabce11a8411e634b4d30eaa3173fbc88742f5d583b07c47b03c48962b38.jpg)  
+Figure 1: Shared parameters induce behaviors (left) and can hardly learn successful policies on the challenging Google Research Football task. Our method learns sophisticated cooperative strategies by trading off diversity and sharing (right).
+
+ties and trajectories. This objective enables each agent to distinguish themselves from others and thus involves the contribution of all agents. Accordingly, we derive an intrinsic reward for motivating diversity and optimize it with the global environmental reward by learning the total Q-function as a combination of individual Q-functions. Structurally, we further decompose individual Q-functions as the sum of shared and non-shared local Q-functions for sharing experiences while maintaining representation diversity. We hope agents can use and expand shared knowledge whenever possible. Thus we introduce L1 regularization on each non-shared Q-function, encouraging agents to share and be diverse when necessary on several critical actions. Combining these novelties achieves a dynamic balance between diversity and homogeneity, efficiently catalyzing adaptive and sophisticated cooperation.
+
+We benchmark our approach on Google Research Football (GRF) [18], and StarCraft II micromanagement tasks (SMAC) [16]. The extraordinary performance of our approach on challenging benchmarking tasks shows that our approach achieve significantly higher coordination capacity than baselines while using diversity as a catalyst for more robust and talent policies. To our best knowledge, our approach achieves state-of-the-art performance on SMAC super hard maps and challenging GRF multi-agent tasks like academy_3_vs_1_with_keeper, academy_counterattack_hard, and a full-field scenario 3_vs_1_with_keeper (full field).
+
+# 2 Background
+
+A fully cooperative multi-agent task can be formulated as a Dec-POMDP [19], which is defined as a tuple $\mathcal { G } = \left. N , S , A , P , R , \bar { O } , \Omega , n , \gamma \right.$ , where $N$ is a finite set of $n$ agents, $s \in S$ is the true state of the environment, $A$ is the set of actions, and $\gamma \in [ 0 , 1 )$ is a discount factor. At each time step, each agent $i \in N$ receives his own observation $o _ { i } \in \Omega$ according to the observation function $O ( s , i )$ , and selects an action $a _ { i } \in A$ , which results in a joint action vector $^ { a }$ . The environment then transitions to a new state $s ^ { \prime }$ based on the transition function $P ( s ^ { \prime } | s , \pmb { a } )$ , and inducing a global reward $r = R ( s , { \pmb a } )$ shared by all the agents. Each agent has its own action-observation history $\tau _ { i } \in \mathcal { T } _ { i } \doteq ( \Omega _ { i } \times A ) ^ { * }$ . Due to partial observability, each agent conditions its policy $\pi _ { i } ( a _ { i } | \tau _ { i } )$ on $\tau _ { i }$ . The joint policy $\pi$ induces the joint action-value function $\begin{array} { r } { Q _ { t o t } ^ { \pi } ( s , \pmb { a } ) = \mathbb { E } _ { s _ { 0 : \infty } , \pmb { a } _ { 0 : \infty } } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } \ | \ s _ { 0 } = s , \pmb { a } _ { 0 } = \pmb { a } , \pi \right] , } \end{array}$ .
+
+# 2.1 Centralized Training with Decentralized Execution
+
+Our method adopts the framework of centralized training with decentralized execution (CTDE) [9, 20, 4, 5, 21, 22, 6, 11]. This framework tackles the exponentially growing joint action space by decentralizing the control policies while adopting centralized training to learn cooperation. Agents learn in a centralized manner with access to global information but execute based on their local action-observation history. One promising approach to implement the CTDE framework is value function factorization. The IGM (individual-global-max) principle [21] guarantees the consistency between the local and global greedy actions. When IGM is satisfied, agents can obtain the optimal global action by simply choosing the local greedy action that maximizes each agent’s individual utility function $Q _ { i }$ . Some algorithms have successfully used the IGM principle [5, 6, 23] to push forward the progress of MARL.
+
+# 3 Method
+
+In this section, we present a novel diversity-driven MARL framework (Fig. 2) that balances each agent’s individuality with group coordination, which is a general approach that can be combined with existing CDTE value factorization methods.
+
+# 3.1 Identity-Aware Diversity
+
+We first introduce how to encourage behavioral diversity by designing intrinsic motivations. Intuitively, to encourage the specialty of individual trajectories, agents need to behave differently to highlight themselves from others, taking different actions and visiting different local observations. To achieve this goal, we use an information-theoretic objective for maximizing the mutual information between individual trajectory and agents’ identity:
+
+![](images/9b016a52d0d5eb6ec1e33bb9b99b38d0b4f9112abfd2063acfcded7806e26532.jpg)  
+Figure 2: Schematics of our approach.
+
+$$
+I ^ { \pi } ( \tau _ { T } ; i d ) = H ( \tau _ { T } ) - H ( \tau _ { T } | i d ) = E _ { i d , \tau _ { T } \sim \pi } \left[ \log \frac { p ( \tau _ { T } | i d ) } { p ( \tau _ { T } ) } \right] ,
+$$
+
+where $\tau _ { T }$ and $i d$ is the random variable for agent’s local trajectory and identity, respectively. $\pi$ is the joint policy. To optimize Eq. 1, we expand $p ( \tau _ { T } )$ as $\begin{array} { r } { p ( o _ { 0 } ) \prod _ { t = 0 } ^ { T - 1 } \dot { p } ( a _ { t } | \tau _ { t } ) p ( o _ { t + 1 } | \tau _ { t } , a _ { t } ) } \end{array}$ , and $p ( \tau _ { T } | i d )$ as $\begin{array} { r } { p ( o _ { 0 } | i d ) \prod _ { t = 0 } ^ { T - 1 } p ( a _ { t } | \tau _ { t } , i d ) p ( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d ) } \end{array}$ . Therefore, the mutual information can be written as:
+
+$$
+I ^ { \pi } ( \tau _ { T } ; i d ) = E _ { i d , \tau } \underbrace { [ \log \frac { p ( o _ { 0 } | i d ) } { p ( o _ { 0 } ) }  } _ { \textcircled { \backslash } } + \underbrace { \sum _ { t = 0 } ^ { T - 1 } \log \frac { p ( a _ { t } | \tau _ { t } , i d ) } { p ( a _ { t } | \tau _ { t } ) } } _ { \textcircled { \emptyset } } + \underbrace { \sum _ { t = 0 } ^ { T - 1 } \log \frac { p ( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \tau _ { t } , a _ { t } ) } } _ { \textcircled { \emptyset } } ] .
+$$
+
+Term $\textcircled{1}$ is determined by the environment, and we can ignore it when optimizing the mutual information. The second term quantifies the information gain about agent’s action selection when the identity is given, which measures action-aware diversity as $I ( a ; i d | \tau )$ . However, $p ( a _ { t } | \tau _ { t } , i d )$ is typically the distribution induced by $\epsilon$ -greedy, which only distinguishes the action with the highest possibility. Therefore, directly optimizing this term conceals most information about the local Qfunctions. To solve this problem, we use the Boltzmann softmax distribution of local Q values to replace $p ( a _ { t } | \tau _ { t } , i d )$ , which forms a lower bound of term $\textcircled{2}$ :
+
+$$
+E _ { i d , \tau } \left[ \log \frac { p ( a _ { t } | \tau _ { t } , i d ) } { p ( a _ { t } | \tau _ { t } ) } \right] \geq E _ { i d , \tau } \left[ \log \frac { \mathrm { S o f t M a x } ( \frac { 1 } { \alpha } Q ( a _ { t } | \tau _ { t } , i d ) ) } { p ( a _ { t } | \tau _ { t } ) } \right] .
+$$
+
+The inequity holds because the KL divergence $D _ { \mathrm { K L } } \big ( p ( \cdot | \tau _ { t } , i d ) \| \mathrm { S o f t M a x } ( \frac { 1 } { \alpha } Q ( \cdot | \tau _ { t } , i d ) ) \big )$ is nonnegative. We maximize this lower bound to optimize Term $\textcircled{2}$ . Inspired by variational inference approaches [24], we derive and optimize a tractable lower bound for Term $\textcircled{3}$ at each timestep by introducing a variational posterior estimator $q _ { \phi }$ parameterized by $\phi$ :
+
+$$
+E _ { i d , \tau } \left[ \log \frac { p ( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \tau _ { t } , a _ { t } ) } \right] \geq E _ { i d , \tau } \left[ \log \frac { q _ { \phi } ( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \tau _ { t } , a _ { t } ) } \right] ,
+$$
+
+Similar to the second term, the inequality holds because for any $q _ { \phi }$ , the $\mathrm { K L }$ divergence $D _ { \mathrm { K L } } ( p ( \cdot | \tau _ { t } , a _ { t } , i d ) | | q _ { \phi } ( \cdot | \tau _ { t } , a _ { t } , i d ) )$ is non-negative. Intuitively, optimizing Eq. 4 encourages agents
+
+to have diverse observations that are distinguishable by agents’ identification and thus measures observation-aware diversity as $I ( o ^ { \prime } ; i d | \tau , \bar { a ) }$ . To tighten the this lower bound, we minimize the KL divergence with respect to the parameters $\phi$ . The gradient for updating $\phi$ is:
+
+$$
+\begin{array} { r l } & { \nabla _ { \phi } \mathcal { L } ( \phi ) = \nabla _ { \phi } \mathbb { E } _ { \tau , a , i d } \left[ D _ { \mathrm { K L } } \left( p \left( \cdot | \tau , a , i d \right) \| q _ { \phi } \left( \cdot | \tau , a , i d \right) \right) \right] = \nabla _ { \phi } \mathbb { E } _ { \tau , a , i d , o ^ { \prime } } \left[ \log \frac { p \left( o ^ { \prime } | \tau , a , i d \right) } { q _ { \phi } \left( o ^ { \prime } | \tau , a , i d \right) } \right] } \\ & { \quad \quad \quad = - \mathbb { E } _ { \tau , a , i d , o ^ { \prime } } \left[ \nabla _ { \phi } \log q _ { \phi } \left( o ^ { \prime } | \tau , a , i d \right) \right] . } \end{array}
+$$
+
+Based on the lower bounds shown in Eq. 3 and Eq. 4, we introduce intrinsic rewards to optimise the information-theoretic objective (Eq. 1) for encouraging diverse behaviors:
+
+$$
+\begin{array} { r } { r ^ { I } = E _ { i d } \left[ \beta _ { 2 } D _ { \mathrm { K L } } ( \mathrm { S o f t M a x } ( \beta _ { 1 } Q ( \cdot | \tau _ { t } , i d ) ) | | p ( \cdot | \tau _ { t } ) ) \right. } \\ { \left. + \beta _ { 1 } \log q _ { \phi } ( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d ) - \log p ( o _ { t + 1 } | \tau _ { t } , a _ { t } ) \right] . } \end{array}
+$$
+
+We introduce two scaling factors $\beta _ { 1 } , \beta _ { 2 } \geq 0$ when calculating intrinsic rewards. When $\beta _ { 1 }$ is $_ 0$ , we only optimize the entropy term $H ( \tau _ { T } )$ in the mutual information objective (Eq. 1). $\beta _ { 2 }$ is used to adjust the importance of policy diversity compared with transition diversity. In Appendix A, we discuss and compare two different approaches for estimating $p \left( \boldsymbol { a } _ { t } | \tau _ { t } \right)$ and $p \left( o _ { t + 1 } \vert \tau _ { t } , a _ { t } \right)$ .
+
+# 3.2 Action-Value Learning for Balancing Diversity and Sharing
+
+In the previous section, we introduce an information-theoretic objective for encouraging each agent to behave differently from general trajectories. However, the shared local Q-function does not have enough capacity to present different policies for each agent. For solving this problem, we additionally equip each agent $i$ with an individual local Q-function $Q _ { i } ^ { I }$ . Defining experiences that need to be shared or exclusively learned is inefficient and usually can not generalize. Therefore, we let agents adaptively decide whether to share experiences by decomposing $Q _ { i }$ as:
+
+$$
+Q _ { i } ( a _ { i } | \tau _ { i } ) = Q ^ { S } ( a _ { i } | \tau _ { i } ) + Q _ { i } ^ { I } ( a _ { i } | \tau _ { i } ) ,
+$$
+
+where $Q ^ { S }$ is the shared Q-function among agents. In its current form, agents may learn to decompose their local Q-function arbitrarily. On the contrary, we expect that agents can share as much knowledge as possible so that we apply an L1 regularization on individual local Q-function $Q ^ { I }$ as shown in Fig.2. Such a regularization can also prevent agents from being too diverse and ignore cooperating to finish the task. In our experiments, we show that the L1 regularization is critical to achieving a balance between diversity and cooperation.
+
+# 3.3 Overall Learning Objective
+
+In this section, we discuss how to use the diversity-encouraging reward to train the proposed learning framework. Since the intrinsic rewards $r ^ { I }$ inevitably involves the influence from all agents, we add $r ^ { I }$ to environment rewards $r ^ { e }$ and use the following TD loss:
+
+$$
+\mathcal { L } _ { T D } ( \theta ) = \left[ r ^ { e } + \beta r ^ { I } + \gamma \operatorname* { m a x } _ { a ^ { \prime } } Q _ { t o t } \left( s ^ { \prime } , a ^ { \prime } ; \theta ^ { - } \right) - Q _ { t o t } ( s , a ; \theta ) \right] ^ { 2 } ,
+$$
+
+where $\theta$ is the parameters in the whole framework, $\theta ^ { - }$ is periodically frozen parameters copied from $\theta$ for a stable update, and $\beta$ is a hyper-parameter adjusting the weight of intrinsic rewards compared with environment rewards. We use QPLEX to decompose $Q _ { t o t }$ as mixing of local Q-functions $Q _ { i }$ and train the framework end-to-end by minimizing the loss:
+
+$$
+\mathcal { L } ( \boldsymbol { \theta } ) = \mathcal { L } _ { T D } ( \boldsymbol { \theta } ) + \lambda \sum _ { i } \mathcal { L } _ { L _ { 1 } } ( Q _ { i } ^ { I } ( \boldsymbol { \theta } _ { i } ^ { I } ) ) ,
+$$
+
+where $\theta _ { i } ^ { I }$ is the parameters of $Q _ { i } ^ { I }$ $, \mathcal { L } _ { L _ { 1 } } ( Q _ { i } ^ { I } )$ is the L1 regularization term for independent Q-functions, and $\lambda$ is a scaling factor.
+
+# 4 Case study: outperforming by being diverse only when necessary
+
+We design Pac-Men shown in Fig. 3 to demonstrate how our approach works. In this task, four agents are initialized at the center room and can only observe a $5 \times 5$ grid around them. Three dots are initialized randomly in each edge room. To make this environment more challenging, paths to different rooms have different lengths, which are $\mathrm { d o w n : l e f t : u p : r i g h t } = 4 : 8 : 1 2 : 8$ . Three out of four paths are outside agents’ observation scope, which brings about the difficulty of exploration. Dots will refresh randomly after all rooms are empty. An ineffective competition between agents occurs when they come together in one room. The total environmental reward is the number of dots eaten in one step or -0.1 if no one eats dots. The time limit of this environment is set to 100 steps.
+
+![](images/c0bb6ec7e349d1443212b8b9018658a6db659bfcb562a7d6a56d1e5d9dfbbe2d.jpg)  
+Figure 3: Why does our method work? The balance between identity-aware diversity and experience sharing encourages sophisticated strategies.
+
+Fig. 3-middle demonstrates the learned strategies of our approach, with a heatmap showing the visitation number. Driven by the objective of mutual information between individual trajectory and identity, agents achieve diversity and scatter in different rooms to eat dots. We further analyze the role of independent and shared Q-functions during different stages in Fig. 3 right. We visualize the value of $S D \dot { ( } Q _ { i } ^ { I } ( \cdot ) ) / S D ( Q ^ { S } ( \cdot ) )$ , where SD denotes the standard deviation (SD) of Q values for different actions. A higher SD ratio indicates the independent Q-functions play a leading role, while a lower SD ratio indicates the shared Q function’s domination.
+
+We notice that the SD ratio is considerably larger in the central room and four paths than in four edge rooms. This observation means that agents use independent Q networks to reach different rooms while use the shared Q network to search for dots in them. The result shows that our method achieves a good balance between diversity and knowledge sharing. Taking this advantage, our approach outperforms baselines (Fig. 3 left, baselines are introduced in Sec. 6). Other methods, such as variational exploration (MAVEN [25]) and individuality emergence (EOI [26]), are slower to learn optimal strategies.
+
+# 5 Related Work
+
+Deep multi-agent reinforcement learning algorithms have witnessed significant advances in recent years. COMA [20], MADDPG [9], PR2 [27], and DOP [10] study the problem of policy-based multi-agent reinforcement learning. They use a (decomposed) centralized critic to calculate gradients for decentralized actors. Value-based algorithms decompose the joint value function into individual utility functions in order to enable efficient optimization and decentralized execution. VDN [4], QMIX [5], and QTRAN [21] progressively expand the representation capabilities of the mixing network. QPLEX [6] implements the full IGM class [21] by encoding the IGM principle into a duplex dueling network architecture. Weighted QMIX [23] proposes weighted projection to decompose any joint action-value functions. There are other works that investigate into MARL from the perspective of coordination graphs [28–30], communication [31, 32, 15], and role-based learning [17, 33].
+
+Knowledge sharing in MARL From IQL [34] to QPLEX, many works focus on designing mixing network structures and have provided promising empirical and theoretical results. For these works, experience sharing among agents has been an important component. Learning from others is one essential skill engraved in humans’ genes to survive in society. Based on the relationship between teachers and students in human society, a series of research work hopes each agent can learn from others or selectively share its knowledge with others [35–37]. But it is challenging to specify knowledge in practice, let alone deciding what to share or learn. SEAC [38] partially solves this problem by sharing trajectories only for off-policy training. NCC [32] maintains cognition consistency by representation alignment between neighbors. Roy et al. [39] force each agent to predict others’ local policies and adds a coach for group experience alignment. Christianos et al. [40] group agents during pre-training and force agents in the same group to use one policy. In this paper, we do not try to let agents choose whether to learn or share experiences. Our neural network structure shown in Fig. 2 can balance group coordination and diversity by gradient backpropagation.
+
+Diversity In single-agent settings, diversity emerges for exploration or solving sparse reward problems. Existing methods such as curiosity-driven algorithms [41–44] or maximising mutual information [45–47] have shown great promise. When encouraging diversity in MARL settings, agents’ coordination must be considered. Several recent works study this problem, such as MAVEN [25], EITI & EDTI [48], and EOI [26]. MAVEN learns a diverse ensemble of monotonic approximations with the help of a latent space to explore. EITI and EDTI consider pairwise mutual influence to encourage the interdependence between agents. EOI combines the gradient from the intrinsic value function (IVF) and the total Q-function to train each agent’s local Q-function. In this paper, we encourage agents to explore unique trajectories by optimizing the mutual information between agent’s identity and trajectory. Moreover, we propose a novel network structure to enable experience sharing or consensus, which combines all agents’ rare ideas, while still maintain independent action-value functions for each agent to behave differently when necessary. Our approach considers the trade-off relationship between knowledge sharing and diversity, and learns to establish a balance and leverage their advantages for joint task solving.
+
+# 6 Experiments
+
+In Sec. 4, we use a toy game to illustrate how our approach adaptively balances experience sharing and identity-aware diversity. In this section, we use challenging tasks from GRF and SMAC benchmark to further demonstrate and illustrate the outperformance of our approach. We compare our approach against multi-agent value-based methods (QMIX [5], QPLEX [6]), variational exploration (MAVEN [25]), and individuality emergence (EOI [26]) methods. Different from baselines, we do not include agents’ identification in inputs when calculating local Q-functions. We show the average and variance of the performance for our method, baselines, and ablations tested with five random seeds.
+
+![](images/091154fc787f0091faf68632c735ae8d245aea38d00d6fbd6c3a01f5a38d37f4.jpg)  
+6.1 Performance on Google Research Football (GRF)   
+Figure 4: Comparison of our approach against baseline algorithms on Google Research Football.
+
+We first benchmark our approach on three challenging Google Research Football (GRF) offensive scenarios academy_3_vs_1_with_keeper, academy_counterattack_hard, and our own designed full-field scenario 3_vs_1_with_keeper (full field). Agents’ initial locations for each scenario are shown in Appendix B.3. In GRF tasks, agents need to coordinate timing and positions for organizing offense to seize fleeting opportunities, and only scoring leads to rewards. In our experiments, we control left-side players (in yellow) except the goalkeeper. The right-side players are rule-based bots controlled by the game engine. Agents have a discrete action space of 19, including moving in eight directions, sliding, shooting, and passing. The observation contains the positions and moving directions of the ego-agent, other agents, and the ball. The $z$ -coordinate of the ball is also included.
+
+We make a small and reasonable change to the half-court offensive scenarios: our players will lose if they or the ball returns to our half-court. All baselines and ablations are tested with this modification. Environmental reward only occurs at the end of the game. They will get $+ 1 0 0$ if they win, else get -1.
+
+We show the performance comparison against baselines in Fig. 4. Our approach outperforms all the scenarios. MAVEN needs more time to explore sophisticated strategies, demonstrating that CDS incentives more efficient exploration. EOI lets each agent consider individuality and cooperation simultaneously by setting local learning objectives but without exclusive Q networks, making cooperation and individuality hard to be persistently coordinated. In comparison, taking advantage of the partially shared network structure, CDS agents learn diverse but coordinated strategies. For example, as shown in Fig. 1, three agents have different behaviors, with the first agent passing the ball, the second scoring, while the third running to threaten. These diverse behaviors closely coordinate, forming a perfect scoring strategy and leading to significant outperformance against EOI.
+
+![](images/1877adb1813ca5c7c8d63f33743d7be7f34b234f741886e53fc2a42499523c27.jpg)  
+6.2 Performance on StarCraft II   
+Figure 5: Comparison of our approach against baseline algorithms on four super hard SMAC maps: corridor, MMM2, 6h_vs_8z, and $3 { \bf s } 5 z \_ { \bf V } { \bf s } \_ 3 { \bf s } 6 z$ and two hard SMAC maps: 5m_vs_6m and 3s_vs_5z.
+
+In this section, we test our approach on the StarCraft II micromanagement (SMAC) benchmarks [16]. This benchmark consists of various maps classified as easy, hard, and super hard. Here we test our method on four super hard maps: corridor, MMM2, 6h_vs_ $_ { 8 z }$ , and $3 { \bf s } 5 z \_ { \bf V } { \bf s } \_ 3 { \bf s } 6 z .$ , and two hard SMAC maps: 5m_vs_6m and 3s_vs_5z. For the four super hard maps, our approach outperforms all baselines with acceptable variance across random seeds, as shown in Fig. 5. The baselines QPLEX and QMIX can achieve satisfactory performance on some challenging benchmarks, such as $ { 3 \mathbf { s } } 5  { \mathbf { z } } _ { - }  { \mathbf { v } }  { \mathbf { s } } _ { - } 3  { \mathbf { s } } 6  { \mathbf { z } }$ and MMM2. But on other maps, they need the proposed diversity-celebrating method to get better performance. Compared with MAVEN and EOI, our approach maintains its out-performance with the balance between diversity and homogeneity for learning sophisticated cooperation. Our approach performs similarly with baselines for the two hard maps, indicating our balancing process may not improve the learning efficiency in environments that require pure homogeneity. But for challenging environments, where sophisticated strategies are laborious to explore, our approach can efficiently search for valuable strategies with stable updates.
+
+# 6.3 Ablations and Visualization
+
+To understand the contribution of each component in the proposed CDS framework, we carry out ablation studies to test the contribution of its three main components: Identity-aware diversity (A) encouragement and partially shared (B) neural network structure with $L I$ regularization (C) on non-shared Q-functions. To test component A, we ablate our intrinsic rewards to four different levels. (1) CDS-Raw ablates all intrinsic rewards by setting $\beta$ in Eq. 8 to zero. (2) CDS-No-Identity ablates
+
+$H \left( \tau _ { T } | i d \right)$ and only optimize $H \left( \tau _ { T } \right)$ in Eq. 1 by setting $\beta _ { 1 }$ in Eq. 6 to zero. (3) CDS-No-Action ablates item $\textcircled{2}$ in Eq. 2 by setting $\beta _ { 2 }$ in Eq. 6 to zero. (4) CDS-No-Obs ablates item $\textcircled{3}$ in Eq. 2 by ablating $\beta _ { 1 } \log q _ { \phi } \left( o _ { t + 1 } | \tau _ { t } , a _ { t } , i d \right) - \log p \left( o _ { t + 1 } | \tau _ { t } , a _ { t } \right)$ in Eq. 6. To test component B, we design CDS-All-Shared, which ablates independent action-value functions together with the L1 loss and, like baselines, adds agents’ identification to the input. To test component C, we design CDS-No-L1, which ablates L1 regularization terms by setting $\lambda$ in Eq. 9 to zero.
+
+![](images/f4f4acff22a099778f471bfeeae5749469d3e867aebfcbd2bc2d49bbd05f0917.jpg)  
+Figure 6: Left. Ablation studies on academy_counterattack_hard. Right. Visualization of trained policies, which achieve complex cooperation with impressive off-the-ball moving strategies.
+
+We first carry out ablation studies on academy_counterattack_hard to analyze which part of our novelties lead to the outstanding performance as shown on the left side of Fig. 6. The ablation of each part of our intrinsic reward will bring a noticeable decrease in performance. Among them, the least impact on performance is the ablation of action-aware diversity. CDS-No-L1 performs similarly to MAVEN, which indicates that unlimited diversity is harmful to cooperation. CDS-All-Shared performs even worse than QPLEX, demonstrating that identity-aware diversity is difficult to emerge without our specially designed network structure.
+
+We further visualize the final trained strategies on the right side of Fig. 6, which shows complex cooperation between agents. Our players first attack down the wing by dribbling and passing the ball. Then one of them draws the attention of the enemy defenders and the goalkeeper, while the ball being passed across the penalty area. Another player catches the ball and completes the shot. The most impressive part of our sophisticated strategies is off-the-ball moving strategies. All agents without the ball try to use their unique and valuable moves to create more scoring opportunities, which shows behavior and position diversity for finishing the goal.
+
+![](images/ad95f5b2204577f5c96d869c4f14a789e84d6d630291fc5993e20a352f009131.jpg)  
+Figure 7: Left. Ablation studies in super hard map corridor. Right. Visualization of the final trained strategies, which achieves a hard-earned victory brought by the sacrifice of a warrior.
+
+We also carry out ablation studies on the super hard map corridor as shown in Fig. 7 left. Same as results on academy_counterattack_hard, the ablation of action-aware diversity causes the least performance gap. Among all the ablations, CDS-No-L1 and CDS-No-Identity perform worst, whose performance is similar to QPLEX. This phenomenon indicates excessive diversity is harmful to the emerge of complex cooperation. CDS-All-Shared achieves acceptable performance, different from the GRF scenario, reflecting the different demand levels for the representation diversity of these two kinds of benchmarks.
+
+To better explain why our approach performs well. On corridor, we also visualize the final strategies in Fig. 7 right. In this super hard map, six friendly Zealots are facing 24 enemy Zerglings. The disparity in quantity means our agents are doomed to lose if they attack together. One Zealot, whose route is highlighted blue, becomes a warrior leaving the team to attract the attention of most enemies in the blue oval. Although doomed to sacrifice, he brings enough time for the team to eliminate a small part of the enemies in the green oval. After that, another Zealot stands out to attract some enemies and enables teammates to eradicate them. These sophisticated strategies reflect the leverage between diversity and homogeneity by encouraging agents to be diverse only when necessary.
+
+# 7 Closing Remarks
+
+Observing that behavioral diversity among agents is essential for many challenging and complex multi-agent tasks, in this paper, we introduce a novel mechanism of being diverse when necessary into shared multi-agent reinforcement learning. The balance between individual diversity and group coordination induced by our CDS approach pushes forward state-of-the-art of deep MARL on challenging benchmark tasks while keeping parameter sharing benefits. We hope that our method can shed light on future works to motivate agents to cooperate with diversity to further explore complex multi-agent coordination problems.
+
+# Acknowledgments
+
+This work was funded by the National Key Research and Development Project of China under Grant 2017YFC0704100 and 2016YFB0901900, in part by the National Natural Science Foundation of China under Grant 61425027 and U1813216, in part by Science and Technology Innovation $2 0 3 0 -$ “New Generation Artificial Intelligence” Major Project (No. 2018AAA0100904), a grant from the Institute of Guo Qiang,Tsinghua University, and a grant from Turing AI Institute of Nanjing.
+
+# References
+
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+        "text": "Celebrating Diversity in Shared Multi-Agent Reinforcement Learning ",
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+        "type": "text",
+        "text": "Chenghao Li, Tonghan Wang, Chengjie Wu, Qianchuan Zhao, Jun Yang∗, Chongjie Zhang∗ ",
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+        "text": "Tsinghua University {lich18, wangth18, wucj19} $@$ mails.tsinghua.edu.cn, {zhaoqc, yangjun603, chongjie}@tsinghua.edu.cn ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "type": "text",
+        "text": "Recently, deep multi-agent reinforcement learning (MARL) has shown the promise to solve complex cooperative tasks. Its success is partly because of parameter sharing among agents. However, such sharing may lead agents to behave similarly and limit their coordination capacity. In this paper, we aim to introduce diversity in both optimization and representation of shared multi-agent reinforcement learning. Specifically, we propose an information-theoretical regularization to maximize the mutual information between agents’ identities and their trajectories, encouraging extensive exploration and diverse individualized behaviors. In representation, we incorporate agent-specific modules in the shared neural network architecture, which are regularized by L1-norm to promote learning sharing among agents while keeping necessary diversity. Empirical results show that our method achieves state-of-the-art performance on Google Research Football and super hard StarCraft II micromanagement tasks†. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Cooperative multi-agent reinforcement learning (MARL) has drawn increasing interest in recent years, which provides a promise for solving many real-world challenging problems, such as sensor networks [1], traffic management [2], and coordination of robot swarms [3]. However, learning effective policies for such complex multi-agent systems remains challenging. One central problem is that the joint action-observation space grows exponentially with the number of agents, which imposes high demand on the scalability of learning algorithms. ",
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+        "type": "text",
+        "text": "To address this scalability challenge, policy decentralization with shared parameters (PDSP) is widely used, where agents share their neural network weights. Parameter sharing significantly improves learning efficiency because it dramatically reduces the total number of policy parameters, while experiences and gradients of one agent can be used to train others. Enjoying these advantages, many advanced deep MARL approaches adopt the PDSP paradigm, including value-based methods [4–8], policy gradients [9–13] and communication learning algorithms [14, 15]. These approaches achieve state-of-the-art performance on tasks such as StarCraft II micromanagement [16]. ",
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+        "type": "text",
+        "text": "While parameter sharing has been proven to accelerate training [17], its drawbacks are also apparent in complex tasks. These tasks typically require substantial exploration and diversified strategies among agents. When parameters are shared, agents tend to acquire homogeneous behaviors because they typically adopt similar actions under similar observations, preventing efficient exploration and the emergence of sophisticated cooperative policies. This tendency becomes particularly problematic for many challenging multi-agent coordination tasks, hindering deep MARL from broader applications. For example, the unsatisfactory performance of state-of-the-art MARL algorithms on Google Research Football (Fig. 1, and [18]) highlights an urgent demand for diverse behaviors. ",
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+        "text": "Notably, sacrificing the merits of parameter sharing for diversity is also unfavorable. Like humans, sharing necessary experience or understanding of tasks can broadly accelerate cooperation learning. Without parameter sharing, agents search in a much larger parameter space, which may be wasteful because they do not need to behave differently all the time. Therefore, the question is how to adaptively trade-off diversity and sharing. In this paper, we solve this dilemma by proposing several structural and learning novelties. ",
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+        "text": "To encourage diversity, we propose a novel information-theoretical objective to maximize the mutual information between agents’ identi",
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+        "image_caption": [
+            "Figure 1: Shared parameters induce behaviors (left) and can hardly learn successful policies on the challenging Google Research Football task. Our method learns sophisticated cooperative strategies by trading off diversity and sharing (right). "
+        ],
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+    {
+        "type": "text",
+        "text": "ties and trajectories. This objective enables each agent to distinguish themselves from others and thus involves the contribution of all agents. Accordingly, we derive an intrinsic reward for motivating diversity and optimize it with the global environmental reward by learning the total Q-function as a combination of individual Q-functions. Structurally, we further decompose individual Q-functions as the sum of shared and non-shared local Q-functions for sharing experiences while maintaining representation diversity. We hope agents can use and expand shared knowledge whenever possible. Thus we introduce L1 regularization on each non-shared Q-function, encouraging agents to share and be diverse when necessary on several critical actions. Combining these novelties achieves a dynamic balance between diversity and homogeneity, efficiently catalyzing adaptive and sophisticated cooperation. ",
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+    {
+        "type": "text",
+        "text": "We benchmark our approach on Google Research Football (GRF) [18], and StarCraft II micromanagement tasks (SMAC) [16]. The extraordinary performance of our approach on challenging benchmarking tasks shows that our approach achieve significantly higher coordination capacity than baselines while using diversity as a catalyst for more robust and talent policies. To our best knowledge, our approach achieves state-of-the-art performance on SMAC super hard maps and challenging GRF multi-agent tasks like academy_3_vs_1_with_keeper, academy_counterattack_hard, and a full-field scenario 3_vs_1_with_keeper (full field). ",
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+        "type": "text",
+        "text": "2 Background ",
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+    },
+    {
+        "type": "text",
+        "text": "A fully cooperative multi-agent task can be formulated as a Dec-POMDP [19], which is defined as a tuple $\\mathcal { G } = \\left. N , S , A , P , R , \\bar { O } , \\Omega , n , \\gamma \\right.$ , where $N$ is a finite set of $n$ agents, $s \\in S$ is the true state of the environment, $A$ is the set of actions, and $\\gamma \\in [ 0 , 1 )$ is a discount factor. At each time step, each agent $i \\in N$ receives his own observation $o _ { i } \\in \\Omega$ according to the observation function $O ( s , i )$ , and selects an action $a _ { i } \\in A$ , which results in a joint action vector $^ { a }$ . The environment then transitions to a new state $s ^ { \\prime }$ based on the transition function $P ( s ^ { \\prime } | s , \\pmb { a } )$ , and inducing a global reward $r = R ( s , { \\pmb a } )$ shared by all the agents. Each agent has its own action-observation history $\\tau _ { i } \\in \\mathcal { T } _ { i } \\doteq ( \\Omega _ { i } \\times A ) ^ { * }$ . Due to partial observability, each agent conditions its policy $\\pi _ { i } ( a _ { i } | \\tau _ { i } )$ on $\\tau _ { i }$ . The joint policy $\\pi$ induces the joint action-value function $\\begin{array} { r } { Q _ { t o t } ^ { \\pi } ( s , \\pmb { a } ) = \\mathbb { E } _ { s _ { 0 : \\infty } , \\pmb { a } _ { 0 : \\infty } } \\left[ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } \\ | \\ s _ { 0 } = s , \\pmb { a } _ { 0 } = \\pmb { a } , \\pi \\right] , } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "2.1 Centralized Training with Decentralized Execution ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our method adopts the framework of centralized training with decentralized execution (CTDE) [9, 20, 4, 5, 21, 22, 6, 11]. This framework tackles the exponentially growing joint action space by decentralizing the control policies while adopting centralized training to learn cooperation. Agents learn in a centralized manner with access to global information but execute based on their local action-observation history. One promising approach to implement the CTDE framework is value function factorization. The IGM (individual-global-max) principle [21] guarantees the consistency between the local and global greedy actions. When IGM is satisfied, agents can obtain the optimal global action by simply choosing the local greedy action that maximizes each agent’s individual utility function $Q _ { i }$ . Some algorithms have successfully used the IGM principle [5, 6, 23] to push forward the progress of MARL. ",
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+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we present a novel diversity-driven MARL framework (Fig. 2) that balances each agent’s individuality with group coordination, which is a general approach that can be combined with existing CDTE value factorization methods. ",
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+        "text": "3.1 Identity-Aware Diversity ",
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+        "type": "text",
+        "text": "We first introduce how to encourage behavioral diversity by designing intrinsic motivations. Intuitively, to encourage the specialty of individual trajectories, agents need to behave differently to highlight themselves from others, taking different actions and visiting different local observations. To achieve this goal, we use an information-theoretic objective for maximizing the mutual information between individual trajectory and agents’ identity: ",
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+        "image_caption": [
+            "Figure 2: Schematics of our approach. "
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+        "img_path": "images/8a0deb2560b493d535eda5fb3c7a5fe78ec7fd3a2c208d631c99c145b5fc4920.jpg",
+        "text": "$$\nI ^ { \\pi } ( \\tau _ { T } ; i d ) = H ( \\tau _ { T } ) - H ( \\tau _ { T } | i d ) = E _ { i d , \\tau _ { T } \\sim \\pi } \\left[ \\log \\frac { p ( \\tau _ { T } | i d ) } { p ( \\tau _ { T } ) } \\right] ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\tau _ { T }$ and $i d$ is the random variable for agent’s local trajectory and identity, respectively. $\\pi$ is the joint policy. To optimize Eq. 1, we expand $p ( \\tau _ { T } )$ as $\\begin{array} { r } { p ( o _ { 0 } ) \\prod _ { t = 0 } ^ { T - 1 } \\dot { p } ( a _ { t } | \\tau _ { t } ) p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } ) } \\end{array}$ , and $p ( \\tau _ { T } | i d )$ as $\\begin{array} { r } { p ( o _ { 0 } | i d ) \\prod _ { t = 0 } ^ { T - 1 } p ( a _ { t } | \\tau _ { t } , i d ) p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d ) } \\end{array}$ . Therefore, the mutual information can be written as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae5f72423fed996511f816eb9a2dd71e770a785cd884fcd343dcb1e1afb75b61.jpg",
+        "text": "$$\nI ^ { \\pi } ( \\tau _ { T } ; i d ) = E _ { i d , \\tau } \\underbrace { [ \\log \\frac { p ( o _ { 0 } | i d ) } { p ( o _ { 0 } ) }  } _ { \\textcircled { \\backslash } } + \\underbrace { \\sum _ { t = 0 } ^ { T - 1 } \\log \\frac { p ( a _ { t } | \\tau _ { t } , i d ) } { p ( a _ { t } | \\tau _ { t } ) } } _ { \\textcircled { \\emptyset } } + \\underbrace { \\sum _ { t = 0 } ^ { T - 1 } \\log \\frac { p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } ) } } _ { \\textcircled { \\emptyset } } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            561,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Term $\\textcircled{1}$ is determined by the environment, and we can ignore it when optimizing the mutual information. The second term quantifies the information gain about agent’s action selection when the identity is given, which measures action-aware diversity as $I ( a ; i d | \\tau )$ . However, $p ( a _ { t } | \\tau _ { t } , i d )$ is typically the distribution induced by $\\epsilon$ -greedy, which only distinguishes the action with the highest possibility. Therefore, directly optimizing this term conceals most information about the local Qfunctions. To solve this problem, we use the Boltzmann softmax distribution of local Q values to replace $p ( a _ { t } | \\tau _ { t } , i d )$ , which forms a lower bound of term $\\textcircled{2}$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/397d6b4d44d4de2f1e3e464bdd9d2afd56cac85068122f76829fd493d02956e4.jpg",
+        "text": "$$\nE _ { i d , \\tau } \\left[ \\log \\frac { p ( a _ { t } | \\tau _ { t } , i d ) } { p ( a _ { t } | \\tau _ { t } ) } \\right] \\geq E _ { i d , \\tau } \\left[ \\log \\frac { \\mathrm { S o f t M a x } ( \\frac { 1 } { \\alpha } Q ( a _ { t } | \\tau _ { t } , i d ) ) } { p ( a _ { t } | \\tau _ { t } ) } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The inequity holds because the KL divergence $D _ { \\mathrm { K L } } \\big ( p ( \\cdot | \\tau _ { t } , i d ) \\| \\mathrm { S o f t M a x } ( \\frac { 1 } { \\alpha } Q ( \\cdot | \\tau _ { t } , i d ) ) \\big )$ is nonnegative. We maximize this lower bound to optimize Term $\\textcircled{2}$ . Inspired by variational inference approaches [24], we derive and optimize a tractable lower bound for Term $\\textcircled{3}$ at each timestep by introducing a variational posterior estimator $q _ { \\phi }$ parameterized by $\\phi$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee51afc487810a2b262a334293dcde3f47e87f4e5401f13a54811b5daf062f02.jpg",
+        "text": "$$\nE _ { i d , \\tau } \\left[ \\log \\frac { p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } ) } \\right] \\geq E _ { i d , \\tau } \\left[ \\log \\frac { q _ { \\phi } ( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d ) } { p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } ) } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            842,
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+            877
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similar to the second term, the inequality holds because for any $q _ { \\phi }$ , the $\\mathrm { K L }$ divergence $D _ { \\mathrm { K L } } ( p ( \\cdot | \\tau _ { t } , a _ { t } , i d ) | | q _ { \\phi } ( \\cdot | \\tau _ { t } , a _ { t } , i d ) )$ is non-negative. Intuitively, optimizing Eq. 4 encourages agents ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "to have diverse observations that are distinguishable by agents’ identification and thus measures observation-aware diversity as $I ( o ^ { \\prime } ; i d | \\tau , \\bar { a ) }$ . To tighten the this lower bound, we minimize the KL divergence with respect to the parameters $\\phi$ . The gradient for updating $\\phi$ is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce24afb95b6157a86374bae0a69dcd36b8a5513750ac7c705c1caa21b8ca1bfa.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\phi } \\mathcal { L } ( \\phi ) = \\nabla _ { \\phi } \\mathbb { E } _ { \\tau , a , i d } \\left[ D _ { \\mathrm { K L } } \\left( p \\left( \\cdot | \\tau , a , i d \\right) \\| q _ { \\phi } \\left( \\cdot | \\tau , a , i d \\right) \\right) \\right] = \\nabla _ { \\phi } \\mathbb { E } _ { \\tau , a , i d , o ^ { \\prime } } \\left[ \\log \\frac { p \\left( o ^ { \\prime } | \\tau , a , i d \\right) } { q _ { \\phi } \\left( o ^ { \\prime } | \\tau , a , i d \\right) } \\right] } \\\\ & { \\quad \\quad \\quad = - \\mathbb { E } _ { \\tau , a , i d , o ^ { \\prime } } \\left[ \\nabla _ { \\phi } \\log q _ { \\phi } \\left( o ^ { \\prime } | \\tau , a , i d \\right) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on the lower bounds shown in Eq. 3 and Eq. 4, we introduce intrinsic rewards to optimise the information-theoretic objective (Eq. 1) for encouraging diverse behaviors: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a3d016d7e4333f3018085fcbbefb20fc900ce6a16d277ff8ad7a6e97640d7fc.jpg",
+        "text": "$$\n\\begin{array} { r } { r ^ { I } = E _ { i d } \\left[ \\beta _ { 2 } D _ { \\mathrm { K L } } ( \\mathrm { S o f t M a x } ( \\beta _ { 1 } Q ( \\cdot | \\tau _ { t } , i d ) ) | | p ( \\cdot | \\tau _ { t } ) ) \\right. } \\\\ { \\left. + \\beta _ { 1 } \\log q _ { \\phi } ( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d ) - \\log p ( o _ { t + 1 } | \\tau _ { t } , a _ { t } ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We introduce two scaling factors $\\beta _ { 1 } , \\beta _ { 2 } \\geq 0$ when calculating intrinsic rewards. When $\\beta _ { 1 }$ is $_ 0$ , we only optimize the entropy term $H ( \\tau _ { T } )$ in the mutual information objective (Eq. 1). $\\beta _ { 2 }$ is used to adjust the importance of policy diversity compared with transition diversity. In Appendix A, we discuss and compare two different approaches for estimating $p \\left( \\boldsymbol { a } _ { t } | \\tau _ { t } \\right)$ and $p \\left( o _ { t + 1 } \\vert \\tau _ { t } , a _ { t } \\right)$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 Action-Value Learning for Balancing Diversity and Sharing ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the previous section, we introduce an information-theoretic objective for encouraging each agent to behave differently from general trajectories. However, the shared local Q-function does not have enough capacity to present different policies for each agent. For solving this problem, we additionally equip each agent $i$ with an individual local Q-function $Q _ { i } ^ { I }$ . Defining experiences that need to be shared or exclusively learned is inefficient and usually can not generalize. Therefore, we let agents adaptively decide whether to share experiences by decomposing $Q _ { i }$ as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/062d3ea8271265cafc0f97a1c091640e39a9e6fc5a4db910ef972d07b42351e5.jpg",
+        "text": "$$\nQ _ { i } ( a _ { i } | \\tau _ { i } ) = Q ^ { S } ( a _ { i } | \\tau _ { i } ) + Q _ { i } ^ { I } ( a _ { i } | \\tau _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $Q ^ { S }$ is the shared Q-function among agents. In its current form, agents may learn to decompose their local Q-function arbitrarily. On the contrary, we expect that agents can share as much knowledge as possible so that we apply an L1 regularization on individual local Q-function $Q ^ { I }$ as shown in Fig.2. Such a regularization can also prevent agents from being too diverse and ignore cooperating to finish the task. In our experiments, we show that the L1 regularization is critical to achieving a balance between diversity and cooperation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 Overall Learning Objective ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we discuss how to use the diversity-encouraging reward to train the proposed learning framework. Since the intrinsic rewards $r ^ { I }$ inevitably involves the influence from all agents, we add $r ^ { I }$ to environment rewards $r ^ { e }$ and use the following TD loss: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/2ac4d3ca3e41ed291730e3cd367252d28506e1c6ac565a675d224a31ca69449c.jpg",
+        "text": "$$\n\\mathcal { L } _ { T D } ( \\theta ) = \\left[ r ^ { e } + \\beta r ^ { I } + \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { t o t } \\left( s ^ { \\prime } , a ^ { \\prime } ; \\theta ^ { - } \\right) - Q _ { t o t } ( s , a ; \\theta ) \\right] ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ is the parameters in the whole framework, $\\theta ^ { - }$ is periodically frozen parameters copied from $\\theta$ for a stable update, and $\\beta$ is a hyper-parameter adjusting the weight of intrinsic rewards compared with environment rewards. We use QPLEX to decompose $Q _ { t o t }$ as mixing of local Q-functions $Q _ { i }$ and train the framework end-to-end by minimizing the loss: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/0dd25515a9463b68d1b3ff05d50d41a177964884dd9974c3178732123984c38f.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } ) = \\mathcal { L } _ { T D } ( \\boldsymbol { \\theta } ) + \\lambda \\sum _ { i } \\mathcal { L } _ { L _ { 1 } } ( Q _ { i } ^ { I } ( \\boldsymbol { \\theta } _ { i } ^ { I } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            766,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta _ { i } ^ { I }$ is the parameters of $Q _ { i } ^ { I }$ $, \\mathcal { L } _ { L _ { 1 } } ( Q _ { i } ^ { I } )$ is the L1 regularization term for independent Q-functions, and $\\lambda$ is a scaling factor. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Case study: outperforming by being diverse only when necessary ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We design Pac-Men shown in Fig. 3 to demonstrate how our approach works. In this task, four agents are initialized at the center room and can only observe a $5 \\times 5$ grid around them. Three dots are initialized randomly in each edge room. To make this environment more challenging, paths to different rooms have different lengths, which are $\\mathrm { d o w n : l e f t : u p : r i g h t } = 4 : 8 : 1 2 : 8$ . Three out of four paths are outside agents’ observation scope, which brings about the difficulty of exploration. Dots will refresh randomly after all rooms are empty. An ineffective competition between agents occurs when they come together in one room. The total environmental reward is the number of dots eaten in one step or -0.1 if no one eats dots. The time limit of this environment is set to 100 steps. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "image",
+        "img_path": "images/c0bb6ec7e349d1443212b8b9018658a6db659bfcb562a7d6a56d1e5d9dfbbe2d.jpg",
+        "image_caption": [
+            "Figure 3: Why does our method work? The balance between identity-aware diversity and experience sharing encourages sophisticated strategies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Fig. 3-middle demonstrates the learned strategies of our approach, with a heatmap showing the visitation number. Driven by the objective of mutual information between individual trajectory and identity, agents achieve diversity and scatter in different rooms to eat dots. We further analyze the role of independent and shared Q-functions during different stages in Fig. 3 right. We visualize the value of $S D \\dot { ( } Q _ { i } ^ { I } ( \\cdot ) ) / S D ( Q ^ { S } ( \\cdot ) )$ , where SD denotes the standard deviation (SD) of Q values for different actions. A higher SD ratio indicates the independent Q-functions play a leading role, while a lower SD ratio indicates the shared Q function’s domination. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We notice that the SD ratio is considerably larger in the central room and four paths than in four edge rooms. This observation means that agents use independent Q networks to reach different rooms while use the shared Q network to search for dots in them. The result shows that our method achieves a good balance between diversity and knowledge sharing. Taking this advantage, our approach outperforms baselines (Fig. 3 left, baselines are introduced in Sec. 6). Other methods, such as variational exploration (MAVEN [25]) and individuality emergence (EOI [26]), are slower to learn optimal strategies. ",
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Deep multi-agent reinforcement learning algorithms have witnessed significant advances in recent years. COMA [20], MADDPG [9], PR2 [27], and DOP [10] study the problem of policy-based multi-agent reinforcement learning. They use a (decomposed) centralized critic to calculate gradients for decentralized actors. Value-based algorithms decompose the joint value function into individual utility functions in order to enable efficient optimization and decentralized execution. VDN [4], QMIX [5], and QTRAN [21] progressively expand the representation capabilities of the mixing network. QPLEX [6] implements the full IGM class [21] by encoding the IGM principle into a duplex dueling network architecture. Weighted QMIX [23] proposes weighted projection to decompose any joint action-value functions. There are other works that investigate into MARL from the perspective of coordination graphs [28–30], communication [31, 32, 15], and role-based learning [17, 33]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Knowledge sharing in MARL From IQL [34] to QPLEX, many works focus on designing mixing network structures and have provided promising empirical and theoretical results. For these works, experience sharing among agents has been an important component. Learning from others is one essential skill engraved in humans’ genes to survive in society. Based on the relationship between teachers and students in human society, a series of research work hopes each agent can learn from others or selectively share its knowledge with others [35–37]. But it is challenging to specify knowledge in practice, let alone deciding what to share or learn. SEAC [38] partially solves this problem by sharing trajectories only for off-policy training. NCC [32] maintains cognition consistency by representation alignment between neighbors. Roy et al. [39] force each agent to predict others’ local policies and adds a coach for group experience alignment. Christianos et al. [40] group agents during pre-training and force agents in the same group to use one policy. In this paper, we do not try to let agents choose whether to learn or share experiences. Our neural network structure shown in Fig. 2 can balance group coordination and diversity by gradient backpropagation. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Diversity In single-agent settings, diversity emerges for exploration or solving sparse reward problems. Existing methods such as curiosity-driven algorithms [41–44] or maximising mutual information [45–47] have shown great promise. When encouraging diversity in MARL settings, agents’ coordination must be considered. Several recent works study this problem, such as MAVEN [25], EITI & EDTI [48], and EOI [26]. MAVEN learns a diverse ensemble of monotonic approximations with the help of a latent space to explore. EITI and EDTI consider pairwise mutual influence to encourage the interdependence between agents. EOI combines the gradient from the intrinsic value function (IVF) and the total Q-function to train each agent’s local Q-function. In this paper, we encourage agents to explore unique trajectories by optimizing the mutual information between agent’s identity and trajectory. Moreover, we propose a novel network structure to enable experience sharing or consensus, which combines all agents’ rare ideas, while still maintain independent action-value functions for each agent to behave differently when necessary. Our approach considers the trade-off relationship between knowledge sharing and diversity, and learns to establish a balance and leverage their advantages for joint task solving. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Sec. 4, we use a toy game to illustrate how our approach adaptively balances experience sharing and identity-aware diversity. In this section, we use challenging tasks from GRF and SMAC benchmark to further demonstrate and illustrate the outperformance of our approach. We compare our approach against multi-agent value-based methods (QMIX [5], QPLEX [6]), variational exploration (MAVEN [25]), and individuality emergence (EOI [26]) methods. Different from baselines, we do not include agents’ identification in inputs when calculating local Q-functions. We show the average and variance of the performance for our method, baselines, and ablations tested with five random seeds. ",
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+    {
+        "type": "image",
+        "img_path": "images/091154fc787f0091faf68632c735ae8d245aea38d00d6fbd6c3a01f5a38d37f4.jpg",
+        "image_caption": [
+            "6.1 Performance on Google Research Football (GRF) ",
+            "Figure 4: Comparison of our approach against baseline algorithms on Google Research Football. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first benchmark our approach on three challenging Google Research Football (GRF) offensive scenarios academy_3_vs_1_with_keeper, academy_counterattack_hard, and our own designed full-field scenario 3_vs_1_with_keeper (full field). Agents’ initial locations for each scenario are shown in Appendix B.3. In GRF tasks, agents need to coordinate timing and positions for organizing offense to seize fleeting opportunities, and only scoring leads to rewards. In our experiments, we control left-side players (in yellow) except the goalkeeper. The right-side players are rule-based bots controlled by the game engine. Agents have a discrete action space of 19, including moving in eight directions, sliding, shooting, and passing. The observation contains the positions and moving directions of the ego-agent, other agents, and the ball. The $z$ -coordinate of the ball is also included. ",
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+        "type": "text",
+        "text": "We make a small and reasonable change to the half-court offensive scenarios: our players will lose if they or the ball returns to our half-court. All baselines and ablations are tested with this modification. Environmental reward only occurs at the end of the game. They will get $+ 1 0 0$ if they win, else get -1. ",
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+    {
+        "type": "text",
+        "text": "We show the performance comparison against baselines in Fig. 4. Our approach outperforms all the scenarios. MAVEN needs more time to explore sophisticated strategies, demonstrating that CDS incentives more efficient exploration. EOI lets each agent consider individuality and cooperation simultaneously by setting local learning objectives but without exclusive Q networks, making cooperation and individuality hard to be persistently coordinated. In comparison, taking advantage of the partially shared network structure, CDS agents learn diverse but coordinated strategies. For example, as shown in Fig. 1, three agents have different behaviors, with the first agent passing the ball, the second scoring, while the third running to threaten. These diverse behaviors closely coordinate, forming a perfect scoring strategy and leading to significant outperformance against EOI. ",
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+        "img_path": "images/1877adb1813ca5c7c8d63f33743d7be7f34b234f741886e53fc2a42499523c27.jpg",
+        "image_caption": [
+            "6.2 Performance on StarCraft II ",
+            "Figure 5: Comparison of our approach against baseline algorithms on four super hard SMAC maps: corridor, MMM2, 6h_vs_8z, and $3 { \\bf s } 5 z \\_ { \\bf V } { \\bf s } \\_ 3 { \\bf s } 6 z$ and two hard SMAC maps: 5m_vs_6m and 3s_vs_5z. "
+        ],
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+        "type": "text",
+        "text": "In this section, we test our approach on the StarCraft II micromanagement (SMAC) benchmarks [16]. This benchmark consists of various maps classified as easy, hard, and super hard. Here we test our method on four super hard maps: corridor, MMM2, 6h_vs_ $_ { 8 z }$ , and $3 { \\bf s } 5 z \\_ { \\bf V } { \\bf s } \\_ 3 { \\bf s } 6 z .$ , and two hard SMAC maps: 5m_vs_6m and 3s_vs_5z. For the four super hard maps, our approach outperforms all baselines with acceptable variance across random seeds, as shown in Fig. 5. The baselines QPLEX and QMIX can achieve satisfactory performance on some challenging benchmarks, such as $ { 3 \\mathbf { s } } 5  { \\mathbf { z } } _ { - }  { \\mathbf { v } }  { \\mathbf { s } } _ { - } 3  { \\mathbf { s } } 6  { \\mathbf { z } }$ and MMM2. But on other maps, they need the proposed diversity-celebrating method to get better performance. Compared with MAVEN and EOI, our approach maintains its out-performance with the balance between diversity and homogeneity for learning sophisticated cooperation. Our approach performs similarly with baselines for the two hard maps, indicating our balancing process may not improve the learning efficiency in environments that require pure homogeneity. But for challenging environments, where sophisticated strategies are laborious to explore, our approach can efficiently search for valuable strategies with stable updates. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "6.3 Ablations and Visualization ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "To understand the contribution of each component in the proposed CDS framework, we carry out ablation studies to test the contribution of its three main components: Identity-aware diversity (A) encouragement and partially shared (B) neural network structure with $L I$ regularization (C) on non-shared Q-functions. To test component A, we ablate our intrinsic rewards to four different levels. (1) CDS-Raw ablates all intrinsic rewards by setting $\\beta$ in Eq. 8 to zero. (2) CDS-No-Identity ablates ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$H \\left( \\tau _ { T } | i d \\right)$ and only optimize $H \\left( \\tau _ { T } \\right)$ in Eq. 1 by setting $\\beta _ { 1 }$ in Eq. 6 to zero. (3) CDS-No-Action ablates item $\\textcircled{2}$ in Eq. 2 by setting $\\beta _ { 2 }$ in Eq. 6 to zero. (4) CDS-No-Obs ablates item $\\textcircled{3}$ in Eq. 2 by ablating $\\beta _ { 1 } \\log q _ { \\phi } \\left( o _ { t + 1 } | \\tau _ { t } , a _ { t } , i d \\right) - \\log p \\left( o _ { t + 1 } | \\tau _ { t } , a _ { t } \\right)$ in Eq. 6. To test component B, we design CDS-All-Shared, which ablates independent action-value functions together with the L1 loss and, like baselines, adds agents’ identification to the input. To test component C, we design CDS-No-L1, which ablates L1 regularization terms by setting $\\lambda$ in Eq. 9 to zero. ",
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+    {
+        "type": "image",
+        "img_path": "images/f4f4acff22a099778f471bfeeae5749469d3e867aebfcbd2bc2d49bbd05f0917.jpg",
+        "image_caption": [
+            "Figure 6: Left. Ablation studies on academy_counterattack_hard. Right. Visualization of trained policies, which achieve complex cooperation with impressive off-the-ball moving strategies. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first carry out ablation studies on academy_counterattack_hard to analyze which part of our novelties lead to the outstanding performance as shown on the left side of Fig. 6. The ablation of each part of our intrinsic reward will bring a noticeable decrease in performance. Among them, the least impact on performance is the ablation of action-aware diversity. CDS-No-L1 performs similarly to MAVEN, which indicates that unlimited diversity is harmful to cooperation. CDS-All-Shared performs even worse than QPLEX, demonstrating that identity-aware diversity is difficult to emerge without our specially designed network structure. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We further visualize the final trained strategies on the right side of Fig. 6, which shows complex cooperation between agents. Our players first attack down the wing by dribbling and passing the ball. Then one of them draws the attention of the enemy defenders and the goalkeeper, while the ball being passed across the penalty area. Another player catches the ball and completes the shot. The most impressive part of our sophisticated strategies is off-the-ball moving strategies. All agents without the ball try to use their unique and valuable moves to create more scoring opportunities, which shows behavior and position diversity for finishing the goal. ",
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+    {
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+        "img_path": "images/ad95f5b2204577f5c96d869c4f14a789e84d6d630291fc5993e20a352f009131.jpg",
+        "image_caption": [
+            "Figure 7: Left. Ablation studies in super hard map corridor. Right. Visualization of the final trained strategies, which achieves a hard-earned victory brought by the sacrifice of a warrior. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also carry out ablation studies on the super hard map corridor as shown in Fig. 7 left. Same as results on academy_counterattack_hard, the ablation of action-aware diversity causes the least performance gap. Among all the ablations, CDS-No-L1 and CDS-No-Identity perform worst, whose performance is similar to QPLEX. This phenomenon indicates excessive diversity is harmful to the emerge of complex cooperation. CDS-All-Shared achieves acceptable performance, different from the GRF scenario, reflecting the different demand levels for the representation diversity of these two kinds of benchmarks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To better explain why our approach performs well. On corridor, we also visualize the final strategies in Fig. 7 right. In this super hard map, six friendly Zealots are facing 24 enemy Zerglings. The disparity in quantity means our agents are doomed to lose if they attack together. One Zealot, whose route is highlighted blue, becomes a warrior leaving the team to attract the attention of most enemies in the blue oval. Although doomed to sacrifice, he brings enough time for the team to eliminate a small part of the enemies in the green oval. After that, another Zealot stands out to attract some enemies and enables teammates to eradicate them. These sophisticated strategies reflect the leverage between diversity and homogeneity by encouraging agents to be diverse only when necessary. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Closing Remarks ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            181,
+            349,
+            199
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Observing that behavioral diversity among agents is essential for many challenging and complex multi-agent tasks, in this paper, we introduce a novel mechanism of being diverse when necessary into shared multi-agent reinforcement learning. The balance between individual diversity and group coordination induced by our CDS approach pushes forward state-of-the-art of deep MARL on challenging benchmark tasks while keeping parameter sharing benefits. We hope that our method can shed light on future works to motivate agents to cooperate with diversity to further explore complex multi-agent coordination problems. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            330,
+            328,
+            348
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work was funded by the National Key Research and Development Project of China under Grant 2017YFC0704100 and 2016YFB0901900, in part by the National Natural Science Foundation of China under Grant 61425027 and U1813216, in part by Science and Technology Innovation $2 0 3 0 -$ “New Generation Artificial Intelligence” Major Project (No. 2018AAA0100904), a grant from the Institute of Guo Qiang,Tsinghua University, and a grant from Turing AI Institute of Nanjing. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            454,
+            266,
+            469
+        ],
+        "page_idx": 8
+    },
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+        "type": "text",
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parse/train/ErivP29kYnx/ErivP29kYnx.md

@@ -0,0 +1,258 @@
+# ReSSL: Relational Self-Supervised Learning with Weak Augmentation
+
+Mingkai Zheng1,2 Shan $\mathbf { Y o u } ^ { 2 , 4 * }$ Fei Wang3 Chen Qian2 Changshui Zhang4 Xiaogang Wang2,5 Chang Xu1
+
+1School of Computer Science, Faculty of Engineering, The University of Sydney 2SenseTime Research 3University of Science and Technology of China 4Department of Automation, Tsinghua University, Institute for Artificial Intelligence, Tsinghua University (THUAI),   
+Beijing National Research Center for Information Science and Technology (BNRist) 5The Chinese University of Hong Kong
+
+# Abstract
+
+Self-supervised Learning (SSL) including the mainstream contrastive learning has achieved great success in learning visual representations without data annotations. However, most of methods mainly focus on the instance level information (i.e., the different augmented images of the same instance should have the same feature or cluster into the same class), but there is a lack of attention on the relationships between different instances. In this paper, we introduced a novel SSL paradigm, which we term as relational self-supervised learning (ReSSL) framework that learns representations by modeling the relationship between different instances. Specifically, our proposed method employs sharpened distribution of pairwise similarities among different instances as relation metric, which is thus utilized to match the feature embeddings of different augmentations. Moreover, to boost the performance, we argue that weak augmentations matter to represent a more reliable relation, and leverage momentum strategy for practical efficiency. Experimental results show that our proposed ReSSL significantly outperforms the previous stateof-the-art algorithms in terms of both performance and training efficiency. Code is available at https://github.com/KyleZheng1997/ReSSL
+
+# 1 Introduction
+
+Recently, self-supervised learning (SSL) has shown its superiority and achieved promising results for unsupervised visual representation learning in computer vision tasks [40, 27, 32, 6, 9, 47, 23, 24]. The purpose of a typical self-supervised learning algorithm is to learn general visual representations from a large amount of data without human annotations, which can be transferred or leveraged in downstream tasks (e.g., classification, detection, and segmentation). Some previous works [5, 23] even have proven that a good unsupervised pretraining can lead to a better downstream performance than supervised pretraining.
+
+Among various SSL algorithms, contrastive learning [47, 45, 6] serves as a state-of-the-art framework, which mainly focuses on learning an invariant feature from different views. For example, instance discrimination is a widely adopted pre-text task as in [6, 24, 47], which utilizes the noisy contrastive estimation (NCE) to encourage two augmented views of the same image to be pulled closer on the embedding space but pushes apart all the other images away. Deep Clustering [4, 48, 5] is an alternative pre-text task that forces different augmented views of the same instance to be clustered into the same class. However, instance discrimination based methods will inevitably induce a class collision problem [1, 36, 10], where similar images should be pulled closer instead of being pushed away. Deep clustering based methods cooperated with traditional clustering algorithms to assign a label for each instance, which relaxed the constraint of instance discrimination, but most of these algorithms adopt a strong assumption, i.e., the labels must induce an equipartition of the data, which might introduce some noise and hurt the learned representations.
+
+In this paper, we introduce a novel Relational Self-Supervised Learning framework (ReSSL), which does not encourage explicitly to push away different instances, but uses relation as a manner to investigate the inter-instance relationships and highlight the intra-instance invariance. Concretely, we aim to maintain the consistency of pairwise similarities among different instances for two different augmentations. For example, if we have three instances $\mathbf { x } ^ { 1 }$ , $\mathbf { x } ^ { 2 }$ , y and $\mathbf { z }$ where $\mathbf { x } ^ { 1 }$ , $\mathbf { x } ^ { 2 }$ are two different augmentations of $\mathbf { x }$ , $\mathbf { y }$ and $\mathbf { z }$ are different samples. Then, if $\mathbf { x } ^ { 1 }$ is similar to y but different to $\mathbf { z }$ , we wish $\bar { \mathbf { x } } ^ { 2 }$ can maintain such relationship and vice versa. In this way, the relation can be modelled as a similarity distribution between a set of augmented images, and then use it as a metric to align the same images with different augmentations, so that the relationship between different instances could be maintained across different views.
+
+However, this simple manner induces unexpectedly horrible performance if we follow the same training recipe as other contrastive learning methods [6, 24]. We argue that construction of a proper relation matters for ReSSL; aggressive data augmentations as in [6, 7, 41] are usually leveraged by default to generate diverse positive pairs that increase the difficulty of the pre-text task. However, this hurts the reliability of the target relation. Views generated by aggressive augmentations might cause the loss of semantic information, so the target relation might be noisy and not that reliable. In this way, we propose to leverage weaker augmentations to represent the relation, since much lesser disturbances provide more stable and meaningful relationships between different instances. Besides, we also sharpen the target distribution to emphasize the most important relationship and utilize the memory buffer with a momentum-updated network to reduce the demand of large batch size for more efficiency. Experimental results on multiple benchmark datasets show the superiority of ReSSL in terms of both performance and efficiency. For example, with 200 epochs of pre-training, our ReSSL achieved $6 9 . 9 \%$ on ImageNet [14] linear evaluation protocol, which is $2 . 4 \%$ higher than our baseline method (MoCoV2 [8]). When working with the Multi-Crop strategy (200 epochs), ReSSL achieved new state-of-the-art $7 4 . 7 \%$ Top-1 accuracy, which is $1 . 4 \%$ higher than CLSA-Multi [46].
+
+Our contributions can be summarized as follows.
+
+• We proposed a novel SSL paradigm, which we term it as relational self-supervised learning (ReSSL). ReSSL maintains the relational consistency between the instances under different augmentations instead of explicitly pushing different instances away. • Our proposed weak augmentation and sharpening distribution strategy provide a stable and high quality target similarity distribution, which makes the framework works well. • ReSSL is a simple and effective SSL framework since it replaces the widely adopted contrastive loss with our proposed relational consistency loss. It achieved state-of-the-art performance under the same training cost.
+
+# 2 Related Work
+
+Self-Supervised Learning. Early works in self-supervised learning methods rely on all sorts of pretext to learn visual representations. For example, colorizing gray-scale images [50], image jigsaw puzzle [39], image super-resolution [34], image inpainting [19], predicting a relative offset for a pair of patches [16], predicting the rotation angle [35], and image reconstruction [2, 22, 3, 17]. Although these methods have shown their effectiveness, they lack the generality of the learned representations.
+
+Instance Discrimination. The recent contrastive learning methods [32, 40, 6, 24, 41, 38, 29, 27, 30] have made a lot of progress in the field of self-supervised learning. Most of the previous contrastive learning methods are based on the instance discrimination [47] task in which positive pairs are defined as different views of the same image, while negative pairs are formed by sampling views from different images. SimCLR [6, 7] shows that image augmentation (e.g.Grayscale, Random Resized Cropping, Color Jittering, and Gaussian Blur), nonlinear projection head and large batch size plays a critical role in contrastive learning. Since large batch size usually requires a lot of GPU memory, which is not very friendly to most of researchers. MoCo [24, 8] proposed a momentum contrast mechanism that forces the query encoder to learn the representation from a slowly progressing key encoder and maintain a memory buffer to store a large number of negative samples. InfoMin [41] proposed a set of stronger augmentation that reduces the mutual information between views while keeping task-relevant information intact. AlignUniform [45] shows that alignment and uniformity are two critical properties of contrastive learning.
+
+Deep Clustering. In contrast to instance discrimination which treats every instance as a distinct class, deep clustering [4] adopts the traditional clustering method (e.g.KMeans) to label each image iteratively. Eventually, similar samples will be clustered into the same class. Simply apply the KMeans algorithm might lead to a degenerate solution where all data points are mapped to the same cluster; SeLa [48] solved this issue by adding the constraint that the labels must induce equipartition of the data and proposed a fast version of the Sinkhorn-Knopp to achieve this. SwAV [5] further extended this idea and proposed a scalable online clustering framework. PCL [36] reveals the class collision problem and simply performed instance discrimination and unsupervised clustering simultaneously; although it gets the same linear classification accuracy with MoCoV2, it has better performance on downstream tasks.
+
+Contrastive Learning Without Negatives. Most previous contrastive learning methods prevent the model collapse in an explicit manner (e.g. push different instances away from each other or force different instances to be clustered into different groups.) BYOL [23] can learn a high-quality representation without negatives. Specifically, it trains an online network to predict the target network representation of the same image under a different augmented view and using an additional predictor network on top of the online encoder to avoiding the model collapse. SimSiam [9] shows that simple Siamese networks can learn meaningful representations even without the use of negative pairs, large batch size, and momentum encoders.
+
+# 3 Methodology
+
+In this section, we will first revisit the preliminary work on contrastive learning; then, we will introduce our proposed relational self-supervised learning framework. After that, the algorithm and the implementation details will also be explained.
+
+# 3.1 Preliminaries on Self-supervised Learning
+
+Given $N$ unlabeled samples $\mathbf { x }$ , we randomly apply a composition of augmentation functions $T ( \cdot )$ to obtain two different views $\mathbf { x } ^ { 1 }$ and $\mathbf { x } ^ { 2 }$ through $T ( \mathbf { x } , \theta _ { 1 } ^ { \top } )$ and $T ( \mathbf { x } , \theta _ { 2 } )$ where $\theta$ is the random seed for $T$ . Then, a convolutional neural network based encoder $\mathcal F ( \cdot )$ is employed to extract the information from these samples, i.e., $\mathbf { h } = \mathcal { F } ( T ( \mathbf { x } , \theta ) )$ . Finally, a two-layer non-linear projection head $g ( \cdot )$ is utilized to map $\mathbf { h }$ into embedding space, which can be written as: ${ \mathbf z } = g ( { \mathbf h } )$ . SimCLR [6] and MoCo [24] style framework adopt the noise contrastive estimation (NCE) objective for discriminating different instances in the dataset. Suppose $\mathbf { z } _ { i } ^ { 1 }$ and $\mathbf { z } _ { i } ^ { 2 }$ are the representations of two augmented views of $\mathbf { x } _ { i }$ and $\mathbf { z } _ { k }$ is a different instance. The NCE objective can be expressed by Eq. (1), where the similarity function $s i m ( \cdot )$ represents the dot product between $L _ { 2 }$ normalized vectors $s i m ( \mathbf { u } , \mathbf { v } ) = \mathbf { u } ^ { T } \mathbf { v } / \| \mathbf { u } \| \| \mathbf { v } \|$ and $\tau$ is the temperature parameter.
+
+$$
+\mathcal { L } _ { N C E } = - \log \frac { \exp ( s i m ( \mathbf { z } ^ { 1 } , \mathbf { z } ^ { 2 } ) / \tau ) } { \exp ( s i m ( \mathbf { z } _ { i } ^ { 1 } , \mathbf { z } _ { i } ^ { 2 } ) / \tau ) + \sum _ { k = 1 } ^ { N } \exp ( s i m ( \mathbf { z } _ { i } ^ { 1 } , \mathbf { z } _ { k } ) / \tau ) } .
+$$
+
+BYOL [23] and SimSiam [9] style framework add an additional non-linear predictor head $q ( \cdot )$ which further maps $\mathbf { z }$ to $\mathbf { p }$ . The model will minimize the negative cosine similarity (equivalent to minimize the L2 distance) between $\mathbf { z }$ to $\mathbf { p }$ .
+
+$$
+\mathcal { L } _ { c o s } = - \frac { \mathbf { p } ^ { 1 } } { \Vert \mathbf { p } ^ { 1 } \Vert } \cdot \frac { \mathbf { z } ^ { 2 } } { \Vert \mathbf { z } ^ { 2 } \Vert } , \qquad \mathcal { L } _ { m s e } = \Vert \mathbf { p } ^ { 1 } - \mathbf { z } ^ { 2 } \Vert _ { 2 } ^ { 2 } .
+$$
+
+Tricks like stop-gradient and momentum teacher are often applied to avoid model collapsing.
+
+# 3.2 Relational Self-Supervised Learning
+
+In classical self-supervised learning, different instances are to be pushed away from each other, and augmented views of the same instance is expected to be of exactly the same features. However, both constrains are too restricted because of the existence of similar samples and the distorted semantic information if aggressive augmentation is adopted. In this way, we do not encourage explicit negative instances (those to be pushed away) for each instance; instead, we leverage the pairwise similarities as a manner to explore their relationships. And we pull the features of two different augmentations in this sense of relation metric. As a result, our method relaxes both (1) and (2), where different instances do not always need to be pushed away from each other; and augmented views of the same instance only need to share the similar but not exactly the same features.
+
+![](images/22fb022bc11cda14fbc85746e62700705dfdf636406ec392110ed5d841f7ee89.jpg)  
+Figure 1: The overall framework of our proposed method. We adopt the student-teacher framework where the student is trained to predict the representation of the teacher, and the teacher is updated with a “momentum update” (exponential moving average) of the student. The relationship consistency is achieve by align the conditional distribution for student and teacher model. Please see more details in our method part.
+
+Concretely, given a image $\mathbf { x }$ in a batch of samples , two different augmented views can be obtained by $\mathbf { x } ^ { 1 } = \mathbf { \check { \Gamma } } T ( \mathbf { x } , \theta _ { 1 } )$ , $\mathbf { x } ^ { 2 } = T ( \mathbf { x } , \theta _ { 2 } )$ and calculate the corresponds embedding $\mathbf { z } ^ { 1 } = g ( \mathcal { F } ( \mathbf { x } ^ { 1 } ) )$ , $\mathbf { z } ^ { 2 } = g ( \mathcal { F } ( \mathbf { x } ^ { 2 } ) )$ . Then, we calculate the similarities between the instances of the first augmented images. Which can be measured by $s i m ( \mathbf { z } ^ { 1 } , \mathbf { z } _ { i } )$ . A softmax layer can be adopted to process the calculated similarities, which then produces a relationship distribution:
+
+$$
+{ \bf p } _ { i } ^ { 1 } = \frac { \exp ( s i m ( { \bf z } ^ { 1 } , { \bf z } _ { i } ) / \tau _ { t } ) } { \sum _ { k = 1 } ^ { K } \exp ( s i m ( { \bf z } ^ { 1 } , { \bf z } _ { k } ) / \tau _ { t } , ) } .
+$$
+
+where $\tau _ { t }$ is the temperature parameter. At the same time, we can calculate the relationship between $\mathbf { x } ^ { 2 }$ and the $i$ -th instance as $s i m ( \mathbf { z } ^ { 2 } , \mathbf { z } _ { i } )$ . The resulting relationship distribution can be written as:
+
+$$
+{ { \bf { p } } _ { i } ^ { 2 } } = \frac { { \exp ( { s i m ( { { \bf { z } } ^ { 2 } } , { \bf { z } } _ { i } ) / { \tau _ { s } } } ) } } { { \sum _ { k = 1 } ^ { K } { \exp ( { s i m ( { { \bf { z } } ^ { 2 } } , { \bf { z } } _ { k } ) / { \tau _ { s } } } , ) } } } .
+$$
+
+where $\tau _ { s }$ is a different temperature parameter. We propose to push the relational consistency between $p _ { i } ^ { 1 }$ and $p _ { i } ^ { 2 }$ by minimizing the Kullback–Leibler divergence, which can be formulated as:
+
+$$
+\mathcal { L } _ { r e l a t i o n } = D _ { K L } ( \mathbf { p } ^ { 1 } | | \mathbf { p } ^ { 2 } ) = H ( \mathbf { p } ^ { 1 } , \mathbf { p } ^ { 2 } ) - H ( \mathbf { p } ^ { 1 } ) .
+$$
+
+Since the $\mathbf { p } ^ { 1 }$ will only be used as a target, we only minimize $H ( \mathbf { p } ^ { 1 } , \mathbf { p } ^ { 2 } )$ in our implementation.
+
+More efficiency with Momentum targets. However, the quality of the target similarity distribution $\mathbf { p } ^ { 1 }$ is crucial, to make the similarity distribution reliable and stable, we usually require a large batch size which is very unfriendly to GPU memories. To resolve this issue, we utilize a “momentum update" network as in [24, 8], and maintain a large memory buffer $\mathcal { Q }$ of $K$ past samples $\{ { \bf { z } } _ { k } | k = $ $1 , . . . , K \}$ (following the FIFO principle) for storing the feature embeddings from the past batches, which can then be used for simulating the large batch size relationship and providing a stable similarity distribution.
+
+$$
+\mathcal { F } _ { t } \gets m \mathcal { F } _ { t } + ( 1 - m ) \mathcal { F } _ { s } , \quad g _ { t } \gets m g _ { t } + ( 1 - m ) g _ { s } ,
+$$
+
+where $\mathcal { F } _ { s }$ and $g _ { s }$ denote the most latest encoder and head, respectively, so we name them as the student model with a subscript $s$ . On the other hand, $\mathcal { F } _ { t }$ and $g _ { t }$ stand for ensembles of the past encoder and head, respectively, so we name them as the teacher model with a subscript $t , m$ represents the momentum coefficient which controls how fast the teacher $\mathcal { F } _ { t }$ will be updated.
+
+Sharper Distribution as Target. Note, the value of $\tau _ { t }$ has to be smaller than $\tau _ { s }$ since $\tau _ { t }$ will be used to generate the target distribution. A smaller $\tau$ will result in a “sharper" distribution which can be interpreted as highlight the most similar feature for $\mathbf { z } ^ { 1 }$ . Align $ { \mathbf { p } } ^ { 2 }$ with $\mathbf { p } ^ { 1 }$ can be regarded as pulling $\mathbf { z } ^ { 2 }$ towards the features that are similar with $\mathbf { z } ^ { 1 }$ .
+
+Weak Augmentation Strategy for Teacher. To further improve the quality and stability of the target distribution, we adopt a weak augmentation strategy for the teacher model since the standard contrastive augmentation is too aggressive, which introduced too many disturbances and will mislead the student network. Please refer to more details in our empirical study.
+
+Compare with SEED and CLSA. SEED [21] follows the standard Knowledge Distillation (KD) paradigm [26, 49, 18] where it aims to distill the knowledge from a larger network into a smaller architecture. The knowledge transfer happens in the same view but between different models. In our framework, we are trying to maintain the relational consistency between different augmentations; the knowledge transfer happens between different views but in the same network. CLSA [46] also introduced the concept of using weak augmentation to guide a stronger augmentation. However, the “weak" augmentation in CLSA is equivalent to the “strong" augmentation in our method (We do not use any stronger augmentations such as [12, 13]). On the other hand, CLSA still adopts the InfoNCE loss (1) for instance discrimination, where our proposed method only utilized the relational consistency loss (5). Finally, CLSA requires at least one additional sample during training, which will slow down the training speed.
+
+# Algorithm 1: Relational Self-supervised Learning with Weak Augmentation (ReSSL)
+
+<table><tr><td colspan="2">the non-linear projection head for teacher and student. Q: the memory buffer while network not converge do for i=1 to step do Fetch x from current batch B z1=gt(Ft(Tw(x,01)）))； z²=gs(Fs(Tc(x,02）)；</td></tr><tr><td colspan="2">p1 = SoftMax(z1QT/ Tt); p² =SoftMax(z²QT / Ts）; // Eq.(3)(4) Calculate Lrelation loss by CrossEntropy(p1,p²) ; // Eq.(5) Update Fs and gs with loss Lrelation Update Ft and gt by Ft ← mFt +(1-m)Fs,gt ←mgt+(1-m)gs ; // Eq.</td></tr></table>
+
+# 4 Empirical Study
+
+In this section, we will empirically study our proposed method on 4 popular self-supervised learning benchmarks and compare to previous state-of-the-art algorithms (SimCLR [6], BYOL [23], SimSiam [9], MoCoV2 [8]).
+
+Small Dataset. CIFAR-10 and CIFAR-100 [31]. The CIFAR-10 dataset consists of $6 0 0 0 0 \ 3 2 { \mathrm { x } } 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. CIFAR-100 is just like the CIFAR-10, except it has 100 classes containing 600 images each. There are 500 training images and 100 testing images per class.
+
+Medium Dataset. STL-10 [11] and Tiny ImageNet [33]. STL10 [11] dataset is composed of $9 6 \mathbf { x } 9 6$ resolution images of 10 classes, 5K labeled training images, 8K validation images, and 100K unlabeled images. The Tiny ImageNet dataset is composed of $6 4 \mathrm { x } 6 4$ resolution images of 200 classes with 100K training images and 10k validation images.
+
+Implementation Details We adopt the ResNet18 [25] as our backbone network. Because most of our dataset contains low-resolution images, we replace the first $7 \mathbf { x } 7$ Conv of stride 2 with $3 { \tt X } 3$ Conv of stride 1 and remove the first max pooling operation for a small dataset. For data augmentations, we use the random resized crops (the lower bound of random crop ratio is set to 0.2), color distortion (strength $= 0 . 5$ ) with a probability of 0.8, and Gaussian blur with a probability of 0.5. The images from the small and medium datasets will be resized to $3 2 \mathrm { x } 3 2 $ and 64x64 resolution respectively. Our method is based on MoCoV2 [8]; in order to simulate the shuffle BN trick on one GPU, we simply divide a batch of data into different groups and then calculate BN statistics within each group. The momentum value and memory buffer size are set to 0.99/0.996 and 4096/16384 for small and medium datasets respectively. Moreover, The model is trained using SGD optimizer with a momentum of 0.9 and weight decay of $5 e ^ { - 4 }$ . We linear warm up the learning rate for 5 epochs until it reaches $0 . 0 6 \times B a t c h S i z e / 2 5 6$ , then switch to the cosine decay scheduler [37].
+
+Table 1: Compare to other SSL algorithms on small and medium dataset.   
+
+<table><tr><td>Method</td><td>BackProp</td><td>EMA</td><td>CIFAR-10</td><td>CIFAR-100</td><td>STL-10</td><td>Tiny ImageNet</td></tr><tr><td>Supervised</td><td>1</td><td>-</td><td>94.22</td><td>74.66</td><td>82.55</td><td>59.26</td></tr><tr><td>SimCLR [6]</td><td>2x</td><td>No</td><td>84.92</td><td>59.28</td><td>85.48</td><td>44.38</td></tr><tr><td>BYOL [23]</td><td>2x</td><td>Yes</td><td>85.82</td><td>57.75</td><td>87.45</td><td>42.70</td></tr><tr><td>SimSiam [9]</td><td>2x</td><td>No</td><td>88.51</td><td>60.00</td><td>87.47</td><td>37.04</td></tr><tr><td>MoCoV2 [8]</td><td>1x</td><td>Yes</td><td>86.18</td><td>59.51</td><td>85.88</td><td>43.36</td></tr><tr><td>ReSSL (Ours)</td><td>1x</td><td>Yes</td><td>90.20</td><td>63.79</td><td>88.25</td><td>46.60</td></tr></table>
+
+Evaluation Protocol. All the models will be trained for 200 epochs. For testing the representation quality, we evaluate the pre-trained model on the widely adopted linear evaluation protocol - We will freeze the encoder parameters and train a linear classifier on top of the average pooling features for 100 epochs. To test the classifier, we use the center crop of the test set and computes accuracy according to predicted output. We train the classifier with a learning rate of 30, no weight decay, and momentum of 0.9. The learning rate will be times 0.1 in 60 and 80 epochs. Note, for STL-10; the pretraining will be applied on both labeled and unlabeled images. During the linear evaluation, only the labeled 5K images will be used.
+
+Result. As we can see the result in Table 1, our proposed method outperforms the previous method on all four benchmarks. Reminder, most of the previous method requires twice back-propagation, which results in a much higher training cost than MoCoV2 and our method.
+
+# 4.1 A Properly Sharpened Relation is A Better Target
+
+The temperature parameter is very crucial in most contrastive learning algorithms. To verify the effective of $\tau _ { s }$ and $\tau _ { t }$ for our proposed method, we fixed $\tau _ { s } = 0 . 1$ or 0.2, and sweep over $\tau _ { t } =$ $\{ 0 . 0 1 , 0 . 0 2 , . . . , 0 . 0 7 \}$ . The result is shown in Table 2. For $\tau _ { t }$ , the optimal value is either 0.04 or 0.05 across all different datasets. As we can see, the performance is increasing when we increase $\tau _ { t }$ from 0 to 0.04 and 0.05. After that, the performance will start to decrease. Note, $\tau _ { t } \to 0$ correspond to the Top-1 or argmax operation which produce a one-hot distribution as the target. On the other hand, when $\tau _ { t }  0 . 1$ , the target will be a much flatter distribution that cannot highlight the most similar features for students. Hence, $\tau _ { t }$ can not be either too small or too large, but it has to be smaller than $\tau _ { s }$ $\mathbf { \bar { p } } ^ { 1 }$ has to be sharper than $ { \mathbf { p } } ^ { 2 }$ ) so that the target distribution can provide effective guidance to the student model.
+
+Table 2: Effect of different $\tau _ { t }$ and $\tau _ { s }$ for ReSSL   
+
+<table><tr><td>Dataset</td><td>Ts</td><td>Tt =0.01</td><td>Tt =0.02</td><td>Tt=0.03</td><td>Tt = 0.04</td><td>Tt=0.05</td><td>Tt =0.06</td><td>Tt = 0.07</td></tr><tr><td>CIFAR-10</td><td>0.1</td><td>89.35</td><td>89.74</td><td>90.09</td><td>90.04</td><td>90.20</td><td>90.18</td><td>88.67</td></tr><tr><td>CIFAR-10</td><td>0.2</td><td>89.52</td><td>89.67</td><td>89.24</td><td>89.50</td><td>89.22</td><td>89.40</td><td>89.50</td></tr><tr><td>CIFAR-100</td><td>0.1</td><td>62.34</td><td>62.79</td><td>62.71</td><td>63.79</td><td>63.46</td><td>63.20</td><td>61.31</td></tr><tr><td>CIFAR-100</td><td>0.2</td><td>60.37</td><td>60.05</td><td>60.24</td><td>60.09</td><td>59.09</td><td>59.12</td><td>59.76</td></tr><tr><td>STL-10</td><td>0.1</td><td>86.65</td><td>86.96</td><td>87.16</td><td>87.32</td><td>88.25</td><td>87.83</td><td>87.08</td></tr><tr><td>STL-10</td><td>0.2</td><td>85.17</td><td>86.12</td><td>85.01</td><td>85.67</td><td>85.21</td><td>85.51</td><td>85.28</td></tr><tr><td>Tiny ImageNet</td><td>0.1</td><td>45.20</td><td>45.40</td><td>46.30</td><td>46.60</td><td>45.08</td><td>45.24</td><td>44.18</td></tr><tr><td>Tiny ImageNet</td><td>0.2</td><td>43.28</td><td>42.98</td><td>43.58</td><td>42.12</td><td>42.70</td><td>42.76</td><td>42.60</td></tr></table>
+
+For $\tau _ { t }$ , it is clearly to see that the result of $\tau _ { s } = 0 . 1$ can always result a much higher performance than $\tau _ { s } = 0 . 2$ , which is different to $\mathbf { M o C o V } 2$ where $\tau _ { s } = 0 . 2$ is the optimal value. According to [43, 44, 15], a greater temperature will result in a larger angular margin in the hypersphere. Since MoCoV2 adopts instance discrimination as the pretext task, a large temperature can enhance the compactness for the same instance and discrepancy for different instances. In contrast to instance discrimination, our method can be interpreted as pulling similar instances closer on the hypersphere; when the ground truth label is not available, the large angular margin might hurt the performance.
+
+![](images/19895d46df03376d75adeaf9e64fbd761eb75bd5154dd10d4a8f311e2c893a33.jpg)  
+Figure 2: Visualization of the 10 nearest neighbour of the query image. The top half is the result when we apply the weak augmentation. The bottom half is the case when the typical contrastive augmentation is adopted. Note, we use the red square to highlight the images that has different ground truth label with the query image.
+
+# 4.2 Weak Augmentation Makes Better Relation
+
+As we have mentioned, the weaker augmentation strategy for the teacher model is the key to the success of our framework. Here, We implement the weak augmentation as a random resized crop (the random ratio is set to $( 0 . 2 , 1 )$ ) and a random horizontal flip. For temperature parameter, we simply adopt the same setting as in Table 2 and report the performance of the best setting. The result is shown in Table 3, as we can see that when we use the weak augmentation for the teacher model, the performance is significantly boosted across all datasets. We believe that this phenomenon is because relatively small disturbances in the teacher model can provide more accurate similarity guidance to the student model. To further verify this hypothesis, we random sampled three image from STL-10 training set as the query images, and then find the 10 nearest neighbour based on the weak / contrastive augmented query. We visualized the result in Figure 2,
+
+Table 3: Effect of weak augmentation guided ReSSL   
+
+<table><tr><td>Teacher Aug</td><td>Student Aug</td><td>CIFAR-10</td><td>CIFAR-100</td><td>STL-10</td><td>Tiny ImageNet</td></tr><tr><td>Contrastive</td><td>Contrastive</td><td>86.17</td><td>57.60</td><td>84.71</td><td>40.38</td></tr><tr><td>Weak</td><td>Contrastive</td><td>90.20</td><td>63.79</td><td>88.25</td><td>46.60</td></tr></table>
+
+# 4.3 More Experiments on Weak Augmentation
+
+Since the weak augmentation for the teacher model is one of the crucial points in ReSSL, we further analyze the effect of applying different augmentations on the teacher model. In this experiment, we simply set $\tau _ { t } = 0 . 0 4$ and report the linear evaluation performance on the Tiny ImageNet dataset. The results are shown in Table 4. The first row is the baseline, where we simply resize all images to the same resolution (no extra augmentation is applied). Then, we applied random resized crops, random flip, color jitter, grayscale, gaussian blur, and various combinations. We empirically find that if we use no augmentation (e.g., no random resized crops) for the teacher model, the performance tends to degrade. This might result from that the gap of features between two views is way too smaller, which undermines the learning of representations. However, too strong augmentations of teacher model will introduce too much noise and make the target distribution inaccurate (see Figure 2). Thus mildly weak augmentations are better option for the teacher, and random resized crops with random flip is the combination with the highest performance as Table 4 shows.
+
+Table 4: Effect of different augmentation for teacher model (Tiny ImageNet)   
+
+<table><tr><td>Random Resized Crops</td><td>Random Flip</td><td>Color Jitter</td><td>GrayScale</td><td>Gaussian Blur</td><td>Acc</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>31.74</td></tr><tr><td>√</td><td></td><td></td><td></td><td></td><td>46.00</td></tr><tr><td></td><td>√</td><td></td><td></td><td></td><td>30.98</td></tr><tr><td></td><td></td><td>√</td><td></td><td></td><td>29.46</td></tr><tr><td></td><td></td><td></td><td>√</td><td></td><td>29.68</td></tr><tr><td></td><td></td><td></td><td></td><td>√</td><td>30.10</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td></td><td>46.60</td></tr><tr><td>√</td><td></td><td>√</td><td></td><td></td><td>44.44</td></tr><tr><td>√</td><td></td><td></td><td>4</td><td></td><td>42.28</td></tr><tr><td>√</td><td></td><td></td><td></td><td>√</td><td>44.88</td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td>43.70</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td></td><td>42.28</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>√</td><td>44.52</td></tr></table>
+
+# 4.4 Dimension of the Relation
+
+Since we also adopt the memory buffer as in MoCo [24], the buffer size will be equivalent to the dimension of the distribution $\mathbf { p } ^ { 1 } \mathbf { p } ^ { 2 }$ . Thus, it will be one of the crucial points in our framework. To verify the effect the memory buffer size, we simply keep $\tau _ { s } = 0 . 1$ and $\tau _ { t } = 0 . 0 4$ , then varying the memory buffer size from 256 to 32768. The result is shown in Table 5, as we can see that a larger memory buffer can significantly boost the performance. However, a further increase in the buffer size can only bring a marginal improvement when the buffer is large enough.
+
+Table 5: Effect of different memory buffer size on small and medium dataset   
+
+<table><tr><td>Dataset (Small) CIFAR-10</td><td>K=256 89.37</td><td>K=512 89.53</td><td>K= 1024 89.83</td><td>K= 4096 90.04</td><td>K=8192 90.15</td><td>K=16384 90.35</td></tr><tr><td>CIFAR-100 Dataset (Medium)</td><td>61.17 K=256</td><td>62.47 K=1024</td><td>63.20 K = 4096</td><td>63.79 K=8192</td><td>63.84 K=16384</td><td>64.06 K=32768</td></tr><tr><td>STL-10</td><td>85.88</td><td>87.23</td><td>87.72</td><td>87.42</td><td>87.32</td><td>87.47</td></tr><tr><td>Tiny ImageNet</td><td>43.08</td><td>45.32</td><td>45.78</td><td>45.42</td><td>46.60</td><td>46.48</td></tr></table>
+
+# 4.5 Visualization of Learned Representations
+
+We also show the t-SNE [42] visualizations of the representations learned by our proposed method and MoCov2 on the training set of CIFAR-10. Our proposed relational consistency loss leads to better class separation than the contrastive loss.
+
+![](images/932d623072d019b5214fba4ff8e7b94ec5d45e73c9a939414ec5d984820fa4c3.jpg)  
+Figure 3: t-SNE visualizations on CIFAR-10. Classes are indicated by colors.
+
+# 5 Results on Large-scale Datasets
+
+We also performed our algorithm on the large-scale ImageNet-1k dataset [14]. In the experiments, we adopt a learning rate of $0 . 0 5 * B a t c h S i z e / 2 5 6$ , a memory buffer size of $1 3 0 \mathrm { k }$ , and a 2-layer non-linear projection head with a hidden dimension 4096 and output dimension 512. For $\tau _ { t }$ and $\tau _ { s }$ , we simply adopt the best setting from Table 2 where $\tau _ { t } = 0 . 0 4$ and $\tau _ { s } = 0 . 1$ .
+
+Linear Evaluation. For the linear evaluation of ImageNet-1k, we strictly follow the setting in SwAV [5]. The results are shown in Table 6. As we can see clearly that ReSSL consistently outperforms previous methods on both $1 \mathbf { x }$ and $2 \mathbf { x }$ backprop setting. (Please noted that the student network will be passed in one $2 2 4 \mathbf { x } 2 2 4$ augmented view and two $2 2 4 \mathbf { x } 2 2 4$ augmented views for 1x backprob and $2 \mathbf { x }$ backprob setting respectively.)
+
+Table 6: Top-1 accuracy under the linear evaluation on ImageNet with the ResNet-50 backbone. The table compares the methods over 200 epochs of pretraining.   
+
+<table><tr><td>Method Supervised</td><td>Arch R50</td><td>Backprop 1x</td><td>EMA No</td><td>Batch Size 256</td><td>Param 24</td><td>Epochs 120</td><td>Top-1 76.5</td></tr><tr><td>1xBackpropMethods</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InstDisc [47]</td><td>R50</td><td>1x</td><td>No</td><td>256</td><td>24</td><td>200</td><td>58.5</td></tr><tr><td>LocalAgg [52]</td><td>R50</td><td>1x</td><td>No</td><td>128</td><td>24</td><td>200</td><td>58.8</td></tr><tr><td>MoCo v2 [8]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>67.5</td></tr><tr><td>MoCHi [30]</td><td>R50</td><td>1x</td><td>Yes</td><td>512</td><td>24</td><td>200</td><td>68.0</td></tr><tr><td>CPC v2 [32]</td><td>R50</td><td>1x</td><td>No</td><td>512</td><td>24</td><td>200</td><td>63.8</td></tr><tr><td>PCL v2 [36]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>67.6</td></tr><tr><td>AdCo [28]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>68.6</td></tr><tr><td>ReSSL (Ours)</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>69.9</td></tr><tr><td>2xBackprop Methods</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLSA-Single [46]</td><td>R50</td><td>2x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>69.4</td></tr><tr><td>SimCLR [6]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>66.8</td></tr><tr><td>SwAV [5]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>69.1</td></tr><tr><td>SimSiam [23]</td><td>R50</td><td>2x</td><td>No</td><td>256</td><td>24</td><td>200</td><td>70.0</td></tr><tr><td>BYOL[23]</td><td>R50</td><td>2x</td><td>Yes</td><td>4096</td><td>24</td><td>200</td><td>70.6</td></tr><tr><td>WCL [51]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>70.3</td></tr><tr><td>ReSSL (Ours)</td><td>R50</td><td>2x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>71.4</td></tr></table>
+
+Working with Multi-Crop Strategy. We also performed ReSSL with Multi-Crop strategy. The result is shown below in Table 7. Specifically, the result of 4 crops is trained with the resolution of $2 2 4 \times 2 2 4$ , $1 6 0 \times 1 6 0$ , $1 2 8 \times 1 2 8$ , $9 6 \times 9 6$ . For the result of 5 crops, we add an additional $1 9 2 \times 1 9 2$ image which is exactly the same with AdCo [28]. As we can see, our proposed ReSSL is significantly better than previous state-of-the-art methods.
+
+Table 7: Working with Multi-Crop Strategy (Linear Evaluation on ImageNet)   
+
+<table><tr><td>Method</td><td>Arch</td><td>EMA</td><td>Batch Size</td><td>Param</td><td>Epochs</td><td>Top-1</td></tr><tr><td>SwAV [5]</td><td>R50</td><td>No</td><td>256</td><td>24</td><td>200</td><td>72.7</td></tr><tr><td>AdCo [28]</td><td>R50</td><td>No</td><td>256</td><td>24</td><td>200</td><td>73.2</td></tr><tr><td>CLSA-Multi [46]</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>73.3</td></tr><tr><td>ReSSL (4 crops)</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>73.8</td></tr><tr><td>ReSSL (5 crops)</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>74.7</td></tr></table>
+
+Working with Smaller Architecture. We also applied our proposed method on the smaller architecture (ResNet-18). The result is shown in Table 8. Following the same training recipe of the ResNet-50 in above, our proposed method has a higher performance than SEED [21] without a larger pretrained teacher network.
+
+Table 8: Experiments on ResNet-18 (Linear Evaluation on ImageNet)   
+
+<table><tr><td>Method</td><td>Epochs</td><td>Student</td><td>Teacher</td><td>Acc</td></tr><tr><td>MoCo v2</td><td>200</td><td>ResNet-18</td><td>EMA</td><td>52.2</td></tr><tr><td>SEED</td><td>200</td><td>ResNet-18</td><td>ResNet-50 (MoCoV2)</td><td>57.6</td></tr><tr><td>ReSSL (1x backprop)</td><td>200</td><td>ResNet-18</td><td>EMA</td><td>58.1</td></tr></table>
+
+Low-shot Classification. We further evaluate the quality of the learned representations by transferring them to other datasets. Following [36], we perform linear classification on the PASCAL
+
+VOC2007 dataset [20]. Specifically, we resize all images to 256 pixels along the shorter side and taking a $2 2 4 \times 2 2 4$ center crop. Then, we train a linear SVM on top of corresponding global average pooled final representations. To study the transferability of the representations in few-shot scenarios, we vary the number of labeled examples $K$ and report the mAP. Table 9 shows the comparison between our method with previous works. We report the average performance over 5 runs (except for $\mathbf { k } =$ full).It’s clearly to see that our proposed method is consistently outperform MoCo v2 and PCL v2 across all different $K$ .
+
+Table 9: Transfer learning on low-shot image classification   
+
+<table><tr><td>Method Random</td><td>Epochs 1</td><td>ImageNet 1</td><td>K=16 10.10</td><td>K=32 11.34</td><td>K=64 11.96</td><td>Full 12.42</td></tr><tr><td>Supervised</td><td>90</td><td>76.1</td><td>82.26</td><td>84.00</td><td>85.13</td><td>87.27</td></tr><tr><td>MoCo V2 [8]</td><td>200</td><td>67.5</td><td>76.14</td><td>79.16</td><td>81.52</td><td>84.60</td></tr><tr><td>PCL V2 [36]</td><td>200</td><td>67.5</td><td>78.34</td><td>80.72 81.96</td><td>82.67</td><td>85.43</td></tr><tr><td>ReSSL (1x backprob)</td><td>200</td><td>69.9</td><td>79.17</td><td></td><td>83.81</td><td>86.31</td></tr></table>
+
+Semi-Supervised Learning. Next, we evaluate the performance obtained when fine-tuning the model representation using a small subset of labeled data. In this experiments, we adopt our 5 crops pre-trained model. The result is shown in Table 10. Notably, with just 200 epochs of pre-training, ReSSL outperforms all previous methods.
+
+Table 10: Semi-supervised Learning   
+
+<table><tr><td>Method</td><td>Epochs</td><td>Linear Eval</td><td>1% Labels</td><td>10% Labels</td></tr><tr><td>SimCLR [6]</td><td>1000</td><td>69.3</td><td>48.3</td><td>65.6</td></tr><tr><td>BYOL[23]</td><td>1000</td><td>74.3</td><td>53.2</td><td>68.6</td></tr><tr><td>SwAV[5]</td><td>800</td><td>75.3</td><td>53.9</td><td>70.2</td></tr><tr><td>ReSSL (5 crops)</td><td>200</td><td>74.7</td><td>57.9</td><td>70.4</td></tr></table>
+
+# 6 Conclusion
+
+In this work, we propose relational self-supervised learning (ReSSL), a new paradigm for unsupervised visual representation learning framework that maintains the relational consistency between instances under different augmentations. Our proposed ReSSL relaxes the typical constraints in contrastive learning where different instances do not always need to be pushed away on the embedding space, and the augmented views do not need to share exactly the same feature. An extensive empirical study shows the effect of each component in our framework. The experiments on large-scaled datasets demonstrate the efficiency and state-of-the-art performance for unsupervised representation learning.
+
+# Broader Impact
+
+This work provides a technical advancement in the field of unsupervised visual representation learning. An immediate application of this work is to give a pre-trained model for the tasks where the data annotation is very hard to collect (e.g.medical images and fine-grained images.) Moreover, the most significant advantage of ReSSL is that we do not need to train the model for a long time as the previous method (generally 800 or 1000 epochs), which will cause a lot of carbon dioxide emissions. We believe ReSSL is a more environment-friendly method since it can achieve a competitive performance with much lesser training costs.
+
+# Acknowledgment
+
+This work is funded by the National Key Research and Development Program of China (No. 2018AAA0100701) and the NSFC 61876095. Chang Xu was supported in part by the Australian Research Council under Projects DE180101438 and DP210101859. Shan You is supported by Beijing Postdoctoral Research Foundation.
+
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+        "text": "ReSSL: Relational Self-Supervised Learning with Weak Augmentation ",
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+        "text": "Mingkai Zheng1,2 Shan $\\mathbf { Y o u } ^ { 2 , 4 * }$ Fei Wang3 Chen Qian2 Changshui Zhang4 Xiaogang Wang2,5 Chang Xu1 ",
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+        "text": "1School of Computer Science, Faculty of Engineering, The University of Sydney 2SenseTime Research 3University of Science and Technology of China 4Department of Automation, Tsinghua University, Institute for Artificial Intelligence, Tsinghua University (THUAI),   \nBeijing National Research Center for Information Science and Technology (BNRist) 5The Chinese University of Hong Kong ",
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+        "text": "Abstract ",
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+        "text": "Self-supervised Learning (SSL) including the mainstream contrastive learning has achieved great success in learning visual representations without data annotations. However, most of methods mainly focus on the instance level information (i.e., the different augmented images of the same instance should have the same feature or cluster into the same class), but there is a lack of attention on the relationships between different instances. In this paper, we introduced a novel SSL paradigm, which we term as relational self-supervised learning (ReSSL) framework that learns representations by modeling the relationship between different instances. Specifically, our proposed method employs sharpened distribution of pairwise similarities among different instances as relation metric, which is thus utilized to match the feature embeddings of different augmentations. Moreover, to boost the performance, we argue that weak augmentations matter to represent a more reliable relation, and leverage momentum strategy for practical efficiency. Experimental results show that our proposed ReSSL significantly outperforms the previous stateof-the-art algorithms in terms of both performance and training efficiency. Code is available at https://github.com/KyleZheng1997/ReSSL ",
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+        "text": "1 Introduction ",
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+        "text": "Recently, self-supervised learning (SSL) has shown its superiority and achieved promising results for unsupervised visual representation learning in computer vision tasks [40, 27, 32, 6, 9, 47, 23, 24]. The purpose of a typical self-supervised learning algorithm is to learn general visual representations from a large amount of data without human annotations, which can be transferred or leveraged in downstream tasks (e.g., classification, detection, and segmentation). Some previous works [5, 23] even have proven that a good unsupervised pretraining can lead to a better downstream performance than supervised pretraining. ",
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+        "type": "text",
+        "text": "Among various SSL algorithms, contrastive learning [47, 45, 6] serves as a state-of-the-art framework, which mainly focuses on learning an invariant feature from different views. For example, instance discrimination is a widely adopted pre-text task as in [6, 24, 47], which utilizes the noisy contrastive estimation (NCE) to encourage two augmented views of the same image to be pulled closer on the embedding space but pushes apart all the other images away. Deep Clustering [4, 48, 5] is an alternative pre-text task that forces different augmented views of the same instance to be clustered into the same class. However, instance discrimination based methods will inevitably induce a class collision problem [1, 36, 10], where similar images should be pulled closer instead of being pushed away. Deep clustering based methods cooperated with traditional clustering algorithms to assign a label for each instance, which relaxed the constraint of instance discrimination, but most of these algorithms adopt a strong assumption, i.e., the labels must induce an equipartition of the data, which might introduce some noise and hurt the learned representations. ",
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+        "type": "text",
+        "text": "In this paper, we introduce a novel Relational Self-Supervised Learning framework (ReSSL), which does not encourage explicitly to push away different instances, but uses relation as a manner to investigate the inter-instance relationships and highlight the intra-instance invariance. Concretely, we aim to maintain the consistency of pairwise similarities among different instances for two different augmentations. For example, if we have three instances $\\mathbf { x } ^ { 1 }$ , $\\mathbf { x } ^ { 2 }$ , y and $\\mathbf { z }$ where $\\mathbf { x } ^ { 1 }$ , $\\mathbf { x } ^ { 2 }$ are two different augmentations of $\\mathbf { x }$ , $\\mathbf { y }$ and $\\mathbf { z }$ are different samples. Then, if $\\mathbf { x } ^ { 1 }$ is similar to y but different to $\\mathbf { z }$ , we wish $\\bar { \\mathbf { x } } ^ { 2 }$ can maintain such relationship and vice versa. In this way, the relation can be modelled as a similarity distribution between a set of augmented images, and then use it as a metric to align the same images with different augmentations, so that the relationship between different instances could be maintained across different views. ",
+        "bbox": [
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+        "type": "text",
+        "text": "However, this simple manner induces unexpectedly horrible performance if we follow the same training recipe as other contrastive learning methods [6, 24]. We argue that construction of a proper relation matters for ReSSL; aggressive data augmentations as in [6, 7, 41] are usually leveraged by default to generate diverse positive pairs that increase the difficulty of the pre-text task. However, this hurts the reliability of the target relation. Views generated by aggressive augmentations might cause the loss of semantic information, so the target relation might be noisy and not that reliable. In this way, we propose to leverage weaker augmentations to represent the relation, since much lesser disturbances provide more stable and meaningful relationships between different instances. Besides, we also sharpen the target distribution to emphasize the most important relationship and utilize the memory buffer with a momentum-updated network to reduce the demand of large batch size for more efficiency. Experimental results on multiple benchmark datasets show the superiority of ReSSL in terms of both performance and efficiency. For example, with 200 epochs of pre-training, our ReSSL achieved $6 9 . 9 \\%$ on ImageNet [14] linear evaluation protocol, which is $2 . 4 \\%$ higher than our baseline method (MoCoV2 [8]). When working with the Multi-Crop strategy (200 epochs), ReSSL achieved new state-of-the-art $7 4 . 7 \\%$ Top-1 accuracy, which is $1 . 4 \\%$ higher than CLSA-Multi [46]. ",
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+        "type": "text",
+        "text": "Our contributions can be summarized as follows. ",
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+        "text": "• We proposed a novel SSL paradigm, which we term it as relational self-supervised learning (ReSSL). ReSSL maintains the relational consistency between the instances under different augmentations instead of explicitly pushing different instances away. • Our proposed weak augmentation and sharpening distribution strategy provide a stable and high quality target similarity distribution, which makes the framework works well. • ReSSL is a simple and effective SSL framework since it replaces the widely adopted contrastive loss with our proposed relational consistency loss. It achieved state-of-the-art performance under the same training cost. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Self-Supervised Learning. Early works in self-supervised learning methods rely on all sorts of pretext to learn visual representations. For example, colorizing gray-scale images [50], image jigsaw puzzle [39], image super-resolution [34], image inpainting [19], predicting a relative offset for a pair of patches [16], predicting the rotation angle [35], and image reconstruction [2, 22, 3, 17]. Although these methods have shown their effectiveness, they lack the generality of the learned representations. ",
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+        "type": "text",
+        "text": "Instance Discrimination. The recent contrastive learning methods [32, 40, 6, 24, 41, 38, 29, 27, 30] have made a lot of progress in the field of self-supervised learning. Most of the previous contrastive learning methods are based on the instance discrimination [47] task in which positive pairs are defined as different views of the same image, while negative pairs are formed by sampling views from different images. SimCLR [6, 7] shows that image augmentation (e.g.Grayscale, Random Resized Cropping, Color Jittering, and Gaussian Blur), nonlinear projection head and large batch size plays a critical role in contrastive learning. Since large batch size usually requires a lot of GPU memory, which is not very friendly to most of researchers. MoCo [24, 8] proposed a momentum contrast mechanism that forces the query encoder to learn the representation from a slowly progressing key encoder and maintain a memory buffer to store a large number of negative samples. InfoMin [41] proposed a set of stronger augmentation that reduces the mutual information between views while keeping task-relevant information intact. AlignUniform [45] shows that alignment and uniformity are two critical properties of contrastive learning. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Deep Clustering. In contrast to instance discrimination which treats every instance as a distinct class, deep clustering [4] adopts the traditional clustering method (e.g.KMeans) to label each image iteratively. Eventually, similar samples will be clustered into the same class. Simply apply the KMeans algorithm might lead to a degenerate solution where all data points are mapped to the same cluster; SeLa [48] solved this issue by adding the constraint that the labels must induce equipartition of the data and proposed a fast version of the Sinkhorn-Knopp to achieve this. SwAV [5] further extended this idea and proposed a scalable online clustering framework. PCL [36] reveals the class collision problem and simply performed instance discrimination and unsupervised clustering simultaneously; although it gets the same linear classification accuracy with MoCoV2, it has better performance on downstream tasks. ",
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+    {
+        "type": "text",
+        "text": "Contrastive Learning Without Negatives. Most previous contrastive learning methods prevent the model collapse in an explicit manner (e.g. push different instances away from each other or force different instances to be clustered into different groups.) BYOL [23] can learn a high-quality representation without negatives. Specifically, it trains an online network to predict the target network representation of the same image under a different augmented view and using an additional predictor network on top of the online encoder to avoiding the model collapse. SimSiam [9] shows that simple Siamese networks can learn meaningful representations even without the use of negative pairs, large batch size, and momentum encoders. ",
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+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we will first revisit the preliminary work on contrastive learning; then, we will introduce our proposed relational self-supervised learning framework. After that, the algorithm and the implementation details will also be explained. ",
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+    {
+        "type": "text",
+        "text": "3.1 Preliminaries on Self-supervised Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given $N$ unlabeled samples $\\mathbf { x }$ , we randomly apply a composition of augmentation functions $T ( \\cdot )$ to obtain two different views $\\mathbf { x } ^ { 1 }$ and $\\mathbf { x } ^ { 2 }$ through $T ( \\mathbf { x } , \\theta _ { 1 } ^ { \\top } )$ and $T ( \\mathbf { x } , \\theta _ { 2 } )$ where $\\theta$ is the random seed for $T$ . Then, a convolutional neural network based encoder $\\mathcal F ( \\cdot )$ is employed to extract the information from these samples, i.e., $\\mathbf { h } = \\mathcal { F } ( T ( \\mathbf { x } , \\theta ) )$ . Finally, a two-layer non-linear projection head $g ( \\cdot )$ is utilized to map $\\mathbf { h }$ into embedding space, which can be written as: ${ \\mathbf z } = g ( { \\mathbf h } )$ . SimCLR [6] and MoCo [24] style framework adopt the noise contrastive estimation (NCE) objective for discriminating different instances in the dataset. Suppose $\\mathbf { z } _ { i } ^ { 1 }$ and $\\mathbf { z } _ { i } ^ { 2 }$ are the representations of two augmented views of $\\mathbf { x } _ { i }$ and $\\mathbf { z } _ { k }$ is a different instance. The NCE objective can be expressed by Eq. (1), where the similarity function $s i m ( \\cdot )$ represents the dot product between $L _ { 2 }$ normalized vectors $s i m ( \\mathbf { u } , \\mathbf { v } ) = \\mathbf { u } ^ { T } \\mathbf { v } / \\| \\mathbf { u } \\| \\| \\mathbf { v } \\|$ and $\\tau$ is the temperature parameter. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e43c6b76d9a4dfe804ae451b5d2ca8fa960bb25d236c9053af00c86bd68208a1.jpg",
+        "text": "$$\n\\mathcal { L } _ { N C E } = - \\log \\frac { \\exp ( s i m ( \\mathbf { z } ^ { 1 } , \\mathbf { z } ^ { 2 } ) / \\tau ) } { \\exp ( s i m ( \\mathbf { z } _ { i } ^ { 1 } , \\mathbf { z } _ { i } ^ { 2 } ) / \\tau ) + \\sum _ { k = 1 } ^ { N } \\exp ( s i m ( \\mathbf { z } _ { i } ^ { 1 } , \\mathbf { z } _ { k } ) / \\tau ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "BYOL [23] and SimSiam [9] style framework add an additional non-linear predictor head $q ( \\cdot )$ which further maps $\\mathbf { z }$ to $\\mathbf { p }$ . The model will minimize the negative cosine similarity (equivalent to minimize the L2 distance) between $\\mathbf { z }$ to $\\mathbf { p }$ . ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/4cf689fe6c568dc79beaafced056a921c00b7b91db05245d867d729ff14a8697.jpg",
+        "text": "$$\n\\mathcal { L } _ { c o s } = - \\frac { \\mathbf { p } ^ { 1 } } { \\Vert \\mathbf { p } ^ { 1 } \\Vert } \\cdot \\frac { \\mathbf { z } ^ { 2 } } { \\Vert \\mathbf { z } ^ { 2 } \\Vert } , \\qquad \\mathcal { L } _ { m s e } = \\Vert \\mathbf { p } ^ { 1 } - \\mathbf { z } ^ { 2 } \\Vert _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Tricks like stop-gradient and momentum teacher are often applied to avoid model collapsing. ",
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+    {
+        "type": "text",
+        "text": "3.2 Relational Self-Supervised Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In classical self-supervised learning, different instances are to be pushed away from each other, and augmented views of the same instance is expected to be of exactly the same features. However, both constrains are too restricted because of the existence of similar samples and the distorted semantic information if aggressive augmentation is adopted. In this way, we do not encourage explicit negative instances (those to be pushed away) for each instance; instead, we leverage the pairwise similarities as a manner to explore their relationships. And we pull the features of two different augmentations in this sense of relation metric. As a result, our method relaxes both (1) and (2), where different instances do not always need to be pushed away from each other; and augmented views of the same instance only need to share the similar but not exactly the same features. ",
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+        "img_path": "images/22fb022bc11cda14fbc85746e62700705dfdf636406ec392110ed5d841f7ee89.jpg",
+        "image_caption": [
+            "Figure 1: The overall framework of our proposed method. We adopt the student-teacher framework where the student is trained to predict the representation of the teacher, and the teacher is updated with a “momentum update” (exponential moving average) of the student. The relationship consistency is achieve by align the conditional distribution for student and teacher model. Please see more details in our method part. "
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+        "type": "text",
+        "text": "Concretely, given a image $\\mathbf { x }$ in a batch of samples , two different augmented views can be obtained by $\\mathbf { x } ^ { 1 } = \\mathbf { \\check { \\Gamma } } T ( \\mathbf { x } , \\theta _ { 1 } )$ , $\\mathbf { x } ^ { 2 } = T ( \\mathbf { x } , \\theta _ { 2 } )$ and calculate the corresponds embedding $\\mathbf { z } ^ { 1 } = g ( \\mathcal { F } ( \\mathbf { x } ^ { 1 } ) )$ , $\\mathbf { z } ^ { 2 } = g ( \\mathcal { F } ( \\mathbf { x } ^ { 2 } ) )$ . Then, we calculate the similarities between the instances of the first augmented images. Which can be measured by $s i m ( \\mathbf { z } ^ { 1 } , \\mathbf { z } _ { i } )$ . A softmax layer can be adopted to process the calculated similarities, which then produces a relationship distribution: ",
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+        "type": "equation",
+        "img_path": "images/5fbe86febb634ab79187498d3727220a09eb4d4d6e0c4deee344853b4971dde0.jpg",
+        "text": "$$\n{ \\bf p } _ { i } ^ { 1 } = \\frac { \\exp ( s i m ( { \\bf z } ^ { 1 } , { \\bf z } _ { i } ) / \\tau _ { t } ) } { \\sum _ { k = 1 } ^ { K } \\exp ( s i m ( { \\bf z } ^ { 1 } , { \\bf z } _ { k } ) / \\tau _ { t } , ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\tau _ { t }$ is the temperature parameter. At the same time, we can calculate the relationship between $\\mathbf { x } ^ { 2 }$ and the $i$ -th instance as $s i m ( \\mathbf { z } ^ { 2 } , \\mathbf { z } _ { i } )$ . The resulting relationship distribution can be written as: ",
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+        "type": "equation",
+        "img_path": "images/95a17ed8378c564b16d92c8548e6becf455c55ed27b4ff72c3760a0846b1f912.jpg",
+        "text": "$$\n{ { \\bf { p } } _ { i } ^ { 2 } } = \\frac { { \\exp ( { s i m ( { { \\bf { z } } ^ { 2 } } , { \\bf { z } } _ { i } ) / { \\tau _ { s } } } ) } } { { \\sum _ { k = 1 } ^ { K } { \\exp ( { s i m ( { { \\bf { z } } ^ { 2 } } , { \\bf { z } } _ { k } ) / { \\tau _ { s } } } , ) } } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\tau _ { s }$ is a different temperature parameter. We propose to push the relational consistency between $p _ { i } ^ { 1 }$ and $p _ { i } ^ { 2 }$ by minimizing the Kullback–Leibler divergence, which can be formulated as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/432d6f1759d5c8a0962984763c8fb7070c57cfa9cf961b9ecaefaf78ad8ec829.jpg",
+        "text": "$$\n\\mathcal { L } _ { r e l a t i o n } = D _ { K L } ( \\mathbf { p } ^ { 1 } | | \\mathbf { p } ^ { 2 } ) = H ( \\mathbf { p } ^ { 1 } , \\mathbf { p } ^ { 2 } ) - H ( \\mathbf { p } ^ { 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since the $\\mathbf { p } ^ { 1 }$ will only be used as a target, we only minimize $H ( \\mathbf { p } ^ { 1 } , \\mathbf { p } ^ { 2 } )$ in our implementation. ",
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+    {
+        "type": "text",
+        "text": "More efficiency with Momentum targets. However, the quality of the target similarity distribution $\\mathbf { p } ^ { 1 }$ is crucial, to make the similarity distribution reliable and stable, we usually require a large batch size which is very unfriendly to GPU memories. To resolve this issue, we utilize a “momentum update\" network as in [24, 8], and maintain a large memory buffer $\\mathcal { Q }$ of $K$ past samples $\\{ { \\bf { z } } _ { k } | k = $ $1 , . . . , K \\}$ (following the FIFO principle) for storing the feature embeddings from the past batches, which can then be used for simulating the large batch size relationship and providing a stable similarity distribution. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ff5683a2bc30110b7363bf2639b345c4773cd36035117ac4cdb1982b245d3980.jpg",
+        "text": "$$\n\\mathcal { F } _ { t } \\gets m \\mathcal { F } _ { t } + ( 1 - m ) \\mathcal { F } _ { s } , \\quad g _ { t } \\gets m g _ { t } + ( 1 - m ) g _ { s } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } _ { s }$ and $g _ { s }$ denote the most latest encoder and head, respectively, so we name them as the student model with a subscript $s$ . On the other hand, $\\mathcal { F } _ { t }$ and $g _ { t }$ stand for ensembles of the past encoder and head, respectively, so we name them as the teacher model with a subscript $t , m$ represents the momentum coefficient which controls how fast the teacher $\\mathcal { F } _ { t }$ will be updated. ",
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+        "type": "text",
+        "text": "Sharper Distribution as Target. Note, the value of $\\tau _ { t }$ has to be smaller than $\\tau _ { s }$ since $\\tau _ { t }$ will be used to generate the target distribution. A smaller $\\tau$ will result in a “sharper\" distribution which can be interpreted as highlight the most similar feature for $\\mathbf { z } ^ { 1 }$ . Align $ { \\mathbf { p } } ^ { 2 }$ with $\\mathbf { p } ^ { 1 }$ can be regarded as pulling $\\mathbf { z } ^ { 2 }$ towards the features that are similar with $\\mathbf { z } ^ { 1 }$ . ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Weak Augmentation Strategy for Teacher. To further improve the quality and stability of the target distribution, we adopt a weak augmentation strategy for the teacher model since the standard contrastive augmentation is too aggressive, which introduced too many disturbances and will mislead the student network. Please refer to more details in our empirical study. ",
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+    {
+        "type": "text",
+        "text": "Compare with SEED and CLSA. SEED [21] follows the standard Knowledge Distillation (KD) paradigm [26, 49, 18] where it aims to distill the knowledge from a larger network into a smaller architecture. The knowledge transfer happens in the same view but between different models. In our framework, we are trying to maintain the relational consistency between different augmentations; the knowledge transfer happens between different views but in the same network. CLSA [46] also introduced the concept of using weak augmentation to guide a stronger augmentation. However, the “weak\" augmentation in CLSA is equivalent to the “strong\" augmentation in our method (We do not use any stronger augmentations such as [12, 13]). On the other hand, CLSA still adopts the InfoNCE loss (1) for instance discrimination, where our proposed method only utilized the relational consistency loss (5). Finally, CLSA requires at least one additional sample during training, which will slow down the training speed. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Relational Self-supervised Learning with Weak Augmentation (ReSSL) ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/80c06fb2f6646aa3ebddea2fc5fe56f39cd535436576463f8000d33efae532d5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">the non-linear projection head for teacher and student. Q: the memory buffer while network not converge do for i=1 to step do Fetch x from current batch B z1=gt(Ft(Tw(x,01)）))； z²=gs(Fs(Tc(x,02）)；</td></tr><tr><td colspan=\"2\">p1 = SoftMax(z1QT/ Tt); p² =SoftMax(z²QT / Ts）; // Eq.(3)(4) Calculate Lrelation loss by CrossEntropy(p1,p²) ; // Eq.(5) Update Fs and gs with loss Lrelation Update Ft and gt by Ft ← mFt +(1-m)Fs,gt ←mgt+(1-m)gs ; // Eq.</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 Empirical Study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we will empirically study our proposed method on 4 popular self-supervised learning benchmarks and compare to previous state-of-the-art algorithms (SimCLR [6], BYOL [23], SimSiam [9], MoCoV2 [8]). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Small Dataset. CIFAR-10 and CIFAR-100 [31]. The CIFAR-10 dataset consists of $6 0 0 0 0 \\ 3 2 { \\mathrm { x } } 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. CIFAR-100 is just like the CIFAR-10, except it has 100 classes containing 600 images each. There are 500 training images and 100 testing images per class. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Medium Dataset. STL-10 [11] and Tiny ImageNet [33]. STL10 [11] dataset is composed of $9 6 \\mathbf { x } 9 6$ resolution images of 10 classes, 5K labeled training images, 8K validation images, and 100K unlabeled images. The Tiny ImageNet dataset is composed of $6 4 \\mathrm { x } 6 4$ resolution images of 200 classes with 100K training images and 10k validation images. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details We adopt the ResNet18 [25] as our backbone network. Because most of our dataset contains low-resolution images, we replace the first $7 \\mathbf { x } 7$ Conv of stride 2 with $3 { \\tt X } 3$ Conv of stride 1 and remove the first max pooling operation for a small dataset. For data augmentations, we use the random resized crops (the lower bound of random crop ratio is set to 0.2), color distortion (strength $= 0 . 5$ ) with a probability of 0.8, and Gaussian blur with a probability of 0.5. The images from the small and medium datasets will be resized to $3 2 \\mathrm { x } 3 2 $ and 64x64 resolution respectively. Our method is based on MoCoV2 [8]; in order to simulate the shuffle BN trick on one GPU, we simply divide a batch of data into different groups and then calculate BN statistics within each group. The momentum value and memory buffer size are set to 0.99/0.996 and 4096/16384 for small and medium datasets respectively. Moreover, The model is trained using SGD optimizer with a momentum of 0.9 and weight decay of $5 e ^ { - 4 }$ . We linear warm up the learning rate for 5 epochs until it reaches $0 . 0 6 \\times B a t c h S i z e / 2 5 6$ , then switch to the cosine decay scheduler [37]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/aaef0a119213b1b37a24ee7cfe86297f554c07a8b756ab4f5bbf81dc8cdc8940.jpg",
+        "table_caption": [
+            "Table 1: Compare to other SSL algorithms on small and medium dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>BackProp</td><td>EMA</td><td>CIFAR-10</td><td>CIFAR-100</td><td>STL-10</td><td>Tiny ImageNet</td></tr><tr><td>Supervised</td><td>1</td><td>-</td><td>94.22</td><td>74.66</td><td>82.55</td><td>59.26</td></tr><tr><td>SimCLR [6]</td><td>2x</td><td>No</td><td>84.92</td><td>59.28</td><td>85.48</td><td>44.38</td></tr><tr><td>BYOL [23]</td><td>2x</td><td>Yes</td><td>85.82</td><td>57.75</td><td>87.45</td><td>42.70</td></tr><tr><td>SimSiam [9]</td><td>2x</td><td>No</td><td>88.51</td><td>60.00</td><td>87.47</td><td>37.04</td></tr><tr><td>MoCoV2 [8]</td><td>1x</td><td>Yes</td><td>86.18</td><td>59.51</td><td>85.88</td><td>43.36</td></tr><tr><td>ReSSL (Ours)</td><td>1x</td><td>Yes</td><td>90.20</td><td>63.79</td><td>88.25</td><td>46.60</td></tr></table>",
+        "bbox": [
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+            212
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Protocol. All the models will be trained for 200 epochs. For testing the representation quality, we evaluate the pre-trained model on the widely adopted linear evaluation protocol - We will freeze the encoder parameters and train a linear classifier on top of the average pooling features for 100 epochs. To test the classifier, we use the center crop of the test set and computes accuracy according to predicted output. We train the classifier with a learning rate of 30, no weight decay, and momentum of 0.9. The learning rate will be times 0.1 in 60 and 80 epochs. Note, for STL-10; the pretraining will be applied on both labeled and unlabeled images. During the linear evaluation, only the labeled 5K images will be used. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Result. As we can see the result in Table 1, our proposed method outperforms the previous method on all four benchmarks. Reminder, most of the previous method requires twice back-propagation, which results in a much higher training cost than MoCoV2 and our method. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 A Properly Sharpened Relation is A Better Target ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            458,
+            560,
+            474
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The temperature parameter is very crucial in most contrastive learning algorithms. To verify the effective of $\\tau _ { s }$ and $\\tau _ { t }$ for our proposed method, we fixed $\\tau _ { s } = 0 . 1$ or 0.2, and sweep over $\\tau _ { t } =$ $\\{ 0 . 0 1 , 0 . 0 2 , . . . , 0 . 0 7 \\}$ . The result is shown in Table 2. For $\\tau _ { t }$ , the optimal value is either 0.04 or 0.05 across all different datasets. As we can see, the performance is increasing when we increase $\\tau _ { t }$ from 0 to 0.04 and 0.05. After that, the performance will start to decrease. Note, $\\tau _ { t } \\to 0$ correspond to the Top-1 or argmax operation which produce a one-hot distribution as the target. On the other hand, when $\\tau _ { t }  0 . 1$ , the target will be a much flatter distribution that cannot highlight the most similar features for students. Hence, $\\tau _ { t }$ can not be either too small or too large, but it has to be smaller than $\\tau _ { s }$ $\\mathbf { \\bar { p } } ^ { 1 }$ has to be sharper than $ { \\mathbf { p } } ^ { 2 }$ ) so that the target distribution can provide effective guidance to the student model. ",
+        "bbox": [
+            173,
+            486,
+            826,
+            625
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/b099d46dd13f01664b90e11575bafac34fd80e46c4981d5a097525d8b9238f93.jpg",
+        "table_caption": [
+            "Table 2: Effect of different $\\tau _ { t }$ and $\\tau _ { s }$ for ReSSL "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Ts</td><td>Tt =0.01</td><td>Tt =0.02</td><td>Tt=0.03</td><td>Tt = 0.04</td><td>Tt=0.05</td><td>Tt =0.06</td><td>Tt = 0.07</td></tr><tr><td>CIFAR-10</td><td>0.1</td><td>89.35</td><td>89.74</td><td>90.09</td><td>90.04</td><td>90.20</td><td>90.18</td><td>88.67</td></tr><tr><td>CIFAR-10</td><td>0.2</td><td>89.52</td><td>89.67</td><td>89.24</td><td>89.50</td><td>89.22</td><td>89.40</td><td>89.50</td></tr><tr><td>CIFAR-100</td><td>0.1</td><td>62.34</td><td>62.79</td><td>62.71</td><td>63.79</td><td>63.46</td><td>63.20</td><td>61.31</td></tr><tr><td>CIFAR-100</td><td>0.2</td><td>60.37</td><td>60.05</td><td>60.24</td><td>60.09</td><td>59.09</td><td>59.12</td><td>59.76</td></tr><tr><td>STL-10</td><td>0.1</td><td>86.65</td><td>86.96</td><td>87.16</td><td>87.32</td><td>88.25</td><td>87.83</td><td>87.08</td></tr><tr><td>STL-10</td><td>0.2</td><td>85.17</td><td>86.12</td><td>85.01</td><td>85.67</td><td>85.21</td><td>85.51</td><td>85.28</td></tr><tr><td>Tiny ImageNet</td><td>0.1</td><td>45.20</td><td>45.40</td><td>46.30</td><td>46.60</td><td>45.08</td><td>45.24</td><td>44.18</td></tr><tr><td>Tiny ImageNet</td><td>0.2</td><td>43.28</td><td>42.98</td><td>43.58</td><td>42.12</td><td>42.70</td><td>42.76</td><td>42.60</td></tr></table>",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For $\\tau _ { t }$ , it is clearly to see that the result of $\\tau _ { s } = 0 . 1$ can always result a much higher performance than $\\tau _ { s } = 0 . 2$ , which is different to $\\mathbf { M o C o V } 2$ where $\\tau _ { s } = 0 . 2$ is the optimal value. According to [43, 44, 15], a greater temperature will result in a larger angular margin in the hypersphere. Since MoCoV2 adopts instance discrimination as the pretext task, a large temperature can enhance the compactness for the same instance and discrepancy for different instances. In contrast to instance discrimination, our method can be interpreted as pulling similar instances closer on the hypersphere; when the ground truth label is not available, the large angular margin might hurt the performance. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/19895d46df03376d75adeaf9e64fbd761eb75bd5154dd10d4a8f311e2c893a33.jpg",
+        "image_caption": [
+            "Figure 2: Visualization of the 10 nearest neighbour of the query image. The top half is the result when we apply the weak augmentation. The bottom half is the case when the typical contrastive augmentation is adopted. Note, we use the red square to highlight the images that has different ground truth label with the query image. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            88,
+            803,
+            395
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Weak Augmentation Makes Better Relation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            469,
+            516,
+            484
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As we have mentioned, the weaker augmentation strategy for the teacher model is the key to the success of our framework. Here, We implement the weak augmentation as a random resized crop (the random ratio is set to $( 0 . 2 , 1 )$ ) and a random horizontal flip. For temperature parameter, we simply adopt the same setting as in Table 2 and report the performance of the best setting. The result is shown in Table 3, as we can see that when we use the weak augmentation for the teacher model, the performance is significantly boosted across all datasets. We believe that this phenomenon is because relatively small disturbances in the teacher model can provide more accurate similarity guidance to the student model. To further verify this hypothesis, we random sampled three image from STL-10 training set as the query images, and then find the 10 nearest neighbour based on the weak / contrastive augmented query. We visualized the result in Figure 2, ",
+        "bbox": [
+            173,
+            494,
+            825,
+            633
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0284098e7abc56a55a5bbb5892febee7364896b695d31610dcb19632db911b95.jpg",
+        "table_caption": [
+            "Table 3: Effect of weak augmentation guided ReSSL "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Teacher Aug</td><td>Student Aug</td><td>CIFAR-10</td><td>CIFAR-100</td><td>STL-10</td><td>Tiny ImageNet</td></tr><tr><td>Contrastive</td><td>Contrastive</td><td>86.17</td><td>57.60</td><td>84.71</td><td>40.38</td></tr><tr><td>Weak</td><td>Contrastive</td><td>90.20</td><td>63.79</td><td>88.25</td><td>46.60</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 More Experiments on Weak Augmentation ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            719,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Since the weak augmentation for the teacher model is one of the crucial points in ReSSL, we further analyze the effect of applying different augmentations on the teacher model. In this experiment, we simply set $\\tau _ { t } = 0 . 0 4$ and report the linear evaluation performance on the Tiny ImageNet dataset. The results are shown in Table 4. The first row is the baseline, where we simply resize all images to the same resolution (no extra augmentation is applied). Then, we applied random resized crops, random flip, color jitter, grayscale, gaussian blur, and various combinations. We empirically find that if we use no augmentation (e.g., no random resized crops) for the teacher model, the performance tends to degrade. This might result from that the gap of features between two views is way too smaller, which undermines the learning of representations. However, too strong augmentations of teacher model will introduce too much noise and make the target distribution inaccurate (see Figure 2). Thus mildly weak augmentations are better option for the teacher, and random resized crops with random flip is the combination with the highest performance as Table 4 shows. ",
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d761d8849933bda1a963a01143188c2a492cd1a48f80457fcb8feafd4f4c634.jpg",
+        "table_caption": [
+            "Table 4: Effect of different augmentation for teacher model (Tiny ImageNet) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Random Resized Crops</td><td>Random Flip</td><td>Color Jitter</td><td>GrayScale</td><td>Gaussian Blur</td><td>Acc</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>31.74</td></tr><tr><td>√</td><td></td><td></td><td></td><td></td><td>46.00</td></tr><tr><td></td><td>√</td><td></td><td></td><td></td><td>30.98</td></tr><tr><td></td><td></td><td>√</td><td></td><td></td><td>29.46</td></tr><tr><td></td><td></td><td></td><td>√</td><td></td><td>29.68</td></tr><tr><td></td><td></td><td></td><td></td><td>√</td><td>30.10</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td></td><td>46.60</td></tr><tr><td>√</td><td></td><td>√</td><td></td><td></td><td>44.44</td></tr><tr><td>√</td><td></td><td></td><td>4</td><td></td><td>42.28</td></tr><tr><td>√</td><td></td><td></td><td></td><td>√</td><td>44.88</td></tr><tr><td>√</td><td>√</td><td>√</td><td></td><td></td><td>43.70</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td></td><td>42.28</td></tr><tr><td>√</td><td>√</td><td></td><td></td><td>√</td><td>44.52</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Dimension of the Relation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            342,
+            393,
+            356
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since we also adopt the memory buffer as in MoCo [24], the buffer size will be equivalent to the dimension of the distribution $\\mathbf { p } ^ { 1 } \\mathbf { p } ^ { 2 }$ . Thus, it will be one of the crucial points in our framework. To verify the effect the memory buffer size, we simply keep $\\tau _ { s } = 0 . 1$ and $\\tau _ { t } = 0 . 0 4$ , then varying the memory buffer size from 256 to 32768. The result is shown in Table 5, as we can see that a larger memory buffer can significantly boost the performance. However, a further increase in the buffer size can only bring a marginal improvement when the buffer is large enough. ",
+        "bbox": [
+            173,
+            367,
+            826,
+            452
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a0f25f8158b9beac25604319ca29d3584d18c9b9cd382c393c1c73844c1b171e.jpg",
+        "table_caption": [
+            "Table 5: Effect of different memory buffer size on small and medium dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset (Small) CIFAR-10</td><td>K=256 89.37</td><td>K=512 89.53</td><td>K= 1024 89.83</td><td>K= 4096 90.04</td><td>K=8192 90.15</td><td>K=16384 90.35</td></tr><tr><td>CIFAR-100 Dataset (Medium)</td><td>61.17 K=256</td><td>62.47 K=1024</td><td>63.20 K = 4096</td><td>63.79 K=8192</td><td>63.84 K=16384</td><td>64.06 K=32768</td></tr><tr><td>STL-10</td><td>85.88</td><td>87.23</td><td>87.72</td><td>87.42</td><td>87.32</td><td>87.47</td></tr><tr><td>Tiny ImageNet</td><td>43.08</td><td>45.32</td><td>45.78</td><td>45.42</td><td>46.60</td><td>46.48</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5 Visualization of Learned Representations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            590,
+            501,
+            606
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also show the t-SNE [42] visualizations of the representations learned by our proposed method and MoCov2 on the training set of CIFAR-10. Our proposed relational consistency loss leads to better class separation than the contrastive loss. ",
+        "bbox": [
+            176,
+            616,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/932d623072d019b5214fba4ff8e7b94ec5d45e73c9a939414ec5d984820fa4c3.jpg",
+        "image_caption": [
+            "Figure 3: t-SNE visualizations on CIFAR-10. Classes are indicated by colors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            671,
+            810,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Results on Large-scale Datasets ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            823,
+            467,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also performed our algorithm on the large-scale ImageNet-1k dataset [14]. In the experiments, we adopt a learning rate of $0 . 0 5 * B a t c h S i z e / 2 5 6$ , a memory buffer size of $1 3 0 \\mathrm { k }$ , and a 2-layer non-linear projection head with a hidden dimension 4096 and output dimension 512. For $\\tau _ { t }$ and $\\tau _ { s }$ , we simply adopt the best setting from Table 2 where $\\tau _ { t } = 0 . 0 4$ and $\\tau _ { s } = 0 . 1$ . ",
+        "bbox": [
+            174,
+            856,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Linear Evaluation. For the linear evaluation of ImageNet-1k, we strictly follow the setting in SwAV [5]. The results are shown in Table 6. As we can see clearly that ReSSL consistently outperforms previous methods on both $1 \\mathbf { x }$ and $2 \\mathbf { x }$ backprop setting. (Please noted that the student network will be passed in one $2 2 4 \\mathbf { x } 2 2 4$ augmented view and two $2 2 4 \\mathbf { x } 2 2 4$ augmented views for 1x backprob and $2 \\mathbf { x }$ backprob setting respectively.) ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/244684bb762d7e7267f23a58f14ac94606cec4d50c9944e69c1904ae620d891d.jpg",
+        "table_caption": [
+            "Table 6: Top-1 accuracy under the linear evaluation on ImageNet with the ResNet-50 backbone. The table compares the methods over 200 epochs of pretraining. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method Supervised</td><td>Arch R50</td><td>Backprop 1x</td><td>EMA No</td><td>Batch Size 256</td><td>Param 24</td><td>Epochs 120</td><td>Top-1 76.5</td></tr><tr><td>1xBackpropMethods</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InstDisc [47]</td><td>R50</td><td>1x</td><td>No</td><td>256</td><td>24</td><td>200</td><td>58.5</td></tr><tr><td>LocalAgg [52]</td><td>R50</td><td>1x</td><td>No</td><td>128</td><td>24</td><td>200</td><td>58.8</td></tr><tr><td>MoCo v2 [8]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>67.5</td></tr><tr><td>MoCHi [30]</td><td>R50</td><td>1x</td><td>Yes</td><td>512</td><td>24</td><td>200</td><td>68.0</td></tr><tr><td>CPC v2 [32]</td><td>R50</td><td>1x</td><td>No</td><td>512</td><td>24</td><td>200</td><td>63.8</td></tr><tr><td>PCL v2 [36]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>67.6</td></tr><tr><td>AdCo [28]</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>68.6</td></tr><tr><td>ReSSL (Ours)</td><td>R50</td><td>1x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>69.9</td></tr><tr><td>2xBackprop Methods</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLSA-Single [46]</td><td>R50</td><td>2x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>69.4</td></tr><tr><td>SimCLR [6]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>66.8</td></tr><tr><td>SwAV [5]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>69.1</td></tr><tr><td>SimSiam [23]</td><td>R50</td><td>2x</td><td>No</td><td>256</td><td>24</td><td>200</td><td>70.0</td></tr><tr><td>BYOL[23]</td><td>R50</td><td>2x</td><td>Yes</td><td>4096</td><td>24</td><td>200</td><td>70.6</td></tr><tr><td>WCL [51]</td><td>R50</td><td>2x</td><td>No</td><td>4096</td><td>24</td><td>200</td><td>70.3</td></tr><tr><td>ReSSL (Ours)</td><td>R50</td><td>2x</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>71.4</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Working with Multi-Crop Strategy. We also performed ReSSL with Multi-Crop strategy. The result is shown below in Table 7. Specifically, the result of 4 crops is trained with the resolution of $2 2 4 \\times 2 2 4$ , $1 6 0 \\times 1 6 0$ , $1 2 8 \\times 1 2 8$ , $9 6 \\times 9 6$ . For the result of 5 crops, we add an additional $1 9 2 \\times 1 9 2$ image which is exactly the same with AdCo [28]. As we can see, our proposed ReSSL is significantly better than previous state-of-the-art methods. ",
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+        "type": "table",
+        "img_path": "images/4d89d7236f6aad3c3f4f99ebf17309eb4e93595d5df7822379aa31b9507810fb.jpg",
+        "table_caption": [
+            "Table 7: Working with Multi-Crop Strategy (Linear Evaluation on ImageNet) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Arch</td><td>EMA</td><td>Batch Size</td><td>Param</td><td>Epochs</td><td>Top-1</td></tr><tr><td>SwAV [5]</td><td>R50</td><td>No</td><td>256</td><td>24</td><td>200</td><td>72.7</td></tr><tr><td>AdCo [28]</td><td>R50</td><td>No</td><td>256</td><td>24</td><td>200</td><td>73.2</td></tr><tr><td>CLSA-Multi [46]</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>73.3</td></tr><tr><td>ReSSL (4 crops)</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>73.8</td></tr><tr><td>ReSSL (5 crops)</td><td>R50</td><td>Yes</td><td>256</td><td>24</td><td>200</td><td>74.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Working with Smaller Architecture. We also applied our proposed method on the smaller architecture (ResNet-18). The result is shown in Table 8. Following the same training recipe of the ResNet-50 in above, our proposed method has a higher performance than SEED [21] without a larger pretrained teacher network. ",
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+        "type": "table",
+        "img_path": "images/7da1031bb457cd051726ce62f8e44e04d07a8e9aa14292aff0922ec7165bcc41.jpg",
+        "table_caption": [
+            "Table 8: Experiments on ResNet-18 (Linear Evaluation on ImageNet) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Epochs</td><td>Student</td><td>Teacher</td><td>Acc</td></tr><tr><td>MoCo v2</td><td>200</td><td>ResNet-18</td><td>EMA</td><td>52.2</td></tr><tr><td>SEED</td><td>200</td><td>ResNet-18</td><td>ResNet-50 (MoCoV2)</td><td>57.6</td></tr><tr><td>ReSSL (1x backprop)</td><td>200</td><td>ResNet-18</td><td>EMA</td><td>58.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Low-shot Classification. We further evaluate the quality of the learned representations by transferring them to other datasets. Following [36], we perform linear classification on the PASCAL ",
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+    {
+        "type": "text",
+        "text": "VOC2007 dataset [20]. Specifically, we resize all images to 256 pixels along the shorter side and taking a $2 2 4 \\times 2 2 4$ center crop. Then, we train a linear SVM on top of corresponding global average pooled final representations. To study the transferability of the representations in few-shot scenarios, we vary the number of labeled examples $K$ and report the mAP. Table 9 shows the comparison between our method with previous works. We report the average performance over 5 runs (except for $\\mathbf { k } =$ full).It’s clearly to see that our proposed method is consistently outperform MoCo v2 and PCL v2 across all different $K$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/bf551b8730fba03247b609fbef4636480e9d4b4c9e51274d7ee48b17908c0d8c.jpg",
+        "table_caption": [
+            "Table 9: Transfer learning on low-shot image classification "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method Random</td><td>Epochs 1</td><td>ImageNet 1</td><td>K=16 10.10</td><td>K=32 11.34</td><td>K=64 11.96</td><td>Full 12.42</td></tr><tr><td>Supervised</td><td>90</td><td>76.1</td><td>82.26</td><td>84.00</td><td>85.13</td><td>87.27</td></tr><tr><td>MoCo V2 [8]</td><td>200</td><td>67.5</td><td>76.14</td><td>79.16</td><td>81.52</td><td>84.60</td></tr><tr><td>PCL V2 [36]</td><td>200</td><td>67.5</td><td>78.34</td><td>80.72 81.96</td><td>82.67</td><td>85.43</td></tr><tr><td>ReSSL (1x backprob)</td><td>200</td><td>69.9</td><td>79.17</td><td></td><td>83.81</td><td>86.31</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Semi-Supervised Learning. Next, we evaluate the performance obtained when fine-tuning the model representation using a small subset of labeled data. In this experiments, we adopt our 5 crops pre-trained model. The result is shown in Table 10. Notably, with just 200 epochs of pre-training, ReSSL outperforms all previous methods. ",
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+    {
+        "type": "table",
+        "img_path": "images/9d5fb81438929188dea718dc7e572f2dd88438b968e548fa0bb651592ebda948.jpg",
+        "table_caption": [
+            "Table 10: Semi-supervised Learning "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Epochs</td><td>Linear Eval</td><td>1% Labels</td><td>10% Labels</td></tr><tr><td>SimCLR [6]</td><td>1000</td><td>69.3</td><td>48.3</td><td>65.6</td></tr><tr><td>BYOL[23]</td><td>1000</td><td>74.3</td><td>53.2</td><td>68.6</td></tr><tr><td>SwAV[5]</td><td>800</td><td>75.3</td><td>53.9</td><td>70.2</td></tr><tr><td>ReSSL (5 crops)</td><td>200</td><td>74.7</td><td>57.9</td><td>70.4</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose relational self-supervised learning (ReSSL), a new paradigm for unsupervised visual representation learning framework that maintains the relational consistency between instances under different augmentations. Our proposed ReSSL relaxes the typical constraints in contrastive learning where different instances do not always need to be pushed away on the embedding space, and the augmented views do not need to share exactly the same feature. An extensive empirical study shows the effect of each component in our framework. The experiments on large-scaled datasets demonstrate the efficiency and state-of-the-art performance for unsupervised representation learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work provides a technical advancement in the field of unsupervised visual representation learning. An immediate application of this work is to give a pre-trained model for the tasks where the data annotation is very hard to collect (e.g.medical images and fine-grained images.) Moreover, the most significant advantage of ReSSL is that we do not need to train the model for a long time as the previous method (generally 800 or 1000 epochs), which will cause a lot of carbon dioxide emissions. We believe ReSSL is a more environment-friendly method since it can achieve a competitive performance with much lesser training costs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work is funded by the National Key Research and Development Program of China (No. 2018AAA0100701) and the NSFC 61876095. Chang Xu was supported in part by the Australian Research Council under Projects DE180101438 and DP210101859. Shan You is supported by Beijing Postdoctoral Research Foundation. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+    },
+    {
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parse/train/HylsTT4FvB/HylsTT4FvB.md

@@ -0,0 +1,418 @@
+# ON THE “STEERABILITY” OF GENERATIVE ADVERSARIAL NETWORKS
+
+Ali Jahanian\*, Lucy Chai\*, & Phillip Isola
+
+Massachusetts Institute of Technology Cambridge, MA 02139, USA {jahanian,lrchai,phillipi}@mit.edu
+
+# ABSTRACT
+
+An open secret in contemporary machine learning is that many models work beautifully on standard benchmarks but fail to generalize outside the lab. This has been attributed to biased training data, which provide poor coverage over real world events. Generative models are no exception, but recent advances in generative adversarial networks (GANs) suggest otherwise – these models can now synthesize strikingly realistic and diverse images. Is generative modeling of photos a solved problem? We show that although current GANs can fit standard datasets very well, they still fall short of being comprehensive models of the visual manifold. In particular, we study their ability to fit simple transformations such as camera movements and color changes. We find that the models reflect the biases of the datasets on which they are trained (e.g., centered objects), but that they also exhibit some capacity for generalization: by “steering” in latent space, we can shift the distribution while still creating realistic images. We hypothesize that the degree of distributional shift is related to the breadth of the training data distribution. Thus, we conduct experiments to quantify the limits of GAN transformations and introduce techniques to mitigate the problem. Code is released on our project page: https://ali-design.github.io/gan_steerability/.
+
+# 1 INTRODUCTION
+
+The quality of deep generative models has increased dramatically over the past few years. When introduced in 2014, Generative Adversarial Networks (GANs) could only synthesize MNIST digits and low-resolution grayscale faces (Goodfellow et al., 2014). The most recent models, however, produce diverse high-resolution images that are often indistinguishable from natural photos (Brock et al., 2018; Karras et al., 2018).
+
+Science fiction has long dreamed of virtual realities filled of synthetic content as rich as, or richer, than the real world (e.g., The Matrix, Ready Player One). How close are we to this dream? Traditional computer graphics can render photorealistic 3D scenes, but cannot automatically generate detailed content. Generative models like GANs, in contrast, can create content from scratch, but we do not currently have tools for navigating the generated scenes in the same kind of way as you can walk through and interact with a 3D game engine.
+
+In this paper, we explore the degree to which you can navigate the visual world of a GAN. Figure 1 illustrates the kinds of transformations we explore. Consider the dog at the top-left. By moving in some direction of GAN latent space, can we hallucinate walking toward this dog? As the figure indicates, and as we will show in this paper, the answer is yes. However, as we continue to zoom in, we quickly reach limits. Once the dog face fills the full frame, continuing to walk in this direction fails to increase the zoom. A similar effect occurs in the daisy example (row 2 of Fig. 1), where a direction in latent space moves the daisy up and down, but cannot move it out of frame.
+
+We hypothesize that these limits are due to biases in the distribution of images on which the GAN is trained. For example, if the training dataset consists of centered dogs and daises, the same may be the case in GAN-generated images. Nonetheless, we find that some degree of transformation is possible. When and why can we achieve certain transformations but not others?
+
+![](images/9a6f038d46959cede0b1cf06d0bfc0bcdc9aabf73443bc7702f2a5b693f231a2.jpg)  
+Figure 1: Learned latent space trajectories in generative adversarial networks correspond to visual transformations like camera shift and zoom. Take the “steering wheel”, drive in the latent space, and explore the natural image manifold via generative transformations!
+
+This paper seeks to quantify the degree to which we can achieve basic visual transformations by navigating in GAN latent space. In other words, are GANs “steerable” in latent space?1 We analyze the relationship between the data distribution on which the model is trained and the success in achieving these transformations. From our experiments, it is possible to shift the distribution of generated images to some degree, but we cannot extrapolate entirely out of the dataset’s support. In particular, attributes can be shifted in proportion to the variability of that attribute in the training data. We further demonstrate an approach to increase model steerability by jointly optimizing the generator and latent direction, together with data augmentation on training images. One of the current criticisms of generative models is that they simply interpolate between datapoints, and fail to generate anything truly new, but our results add nuance to this story. It is possible to achieve distributional shift, but the ability to create realistic images from a modified distributions relies on sufficient diversity in the dataset along the dimension that we vary.
+
+Our main findings are:
+
+A simple walk in the latent space of GANs achieves camera motion and color transformations in the output image space. These walks are learned in self-supervised manner without labeled attributes or distinct source and target images. The linear walk is as effective as more complex non-linear walks, suggesting that the models learn to roughly linearize these operations without being explicitly trained to do so. The extent of each transformation is limited, and we quantify a relationship between dataset variability and how much we can shift the model distribution. The transformations are a general-purpose framework that work with different model architectures, e.g. BigGAN, StyleGAN, and DCGAN, and illustrate different disentanglement properties in their respective latent spaces.   
+• Data augmentation improves steerability, as does jointly training the walk trajectory and the generator weights, which allows us to achieve larger transformation effects.
+
+# 2 RELATED WORK
+
+Latent space manipulations can be seen from several perspectives – how we achieve it, what limits it, and what it enables us to do. Our work addresses these three aspects together, and we briefly refer to each one in related work.
+
+Interpolations in latent space Traditional approaches to image editing with GAN latent spaces find linear directions that correspond to changes in labeled attributes, such as smile-vectors and gender-vectors for faces (Radford et al., 2015; Karras et al., 2018). However these manipulations are not exclusive to GANs; in flow-based generative models, linearly interpolating between two encoded images allow one to edit a source image toward attributes of the target (Kingma & Dhariwal, 2018). Mollenhoff & Cremers (2019) proposes a modified GAN formulation by treating data ¨ as directional $k$ -currents, where moving along tangent planes naturally corresponds to interpretable manipulations. Upchurch et al. (2017) removes the generative model entirely and instead interpolates in the intermediate feature space of a pretrained classifier, again using feature mappings of source and target sets to determine an edit direction. Unlike these approaches, we learn our latentspace trajectories in a self-supervised manner without labeled attributes or distinct source and target images. Instead, we learn to approximate editing operations on individual source images. We find that linear trajectories in latent space can capture simple image manipulations, e.g., zoom-vectors and shift-vectors, although we also obtain similar results using nonlinear trajectories.
+
+Dataset bias Biases from training data and network architecture both impact the generalization capacity of learned models (Torralba & Efros, 2011; Geirhos et al., 2018; Amini et al.). Dataset biases partly comes from human preferences in taking photos: we tend to take pictures in specific “canonical” views that are not fully representative of the entire visual world (Mezuman & Weiss, 2012; Jahanian et al., 2015). Consequently, models trained with these datasets inherit their biases. This may result in models that misrepresent the given task – such as tendencies towards texture bias rather than shape bias on ImageNet classifiers (Geirhos et al., 2018) – and in turn limits their generalization performance on similar objectives (Azulay & Weiss, 2018). Our latent space trajectories transform the output corresponding to various image editing operations, but ultimately we are constrained by biases in the data and cannot extrapolate arbitrarily far beyond the data’s support.
+
+Generative models for content creation The recent progress in generative models has opened interesting avenues for content creation (Brock et al., 2018; Karras et al., 2018), including applications that enable users to fine-tune the generated output (Simon; Zhu et al., 2016; Bau et al., 2018). A by-product the current work is enable users to modify image properties by turning a single knob – the magnitude of the learned transformation in latent space. We further demonstrate that these image manipulations are not just a simple creativity tool; they also provide us with a window into biases and generalization capacity of these models.
+
+Applications of latent space editing Image manipulations using generative models suggest several interesting downstream applications. For example, Denton et al. (2019) learns linear walks corresponding to various facial characteristics – they use these to measure biases in facial attribute detectors, whereas we study biases in the generative model that originate from training data. Shen et al. (2019) also assumes linear latent space trajectories and learns paths for face attribute editing according to semantic concepts such as age and expression, thus demonstrating disentanglement of the latent space. White (2016) suggests approaches to improve the learned manipulations, such as using spherical linear interpolations, resampling images to remove biases in attribute vectors, and using data augmentation as a synthetic attribute for variational autoencoders. Goetschalckx et al. (2019) applies a linear walk to achieve transformations corresponding to cognitive properties of an image such as memorability, aesthetics, and emotional valence. Unlike these works, we do not require an attribute detector or assessor function to learn the latent space trajectory, and therefore our loss function is based on image similarity between source and target images. In addition to linear walks, we explore using non-linear walks parametrized by neural networks for editing operations.
+
+# 3 METHOD
+
+Generative models such as GANs (Goodfellow et al., 2014) learn a mapping function $G$ such that $G : z  x$ . Here, $z$ is the latent code drawn from a Gaussian density and $x$ is an output, e.g., an image. Our goal is to achieve transformations in the output space by moving in latent space, as shown in Fig. 2. In general, this goal also captures the idea in equivariance, in which transformations in the input space result in equivalent transformations in the output space (c.f. Hinton et al. (2011); Cohen et al. (2019); Lenc & Vedaldi (2015)).
+
+Objective We want to learn an $N$ -dimensional vector representing the optimal path in latent space for a given transformation. The vector is multiplied with continuous parameter $\alpha$ which signifies the step size: large $\alpha$ values correspond to a greater degree of transformation, while small $\alpha$ values correspond to a lesser degree. Formally, we learn the walk $w$ by minimizing the objective function:
+
+$$
+\boldsymbol { w } ^ { * } = \underset { \boldsymbol { w } } { \arg \operatorname* { m i n } } \mathbb { E } _ { z , \boldsymbol { \alpha } } [ \mathcal { L } ( \boldsymbol { G } ( \boldsymbol { z } + \boldsymbol { \alpha } \boldsymbol { w } ) , \mathrm { e d i t } ( \boldsymbol { G } ( \boldsymbol { z } ) , \boldsymbol { \alpha } ) ) ] .
+$$
+
+![](images/c90dec3c4443b64bcda2f6ce13adf0345d8531a294a1420764cfaccb5ca531f6.jpg)  
+Figure 2: We aim to find a path in $z$ space to transform the generated image $G ( z )$ to its edited version edit $\left( G ( z , \alpha ) \right)$ , e.g., an $\alpha \times$ zoom. This walk results in the generated image $G ( z + \alpha w )$ when we choose a linear walk, or $G ( f ( f ( . . . ( z ) ) )$ when we choose a non-linear walk.
+
+Here, $\mathcal { L }$ measures the distance between the generated image after taking an $\alpha$ -step in the latent direction $G ( z + \alpha w )$ and the target edit $( G ( z ) , \alpha )$ derived from the source image $\overset { \cdot } { G } ( z )$ . We use $L 2$ loss as our objective $\mathcal { L }$ , however we also obtain similar results when using the LPIPS perceptual image similarity metric (Zhang et al., 2018) (see Appendix B.4.1). Note that we can learn this walk in a fully self-supervised manner – we perform the edit(·) operation on an arbitrary generated image and subsequently the vector to minimize the objective. Let model $( \alpha )$ denote the optimized transformation vector $w ^ { * }$ with the step size $\alpha$ , defined as model $( \alpha ) = G \big ( z + \alpha w ^ { * } \big )$ .
+
+The previous setup assumes linear latent space walks, but we can also learn non-linear trajectories in which the walk direction depends on the current latent space position. For the non-linear walk, we learn a function, $f ^ { \ast } ( z )$ , which corresponds to a small $\epsilon$ -step transformation $\mathtt { e d i t } ( G ( z ) , \epsilon )$ . To achieve bigger transformations, we apply $f$ recursively, mimicking discrete Euler ODE approximations. Formally, for a fixed $\epsilon$ , we minimize
+
+$$
+\mathcal { L } = \mathbb { E } _ { z , n } [ | | G ( f ^ { n } ( z ) ) - \mathrm { e d i t } ( G ( z ) , n \epsilon ) ) | | ] ,
+$$
+
+where $f ^ { n } ( \cdot )$ is an $n$ th-order function composition $f ( f ( f ( \dots ) ) )$ , and $f ( z )$ is parametrized with a neural network. We discuss further implementation details in Appendix A.4. We use this function composition approach rather than the simpler setup of $G ( z + \alpha \mathbf { N } \mathbf { N } ( z ) )$ because the latter learns to ignore the input $z$ when $\alpha$ takes on continuous values, and is thus equivalent to the previous linear trajectory (see Appendix A.3 for further details).
+
+Quantifying Steerability We further seek to quantify how well we can achieve desired image manipulations under each transformation. To this end, we compare the distribution of a given attribute, e.g., “luminance”, in the dataset versus in images generated after walking in latent space.
+
+For color transformations, we consider the effect of increasing or decreasing the $\alpha$ coefficient corresponding to each color channel. To estimate the color distribution of model-generated images, we randomly sample $N = 1 0 0$ pixels per image both before and after taking a step in latent space. Then, we compute the pixel value for each channel, or the mean RGB value for luminance, and normalize the range between 0 and 1.
+
+For zoom and shift transformations, we rely on an object detector which captures the central object in the image class. We use a MobileNet-SSD v1 (Liu et al., 2016) detector to estimate object bounding boxes, and average over image classes recognizable by the detector. For each successful detection, we take the highest probability bounding box corresponding to the desired class and use that to quantify the amount of transformation. For the zoom operation, we use the area of the bounding box normalized by the area of the total image. For shift in the X and Y directions, we take the center X and Y coordinates of the bounding box, and normalize by image width or height.
+
+Truncation parameters in GANs (as used in Brock et al. (2018); Karras et al. (2018)) trade off between the diversity of the generated images and sample quality. When comparing generated images to the dataset distribution, we use the largest possible truncation for the model and perform similar cropping and resizing of the dataset as done during model training (see Brock et al. (2018)). When comparing the attributes of generated distributions under different $\alpha$ magnitudes to each other but not to the dataset, we reduce truncation to 0.5 to ensure better performance of the object detector.
+
+Reducing Transformation Limits Equations 1 and 2 learn a latent space walk assuming a pretrained generative model, thus keeping the model weights fixed. The previous approach allows us to understand the latent space organization and limitations in the model’s transformation capacity. To overcome these limits, we explore adding data augmentation by editing the training images with each corresponding transformation, and train the generative model with this augmented dataset. We also introduce a modified objective function that jointly optimizes the generator weights and a linear walk vector:
+
+$$
+G ^ { * } , w ^ { * } = \arg \operatorname* { m i n } _ { G , w } \left( \mathcal { L } _ { e d i t } + \mathcal { L } _ { G A N } \right) ,
+$$
+
+where the edit loss encourages low $L 2$ error between learned transformation and target image:
+
+$$
+\mathcal { L } _ { e d i t } = L 2 \left( G ( z + \alpha w ) - \mathsf { e d i t } ( G ( z ) , \alpha ) \right) .
+$$
+
+The GAN loss optimizes for discriminator error:
+
+$$
+\mathcal { L } _ { G A N } = \operatorname* { m a x } _ { D } \left( \mathbb { E } _ { z , \alpha } [ D ( G ( z + \alpha w ) ) ] - \mathbb { E } _ { x , \alpha } [ D ( \mathsf { e d i t } ( x , \alpha ) ) ] \right) ,
+$$
+
+where we draw images $x$ from the training dataset and perform data augmentation by applying the edit operation on them. This optimization approach encourages the generator to organize its latent space so that the transformations lie along linear paths, and when combined with data augmentation, results in larger transformation ranges which we demonstrate in Sec. 4.4
+
+# 4 EXPERIMENTS
+
+We demonstrate our approach using BigGAN (Brock et al., 2018), a class-conditional GAN trained on 1000 ImageNet categories. We learn a shared latent space walk by averaging across the image categories, and further quantify how this walk affects each class differently. We focus on linear walks in latent space for the main text, and show additional results on nonlinear walks in Sec. 4.3 and Appendix B.4.2. We also conduct experiments on StyleGAN (Karras et al., 2018), which uses an unconditional style-based generator architecture in Sec. 4.3 and Appendix B.5.
+
+# 4.1 WHAT IMAGE TRANSFORMATIONS CAN WE ACHIEVE IN LATENT SPACE?
+
+![](images/c24cf1a1a51ac9cbc73b9152b59334f6d3d0e11e3ba2407384c6be50649a8ae3.jpg)  
+Figure 3: Transformation limits. As we increase the magnitude of $w ^ { * }$ , the operation either does not transform the image any further, or the image becomes unrealisitic. Below each figure we also indicate the average LPIPS perceptual distance between 200 sampled image pairs of that category. Perceptual distance decreases as we move farther from the source (center image), which indicates that the images are converging.
+
+We show qualitative results of the learned transformations in Fig. 1. By steering in the generator latent space, we learn a variety of transformations on a given source image (shown in the center panel of each transformation). Interestingly, several priors come into play when learning these image transformations. When we shift a daisy downwards in the Y direction, the model hallucinates that the sky exists on the top of the image. However, when we shift the daisy up, the model inpaints the remainder of the image with grass. When we alter the brightness of a image, the model transitions between nighttime and daytime. This suggests that the model can extrapolate from the original source image, and still remain consistent with the image context.
+
+![](images/ec6ad021ee1ff39e44bf5c028d0456cb926b15a38067de6239e162f9e5ab6859.jpg)  
+Figure 4: Each row shows how a single latent direction $w ^ { * }$ affects two different ImageNet classes. We observe that changes are consistent with semantic priors (e.g., “Volcanoes” explode, “Alps” do not). Boxplots show the LPIPS perceptual distance before and after transformation for 200 samples per class.
+
+However, when we increase the step size of $\alpha$ , we observe that the degree to which we can achieve each transformation is limited. In Fig. 3 we observe two potential failure cases: one in which the the image becomes unrealistic, and the other in which the image fails to transform any further. When we try to zoom in on a Persian cat, we observe that the cat no longer increases in size beyond some point, and in fact consistently undershoots the target zoom. On the other hand, when we try to zoom out on the cat, we observe that it begins to fall off the image manifold, and does not become any smaller after some point. Indeed, the perceptual distance (using LPIPS) between images decreases as we push $\alpha$ towards the transformation limits. Similar trends hold with other transformations: we are able to shift a lorikeet up and down to some degree until the transformation yields unrealistic output, and despite adjusting $\alpha$ on the rotation vector, we are unable to rotate a pizza. Are the limitations to these transformations governed by the training dataset? In other words, are our latent space walks limited because in ImageNet photos the cats are mostly centered and taken within a certain size? We seek to investigate and quantify these biases in the next sections.
+
+An intriguing characteristic of the learned trajectory is that the amount it affects the output depends on the image class. In Fig. 4, we investigate the impact of the walk for different image categories under color transformations. By moving in the direction of a redness vector, we are able to successfully recolor a jellyfish, but we are unable to change the color of a goldfinch, which remains yellow which slight changes in background textures. Likewise, increasing brightness changes an erupting volcano to a dormant one, but does not have much effect on Alps, which only transitions between night and day. In the third example, we use our latent walk to turn red sports cars to blue, but it cannot recolor firetrucks. Again, perceptual distance over image samples confirms these qualitative observations: a 2-sample $t$ -test yields $t = 2 0 . 7 7$ , $p < 0 . 0 0 1$ for jellyfish/goldfinch, $t = 8 . 1 4$ , $p < 0 . 0 0 1$ for volcano/alp, and $t = 6 . 8 4$ , $p < 0 . 0 0 1$ for sports car/fire engine. We hypothesize that the different impact of the shared transformation on separate image classes relates to the variability in the underlying dataset. The overwhelming majority of firetrucks are red2, but sports cars appear in a variety of colors. Therefore, our color transformation is constrained by the dataset biases of individual classes.
+
+With shift, we can move the distribution of the center object by varying $\alpha$ . In the underlying model, the center coordinate of the object is most concentrated at half of the image width and height, but after applying the shift in X and shift in Y transformation, the mode of the transformed distribution varies between 0.3 and 0.7 of the image width/height. To quantify the distribution changes, we compute the area of intersection between the original model distribution and the distribution after applying each transformation and observe that the intersection decreases as we increase or decrease the magnitude of $\alpha$ . However, our transformations are limited to a certain extent – if we increase $\alpha$ beyond 150 pixels for vertical shifts, we start to generate unrealistic images, as evidenced by a sharp rise in FID and converging modes in the transformed distributions (Fig. 5 columns 2 & 3).
+
+![](images/8bd7ec7105ec8c4c2fafaaf73ce8b8a65f7d8363ca24419c0a021b2b31f9d9a0.jpg)  
+Figure 5: Quantifying the extent of transformations. We compare the attributes of generated images under the raw model output $G ( z )$ , compared to the distribution under a learned transformation model $( \alpha )$ . We measure the intersection between $G ( z )$ and $\scriptstyle { \mathrm { m o d e l } } ( \alpha )$ , and also compute the FID on the transformed image to limit our transformations to the natural image manifold.
+
+We perform a similar procedure for zoom, by measuring the area of the bounding box for the detected object under different magnitudes of $\alpha$ . Like shift, we observe that subsequent increases in $\alpha$ magnitude start to have smaller and smaller effects on the mode of the resulting distribution (Fig. 5 last column). Past an ${ 8 } \mathbf { x }$ zoom in or out, we observe an increase in the FID signifying decreasing image quality. Interestingly for zoom, the FID under zooming in and zooming out is anti-symmetric, indicating that how well we can zoom-in and retain realisitic images differs from that of zooming out. These trends are consistent with the plateau in transformation behavior that we qualitatively observe in Fig. 3. Although we can arbitrarily increase the $\alpha$ step size, after some point we are unable to achieve further transformation and risk deviating from the natural image manifold.
+
+# 4.2 HOW DOES THE DATA AFFECT THE TRANSFORMATIONS?
+
+Is the extent to which we can transform each class, as we observed in Fig. 4, due to limited variability in the underlying dataset for each class? One way of quantifying this is to measure the difference in transformed model means, model $( + \alpha )$ and model $( - \alpha )$ , and compare it to the spread of the dataset distribution. For each class, we compute standard deviation of the dataset with respect to our statistic of interest (pixel RGB value for color, and bounding box area and center value for zoom and shift transformations respectively). We hypothesize that if the amount of transformation is biased depending on the image class, we will observe a correlation between the distance of the mean shifts and the standard deviation of the data distribution.
+
+More concretely, we define the change in model means under a given transformation as:
+
+$$
+\Delta \mu _ { k } = \mu _ { k , \mathrm { m o d e l } ( + \alpha ^ { * } ) } - \mu _ { k , \mathrm { m o d e l } ( - \alpha ^ { * } ) }
+$$
+
+for a given class $k$ and we set $\alpha ^ { * }$ to be largest and smallest $\alpha$ values used in training. The degree to which we achieve each transformation is a function of $\alpha$ , so we use the same $\alpha$ value for all classes – one that is large enough to separate the means of $\mu _ { k , \mathrm { m o d e 1 } ( \alpha ^ { * } ) }$ and $\mu _ { k , \mathrm { m o d e 1 } ( - \alpha ^ { * } ) }$ under transformation, but also for which the FID of the generated distribution remains below a threshold $T$ of generating reasonably realistic images (for our experiments we use $T = 2 2$ ).
+
+![](images/df44290e914f0b4c7a428d057e9a0755a8b43017c6c995e27629f24ea7659497.jpg)  
+Figure 6: Understanding per-class biases. We observe a correlation between the variability in the training data for ImageNet classes, and our ability to shift the distribution under latent space transformations. Classes with low variability (e.g., robin) limit our ability to achieve desired transformations, in comparison to classes with a broad dataset distribution (e.g., laptop). To the right, we show the distribution of the zoom attribute in the dataset (black) and under $+ \alpha$ (red) and $- \alpha$ (green) transformations for these two examples.
+
+In Fig. 6 we plot the standard deviation $\sigma$ of the dataset on the $\mathbf { X }$ -axis, and the model $\Delta \mu$ under a $+ \alpha ^ { * }$ and $- \alpha ^ { * }$ transformation on the y-axis, as defined in Eq. 6. We sample randomly from 100 classes for the color, zoom and shift transformations, and generate 200 samples of each class under the positive and negative transformations. We use the same setup of drawing samples from the model and dataset and computing the statistics for each transformation as described in Sec. 4.1.
+
+Indeed, we find that the width of the dataset distribution, captured by the standard deviation of random samples drawn from the dataset for each class, relates to how much we can transform. There is a positive correlation between the spread of the dataset and the magnitude of $\Delta \mu$ observed in the transformed model distributions, and the slope of all observed trends differs significantly from zero $\mathit { p } < 0 . 0 0 1$ for all transformations). For the zoom transformation, we show examples of two extremes along the trend. For the “robin” class the spread $\sigma$ in the dataset is low, and subsequently, the separation $\Delta \mu$ that we are able to achieve by applying $+ \alpha ^ { * }$ and $- \alpha ^ { * }$ transformations is limited. On the other hand, for “laptops”, the dataset spread is broad; ImageNet contains images of laptops of various sizes, and we are able to attain wider shifts in the model distribution.
+
+From these results, we conclude that the amount of transformation we can achieve relates to the dataset variability. Consistent with our qualitative observations in Fig. 4, we find that if the images for a particular class have adequate coverage over the entire range of a given transformation, then we are better able to move the model distribution to both extremes. On the other hand, if the images for a given class are less diverse, the transformation is limited by this dataset bias.
+
+# 4.3 ALTERNATIVE ARCHITECTURES AND WALKS
+
+We ran an identical set of experiments using the nonlinear walk in the BigGAN latent space (Eq 2) and obtained similar quantitative results. To summarize, the Pearson’s correlation coefficient between dataset $\sigma$ and model $\Delta \mu$ for linear walks and nonlinear walks is shown in Table 1, and full results in Appendix B.4.2. Qualitatively, we observe that while the linear trajectory undershoots the targeted level of transformation, it is able to preserve more realistic-looking results (Fig. 7). The transformations involve a trade-off between minimizing the loss and maintaining realistic output, and we hypothesize that the linear walk functions as an implicit regularizer that corresponds well with the inherent organization of the latent space.
+
+![](images/64515b5dbabd6d8c2dac3efe5583fc44cfb29cefa46f828fca9372ccbecbfa6a.jpg)  
+Figure 7: Comparison of linear and nonlinear walks for the zoom operation. The linear walk undershoots the targeted level of transformation, but maintains more realistic output.
+
+<table><tr><td></td><td>Luminance</td><td>Shift X</td><td>Shift Y</td><td>Zoom</td></tr><tr><td>Linear</td><td>0.59</td><td>0.28</td><td>0.39</td><td>0.37</td></tr><tr><td>Non-linear</td><td>0.49</td><td>0.49</td><td>0.55</td><td>0.60</td></tr></table>
+
+Table 1: Pearson’s correlation coefficient between dataset $\sigma$ and model $\Delta \mu$ for measured attributes.   
+p-value for slope $< 0 . 0 0 1$ for all transformations.
+
+![](images/e0ca9735d3bc0e89a112fa2709410145ba594b6a2656d03ba8f7200f4349918e.jpg)  
+Figure 8: Distribution for luminance transformation learned from the StyleGAN cars generator, and qualitative examples of color transformations on various datasets using StyleGAN.
+
+To test the generality of our findings across model architecture, we ran similar experiments on StyleGAN, in which the latent space is divided into two spaces, $z$ and $W$ . As Karras et al. (2018) notes that the $W$ space is less entangled than $z$ , we apply the linear walk to $W$ and show results in Fig. 8 and Appendix B.5. One interesting aspect of StyleGAN is that we can change color while leaving other structure in the image unchanged. In other words, while green faces do not naturally exist in the dataset, the StyleGAN model is still able to generate them. This differs from the behavior of BigGAN, where changing color results in different semantics in the image, e.g., turning a dormant volcano to an active one. StyleGAN, however, does not preserve the exact geometry of objects under other transformations, e.g., zoom and shift (see Appendix B.5).
+
+# 4.4 TOWARDS STEERABLE GANS
+
+So far, we have frozen the parameters of the generative model when learning a latent space walk for image editing, and observe that the transformations are limited by dataset bias. Here we investigate approaches to overcome these limitations and increase model steerability. For these experiments, we use a class-conditional DCGAN model (Radford et al., 2015) trained on MNIST digits (LeCun, 1998).
+
+To study the effect of dataset biases, we train (1) a vanilla DCGAN and (2) a DCGAN with data augmentation, and then learn the optimal walk in Eq. 1 after the model has been trained – we refer to these two approaches in Fig. 9 as argmin $W$ and argmin $W + a u g$ , respectively. We observe that adding data augmentation yields transformations that better approximate the target image and attain lower $L 2$ error than the vanilla DCGAN (blue and orange curves in Fig. 9). Qualitatively, we observe that transformations using the vanilla GAN (argmin W) become patchy and unrealistic as we increase the magnitude of $\alpha$ , but when the model is trained with data augmentation (argmin $W +$ aug), the digits retain their structural integrity.
+
+Rather than learning the walk vector $w$ assuming a frozen generator, we may also jointly optimize the model and linear walk parameter together, as we formalized in Eq. 3. This allows the model to learn an equivariance between linear directions in the latent space and the corresponding image transformations. We refer to this model as argmin $G , W$ in Fig. 9. Compared to the frozen generator (in argmin $W$ and argmin $W + a u g$ ), the joint objective further decreases $L 2$ error (green curve in Fig. 9). We show additional qualitative examples in Appendix B.8. The steerable range of the generator increases with joint optimization and data augmentation, which provides additional evidence that training data bias impacts the models’ steerability and generalization capacity. We tried DCGAN on CIFAR10 as a more complicated dataset, however were unable to get steering to be effective – all three methods failed to produce realistic transformations and joint training in fact performed the worst. Finding the right steering implementation per GAN and dataset, especially for joint training, may be a difficult problem and an interesting direction for future work.
+
+![](images/3f8dbe9edc869688364454738a81a70f1c7a8a15e8ec06f21bea0d2ff46ed19c.jpg)  
+Figure 9: Reducing the effect of transformation limits. Using a DCGAN model on MNIST digits, we compare the $L 2$ reconstruction errors on latent space walks for models trained with vanilla GANs without (argmin W) and with data augmentation (argmin $W + a u g$ ). We also compare to jointly optimizing the generator and the walk parameters with data augmentation (argmin $G , W )$ , which achieves the lowest $L 2$ error.
+
+# 5 CONCLUSION
+
+GANs are powerful generative models, but are they simply replicating the existing training datapoints, or can they to generalize beyond the training distribution? We investigate this question by exploring walks in the latent space of GANs. We optimize trajectories in latent space to reflect simple image transformations in the generated output, learned in a self-supervised manner. We find that the model is able to exhibit characteristics of extrapolation – we are able to “steer” the generated output to simulate camera zoom, horizontal and vertical movement, camera rotations, and recolorization. However, our ability to naively move the distribution is finite: we can transform images to some degree but cannot extrapolate entirely outside the support of the training data. To increase model steerability, we add data augmentation during training and jointly optimize the model and walk trajectory. Our experiments illustrate the connection between training data bias and the resulting distribution of generated images, and suggest methods for extending the range of images that the models are able to create.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Quang H Le, Lore Goetschalckx, Alex Andonian, David Bau, and Jonas Wulff for helpful discussions. This work was supported by a Google Faculty Research Award to P.I., and a U.S. National Science Foundation Graduate Research Fellowship to L.C.
+
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+
+# A METHOD DETAILS
+
+# A.1 OPTIMIZATION FOR THE LINEAR WALK
+
+We learn the walk vector using mini-batch stochastic gradient descent with the Adam optimizer (Kingma & Ba, 2014) in tensorflow, trained on 20000 unique samples from the latent space $z$ . We share the vector $w$ across all ImageNet categories for the BigGAN model.
+
+# A.2 IMPLEMENTATION DETAILS FOR LINEAR WALK
+
+We experiment with a number of different transformations learned in the latent space, each corresponding to a different walk vector. Each of these transformations can be learned without any direct supervision, simply by applying our desired edit to the source image. Furthermore, the parameter $\alpha$ allows us to vary the extent of the transformation. We found that a slight modification to each transformation improved the degree to which we were able to steer the output space: we scale $\alpha$ differently for the learned transformation $G ( z + \alpha _ { g } w )$ , and the target edit edit $\left( G ( z ) , \alpha _ { t } \right)$ . We detail each transformation below:
+
+Shift. We learn transformations corresponding to shifting an image in the horizontal X direction and the vertical Y direction. We train on source images that are shifted $- \alpha _ { t }$ pixels to the left and $\alpha _ { t }$ pixels to the right, where we set $\alpha _ { t }$ to be between zero and one-half of the source image width or height $D$ . When training the walk, we enforce that the $\alpha _ { g }$ parameter ranges between -1 and 1; thus for a random shift by $t$ pixels, we use the value $\alpha _ { g } = \alpha _ { t } / D$ . We apply a mask to the shifted image, so that we only apply the loss function on the visible portion of the source image. This forces the generator to extrapolate on the obscured region of the target image.
+
+Zoom. We learn a walk which is optimized to zoom in and out up to four times the original image. For zooming in, we crop the central portion of the source image by some $\alpha _ { t }$ amount, where $0 . 2 5 <$ $\alpha _ { t } < 1$ and resize it back to its original size. To zoom out, we downsample the image by $\alpha _ { t }$ where $1 < \alpha _ { t } < 4$ . To allow for both a positive and negative walk direction, we set $\alpha _ { g } = \log ( \alpha _ { t } )$ . Similar to shift, a mask applied during training allows the generator to inpaint the background scene.
+
+Color. We implement color as a continuous RGB slider, e.g., a 3-tuple $\alpha _ { t } = ( \alpha _ { R } , \alpha _ { G } , \alpha _ { B } )$ , where each $\alpha _ { R }$ , $\alpha _ { G }$ , $\alpha _ { B }$ can take values between $[ - 0 . 5 , 0 . 5 ]$ in training. To edit the source image, we simply add the corresponding $\alpha _ { t }$ values to each of the image channels. Our latent space walk is parameterized as $z + \alpha _ { g } w = z + \alpha _ { R } w _ { R } + \alpha _ { G } w _ { G } + \alpha _ { B } w _ { B }$ where we jointly learn the three walk directions $w _ { R } , w _ { G }$ , and $w _ { B }$ .
+
+Rotate in 2D. Rotation in 2D is trained in a similar manner as the shift operations, where we train with $- 4 5 \leq \alpha _ { t } \leq 4 5$ degree rotation. Using $R = 4 5$ , scale $\alpha _ { g } = \alpha _ { t } / R$ . We use a mask to enforce the loss only on visible regions of the target.
+
+Rotate in 3D. We simulate a 3D rotation using a perspective transformation along the $\mathrm { _ { Z } }$ -axis, essentially treating the image as a rotating billboard. Similar to the 2D rotation, we train with $- 4 5 \leq \alpha _ { t } \leq 4 5$ degree rotation, we scale $\alpha _ { g } = \alpha _ { t } / R$ where $R = 4 5$ , and apply a mask during training.
+
+# A.3 LINEAR $\operatorname { N N } ( z )$ WALK
+
+Rather than defining $w$ as a vector in $z$ space (Eq. 1), one could define it as a function that takes a $z$ as input and maps it to the desired $z ^ { \prime }$ after taking a variable-sized step $\alpha$ in latent space. In this case, we may parametrize the walk with a neural network $w = \Nu \Nu ( z )$ , and transform the image using $G ( z + \mathrm { \bar { \alpha } N N } ( z ) )$ . However, as we show in the following proof, this idea will not learn to let $w$ be a function of $z$ .
+
+Proof. For simplicity, let $\begin{array} { r l r } { w } & { { } \ = \ } & { F ( z ) } \end{array}$ . We optimize for $\begin{array} { r l } { J ( w , \alpha ) \quad } & { { } = } \end{array}$ $\mathbb { E } _ { z } \left[ \dot { \mathcal { L } } ( G ( z + \alpha w ) , \mathrm { e d i t } ( G ( z ) , \alpha ) ) \right]$ where $\alpha$ is an arbitrary scalar value. Note that for the target image, two equal edit operations is equivalent to performing a single edit of twice the size (e.g., shifting by $1 0 \mathrm { p x }$ the same as shifting by 5px twice; zooming by $4 \mathbf { x }$ is the same as zooming by $2 \mathbf { x }$ twice). That is,
+
+$$
+\mathsf { e d i t } ( G ( z ) , 2 \alpha ) = \mathsf { e d i t } ( \mathsf { e d i t } ( G ( z ) , \alpha ) , \alpha ) .
+$$
+
+To achieve this target, starting from an initial $z$ , we can take two steps of size $\alpha$ in latent space as follows:
+
+$$
+\begin{array} { l } { { z _ { 1 } = z + \alpha F ( z ) } } \\ { { z _ { 2 } = z _ { 1 } + \alpha F ( z _ { 1 } ) } } \end{array}
+$$
+
+However, because we let $\alpha$ take on any scalar value during optimization, our objective function enforces that starting from $z$ and taking a step of size $2 \alpha$ equals taking two steps of size $\alpha$ :
+
+$$
+z + 2 \alpha F ( z ) = z _ { 1 } + \alpha F ( z _ { 1 } )
+$$
+
+Therefore:
+
+$$
+\begin{array} { c } { { z + 2 \alpha F ( z ) = z + \alpha F ( z ) + \alpha F ( z _ { 1 } ) \Rightarrow } } \\ { { \alpha F ( z ) = \alpha F ( z _ { 1 } ) \Rightarrow } } \\ { { F ( z ) = F ( z _ { 1 } ) . } } \end{array}
+$$
+
+Thus $F ( \cdot )$ simply becomes a linear trajectory that is independent of the input $z$
+
+# A.4 OPTIMIZATION FOR THE NON-LINEAR WALK
+
+Given the limitations of the previous walk, we define our nonlinear walk $F ( z )$ using discrete step sizes $\epsilon$ . We define $F ( z )$ as $z + \mathrm { N N } ( z )$ , where the neural network NN learns a fixed $\epsilon$ step transformation, rather than a variable $\alpha$ step. We then renormalize the magnitude $z$ . This approach mimics the Euler method for solving ODEs with a discrete step size, where we assume that the gradient of the transformation in latent space is of the form $\begin{array} { r } { \epsilon \frac { d z } { d t } = \mathbf { \hat { N } } \mathbf { N } ( z ) } \end{array}$ and we approximate $\begin{array} { r } { z _ { i + 1 } = z _ { i } + \epsilon \frac { d z } { d t } | _ { z _ { i } } } \end{array}$ . The key difference from A.3 is the fixed step size, which avoids optimizing for the equality in (7).
+
+We use a two-layer neural network to parametrize the walk, and optimize over 20000 samples using the Adam optimizer as before. Positive and negative transformation directions are handled with two neural networks having identical architecture but independent weights. We set $\epsilon$ to achieve the same transformation ranges as the linear trajectory within 4-5 steps.
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 MODEL AND DATA DISTRIBUTIONS
+
+How well does the model distribution of each property match the dataset distribution? If the generated images do not form a good approximation of the dataset variability, we expect that this would also impact our ability to transform generated images. In Fig. 10 we show the attribute distributions of the BigGAN model $G ( z )$ compared to samples from the ImageNet dataset. We show corresponding results for StyleGAN and its respective datasets in Appendix B.5. While there is some bias in how well model-generated images approximate the dataset distribution, we hypothesize that additional biases in our transformations come from variability in the training data.
+
+# B.2 QUANTIFYING TRANSFORMATION LIMITS
+
+We observe that when we increase the transformation magnitude $\alpha$ in latent space, the generated images become unrealistic and the transformation ceases to have further effect. We show this qualitatively in Fig. 3. To quantitatively verify this trends, we can compute the LPIPS perceptual distance of images generated using consecutive pairs of $\alpha _ { i }$ and $\alpha _ { i + 1 }$ . For shift and zoom transformations, perceptual distance is larger when $\alpha$ (or $\log ( \alpha )$ for zoom) is near zero, and decreases as the the magnitude of $\alpha$ increases, which indicates that large $\alpha$ magnitudes have a smaller transformation effect, and the transformed images appear more similar. On the other hand, color and rotate in 2D/3D exhibit a steady transformation rate as the magnitude of $\alpha$ increases.
+
+Note that this analysis does not tell us how well we achieve the specific transformation, nor whether the latent trajectory deviates from natural-looking images. Rather, it tells us how much we manage to change the image, regardless of the transformation target. To quantify how well each transformation is achieved, we rely on attribute detectors such as object bounding boxes (see B.3).
+
+# B.3 DETECTED BOUNDING BOXES
+
+To quantify the degree to which we are able to achieve the zoom and shift transformations, we rely on a pre-trained MobileNet- $S S D \nu I ^ { 3 }$ object detection model. In Fig. 12 and 13 we show the results of applying the object detection model to images from the dataset, and images generated by the model under the zoom, horizontal shift, and vertical shift transformations for randomly selected values of $\alpha$ , to qualitatively verify that the object detection boundaries are reasonable. Not all ImageNet images contain recognizable objects, so we only use ImageNet classes containing objects recognizable by the detector for this analysis.
+
+# B.4 ALTERNATIVE WALKS IN BIGGAN
+
+# B.4.1 LPIPS OBJECTIVE
+
+In the main text, we learn the latent space walk $w$ by minimizing the objective function:
+
+$$
+J ( w , \alpha ) = \mathbb { E } _ { z } \left[ \mathcal { L } ( G ( z + \alpha w ) , \mathsf { e d i t } ( G ( z ) , \alpha ) ) \right] .
+$$
+
+using a Euclidean loss for $\mathcal { L }$ . In Fig. 14 we show qualitative results using the LPIPS perceptual similarity metric (Zhang et al., 2018) instead of Euclidean loss. Walks were trained using the same parameters as those in the linear-L2 walk shown in the main text: we use $2 0 \mathrm { k }$ samples for training, with Adam optimizer and learning rate 0.001 for zoom and color, 0.0001 for the remaining edit operations (due to scaling of $\alpha$ ).
+
+# B.4.2 NON-LINEAR WALKS
+
+Following B.4.2, we modify our objective to use discrete step sizes $\epsilon$ rather than continuous steps. We learn a function $F ( z )$ to perform this $\epsilon$ -step transformation on given latent code $z$ , where $F ( z )$ is parametrized with a neural network. We show qualitative results in Fig. 15. We perform the same set of experiments shown in the main text using this nonlinear walk in Fig. 16. These experiments exhibit similar trends as we observed in the main text – we are able to modify the generated distribution of images using latent space walks, and the amount to which we can transform is related to the variability in the dataset. However, there are greater increases in FID when we apply the non-linear transformation, suggesting that these generated images deviate more from natural images and look less realistic.
+
+# B.4.3 ADDITIONAL QUALITATIVE EXAMPLES
+
+We show qualitative examples for randomly generated categories for BigGAN linear-L2, linear LPIPS, and nonlinear trajectories in Figs. 17, 18, 19 respectively.
+
+# B.5 WALKS IN STYLEGAN
+
+We perform similar experiments for linear latent space walks using StyleGAN models trained on the LSUN cat, LSUN car, and FFHQ face datasets. As suggested by Karras et al. (2018), we learn the walk vector in the intermediate $W$ latent space due to improved attribute disentanglement in $W$ . We show qualitative results for color, shift, and zoom transformations in Figs. 20, 22, 24 and corresponding quantitative analyses in Figs. 21, 23, 25. We show qualitative examples for the comparison of optimizing in the $W$ and $z$ latent spaces in Stylegan in 28.
+
+# B.6 WALKS IN PROGRESSIVE GAN
+
+We also experiment with the linear walk objective in the latent space of Progressive GAN Karras et al. (2017). One interesting property of the Progressive GAN interpolations is that they take much longer to train to have a visual effect – for example for color, we could obtain drastic color changes in Stylegan W latent space using as few as 2k samples, but with progressive gan, we used $6 0 \mathrm { k }$ samples and still did not obtain as strong of an effect. This points to the Stylegan w latent space being more “flexible” and generalizable for transformation, compared to the latent space of progressive GAN. Moreover, we qualitatively observe some entanglement in the progressive gan transformations – for example, changing the level of zoom also changes the lighting. We did not observe big effects in the horizontal and vertical shift transformations. Qualitative examples and quantitative results are shown in Figs. 26, 27.
+
+# B.7 QUALITATIVE EXAMPLES FOR ADDITIONAL TRANSFORMATIONS
+
+Since the color transformation operates on individual pixels, we can optimize the walk using a segmented target – for example when learning a walk for cars, we only modify pixels in segmented car region when generating edit $( G ( z ) , \alpha )$ . StyleGAN is able to roughly localize the color transformation to this region, suggesting disentanglement of different objects within the $W$ latent space (Fig. 29 left) as also noted in Karras et al. (2018); Shen et al. (2019). We also show qualitative results for adjust image contrast (Fig. 29 right), and for combining zoom, shift X, and shift Y transformations (Fig. 30).
+
+# B.8 ADDITIONAL RESULTS FOR IMPROVING MODEL STEERABILITY
+
+We further test the hypothesis that dataset variability impacts the amount we are able to transform by comparing DCGAN models trained with and without data augmentation. Namely, with data augmentation, the discriminator is able to see edited versions of the real images. We also jointly train the model and the walk trajectory which encourages the model to learn linear walks. For zoom, horizontal shift, and 2D rotate transformations, additional samples for three training approaches – without data augmentation, with data augmentation, and joint optimization – appear in Fig. 31-33. Qualitatively, transformations using the model trained without data augmentation degrade the digit structure as $\alpha$ magnitude increases, and may even change one digit to another. Training with data augmentation and joint optimization better preserves digit structure and identity.
+
+![](images/e41029fa602863dc647412424048771a3b92bd9faaaf450a712e58e7764222c2.jpg)  
+Figure 10: Comparing model versus dataset distribution. We plot statistics of the generated under the color (luminance), zoom (object bounding box size), and shift operations (bounding box center), and compare them to the statistics of images in the training dataset.
+
+![](images/0c92472901851739dc643edcba539263bd58ca6d1d529611e7eb2633ac3feb3b.jpg)  
+Figure 11: LPIPS Perceptual distances between images generated from pairs of consecutive $\alpha _ { i }$ and $\alpha _ { i + 1 }$ . We sample 1000 images from randomly selected categories using BigGAN, transform them according to the learned linear trajectory for each transformation. We plot the mean perceptual distance and one standard deviation across the 1000 samples (shaded area), as well as 20 individual samples (scatterplot). Because the Rotate 3D operation undershoots the targeted transformation, we observe more visible effects when we increase the $\alpha$ magnitude.
+
+![](images/eea50fb5004d019add5a055a1a93fde4c22e8c6701ea8b94081002323847019f.jpg)  
+Figure 12: Bounding boxes for random selected classes using ImageNet training images.
+
+![](images/6c68b4b46792bf3bc00cdbc6ecb14c665b65630be342c7764fea947301833440.jpg)  
+Figure 13: Bounding boxes for random selected classes using model-generated images for zoom and horizontal and vertical shift transformations under random values of $\alpha$ .
+
+![](images/12aa379ed500808602faa884b5f82dc931805797c2fc040edee033f01403ada0.jpg)  
+Figure 14: Linear walks in BigGAN, trained to minimize LPIPS loss. For comparison, we show the same samples as in Fig. 1 (which used a linear walk with L2 loss).
+
+![](images/6dbe5fc65dd5b5af816522a227fd49d7c87905fc7aabcebbdd49e6315bbbfdb6.jpg)  
+Figure 15: Nonlinear walks in BigGAN, trained to minimize L2 loss for color and LPIPS loss for the remaining transformations. For comparison, we show the same samples in Fig. 1 (which used a linear walk with L2 loss), replacing the linear walk vector $w$ with a nonlinear walk.
+
+![](images/7c4d270412bb6f2ab30fc27925e4d6d349c822312214a5343a9c70b0b05d139e.jpg)  
+Figure 16: Quantitative experiments for nonlinear walks in BigGAN. We show the attributes of generated images under the raw model output $G ( z )$ , compared to the distribution under a learned transformation $\scriptstyle { \mathrm { m o d e l } } ( \alpha )$ , the intersection area between $G ( z )$ and $\scriptstyle { \mathrm { m o d e l } } ( \alpha )$ , FID score on transformed images, and scatterplots relating dataset variability to the extent of model transformation.
+
+![](images/9ab9425f935c33817016d392abd7d2682231fef34f676cbe5332110e0903ef6b.jpg)  
+Figure 17: Qualitative examples for randomly selected categories in BigGAN, using the linear trajectory and L2 objective.
+
+![](images/deae6cc5e95f6c1e75720b2354863f49209b149757e64dbb59ca7a822a4acb06.jpg)  
+Figure 18: Qualitative examples for randomly selected categories in BigGAN, using the linear trajectory and LPIPS objective.
+
+![](images/e15a3655ed50f7c478cc1272004b93864f80910b37653d0831eab4b4afcbdfaf.jpg)  
+Figure 19: Qualitative examples for randomly selected categories in BigGAN, using a nonlinear trajectory.
+
+![](images/146934017a7142841d7a93ebf6fc22706d9be97ede0f1e084581221c63326148.jpg)  
+Figure 20: Qualitative examples for learned transformations using the StyleGAN car generator.
+
+![](images/975886811eb28c8e433ec717b6fdaa9a77ff3a386ff1eb274237715f9226a407.jpg)  
+Figure 21: Quantitative experiments for learned transformations using the StyleGAN car generator.
+
+![](images/6d0a7ca4447a537f62adab0b8c61f2f09ebd8dd00c4baf7f32b8fe0a98a9cbe3.jpg)  
+Figure 22: Qualitative examples for learned transformations using the StyleGAN cat generator.
+
+![](images/a5b32d3a6ad8598caff6a0c742cbadd173fed0a4c52e17fd13f6fb94ac41d633.jpg)  
+Figure 23: Quantitative experiments for learned transformations using the StyleGAN cat generator.
+
+![](images/1692af4eefb460c5de2a2adef41724dd2f377de835d5ddd23984c0fa20181526.jpg)  
+Figure 24: Qualitative examples for learned transformations using the StyleGAN FFHQ face generator.
+
+![](images/02bf3d1d7c6eef3106197f13885042b61915930d0fb97d56e15e76083cf596d9.jpg)  
+Figure 25: Quantitative experiments for learned transformations using the StyleGAN FFHQ face generator. For the zoom operation not all faces are detectable; we plot the distribution as zeros for $\alpha$ values in which no face is detected. We use the dlib face detector (King, 2009) for bounding box coordinates.
+
+![](images/421486bf09c16396dad4e6d8a6dadaf3c3e06424875fd10c7f45838084ccfc45.jpg)  
+Figure 26: Qualitative examples for learned transformations using the Progressive GAN CelebaAHQ face generator.
+
+![](images/d55a27885ee5c2c6dcb6cd26f5c0cf8bbf8f96f08bb5771d24ce80e4e1fb53a9.jpg)  
+Figure 27: Quantitative experiments for learned transformations using the Progressive GAN CelebA-HQ face generator.
+
+![](images/b33d6c74e470c4fd2577ecd03b17395e5769a518aa704500950b590047a36024.jpg)  
+Figure 28: Comparison of optimizing for color transformations in the Stylegan w and $\mathbf { Z }$ latent spaces.
+
+![](images/5feb1e5ab47401f29f016cab8459d188bad4aecdc9b724889e9c322cb44c6cdb.jpg)  
+Figure 29: Qualitative examples of optimizing for a color walk with a segmented target using StyleGAN in left column and a contrast walk for both BigGAN and StyleGAN in the right column.
+
+![](images/b30fd61dacd0c95b6f9c53434e95d3b0e922cb75a0e5ed17cfba5522f15d2af8.jpg)  
+Figure 30: Qualitative examples of a linear walk combining the zoom, shift X, and shift Y transformations. First row shows the target image, second row shows the result of learning a walk for the three transformations jointly, and the third row shows results for combining the separately trained walks. Green vertical line denotes image center.
+
+![](images/7aa219a718217fc5d7fa8465a84c850bcf10eb9248976e864ecdb22750285475.jpg)  
+Figure 31: Quantitative experiments on steerability with an MNIST DCGAN for the Zoom transformation. Odd rows are the target images and even rows are the learned transformations.
+
+![](images/be0d9c98f8f648a0ecc1f19b36c0f32d37343c70cce2fadfa56aefce35f5b783.jpg)  
+Figure 32: Quantitative experiments on steerability with an MNIST DCGAN for the Shift X transformation. Odd rows are the target images and even rows are the learned transformations.
+
+![](images/d9b9d601687143c26a5f8d663861dd9433e2d55f1b8d7edbc60377969d6698d6.jpg)  
+Figure 33: Quantitative experiments on steerability with an MNIST DCGAN for the Rotate 2D transformation. Odd rows are the target images and even rows are the learned transformations.

parse/train/Hyxu6oAqYX/Hyxu6oAqYX.md

@@ -0,0 +1,478 @@
+# AN ENERGY-BASED FRAMEWORK FOR ARBITRARY LABEL NOISE CORRECTION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose an energy-based framework for correcting mislabelled training examples in the context of binary classification. While existing work addresses random and class-dependent label noise, we focus on feature dependent label noise, which is ubiquitous in real-world data and difficult to model. Two elements distinguish our approach from others: 1) instead of relying on the original feature space, we employ an autoencoder to learn a discriminative representation and 2) we introduce an energy-based formalism for the label correction problem. We prove that a discriminative representation can be learned by training a generative model using a loss function comprised of the difference of energies corresponding to each class. The learned energy value for each training instance is compared to the original training labels and contradictions between energy assignment and training label are used to correct labels. We validate our method across eight datasets, spanning synthetic and realistic settings, and demonstrate the technique’s state-of-the-art label correction performance. Furthermore, we derive analytical expressions to show the effect of label noise on the gradients of empirical risk.
+
+# 1 INTRODUCTION
+
+Machine learning algorithms depend on reliable training labels or, when applicable, sufficiently robust learning models in order to produce generalizable predictions (Zhu & Wu, 2004; Frenay & ´ Verleysen, 2014). Many empirical datasets suffer from training label corruption, which can result from annotation error, human bias, or a noisy process for generating labels (Smyth, 1996; Brodley & Friedl, 1999). In practice, it can be costly to obtain noise-free labels; hence, research on reducing the effects of label noise on learning has received considerable attention.
+
+Most work, however, focuses on label noise that is independent of the input features (e.g., random label noise or class conditional random noise). The introduction of feature dependency significantly complicates mathematical analysis (Natarajan et al., 2013; Liu & Tao, 2014; Northcutt et al., 2017). Although work in this domain has yielded impressive results and improved generalization (Rebbapragada & Brodley, $2 0 0 7 \mathrm { a }$ ; Natarajan et al., 2013; Patrini et al., 2017; Rolnick et al., 2017; Ren et al., 2018), the simplicity of these noise assumptions fails to capture crucial mislabelling processes that arise in practice. In this work, we propose a semi-supervised framework for correcting the effects of feature dependent label noise on supervised learning algorithms.
+
+Label noise processes are categorized into three types, as illustrated in Figure 1. Type I refers to a noise model where any label in the training data is incorrect with probability $\gamma$ . In the case of type II noise, the probability of label corruption is conditioned on the class, such that we have different noise rate $\gamma _ { 0 }$ and $\gamma _ { 1 }$ for each class. Finally, type III noise models describe how label noise depends explicitly on input features. Feature dependent label noise or equivalently type III noise is a more realistic type of label noise that is ubiquitous in empirical datasets (Lachenbruch, 1974; Schafer & Graham, 2002; Frenay & Verleysen, 2014). As an example of type III noise, consider the ´ diagnostic labels of Alzheimer’s disease. In this setting, the probability of label noise depends on both age and sex (i.e., younger male patients are harder to diagnose) (Khachaturian, 1985; Murray et al., 2016). Type III noise also includes the prevalence of unreliable labels for training instances in low density regions of feature space (Denoeux, 1995; 1997; 2000) or near classification boundaries (Lachenbruch, 1966; 1974; Chhikara & McKeon, 1984; Cohen, 1997; Beigman & Klebanov, 2009; Beigman Klebanov & Beigman, 2009; Kolcz & Cormack, 2009).
+
+![](images/17ff1ad04f2044ce9bde008ce9a26958e859a944f3d9d177a1fe68e09fa26b28.jpg)  
+Figure 1: Different categories of label noise and their statistical dependencies, as depicted by the red arrow. In type I noise, all instances are equally likely to be mislabelled base on some probability $\gamma \in \ [ 0 , \frac { 1 } { 2 } ]$ . In the case of type II, this probability is different for each class: $\gamma _ { 0 } \ \in [ 0 , \frac { 1 } { 2 } ]$ and $\gamma _ { 1 } \in [ 0 , \frac { 1 } { 2 } ]$ . Type III label noise is the most realistic model and yet the least studied. In this case the probability of an error is a function of the input features: i.e. $p _ { \mathrm { e r r o r } } \sim f ( \pmb { x } )$ .
+
+In this paper, we present a practical framework for correcting label noise that extends beyond type I and type II noise. We train an energy-based generative model on a small subset of the training data with reliable labels, either manually annotated or automatically learned. This model is used to identify and correct mislabelled examples based on the whether the assigned energy of a given point is compatible with its original label. We empirically demonstrate our framework’s ability to handle input dependent label noise on both simulated and real datasets.
+
+# 2 RELATED WORK
+
+The full body of work on the problem of label noise is too extensive to review here; however, we direct interested readers to the detailed review by Frenay & Verleysen (2014). In this section, we ´ highlight key contributions in the label noise literature. We then briefly discuss existing theoretical work.
+
+# 2.1 LABEL NOISE CORRECTION
+
+Existing work in label noise correction is designed for Type I and Type II noise and falls into three categories: relabeling, learning procedures and loss functions. We expand on each area below.
+
+Relabeling: Earlier approaches to label noise focus on relabeling the noisy training set. Brodley & Friedl (1999) use the output of an ensemble of classifiers to identify mislabeled training examples. Sun et al. (2007) identify mislabeled instances based on the entropy of class probabilities outputted by a Bayesian classifier. These approaches, while robust to simpler label noise models, do not account for Type III noise.
+
+Learning Procedures: Methods modifying the learning procedure include the perceptron algorithm with margin (PAM) Frenay & Verleysen (2014). Crammer & Lee (2010) use a velocity-based learn- ´ ing procedure to learn the weight vector distribution, termed gaussian herding (NHERD). Sukhbaatar et al. (2014) introduce a noise layer to a neural network architecture to learn the noise function, while Rebbapragada & Brodley (2007b) weight each example by class confidence in the training procedure. These methods commonly suffer from overfitting to the noise and again, primarily focus on Type I and Type II noise.
+
+Loss Functions: Long & Servedio (2010) have shown classification algorithms that optimize a convex potential over a linear class are not robust to random label noise. This has led to work which aims to modify convex loss functions in order to make them noise-tolerant in the presence of type I and type II noise. Ghosh et al. (2015) derived sufficient conditions for classification losses that render them robust to random noise; namely, the components of a loss (where each component is defined over a given class) must sum to a constant value. Rooyen et al. (2015) proposed a convex loss that avoids the negative result of Long & Servedio (2010) for type I noise by virtue of being negatively unbounded. Natarajan et al. (2013) have shown that risk minimization over the corrupted data is consistent with risk minimization over clean data provided that the standard convex loss (e.g., binary cross entropy) is modified appropriately to produce an unbiased loss estimator (ULE). Each of these loss functions is presented for and validated on Type I and Type II noise.
+
+# 2.2 EXISTING THEORETICAL GUARANTEES
+
+(Bylander, 1994; Blum et al., 1998; Blum & Mitchell, 1998) offer guarantees for hypothesis generalization in the face of Type I and Type II noise. Angluin & Laird (1988); Bylander (1997; 1998); Servedio (1999) present guarantees for type III noise, where the probability of error depends on the distance to the margin. We introduce a gradient-based interpretation of label noise to these existing results.
+
+# 3 BINARY CLASSIFICATION IN THE PRESENCE OF LABEL NOISE
+
+# 3.1 PROBLEM SETUP AND NOTATION
+
+Let $P$ denote the true distribution from which $n$ i.i.d. training examples $( \pmb { x } _ { 1 } , y _ { 1 } ) , ( \pmb { x } _ { 2 } , y _ { 2 } ) , \dots , ( \pmb { x } _ { n } , y _ { n } )$ have been drawn, where $( \pmb { x } _ { i } , y _ { i } ) ~ \in ~ \mathbb { R } ^ { d } \times \{ 0 , 1 \}$ . The clean and corrupted training datasets are denoted as
+
+$$
+\begin{array} { r } { T = \{ ( \boldsymbol { x } _ { i } , y _ { i } ) \mathrm { ~ f o r ~ } i = 1 , 2 , \ldots , n \} \quad \& \quad \tilde { T } = \{ ( \boldsymbol { x } _ { i } , \tilde { y } _ { i } ) \mathrm { ~ f o r ~ } i = 1 , 2 , \ldots , n \} , } \end{array}
+$$
+
+respectively, where due to some label noise process $y _ { i } \to \tilde { y } _ { i }$ . Hence, we have access to the noisy data $\tilde { \tau }$ during training instead of the clean data $\tau$ . It can be assumed that the corrupted samples $( { \pmb x } _ { i } , \tilde { y } _ { i } )$ are drawn from a noisy distribution $\tilde { P }$ .
+
+In supervised binary classification, we aim to learn a discriminator $f : \mathbb { R } ^ { d }  \mathbb { R }$ that minimizes the risk with respect to a given loss function1. Formally, we want to minimize the empirical risk is $\begin{array} { r } { \hat { R } \left[ \ell , f , \mathcal { T } \right] = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \ell \left( f ( x _ { i } , \pmb { \theta } ) , y _ { i } \right) } \end{array}$ , where the $m , \ell , \theta$ denote the size of the mini-batch size, loss function, and the learnable model parameters respectively. Gradient based learning algorithms compute $\nabla _ { \pmb { \theta } } ( \hat { R } [ \ell , f , \mathcal { T } ] )$ and update model parameters. For example, as in mini-batch gradient descent,
+
+$$
+\pmb { \theta } _ { t + 1 } = \pmb { \theta } _ { t } - \eta \nabla _ { \pmb { \theta } } \left( \hat { R } \left[ \ell , f , \mathcal { T } \right] \right) ,
+$$
+
+where $\eta$ is the learning rate. In practice, we may not have access to the clean data $\tau$ . Instead we must learn the discriminator using the noisy data $\tilde { \tau }$ . Thus, the second term in Eq. 2 becomes $\nabla _ { \pmb { \theta } } ( \hat { R } [ \ell , f , \tilde { \mathcal { T } } ] )$ , which denotes the risk with respect to the noisy training data.
+
+# 3.2 LABEL NOISE EFFECT ON RISK MINIMIZATION
+
+We can describe the effect of label noise on learning by expressing the noisy gradient $\nabla _ { \pmb { \theta } } ( \hat { R } [ \ell , f , \tilde { \mathcal { T } } ] )$ in terms of the true gradient $\nabla _ { \pmb { \theta } } ( \hat { R } [ \ell , f , \mathcal { T } ] )$ and a noise term. In the case of type I noise where any label $y _ { i }$ can be flipped to $1 - y _ { i }$ with probability $\gamma \in [ 0 , 1 / 2 ]$ , we have
+
+$$
+\mathbb { E } _ { ( \pmb { x } , \tilde { y } ) \sim \tilde { P } } \left[ \nabla _ { \pmb { \theta } } \hat { R } [ \ell , \tilde { \mathcal { T } } ] \right] = ( 1 - 2 \gamma ) \mathbb { E } _ { \mathrm { t r u e } } + 2 \gamma \mathbb { E } _ { \mathrm { r a n d } } ,
+$$
+
+where $\mathbb { E } _ { \mathrm { t r u e } } \equiv \mathbb { E } _ { ( \pmb { x } , \tilde { \pmb { y } } ) \sim \tilde { P } } [ \nabla _ { \pmb { \theta } } \hat { R } [ \ell , \mathcal { T } ] ]$ is the true (i.e. noise-free) gradient, and $\mathbb { E } _ { \mathrm { r a n d } }$ is the completely noisy term obtained by replacing all targets $y _ { i }$ with a random value of either 0 or 1. For type II noise, we have an equation analogous to Eq. 3, where $\gamma$ is replaced by the average $\bar { \gamma } = ( \gamma _ { 0 } + \gamma _ { 1 } ) / 2$ noise rate and we have an additional term reflecting the noise imbalance between each class when $\gamma _ { 0 } \neq \gamma _ { 1 }$ . For more information on this and derivations of these equations, see Appendix A.
+
+From Eq. 3 we deduce that for small $\gamma$ the true gradients dominate the dynamics of searching $\pmb \theta$ space for optimal parameters. As $\gamma$ grows, the true gradients are scaled down by the factor $( 1 - 2 \gamma )$ , which means that the contribution of ${ \mathbb E } _ { \mathrm { t r u e } }$ diminishes and the random perturbation term $\mathbb { E } _ { \mathrm { r a n d } }$ starts to dominate as $\gamma$ grows. A similar conclusion can be reached for type II noise (replacing $\gamma$ with $\bar { \gamma }$ ) where we have an additional term modifying the true gradients based on the class imbalance (see A.2).
+
+The effect of $\mathbb { E } _ { \mathrm { r a n d } }$ can be negated by increasing the amount of training data proportionately with $\bar { \gamma }$ as discussed by Rolnick et al. (2017): i.e. as the probability of a random error occurring increases, the model needs more data to make progress in the direction of the true gradient. We can neutralize the effects (i.e. suppress $\mathbb { E } _ { \mathrm { r a n d } , }$ ) of random noise (type I) or class conditional random noise (type II) by increasing the number of training examples and by utilizing a robust loss function when possible (Natarajan et al., 2013).
+
+![](images/5e5c17f196d7d191c01059fb3248fddf90ca0747e558aa41789ce0991e26915a.jpg)  
+Figure 2: Plots showing the path traversed in $\pmb \theta$ space by a supervised model as learnable parameters are updated via a gradient based optimizer. The gold line shows the path followed by a noise-free model and the blue lines show the modified paths due to label noise. In the case of random label noise (left plot), the perturbations $\mathbb { E } _ { \mathrm { r a n d } }$ to the true gradient direction are curbed by an increase in training set size. In contrast, for input dependent noise (right plot), supplying the model with more data perpetuates the label bias.
+
+In contrast, Type III noise is substantially more difficult to address because the available training data $\tilde { \tau }$ can have optimal discriminators that are significantly different from the true discriminator corresponding to the clean data $\tau$ . Separating Type III noise into the true term and the perturbation term necessitates restrictive assumptions on the noise model (See Appendix A. In Figure 2, we show this by visualizing SGD over the true training set (gold lines) and the noise training set (blue lines). As shown in Figure 2, increasing the size of the training set is effective in address Type II noise. This strategy is not only ineffective for addressing type III noise, but they can also lead to our model learning the biased discriminator; thus, hindering any possibility of generalization.
+
+# 4 AN ENERGY-BASED FRAMEWORK FOR BIAS CORRECTION
+
+Type III noise is particularly challenging because noise can depend on the input features in an arbitrarily complex way. Modeling this requires significant information about the noise process, which can be difficult to obtain in practice. In order to deal with this challenge we will avoid training directly on the mislabelled instances. Instead, we start our training procedure by assuming there exists a small subset of $\tilde { \tau }$ that is noise-free (e.g. $1 \%$ of the data is sufficient the MNIST dataset). Such a set can be obtained via a manual labelling process or by using domain knowledge to identify the appropriate data points2.
+
+Our objective is to correct mislabelled training instances such that the corrected data leads to the improved generalization of a trained model. We will demonstrate in Section 5 that our framework extends beyond Type I and Type $\mathrm { I I }$ noise to Type III noise, in realistic settings. In particular, we demonstrate the method’s strength in correcting algorithmically assigned labels.
+
+The proposed correction framework consists of three steps:
+
+1. Obtain known labels: There are two ways of identifying instances that have correct labels. The first approach is to use domain knowledge – frequently, one has access to a subset of clean labels and their instances. We can use this subset as our clean data for step 2. If this set is unknown or too small, then we can inference which examples are likely to be correctly labeled, as is done in Ding et al. (2018). We report results on both approaches in Section 5.
+
+2. Train semi-supervised model: The framework uses the filtered data from the previous step to train a semi-supervised algorithm that learns to classify instances based on similarity between features. In contrast with supervised learning, where the goal is to learn $p ( y | \mathbf { \boldsymbol { x } } )$ , we want to learn which features are most compatible with a given label: i.e. $p ( { \pmb x } | { \pmb y } )$ . Thus, we train a generative model (e.g. energy-based autoencoder) to use feature similarity (as quantified by an energy function) to identify which instances belong to the specific class.
+
+3. Identify and correct mislabelled instances: The features of the full training data (with mislabeled targets) are injected into the previously trained semi-supervised model resulting in each instance being assigned an energy value based on the output of the model. The energy corresponding to each instance serves as a proxy for class assignment (i.e. low energy corresponds to $y = 0$ whereas high energy corresponds to $y = 1$ ). Contradictions between energy assignment and training labels are used to correct the training data such that all labels are compatible with their assigned energy.
+
+We first motivate the design decisions involved in this framework. Namely, we explore our choice of contrastive divergence as a loss function and empirically justify our use of an autoencoder. In the subsequent section, we elaborate on our implementation of the framework.
+
+# 4.1 FRAMEWORK MOTIVATION
+
+Given a small, clean dataset $\mathcal { T } \subset \tilde { \mathcal { T } }$ , we aim to train a model which can be used to determine labels on the remaining, noisily labelled dataset. One obvious approach is to train a binary classifier directly and use it to correct labels. However, as discussed by Berthelot et al. (2017), binary classification provides a relatively weak training signal which largely ignores the intricacies of the input feature distribution. See Fig. 3 where we compare the class separation achieved by a few standard binary classifiers with our proposed approach. Alternatively, one could train class conditional models. However, such models are trained without knowledge of the primary task, differentiating classes. Further, traditional unsupervised training regimes typically use a maximum likelihood formulation (i.e. forward KL divergence) which is prone to distributing probability mass broadly (the so-called “mean-seeking behaviour” (Murphy, 2012)) and further limit the discriminative ability of the learned model.
+
+Instead, we prioritize learning to discriminate the underlying classes based on aspects of the input feature distributions. To do this, we propose to use a contrastive divergence (Carreira-Perpinan $\&$ Hinton, 2005) training loss:
+
+$$
+\mathrm { C D } \left( \pmb { \theta } \right) = \mathrm { K L } \left[ p _ { 0 } \left( \pmb { x } \right) \| \boldsymbol { q } \left( \pmb { x ; \theta } \right) \right] - \mathrm { K L } \left[ p _ { 1 } \left( \pmb { x } \right) \| \boldsymbol { q } \left( \pmb { x ; \theta } \right) \right] .
+$$
+
+where $p _ { 0 } ( { \pmb x } )$ and $p _ { 1 } ( { \pmb x } )$ denote the target probability distributions conditioned on class 0 and 1 respectively and $\boldsymbol { q } ( \boldsymbol { \pmb x } ; \boldsymbol { \theta } )$ denotes the learned model. Thus, training aims to produce a model ${ \boldsymbol { q } } \left( { \pmb { x } } ; { \pmb { \theta } } \right)$ which assigns high probability to samples from class 0, while giving low probability to samples from class 1. In theorem 1, we show that a finite sample approximation of the optimization objective in Eq. 4 can be computed as
+
+$$
+\mathcal { L } \left( \pmb { \theta } \right) = \frac { 1 } { \vert \mathcal { T } _ { 0 } \vert } \sum _ { \pmb { x } _ { 0 } \in \mathcal { T } _ { 0 } } E \left( \pmb { x } _ { 0 } ; \pmb { \theta } \right) - \frac { 1 } { \vert \mathcal { T } _ { 1 } \vert } \sum _ { \pmb { x } _ { 1 } \in \mathcal { T } _ { 1 } } E \left( \pmb { x } _ { 1 } ; \pmb { \theta } \right) ,
+$$
+
+where $\mathcal { T } _ { y }$ , for $y ~ \in ~ \{ 0 , 1 \}$ , contains instances from class $y$ and $\mathcal { T } _ { y } \subseteq \mathcal { T } _ { \mathfrak { s } }$ , $| \mathcal { T } _ { y } |$ denotes the number of elements in $\mathcal { T } _ { y }$ and $E ( \pmb { x } ; \pmb { \theta } )$ is the energy of the model ${ \bf { \nabla } } q \left( { { \bf { x } } ; \theta } \right)$ , i.e., $q \left( { { \pmb x } ; { \pmb \theta } } \right) \ =$ $\exp \left( - E ( \pmb { x } ; \pmb { \theta } ) \right) / Z ( \pmb { \theta } )$ where $Z ( \pmb \theta )$ is the normalizing partition function. Analogously, we can define Eq. 5 over each mini-batch. This contrastive loss have been applied successfully in adversarial training, resulting in faster and improved stability of training, enhanced generator quality, and improved generator diversity (Zhao et al., 2016; Berthelot et al., 2017). As described in section 2.2 of (LeCun et al., 2006), the negative log-likelihood function, $E ( { \pmb x } ; { \pmb \theta } ) = - \log q \left( { \pmb x } ; { \pmb \theta } \right)$ , constitute a valid energy which has been used in numerous energy-based models [e.g. (Bengio et al., 2003; Zhao et al., 2016; Berthelot et al., 2017)].
+
+Theorem 1. The contrastive divergence $\mathrm { C D } \left( \theta \right)$ in Equation 4 has the finite approximation $\mathcal { L } \left( \pmb { \theta } \right) u p$ to some constant $k$ which is independent of $\pmb { \theta }$ .
+
+Proof. Using the definition of KL divergence, we expand Equation 4:
+
+$$
+\mathrm { C D } \left( \pmb { \theta } \right) = - \int _ { \pmb { x } } p _ { 0 } \left( \pmb { x } \right) \log q \left( \pmb { x ; \theta } \right) d \pmb { x } + \int _ { \pmb { x ^ { \prime } } } p _ { 1 } \left( \pmb { x ^ { \prime } } \right) \log q \left( \pmb { x ^ { \prime } ; \theta } \right) d \pmb { x ^ { \prime } } + \boldsymbol { k }
+$$
+
+![](images/a982af980436e8a40f9e96bbc10fc1f295b6db842ce219f9f63fffda66101b71.jpg)  
+Figure 3: In the first row, we visualize the separation achieved by classifiers trained on a training set with noisy labels. From left to right, we show the results of logistic regression, an SVM, a 1 layer feed-forward neural network and the proposed technique. In the second row, we visualize the training data. From left to right, we show the clean data, the noisy data and the autoencodercorrected data. We observe that standard binary classifiers are unable to achieve satisfactory class separation. The 1 hidden layer NN classifier accomplishes significant separation but it incorrectly identifies mislabelled instances. The proposed model is able to identify mislabelled instances based on the energy distribution of the training data.
+
+where the two entropy terms $\begin{array} { r } { \int _ { \pmb { x } } p _ { 0 } \left( \pmb { x } \right) \log p _ { 0 } \left( \pmb { x } \right) d \pmb { x } } \end{array}$ and $\begin{array} { r } { \int _ { \pmb { x } } p _ { 1 } \left( \pmb { x } \right) \log p _ { 1 } \left( \pmb { x } \right) d \pmb { x } } \end{array}$ have been grouped in the constant $k$ as they do not depend on the optimization parameters $\pmb \theta$ . Assume the Gibbs form for the model distribution,
+
+$$
+q \left( { { \bf { x } } ; \pmb { \theta } } \right) = \frac { { { e ^ { - E \left( { { \bf { x } } ; \pmb { \theta } } \right) } } } } { { { Z ( \pmb { \theta } } ) } } ,
+$$
+
+where $E ( \pmb { x } ; \pmb { \theta } )$ is the energy and $\begin{array} { r } { Z ( \pmb { \theta } ) = \int _ { \pmb { x } } e ^ { - E ( \pmb { x } ; \pmb { \theta } ) } d \pmb { x } } \end{array}$ is the partition function. Then, Equation 6 becomes
+
+$$
+\begin{array} { l l l } { { \displaystyle \mathrm { { C D } } \left( \theta \right) } } & { { = } } & { { \displaystyle \int _ { x } p _ { 0 } \left( { \pmb x } \right) E ( { \pmb x } ; \theta ) d { \pmb x } - \int _ { { \pmb x } ^ { \prime } } p _ { 1 } \left( { \pmb x } ^ { \prime } \right) E ( { \pmb x } ; \theta ) d { \pmb x } ^ { \prime } } } \\ { { \displaystyle } } & { { - } } & { { \displaystyle \left( \int _ { \pmb x } p _ { 0 } \left( { \pmb x } \right) d { \pmb x } \right) Z ( \pmb \theta ) + \left( \int _ { \pmb x } p _ { 1 } \left( { \pmb x } \right) d { \pmb x } \right) Z ( \pmb \theta ) + k , } } \end{array}
+$$
+
+where the partition function terms conveniently cancel out due to the normalization of $p _ { 0 }$ and $p _ { 1 }$ Therefore,
+
+$$
+\begin{array} { r } { \mathrm { C D } \left( \pmb { \theta } \right) = \mathbb { E } _ { p _ { 0 } \left( \pmb { x } \right) } \left[ E \left( \pmb { x } \right) \right] - \mathbb { E } _ { p _ { 1 } \left( \pmb { x } \right) } \left[ E \left( \pmb { x } \right) \right] + k } \end{array}
+$$
+
+and we conclude that $\mathrm { C D } \left( \theta \right)$ has the finite approximation ${ \mathcal { L } } \left( \theta \right)$ as defined in Equation 5, up to a constant $k$ . 
+
+# 4.2 FRAMEWORK IMPLEMENTATION: CONTRASTIVE AUTOENCODER
+
+Using the clean subset $\mathcal { T }$ , we implement step 2 of our framework by using an energy-based autoencoder (analogous to Zhao et al. (2016)). In order to ensure stable training and to prevent the second term of Equation 5 from diverging, we balance our mini-batch to have an equal number of positive and negative class samples. After training is complete, we process the original training data $\tilde { \tau }$ using the trained autoencoder. Specifically, we compute the energy using the resulting outputs: $E \left( { \pmb x } ; { \pmb \theta } \right) = - \log q \left( { \pmb x } ; { \pmb \theta } \right)$ , which reduces to the reconstruction loss corresponding to the autoencoder (Goodfellow et al., 2016). For example, if the underlying distribution is assumed to be Gaussian then, we have $E \left( { \pmb x } ; { \pmb \theta } \right) = \| { \pmb x } - \hat { \pmb x } ( { \pmb \theta } ) \| ^ { 2 }$ , where $\hat { \mathbf { x } } ( \overset { \cdot } { \boldsymbol { \theta } } )$ is the output of the autoencoder3. Whereas, if the underlying distribution is assumed to be Bernoulli, we obtain $E \left( \pmb { x } ; \pmb { \theta } \right) = - \left[ \pmb { x } \log \hat { \pmb { x } } ( \pmb { \theta } ) + \left( 1 - \pmb { x } \right) \log ( \bar { 1 } - \bar { \pmb { x } } ( \pmb { \theta } ) ) \right]$ .
+
+To execute our label correction protocol, we impose the following two conditions. (i) if $E ( \pmb { x } ; \pmb { \theta } ) >$ $\bar { E } _ { 1 } - a$ for $\pmb { x } \in \tilde { \mathcal { T } } _ { 0 }$ , then change label of $_ { \textbf { \em x } }$ . Here we denote $\tilde { \tau } _ { 0 }$ as the originals negative samples, $\bar { E } _ { 1 }$ as the mean energy over the set of clean positive samples $\mathcal { T } _ { 1 }$ , and $a$ is a tuneable hyperparameter that determines how aggressively we want to change $0  1$ . Analogously, we impose condition (ii) if $E ( \pmb { x } ; \pmb { \theta } ) < \bar { E } _ { 0 } - b$ for $\pmb { x } \in \tilde { \mathcal { T } } _ { 1 } ^ { }$ , then change label of $_ { \textbf { \em x } }$ . The rationale is to modify samples where the original label $\tilde { y }$ contradicts the energy assignment. When one is not able to tune the threshold hyperparameters $a$ and $b$ , an empirical heuristic is to set them equal to the standard deviation about the means $\bar { E } _ { 1 }$ and $\bar { E } _ { 0 }$ respectively. See Appendix B details and pseudo-code.
+
+# 5 EXPERIMENTS
+
+We conduct experiments with simulated data and real-world data where three different type III noise models are studied: (i) Linear noise $( p _ { \mathrm { e r r o r } } \sim \alpha x _ { i } )$ ): the probability of an error occurring depends linearly on a feature (Table 1). (ii) Quadratic noise $( p _ { \mathrm { e r r o r } } \sim \dot { \alpha } \dot { x } _ { i } ^ { 2 } )$ : the probability of an error occurring depends on the square of a feature. We control the amount of noise added with $\alpha$ and we select $x _ { i }$ if it has sufficient variance to make the noise model distinct from random noise, i.e. 0 variance implies random noise (Table 2). (iii) Boundary noise: the probability of label error depends on the distance from the class boundary, which is determined using the noise-free data. We report the average class-weighted F1-score along with the standard error over 10 runs with random train-test (80:20) splits (Table 3).
+
+UCI benchmark datasets: We train logistic regression on the corrected dataset resulting from the proposed algorithm. If a small clean dataset is not initially provided, then our results are labelled as AE (learned), otherwise they are labelled at AE (known) – see step 1 in Sec. 4. We compare to noise robust algorithms: NHERD Crammer & Lee (2010), PAM Frenay & Verleysen (2014), and ´ ULE Natarajan et al. (2013). Additionally, we report upper and lower bounds by training logistic regression on the noisy data (LR-N) and on the clean data (LR-C). All algorithms are tested on noisefree data. Even without access to any clean labels (i.e., we learn which labels are likely clean), our framework generally outperforms the other algorithms in the presence of linear noise and quadratic noise. In general, learning which labels are noisy following the method in Ding et al. (2018) is not possible for arbitrary label noise processes: e.g., when the optimal discriminators of $\tilde { \tau }$ are very different than the optimal discriminators of $\tau$ . For boundary noise, we show that using a small clean dataset $\mathcal { T }$ enables our methods to learn in the presence of boundary noise, where the other methods generally breakdown (Table 3). For a fair evaluation, the benchmark algorithms (which don’t have access to clean data) should be compared to AE (learned).
+
+MNIST: Next, we compare our method with another state-of-the-art noise robust algorithm that relies on clean data, i.e., the loss weighting scheme of Ren et al. (2018). Both methods are given the same percentage of clean data for each noise setting. The task is to classify the distinguish $\mathbf { \ddot { 3 } } \mathbf { \ " }$ (class 0) vs. other digits (class 1). The noise process is input dependent as it changes $4 , 5 , 6$ to class 0 depending on a specified noise rate $( \alpha )$ . We demonstrate (Table 4) that our method generalizes better in the presence of type III noise, even when starting with only $1 \%$ of the data that is clean.
+
+Arrhythmia We use the MIT-BIH Arrhythmia dataset Goldberger et al. (2000) to evaluate the proposed method’s ability to correct algorithmically assigned labels. Algorithmically-assigned labels (AALs) are prevalent in domains with abundant unlabeled data and high labelling costs. Common applications include web page annotation, but the value of this approach extends to automatic annotation of images and natural language. We employ the data preprocessing procedure described by Mondejar-Guerra et al. (2019), where each arrhythmia is described by 59 features. ´
+
+We divide the dataset into three parts: $T _ { 1 }$ , $T _ { 2 }$ and $T _ { 3 }$ . $T _ { 1 }$ and $T _ { 2 }$ serve as training sets and $T _ { 3 }$ is the test set. We train a classifier on $T _ { 1 }$ and subsequently use that classifier to generate AALs for $T _ { 2 }$ . Thus, the predictions of the trained classifier become $\tilde { y }$ and the original expert labels remain $y$ . As demonstrated in Table 1, the proposed technique achieves a higher F1 score than competing
+
+<table><tr><td>Dataset</td><td>Noise Parameters (col, α)</td><td>LR-N</td><td>NHERD</td><td>PAM</td><td>ULE</td><td>AE (learned)</td><td>AE(known)</td><td>LR-C</td></tr><tr><td>Lin-Sep</td><td>1,1.2</td><td>0.86±0.03</td><td>0.91±0.01</td><td>0.76±0.01</td><td>0.90±0.02</td><td>0.94±0.01</td><td>0.95±0.01</td><td>0.96±0.01</td></tr><tr><td>Diabetes</td><td>5,1.0</td><td>0.60±0.03</td><td>0.50±0.01</td><td>0.48±0.03</td><td>0.60±0.03</td><td>0.64±0.01</td><td>0.70±0.02</td><td>0.72±0.02</td></tr><tr><td>German</td><td>1, 1.2</td><td>0.67±0.03</td><td>0.72 ± 0.01</td><td>0.49 ± 0.03</td><td>0.63 ± 0.01</td><td>0.67± 0.03</td><td>0.73 ±0.01</td><td>0.76 ±0.02</td></tr><tr><td>Image</td><td>1,0.7</td><td>0.61± 0.03</td><td>0.63 ±0.01</td><td>0.54 ± 0.01</td><td>0.61± 0.02</td><td>0.67 ± 0.01</td><td>0.77±0.01</td><td>0.77 ± 0.01</td></tr><tr><td>Twonorm</td><td>1, 1.2</td><td>0.68 ±0.04</td><td>0.50 ±0.02</td><td>0.43 ±0.03</td><td>0.82 ± 0.04</td><td>0.88 ± 0.01</td><td>0.91 ± 0.02</td><td>0.98 ± 0.01</td></tr><tr><td>Breast Cancer</td><td>5,1.0</td><td>0.66 ± 0.02</td><td>0.67±0.02</td><td>0.52 ±0.03</td><td>0.60±0.03</td><td>0.60±0.02</td><td>0.71 ± 0.02</td><td>0.70 ± 0.01</td></tr><tr><td>Arrhythmia</td><td>-</td><td>0.79±0.01</td><td>0.83±0.01</td><td>0.81±0.02</td><td>0.66±0.04</td><td>0.85±0.02</td><td>0.85±.01</td><td>0.85±0.02</td></tr></table>
+
+Table 1: Noise model: probability of an error occurring depends linearly on an input feature. We report the class weighted mean f1 score on the noise-free test set along with the standard error.
+
+<table><tr><td>Dataset</td><td>Noise Parameters (col,α)</td><td>LR-N</td><td>NHERD</td><td>PAM</td><td>ULE</td><td>AE (learned)</td><td>AE (known)</td><td>LR-C</td></tr><tr><td>Lin-Sep</td><td>1,1.2</td><td>0.40±0.01</td><td>0.53± 0.04</td><td>0.62±0.03</td><td>0.61 ±0.02</td><td>0.93±0.01</td><td>0.96±0.01</td><td>0.96±0.01</td></tr><tr><td>Diabetes</td><td>5,1.2</td><td>0.62±0.01</td><td>0.59±0.01</td><td>0.67±0.01</td><td>0.71±0.01</td><td>0.66±0.03</td><td>0.69±0.01</td><td>0.72±0.02</td></tr><tr><td>German</td><td>1, 1.2</td><td>0.60±0.02</td><td>0.72 ± 0.01</td><td>0.60 ± 0.01</td><td>0.67 ± 0.01</td><td>0.68 ±0.03</td><td>0.73±0.03</td><td>0.76 ± 0.02</td></tr><tr><td>Image</td><td>1,0.7</td><td>0.55 ± 0.02</td><td>0.64 ± 0.01</td><td>0.57 ± 0.01</td><td>0.60 ±0.03</td><td>0.74 ± 0.01</td><td>0.77 ± 0.01</td><td>0.77 ± 0.01</td></tr><tr><td>Twonorm</td><td>1, 1.2</td><td>0.90±0.03</td><td>0.55 ± 0.02</td><td>0.86±0.01</td><td>0.94 ± 0.01</td><td>0.96±0.01</td><td>0.97 ± 0.01</td><td>0.98 ± 0.01</td></tr><tr><td>Breast Cancer</td><td>5,1.0</td><td>0.60±0.02</td><td>0.60 ±0.02</td><td>0.63±0.02</td><td>0.63±0.03</td><td>0.65 ±0.03</td><td>0.66 ±0.01</td><td>0.70± 0.01</td></tr></table>
+
+Table 2: Noise model: probability of an error occurring depends quadratically on an input feature.   
+We report the class weighted mean f1 score on the noise-free test set along with the standard error.
+
+<table><tr><td>Dataset</td><td>Noise Parameters (α)</td><td>LR-N</td><td>NHERD</td><td>PAM</td><td>ULE</td><td>AE(known)</td><td>LR-C</td></tr><tr><td>Lin-Sep</td><td>0.7</td><td>0.39±0.01</td><td>0.41 ± 0.01</td><td>0.53±0.02</td><td>0.85±0.01</td><td>0.94±0.01</td><td>0.96±0.01</td></tr><tr><td>Diabetes</td><td>0.7</td><td>0.56±0.01</td><td>0.53±0.03</td><td>0.50±0.02</td><td>0.55±0.02</td><td>0.68±0.02</td><td>0.72 ±0.02</td></tr><tr><td>German</td><td>0.7</td><td>0.57± 0.01</td><td>0.70±0.01</td><td>0.47 ± 0.01</td><td>0.60 ± 0.01</td><td>0.72 ± 0.01</td><td>0.76±0.02</td></tr><tr><td>Image</td><td>0.7</td><td>0.43 ± 0.01</td><td>0.61 ± 0.01</td><td>0.45 ±0.02</td><td>0.43 ± 0.01</td><td>0.74±0.03</td><td>0.77 ± 0.01</td></tr><tr><td>Twonorm</td><td>0.5</td><td>0.52 ±0.03</td><td>0.51±0.03</td><td>0.52 ±0.03</td><td>0.51 ± 0.04</td><td>0.88±0.03</td><td>0.98 ±0.01</td></tr><tr><td>Breast Cancer</td><td>0.7</td><td>0.62 ± 0.02</td><td>0.62 ±0.02</td><td>0.46 ±0.02</td><td>0.63±0.02</td><td>0.66 ±0.04</td><td>0.70± 0.01</td></tr></table>
+
+Table 3: Noise model: probability of an error occurring depends linearly on the distance from class boundary (which is defined by the clean dataset before noise is synthetically introduced). We report the class weighted mean f1 score on the noise-free test set along with the standard error.
+
+Table 4: We compare the label re-weighting scheme of Ren et al. (2018) with our proposed model. The models compared for different sizes of clean dataset and different noise rates $\alpha$ . The base model in each case is a standard LeNet as in Ren et al. (2018)   
+
+<table><tr><td>% Clean data</td><td>Noise Parameter (α)</td><td>AE(known)</td><td>Ren et al. (2018)</td><td>No noise model</td></tr><tr><td>1</td><td>0.1</td><td>98.28±0.01</td><td>93.27±0.01</td><td>90.87±0.05</td></tr><tr><td>1</td><td>0.8</td><td>98.14± 0.00</td><td>91.27 ± 0.01</td><td>87.32 ± 0.03</td></tr><tr><td>5</td><td>0.1</td><td>98.77 ± 0.00</td><td>93.44 ± 0.01</td><td>91.01 ± 0.02</td></tr><tr><td>5</td><td>0.8</td><td>98.65 ± 0.00</td><td>91.29 ± 0.01</td><td>88.85 ± 0.02</td></tr><tr><td>10</td><td>0.1</td><td>98.93 ± 0.00</td><td>94.87 ± 0.03</td><td>90.99 ±0.02</td></tr><tr><td>10</td><td>0.8</td><td>98.54 ± 0.01</td><td>91.77 ± 0.02</td><td>90.16 ± 0.03</td></tr></table>
+
+methods. The success of the model in this setting suggests its unique robustness to feature-dependent noise.
+
+# 6 CONCLUSION
+
+We have proposed an energy-based framework to correct mislabelled training instances. By minimizing a contrastive loss, the proposed method learns a representation that is valuable in relabeling noisy training sets. We evaluate the proposed model across six datasets and three noise models to demonstrate the method’s empirical value in correcting feature-dependent label noise. Furthermore, we demonstrate our method’s improvement upon existing work in making machine learning more robust to label noise processes.
+
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+
+# APPENDICES
+
+# A EFFECT OF LABEL NOISE ON EMPIRICAL RISK
+
+Here we show how label noise affects optimization of empirical risk. Recall that our task is to improve the labels of a training set in order to enable effective learning. In the following sections, we explore the Type I, Type II, and Type III noise. We first introduce the key variables.
+
+We focus on binary classification, where the probability of class 0 is $p$ and consequently, the probability of class 1 is $( 1 - p )$ :
+
+$$
+\begin{array} { l c l } { { p ( y = 0 ) } } & { { = } } & { { p } } \\ { { p ( y = 1 ) } } & { { = } } & { { 1 - p } } \end{array}
+$$
+
+$\tau$ represents the training set with correct labels, while $\tilde { \tau }$ represents the training set with label noise. In our setting, we are given $\tilde { \tau }$ and propose a method to transform $\tilde { \tau }$ into $\tau$ . We will call $D$ the true label distribution and $\tilde { P }$ the noisy label distribution.
+
+$$
+\begin{array} { r l r } { \mathcal { T } } & { = } & { \{ ( { \bf X } _ { i } , Y _ { i } ) \mathrm { f o r } i = 1 , 2 , \ldots , n \} } \\ { \tilde { \mathcal { T } } } & { = } & { \{ ( { \bf X } _ { i } , \tilde { Y } _ { i } ) \mathrm { f o r } i = 1 , 2 , \ldots , n \} } \end{array}
+$$
+
+In the following calculations, we describe the difference in empirical risks over each training set and the subsequent effect on stochastic gradient descent. We refer to the empirical risk over the clean training set as $\hat { R } \left[ \ell , \mathcal { T } \right]$ . Similarly, we refer to the empirical risk over the corrupted training set as ${ \hat { R } } \left[ \ell , { \tilde { \mathcal { T } } } \right]$ .
+
+$$
+{ \hat { R } } \left[ \ell , { \mathcal { T } } \right] = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \ell \left( \Phi ( \mathbf { X } _ { i } , \pmb { \theta } ) , Y _ { i } \right) ,
+$$
+
+In order to explain the effects of label noise on stochastic gradient descent (SGD), we are primarily interested in the gradient of the empirical risk with respect to $\theta$ . Below, we break the empirical risk term into two class-dependent terms and introduce short-hand to represent each of these terms.
+
+$$
+\begin{array} { r c l } { \Xi _ { ( \mathbf x , \nabla ) \times \big [ } \nabla _ { \mathbf e } \hat { R } _ { \xi } , \mathcal { t } \big ] } & { = } & { \Xi _ { ( \mathbf x , \nabla ) \times \big [ } \nabla _ { \mathbf e } \frac { 1 } { \omega _ { \mathbf x } } \frac { \cos \big [ \hat { \pi } _ { \mathbf e } \big ] } { \omega _ { \mathbf x } } \big [ \langle \mathbf i \nabla _ { \mathbf e } \hat { R } _ { \xi } , \theta \rangle , \mathcal { t } \big ] } \\ & { = } & { \displaystyle \nabla _ { \mathbf e } \frac { 1 } { \omega _ { \mathbf x } } \frac { \cos \big [ \hat { \pi } _ { \mathbf e } \hat { \mathbf \nu } _ { \xi } \big ] } { \omega _ { \mathbf x } - 1 } \mathbb E _ { \mathbf x \nabla \cdot \nabla \cdot \nabla \cdot \big [ \hat { \pi } _ { \mathbf e } \hat { R } _ { \xi } \big ] } \big [ \langle \mathbf i \nabla _ { \mathbf e } \hat { R } _ { \xi } , \theta \rangle , \mathcal { t } \big ] } \\ & { = } & { \displaystyle \nabla _ { \mathbf e } \left[ \frac { 1 } { \omega _ { \mathbf x } } \frac { \cos \big [ \hat { \pi } _ { \mathbf e } \hat { \mathbf \nu } _ { \xi } \big ] } { \omega _ { \mathbf x } - 1 } \int \rho ( \mathbf i \nabla _ { \mathbf x } \cdot \nabla ) _ { \mathbf e } \hat { \nu } ( \langle \mathbf i \nabla _ { \mathbf e } \hat { \mathbf a } , \theta \rangle , \mathcal { t } ) \mathbb { A } _ { \mathbf x } \mathcal { A } _ { \mathbf x } \right] } \\ & { = } & { \displaystyle \nabla _ { \mathbf e } \left[ \frac { 1 } { \omega _ { \mathbf x } } \frac { \sin } { \omega _ { \mathbf x } } \int \rho ( \mathbf i \kappa _ { \xi } | \delta ) \mathbf j \cdot \nabla \langle \mathbf i \nabla _ { \mathbf e } \hat { \mathbf a } , \theta \rangle , \mathcal { t } \right] \mathbb { A } _ { \mathbf x } \mathcal { A } _ { \mathbf x } \mathcal { A } _ { \mathbf y } } \\ & { = } & { \displaystyle \rho ( \hat { \nu } _ { \mathbf e } \hat { \mathbf \nu } _ { \xi } ( \mathbf i ) ) \nabla _ { \mathbf e } \left[ \frac { 1 } { \kappa _ { \mathrm { d e f . } } } \int \gamma ( \mathbf s _ { \mathbf x } | \delta ) j _ { \mathbf x } ( \hat { \nu } _ { \mathbf e } \hat { \mathbf a } ) \hat { \nu } _ { \mathbf e } \right] } \\ &  + \end{array}
+$$
+
+We define shorthand based on Equation (5):
+
+$$
+\begin{array} { r l r } & { } & { \beta _ { 0 } = \nabla _ { \pmb { \theta } } \left[ \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \int p ( \mathbf { x } _ { i } | 0 ) \ell _ { 0 } \left( \Phi ( \mathbf { x } _ { i } , \pmb { \theta } ) \right) d \mathbf { x } _ { i } \right] } \\ & { } & { \beta _ { 1 } = \nabla _ { \pmb { \theta } } \left[ \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \int p ( \mathbf { x } _ { i } | 1 ) \ell _ { 1 } \left( \Phi ( \mathbf { x } _ { i } , \pmb { \theta } ) \right) d \mathbf { x } _ { i } \right] . } \end{array}
+$$
+
+Rewriting the original equation in terms of the above, we arrive at the following form for empirical risk:
+
+$$
+\begin{array} { r } { \mathbb { E } _ { ( \mathbf { X } , Y ) \sim D } \left[ \nabla _ { \theta } \hat { R } [ \ell , \mathcal { T } ] \right] = p ( y _ { i } = 0 ) \beta _ { 0 } + p ( y _ { i } = 1 ) \beta _ { 0 } } \end{array}
+$$
+
+We revisit Equation 8 in the follow sections to interpret empirical risk under varying noise models. Note that $\beta _ { 0 }$ and $\beta _ { 1 }$ are the same in both the expected empirical risk with respect to the true label distribution $( D )$ and the corrupted label distribution $( \tilde { P } )$ for Type I and Type II noise. Additionally, we introduce shorthand for the true expectation of empirical risk $( \mathbb { E } _ { t r u e } )$ and the expectation of empirical risk when labels are assigned at random $( \mathbb { E } _ { r a n d } )$ .
+
+$$
+\begin{array} { r c l } { { \mathbb { E } _ { t r u e } } } & { { = } } & { { p \beta _ { 0 } + ( 1 - p ) \beta _ { 1 } } } \\ { { \mathbb { E } _ { r a n d } } } & { { = } } & { { . 5 \beta _ { 0 } + . 5 \beta _ { 1 } } } \end{array}
+$$
+
+# A.1 TYPE I: RANDOM NOISE
+
+In the random noise scenario, each label has an equal probability of being flipped. We parameterize this noise by $\gamma$ , resulting in :
+
+$$
+\gamma = p ( \tilde { y } _ { i } \ne y _ { i } )
+$$
+
+We may define both $p ( \tilde { y } _ { i } = 0 )$ in terms of $\gamma$ and $p$ :
+
+$$
+\begin{array} { l c l } { { p ( \tilde { y } _ { i } = 0 ) } } & { { = } } & { { p ( \tilde { y } _ { i } = 0 | y _ { i } = 0 ) + p ( \tilde { y } _ { i } = 0 | y _ { i } = 1 ) } } \\ { { p ( \tilde { y } _ { i } = 0 ) } } & { { = } } & { { ( 1 - \gamma ) p + \gamma ( 1 - p ) } } \end{array}
+$$
+
+Likewise, we define $p ( \tilde { y } _ { i } = 1 )$ as:
+
+$$
+\begin{array} { l c l } { { p ( \tilde { y } _ { i } = 1 ) } } & { { = } } & { { p ( \tilde { y } _ { i } = 1 | y _ { i } = 0 ) + p ( \tilde { y } _ { i } = 1 | y _ { i } = 1 ) } } \\ { { p ( \tilde { y } _ { i } = 1 ) } } & { { = } } & { { \gamma p + ( 1 - \gamma ) ( 1 - p ) } } \end{array}
+$$
+
+Below, we substitute these expressions into Equation 8:
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { ( \mathbf { X } , \widetilde { Y } ) \sim \widetilde { P } } \left[ \nabla _ { \theta } \hat { R } [ \ell , \widetilde { \mathcal { T } } ] \right] } & { = \ p ( \tilde { y } _ { i } = 0 ) \beta _ { 0 } + p ( \tilde { y } _ { i } = 1 ) \beta _ { 0 } } \\ & { = \ ( ( 1 - \gamma ) p + \gamma - \gamma p ) \beta _ { 0 } + \left( ( 1 - \gamma ) \left( 1 - p \right) + \gamma p \right) \beta _ { 1 } } \\ & { = \ p \ \beta _ { 0 } - 2 \gamma p \beta _ { 0 } + \gamma \beta _ { 0 } + \beta _ { 1 } - \gamma \beta _ { 1 } - p \beta _ { 1 } + 2 \gamma p \beta _ { 1 } } \\ & { = \ p \beta _ { 0 } + \beta _ { 1 } - p \beta _ { 1 } - 2 \gamma p \beta _ { 0 } - \gamma \beta _ { 1 } + 2 \gamma p \beta _ { 1 } + \gamma \beta _ { 0 } } \\ & { = \ \mathbb { E } _ { t r u e } - 2 \gamma p \beta _ { 0 } - \gamma \beta _ { 1 } + 2 \gamma p \beta _ { 1 } + \gamma \beta _ { 0 } } \\ & { = \ \mathbb { E } _ { t r u e } - 2 \gamma p \beta _ { 0 } - \gamma \beta _ { 1 } + 2 \gamma p \beta _ { 1 } + \gamma \beta _ { 0 } - \gamma \beta _ { 1 } + \gamma \beta _ { 1 } } \\ & { = \ \mathbb { E } _ { t r u e } - 2 \gamma p \beta _ { 0 } - 2 \gamma \beta _ { 1 } + 2 \gamma p \beta _ { 1 } + \gamma \beta _ { 0 } + \gamma \beta _ { 1 } } \\ & { = \ \mathbb { E } _ { t r u e } - 2 \gamma p \beta _ { 0 } - 2 \gamma \beta _ { 1 } + 2 \gamma p \beta _ { 1 } + \gamma \beta _ { 0 } + \gamma \beta _ { 1 } } \\ & { = \ \mathbb { E } _ { t r u e } - 2 \gamma \left( p \beta _ { 0 } + \beta _ { 1 } - p \beta _ { 1 } \right) + \gamma \beta _ { 1 } + \gamma \beta _ { 0 } } \\ & { = \ ( 1 - 2 \gamma ) \mathbb { E } _ { t r u e } + 2 \gamma \mathbb { E } _ { n a d } } \end{array}
+$$
+
+# A.2 TYPE II NOISE
+
+Type II noise implies class-dependent label noise. Thus, we define $\gamma _ { 0 }$ and $\gamma _ { 1 }$ , the class-dependent noise rates. In addition, we define $\gamma ^ { * }$ as the average class-dependent noise rate and $\epsilon$ as the absolute distance of $\gamma _ { 0 }$ and $\gamma _ { 1 }$ from $\gamma ^ { * }$ . Without loss of generality, we assume $\gamma _ { 0 } > \gamma _ { 1 }$ .
+
+$$
+\begin{array} { r c l } { { \gamma _ { 0 } } } & { { = } } & { { p ( \tilde { y } _ { i } = 1 | y _ { i } = 0 ) } } \\ { { \gamma _ { 1 } } } & { { = } } & { { p ( \tilde { y } _ { u } = 0 | y _ { i } = 1 ) } } \\ { { \gamma ^ { * } } } & { { = } } & { { \displaystyle \frac { \gamma _ { 0 } + \gamma _ { 1 } } { 2 } } } \\ { { \epsilon } } & { { = } } & { { \displaystyle \frac { \gamma _ { 0 } - \gamma _ { 1 } } { 2 } } } \end{array}
+$$
+
+We may redefine $p ( \tilde { y } _ { i } = 0 )$ and $p ( \tilde { y } _ { i } = 1 )$ using these terms:
+
+$$
+\begin{array} { r c l } { p ( \tilde { y } _ { i } = 0 ) } & { = } & { \left( \left( 1 - \gamma _ { 0 } \right) p + \gamma _ { 1 } \left( 1 - p \right) \right) } \\ & { = } & { \left( \left( 1 - \gamma ^ { * } - \epsilon \right) p + \left( \gamma ^ { * } - \epsilon \right) \left( 1 - p \right) \right) } \\ & { = } & { \left( p - \gamma ^ { * } p - \epsilon p + \gamma ^ { * } - \gamma ^ { * } p - \epsilon + \epsilon p \right) } \end{array}
+$$
+
+$$
+\begin{array} { l c l } { p ( \tilde { y } _ { i } = 1 ) } & { = } & { \left( \left( 1 - \gamma _ { 1 } \right) \left( 1 - p \right) + \gamma _ { 0 } p \right) } \\ & { = } & { \left( \left( 1 - \gamma ^ { * } + \epsilon \right) \left( 1 - p \right) + \left( \gamma ^ { * } + \epsilon \right) p \right) } \\ & { = } & { \left( 1 - \gamma ^ { * } + \epsilon - p + \gamma ^ { * } p - \epsilon p + \gamma ^ { * } p + \epsilon p \right) } \end{array}
+$$
+
+Plugging this into Equation 8,
+
+$$
+\begin{array} { r c l } { \mathbb { E } _ { ( { \bf X } , \tilde { Y } ) \sim \tilde { P } } \left[ \nabla _ { \theta } \hat { R } [ \ell , \tilde { \mathcal { T } } ] \right] } & { = } & { ( p - \gamma ^ { * } p - \epsilon p + \gamma ^ { * } - \gamma ^ { * } p - \epsilon + \epsilon p ) \beta _ { 0 } } \\ & { + } & { ( 1 - \gamma ^ { * } + \epsilon - p + \gamma ^ { * } p - \epsilon p + \gamma ^ { * } p + \epsilon p ) \beta _ { 1 } } \\ & { = } & { ( p - 2 \gamma ^ { * } p + \gamma ^ { * } - \epsilon ) \beta _ { 0 } } \\ & { + } & { ( 1 - 2 \gamma ^ { * } + \epsilon - p + 2 \gamma ^ { * } p + \gamma ^ { * } ) \beta _ { 1 } } \end{array}
+$$
+
+Similar to our approach with Type I noise, we aim to rephrase this equation in terms of $\mathbb { E } _ { t r u e }$ and Erand.
+
+$$
+\begin{array} { r l } { = } & { { } \left( 1 - 2 \gamma ^ { * } \right) p \beta _ { 0 } + \left( \gamma ^ { * } - \epsilon \right) \beta _ { 0 } + \left( 1 - 2 \gamma ^ { * } \right) \left( 1 - p \right) \beta _ { 1 } + \left( \epsilon + \gamma ^ { * } \right) \beta _ { 1 } } \\ { = } & { { } \left( 1 - 2 \gamma ^ { * } \right) \mathbb { E } _ { t r u e } + \left( \gamma ^ { * } - \epsilon \right) \beta _ { 0 } + \left( \epsilon + \gamma ^ { * } \right) \beta _ { 1 } } \\ { = } & { { } \left( 1 - 2 \gamma ^ { * } \right) \mathbb { E } _ { t r u e } + 2 \gamma ^ { * } \mathbb { E } _ { r a n d } + \epsilon \left( \beta _ { 1 } - \beta _ { 0 } \right) } \end{array}
+$$
+
+# A.3 TYPE III NOISE
+
+In this section, we analyze a specific case of type III noise where label corruption is restricted to a given region of the feature space. As above, we want to study the effect on learning by determining how the true gradient is modified in the presence of noise. We assume a scalar input space and define the feature dependent noise as follows:
+
+$$
+p ( \tilde { y } _ { i } | x ) = \left\{ \begin{array} { l l } { \gamma } & { x \in [ a , b ] } \\ { 0 } & { x \notin [ a , b ] } \end{array} \right\}
+$$
+
+The noise model above asserts that label noise depends only on the feature space and is labelagnostic - thus, examples from both class 1 and class 0 with $x \in [ a , b ]$ are equally likely to suffer from a label flip.
+
+Revisiting
+
+$$
+\begin{array} { r c l } { \mathbb { E } _ { ( \mathbf { X } , \boldsymbol { \tilde { Y } } ) \sim \tilde { P } } \left[ \nabla _ { \theta } \hat { R } [ \ell , \tilde { T } ] \right] } & { = } & { \nabla _ { \theta } \left[ \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int p ( \tilde { y } _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb { \theta } ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right] , } \\ & { = } & { \nabla _ { \theta } \left[ \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } p ( \tilde { y } _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb { \theta } ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right] } \\ & { + } & { \nabla _ { \theta } \left[ \displaystyle \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { x \not \in [ a , b ] } p ( y _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb { \theta } ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right] . } \end{array}
+$$
+
+Now we add and subtract the following term:
+
+$$
+\nabla _ { \pmb \theta } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } p ( y _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb \theta ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right]
+$$
+
+In order to obtain:
+
+$$
+\begin{array} { r c l } { \displaystyle \Xi _ { ( \mathbf { X } , \tilde { Y } ) \sim \tilde { P } } \left[ \nabla _ { \theta } \hat { R } [ \ell , \tilde { \mathcal { T } } ] \right] } & { = } & { \displaystyle \mathbb { E } _ { t r u e } + \nabla _ { \theta } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } p \big ( \tilde { y } _ { i } | x _ { i } \big ) p \big ( x _ { i } \big ) \ell \big ( \Phi ( x _ { i } , \theta ) , \tilde { y } _ { i } \big ) \ d x _ { i } d \tilde { y } _ { i } \right] } \\ & { - } & { \displaystyle \nabla _ { \theta } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } p ( y _ { i } | x _ { i } ) p \big ( x _ { i } \big ) \ell \big ( \Phi ( x _ { i } , \theta ) , \tilde { y } _ { i } \big ) \ d x _ { i } d \tilde { y } _ { i } \right] } \\ & { = } & { \displaystyle \mathbb { E } _ { t r u e } + \nabla _ { \theta } \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } \big ( p \big ( \tilde { y } _ { i } | x _ { i } \big ) - p \big ( y _ { i } | x _ { i } \big ) \big ) p \big ( x _ { i } \big ) \ell \big ( \Phi ( x _ { i } , \theta ) , \tilde { y } _ { i } \big ) \ d x _ { i } \right] } \end{array}
+$$
+
+Where we have used the fact that:
+
+$$
+\begin{array} { l l l } { \mathbb { E } _ { t r u e } } & { = } & { \nabla _ { \pmb { \theta } } \left[ \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \int _ { a } ^ { b } p ( y _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb { \theta } ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right] } \\ & { + } & { \nabla _ { \pmb { \theta } } \left[ \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \int _ { x \not \in [ a , b ] } p ( y _ { i } | x _ { i } ) p ( x _ { i } ) \ell \left( \Phi ( x _ { i } , \pmb { \theta } ) , \tilde { y } _ { i } \right) d x _ { i } d \tilde { y } _ { i } \right] } \end{array}
+$$
+
+# B ALGORITHM PSEUDO-CODE
+
+In this section, we include a more detailed form of the algorithms proposed in the paper.
+
+# Algorithm 1 Train energy-based autoencoder
+
+Require: $\tilde { \tau }$   
+Require: $\mathcal { T }$   
+Require: $\eta$   
+Require: Forward $( x , \theta )$   
+Require: $E \left( \hat { \pmb x } \right)$   
+Require: $\mathcal { L } \left( \boldsymbol { B } , \boldsymbol { \mathcal { O } } ; \boldsymbol { \theta } \right)$   
+Require: GetBalancedBatch $( \mathcal { T } )$
+
+1: for $i$ in $\{ 1 , 2 , \ldots , m \}$ do   
+2: $B _ { i } = \{ x _ { j } , y _ { j } \}  \mathbf { G e t B } { \mathrm { : } }$ alancedBatch $( \mathcal { T } )$   
+3: $\mathcal { O } _ { i } = \{ \hat { \pmb { x } } _ { j } \} $ Forward $( B _ { i } , \pmb \theta _ { i } )$   
+4: $\ell _ { i } \gets \mathcal { L } \left( \mathbf { \bar { \boldsymbol { B } } } _ { i } , \mathcal { O } _ { i } ; \mathbf { \boldsymbol { \theta } } _ { i } \right)$   
+5: $\nabla _ { \pmb { \theta } } \left( \ell _ { i } \right) \gets \mathrm { B a c k w a r d } \left( \ell _ { i } ; \pmb { \theta } _ { i } \right)$   
+6: $\pmb { \theta } _ { i + 1 }  \mathrm { S t e p } ( \pmb { \theta } _ { i } , \eta , \nabla _ { \pmb { \theta } } ( \ell _ { i } ) )$   
+7: end for   
+8: Initialize: ${ \mathcal { E } } = \{ \}$   
+9: for $\{ x _ { i } , \tilde { y } _ { i } \}$ in $\tilde { \tau }$ do   
+10: $\hat { \mathbf { { x } } } _ { i } \gets$ Forward $( { \pmb x } _ { i } ; { \pmb \theta } ^ { \star } )$   
+11: $e _ { i } \gets E \left( \hat { \pmb x } _ { i } \right)$   
+12: $\mathcal { E } [ i ]  e _ { i }$   
+13: end for
+
+# Algorithm 2 Correct training data
+
+Require: $\tilde { \tau }$   
+Require: $\mathcal { T }$   
+Require: $\mathcal { E }$   
+Require: $E \left( { \hat { \pmb x } } \right)$   
+Require: GetNegativeSamples $( \mathcal { T } )$   
+Require: GetPositiveSamples $( \mathcal { T } )$ )
+
+1: $\mathcal { T } _ { 0 } \gets$ GetNegativeSamples $( \mathcal { T } )$   
+2: $\mathcal { T } _ { 1 } $ GetPositiveSamples $( \mathcal { T } )$   
+3: $\begin{array} { r } { \bar { E } _ { 0 }  \frac { 1 } { | \mathcal { T } _ { 0 } | } \sum _ { \pmb { x } \in \mathcal { T } _ { 0 } } E \dot { ( \pmb { x } ) } } \end{array}$   
+4: $\begin{array} { r } { \bar { E } _ { 1 }  \frac { 1 } { | \mathcal { T } _ { 1 } | } \sum _ { \pmb { x } \in \mathcal { T } _ { 1 } } E ( \pmb { x } ) } \end{array}$   
+5: $\begin{array} { r } { \sigma _ { 0 } \gets \left[ \sum _ { \pmb { x } \in \mathcal { T } _ { 0 } } \frac { \left( E ( \pmb { x } ; \pmb { \theta } ) - \bar { E } _ { 0 } \right) ^ { 2 } } { | \mathcal { T } _ { 0 } | - 1 } \right] ^ { \frac { 1 } { 2 } } } \end{array}$ 12   
+6: σ1 ← Px∈T1 (E(x;θ)−E¯1)2|T1|−1 12   
+7: T c ← T˜ T   
+8: for $\{ x _ { i } , y _ { i } \}$ in ${ \mathcal { T } } ^ { \mathrm { c } }$ do   
+9: $e _ { i } \gets \mathcal { E } [ i ]$   
+10: if $y _ { i } = 1$ and $e _ { i } < \bar { E } _ { 0 } + \sigma _ { 0 }$ then   
+11: $y _ { i } \gets 0$   
+12: else if $y _ { i } = 0$ and $e _ { i } > \bar { E } _ { 1 } - \sigma _ { 0 }$ then   
+13: yi ← 1   
+14: end if   
+15: end for
+
+. full training data . training data with known labels . energies from algorithm 1 . energy function, i.e. reconstruction loss . get all samples from class 0 . get all samples from class 1 $\triangleright$ get mean w.r.t class 0 $\triangleright$ get mean w.r.t class 1 $\triangleright$ get standard deviation w.r.t class 0 $\triangleright$ get standard deviation w.r.t class 1 $\triangleright$ i.e. $\mathcal { T } ^ { \mathrm { c } } = \{ ( \mathbf { x } _ { i } , y _ { i } ) \in \tilde { \mathcal { T } } \| ( \mathbf { x } _ { i } , y _ { i } ) \notin \mathcal { T } \}$ $\triangleright$ get energy corresponding to $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$

parse/train/MBH29cOeohr/MBH29cOeohr.md

@@ -0,0 +1,484 @@
+# Fact-driven Logical Reasoning
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Logical reasoning deeply relies on accurate, clearly presented clue forms which   
+2 are usually modeled as entity-like knowledge in existing studies. However, in   
+3 real hierarchical reasoning motivated machine reading comprehension (MRC),   
+4 such one-side modeling are insufficient for those indispensable local complete   
+5 facts or events when only "global" knowledge is really paid attention to. Thus, in   
+6 view of language being a complete knowledge/clue carrier, we propose a general   
+7 formalism to support representing logic units by extracting backbone constituents   
+8 of the sentence such as the subject-verb-object formed "facts", covering both global   
+9 and local knowledge pieces that are necessary as the basis for logical reasoning.   
+10 Beyond building the ad-hoc graphs, we propose a more general and convenient   
+11 fact-driven approach to construct a supergraph on top of our newly defined fact   
+12 units, and enhance the supergraph with further explicit guidance of local question   
+13 and option interactions. Experiments on two challenging logical reasoning MRC   
+14 benchmarks show that our proposed model, FOCAL REASONER, outperforms the   
+15 baseline models dramatically.
+
+# 16 1 Introduction
+
+17 Machine reading comprehension (MRC) requires machine to answer question according to given   
+18 passage [1, 2, 2, 3, 4]. Logical reasoning [5] from MRC accounts for human intuition about entailment   
+19 of sentences and reflects the semantic relations between sentential constituents [6]. Recently, there is   
+20 a surging trend of research into logical reasoning ability, among which ReClor [7] and LogiQA [5] are   
+21 two representative datasets introduced to promote the development of logical reasoning, where logical   
+22 reasoning questions are selected from standardized exams such as GMAT1, requiring models to read   
+23 and comprehend the complicated logical relationships. Similar to the standard question-answering   
+24 (QA)-based MRC tasks in form, our concerned logical reasoning QA tasks contain three elements:   
+25 passage, question and the candidate options as examples shown in Figure 1.
+
+MRC models usually exploit a pre-trained language model (PrLM) as a key encoder for effective contextualized representation. Meanwhile, the major challenge of logical reasoning is to uncover logical structures, and reasoning with the candidate options and questions to predict the correct answer. However, it is difficult for PrLMs to capture the logical structure inherent in the texts since logical supervision is rarely available during pre-training. Existing logical reasoning has shown serious dependence on knowledge-like clues. This is due to the lengthy, noisy text in human language which is though a natural carrier of knowledge but does not provide a clean, exact knowledge form. Thus, an increasing interest is using graph networks to model the entity-aware relationships in the passages [8, 9, 10, 11]. However, all these methods may insufficiently capture indispensable logical units from two perspectives. First, they mostly focus on entity-aware commonsense knowledge, but pay little attention to those non-entity, non-commonsense clues [12]. Second, when existing models
+
+Figure 1: Two examples from LogiQA and ReClor respectively are illustrated. There are arguments and relations between arguments. Both are emphasized by different colors: arguments, relations. Key words in questions are highlighted in Purple. Key options are highlighted in gray.   
+
+<table><tr><td>Question</td><td>Passage</td><td>Answer</td></tr><tr><td>iExample1i From this we know</td><td>Xiao Wang is taller than Xiao Li, Xiao Zhao is taller than Xiao Qian, Xiao Li isshorter than Xiao Sun,and</td><td>A. Xiao Li is shorter than Xiao Zhao. B. Xiao Wang is taller than Xiao Zhao. C. Xiao Sun is shorter than Xiao Wang.</td></tr><tr><td>iExample 21</td><td>Xiao Sun is shorter than Xiao Qian. .... .A large enough comet colliding</td><td>D. Xiao Sun is taller than Xiao Zhao. A. Many other animal species from same era did not become extinct at the same time the dinosaurs did.</td></tr><tr><td>Which one of the follow- ing statements,most seriously weakens theargument?</td><td>with Earth could have caused a cloud of dust that enshrouded the planet and cooled the climate long enough to result in the dinosaurs&#x27; demise.</td><td>B. It cannot be determined from dinosaur skeletons whether the animals died from the effects of a dust cloud. C.The consequences for vegetation and animals of a comet colliding with Earth are not fully understood. D.Various species of animals from the same era and similar to them in habitat and physiology did not become extinct.</td></tr></table>
+
+37 extract predicate logic inside language into knowledge, they only exploit quite limited predicates like   
+38 hasA and isA but ignore a broad range of predicates in real language. From either of the perspectives,   
+39 the existing methods actually only concern about those "global" knowledge that keeps valid across   
+40 the entire data, without sufficient "local" perception of complete facts or events in the given specific   
+41 part of MRC task. We argue such insufficient modeling on logic units roots from the ignorance of   
+42 language itself being the complete knowledge/clue carrier. Thus, we propose extracting a kind of   
+43 broad facts according to backbone constituents of a sentence to effectively cover such indispensable   
+44 logic reasoning basis, filling the gap of local, non-commonsense, non-entity, or even non-knowledge   
+45 clues in existing methods as shown in Figure 2. For example, these units may reflect the facts of who   
+46 did what to whom, or who is what in Figure 3. Such groups can be defined as "fact unit" following   
+47 [13] in Definition 1. The fact units are further organized into a supergraph following Definition 2.   
+48 Definition 1 (Fact Unit) Given an triplet $T ~ = ~ \{ E _ { 1 } , P , E _ { 2 } \}$ , where $E _ { 1 }$ and $E _ { 2 }$ are arguments   
+49 (including entity and non-entity), $P$ is the predicate between them, a fact unit $F$ is the set of all   
+50 entities in $T$ and their corresponding relations.
+
+Definition 2 (Supergraph) A supergraph is a structure made of fact units (regarded as subgraphs) 52 as the vertices, and the relations between fact units as undirected edges.
+
+53 As shown in Figure 2, we regard the defined   
+54 fact as the results of syntactic processing, rather   
+55 than those from semantic role labeling (SRL) as   
+56 in previous study, thus the proposed fact also   
+57 extends the processing means in existing work.   
+58 Correspondingly, in this work, we propose   
+59 a fact-driven logical reasoning model, called   
+60 FOCAL REASONER, which builds supergraphs   
+61 on top of fact units as the basis for logical   
+62 reasoning, to capture both global connections   
+63 between facts and the local concepts or actions   
+64 inside the fact. In addition, we strengthen our   
+65 model by the question-option-aware interaction.   
+66 Specifically, we explicitly reformulate questions   
+67 with negation expressions to compensate for the   
+68 insensitiveness of PrLMs, all of which are interacted in our supergraph. Such resulted FOCAL   
+69 REASONER is evaluated on two challenging logical reasoning benchmarks including ReClor, LogiQA,   
+70 and one dialogue reasoning dataset Mutual for generalizability, achieving new state-of-the-art results.
+
+![](images/ffa5bfaeb784b39a55c9ed091fa012d644b65aa9c5eb5b2eac0da164100ea548.jpg)  
+Figure 2: Our "fact" V.S. existing approaches.
+
+# 71 2 Related Work
+
+72 Machine Reading Comprehension Recent years have witnessed massive researches on Machine   
+73 Reading Comprehension, which has become one of the most important areas of NLP [14, 15, 16,   
+74 17, 18, 19, 20, 21, 22]. Despite the success of MRC models on various datasets such as CNN/Daily   
+75 Mail [1], SQuAD [2], RACE [3] and so on, researchers began to rethink to what extent does the   
+76 problem been solved. Nowadays, there are massive researches into the reasoning ability of machines.   
+77 According to [23, 24, 25], reasoning abilities can be broadly categorized into (1) commonsense   
+78 reasoning [26, 27, 28, 29]; (2) numerical reasoning [30]; (3) multi-hop reasoning [31] and (4) logical   
+79 reasoning [5, 7], among which logical reasoning is essential in human intelligence but has merely   
+80 been delved into. Natural Language Inference (NLI) [32, 33, 34] is a task closely related to logical   
+81 reasoning. However, it has two obvious drawbacks in measuring logical reasoning abilities. One is   
+82 that it only has three logical types which are entailment, contradiction and neutral. The other is its   
+83 limitation on sentence-level reasoning. Hence, it is important to research more comprehensive and   
+84 deeper logical reasoning abilities.   
+85 Logical Reasoning in MRC There are two   
+86 main kinds of features in language data that   
+87 would be the necessary basis for logical   
+88 reasoning: 1) knowledge: global facts that   
+89 keep consistency regardless of the context,   
+90 such as commonsense, mostly derived from   
+91 named entities; 2) non-knowledge: local facts   
+92 or events that may be sensitive to the context,   
+93 mostly derived from detailed language. Existing   
+94 works have made progress in improving logical   
+95 reasoning ability [8, 9, 10, 11, 12, 38]. However,   
+96 these approaches are barely satisfactory as they   
+97 mostly focus on the global facts such as typical   
+98 entity or sentence-level relations, which are   
+99 obviously not sufficient. In this work, we   
+100 strengthen the basis for logical reasoning by   
+101 unifying both types of the features as "facts".   
+102 Different from previous studies that focus on   
+103 the knowledge components, we propose a fact  
+104 driven logical reasoning framework that builds   
+105 supergraphs on top of fact units to capture both
+
+![](images/a9e163928e4ba3e276dd99d23f68db19edfdb59b8eb6053a7b621407621dc4a4.jpg)  
+Figure 3: An example of constructed supergraph. In contrast, the dotted vertices and edges are focused in most existing studies [35, 36, 37].
+
+06 global connections between entity-aware facts and the local concepts or events inside the fact.
+
+# 107 3 Approaches
+
+108 In this section, we will describe our method in detail. The overall architecture of the model is shown   
+109 in Figure 4 . We first construct a supergraph from the raw text based on the fact units extracted.   
+110 Then we conduct reasoning over the supergraph with question-option guided approaches to learn and   
+111 update the features, which are further incorporated in answer prediction.
+
+# 112 3.1 Supergraph Construction
+
+13 Figure 5 illustrates our method for constructing a supergraph from raw text inputs. The first step is   
+14 to obtain triplets that constitute a fact unit. To keep the framework generic, we use a fairly simple   
+15 fact unit extractor based on the syntactic relations. Given a context consisting multiple sentences, we   
+116 first conduct dependency parsing of each sentence. After that, we extract the subject, the predicate,   
+117 and the object tokens to get the "Argument-Predicate-Argument" triplets corresponding to   
+18 each sentence in the context.   
+119 With the obtained triplets, the fact units are organized in the form of Levi graph [39], which turns   
+120 arguments and predicates all into nodes. An original fact unit is in the form of $F = ( V , E , R )$ ,   
+121 where $V$ is the set of the arguments, $E$ is the set of edges connected between arguments, and $R$ is   
+122 the relations of each edge which are predicates here. The corresponding Levi graph is denoted as   
+123 $F _ { l } = \left( V _ { L } , E _ { L } , R _ { L } \right)$ where $V _ { L } = V \cup R$ , which makes the originally directly connected arguments   
+124 be intermediately connected via relations. As for $R _ { L }$ , previous works such as [40, 41] designed three   
+125 types of edges $\dot { R } _ { L } = \{ d e f a u l t , r e v e r s e , s e l f \}$ to enhance information flow. Here in our settings,   
+126 we extend it into five types: default-in, default-out, reverse-in, reverse-out, self, corresponding to the   
+127 directions of edges towards the predicates.   
+128 We construct the supergraph by making connections between fact units $F _ { l }$ . In particular, we take   
+129 three strategies according to question-option, identical concept and co-reference information. (1) For   
+130 question-option pair, We initialize a global node $V _ { g }$ with its representation and connect it to all the fact   
+131 unit nodes. The edge type are set as global. The global node ensures that all fact units are connected   
+132 so that information can be exchanged during graph encoding. (2) There can be identical mentions   
+133 in different sentences, resulting in repeated nodes in fact units. We connect nodes corresponding   
+134 to the same non-pronoun arguments by edges with edge type same. (3) We conduct co-reference   
+135 resolution on context using an off-to-shelf model2 in order to identify arguments in fact units that   
+136 refer to the same one. We add edges with type coref between them. The final supergraph is denoted   
+137 as $S = ( F _ { l } \cup V _ { g } , E )$ where $E$ is the set of edges added with the previous three strategies.
+
+![](images/2bade05cfa2d46e73c42e5a96b7d5d65f6d09a612dc3e4d6c20f44c92dfa4222.jpg)  
+Figure 4: The framework or our model. For supergraph reasoning, in each iteration, each node selectively receives the message from the neighboring nodes to update its representation. The dashed circle means zero vector.
+
+# 3.2 Encoder
+
+# 3.2.1 Context Encoder
+
+140 Our context encoder $F _ { C } ( . )$ is initialized with a pre-trained language model, i.e., RoBERTa-large   
+141 [42]. Question, context and option are concatenated and then fed into the encoder. If the question is   
+142 detected to contain negative meanings, we add a special token <pos> before the question, else we add   
+143 <neg>. In a whole, we get the hidden representation as following:
+
+$\{ h _ { c , 0 } , . . . , h _ { c , l _ { c } + 1 } , h _ { q , 1 } , . . . , h _ { o , 1 } , . . . , h _ { o , l _ { o } + 1 } \} = F _ { C } ( \{ x _ { c , 0 } , . . . , x _ { c , l _ { c } + 1 } , x _ { q , 0 } , . . . , x _ { o , 1 } , . . . , x _ { o , l _ { o } + 1 } \} ) ,$ (1) where 44 $x _ { c , 0 } = < s >$ , $x _ { c , l _ { c } + 1 } = x _ { o , l _ { o } + 1 } = < / \mathrm { s } >$ , $x _ { q , 0 } = < \mathrm { p o s } > / < \mathrm { n e g } >$ and $h _ { i } \in \mathbb { R } ^ { d }$ , $d$ is the hidden size.
+
+# 3.2.2 Supegraph Encoder
+
+Graph Initialization $F _ { C } ( . )$ encodes each token in nodes $V _ { L }$ , and then the averaged hidden state is used as the initial representation of the original word of each node, because PrLMs like RoBERTa take subwords as input while our triplets extraction performs in word-level. For the global QA-context node, we averaged the embeddings of tokens in question and option for initialization. We also use a one-hot embedding layer to encode the relations between two nodes.
+
+![](images/97f35831f1809843a8867e5d46260e7f36caa3817533d4bde6170c05f3d14c45.jpg)  
+Which one of the following ......, most seriously weakens the argument? Various species of animals from the same era as dinosaurs and similar to them ... did not become extinct when the dinosaurs did.   
+Figure 5: The process of constructing the fact chain and its corresponding Levi graph form of an example in Figure 1. Entities and relations are illustrated in its corresponding color.
+
+151 Graph Attention Network Based on the relational graph convolutional network [43] and given   
+152 the initial representation $h _ { i } ^ { 0 }$ for every node $v _ { i }$ , the feed-forward or the message-passing process with   
+153 information control can be written as:
+
+$$
+h _ { i } ^ { ( l + 1 ) } = \mathrm { R e L U } ( \sum _ { r \in R _ { L } } \sum _ { v _ { j } \in \mathcal { N } _ { r } ( v _ { i } ) } g _ { q } ^ { ( l ) } \frac { 1 } { c _ { i , r } } w _ { r } ^ { ( l ) } h _ { j } ^ { ( l ) } ) ,
+$$
+
+154 where $\mathcal { N } _ { r } ( v _ { i } )$ denotes the neighbors of node $v _ { i }$ under relation $r$ and $c _ { i , r }$ is the number of those nodes.   
+155 $w _ { r } ^ { ( l ) }$ is the learnable parameters of layer $l$ . $g _ { q } ^ { ( l ) }$ is a gated value between 0 and 1.
+
+6 Through the graph encoder $F _ { G } ( . )$ , we then obtain the hidden representations of nodes in fact units as:
+
+$$
+\{ h _ { 0 } ^ { F } , . . . h _ { m } ^ { F } \} = F _ { G } ( \{ v _ { L , 0 } , . . . v _ { L , m } \} , E _ { L } ) .
+$$
+
+158 These features are further concatenated to get the final node representation of the supergraph:
+
+$$
+\{ h _ { 0 } ^ { S } , . . . h _ { m } ^ { S } \} = F _ { G } ( \{ h _ { 0 } ^ { F } , . . . h _ { m } ^ { F } \} , E _ { C } ) .
+$$
+
+159 For node features on the supergraph, it is fused via the attention and gating mechanisms with the   
+160 original representations of the context encoder. Specifically, denoting the original whole sequence   
+161 representation after context encoder as $H ^ { C }$ , we apply attention mechanism to append the supergraph   
+162 representation to the original one:
+
+$$
+\tilde { H } = \mathrm { A t t n } ( H ^ { c } , K _ { f } , V _ { f } ) ,
+$$
+
+163 where $\{ K _ { f } , V _ { f } \}$ are packed from the learned representations of the supergraph. We compute   
+164 $\lambda \in [ 0 , \bar { 1 } ]$ to weigh the expected importance of supergraph representation of each source word:
+
+$$
+\lambda _ { 1 } = \sigma ( W _ { \lambda } \tilde { H } + U _ { \lambda } H ^ { C } ) ,
+$$
+
+where $W _ { \lambda }$ and $U _ { \lambda }$ are learnable parameters. $H ^ { C }$ and $\tilde { H }$ are then fused for an effective representation:
+
+$$
+H = H ^ { C } + \lambda \tilde { H } \in \mathbb { R } ^ { 4 \times d } .
+$$
+
+# 3.2.3 Question-Option-aware Interaction
+
+Options have their inherent logical relations, which can be leveraged to aid answer prediction. Inspired by [44], we use an attention-based mechanism to gather option correlation information.
+
+Specifically for an option 0 $O _ { i }$ , the information it get by interaction with option $O _ { j }$ is calculated as:
+
+$$
+O _ { i } ^ { ( j ) } = [ O _ { i } ^ { q } - O _ { i } ^ { q } \mathrm { A t t n } ( O _ { i } ^ { q } , O _ { j } ^ { q } ; v ) ; O _ { i } ^ { q } \circ O _ { i } ^ { q } \mathrm { A t t n } ( O _ { i } ^ { q } , O _ { j } ^ { q } ; v ) ] ,
+$$
+
+171 where $O _ { i } ^ { q }$ is the representation of the concatenation for the $i$ -th option and question after the context   
+172 encoder. Then the option-wise information are gathered to fuse the option correlation information:
+
+$$
+\hat { O } _ { i } = \operatorname { t a n h } ( W _ { c } [ O _ { i } ^ { q } ; \{ O _ { i } ^ { ( j ) } \} _ { i \neq j } ] + b _ { c } ) ,
+$$
+
+where 173 $\mathbf { W } _ { c } \in \mathbb { R } ^ { d \times 7 d }$ and $b _ { c } \in \mathbb { R } ^ { d }$ . Finally, a gating mechanism is used to fuse the option features:
+
+$$
+O _ { i , : k } ^ { q } = g _ { i , : k } \circ O _ { i , : k } ^ { q } + ( 1 - g _ { i , : k } ) \circ \hat { O } _ { i , : k } ,
+$$
+
+where the 174 $g _ { i , : k } = \sigma ( W _ { g } [ O _ { i , : k } ; O _ { i , : k } ^ { \widehat { q } } ; \widetilde { Q } ] + b _ { g } ) \in \mathbb { R } ^ { d }$ is the $i$ -th column of gate $g$
+
+# 3.3 Hierarchical Decoder
+
+176 To better incorporate the information obtained above, apart from getting the original pooled context  
+177 attended representation $h ^ { C } \in \mathbb { R } ^ { 4 \times d }$ , we combine the attended vectors ${ \bf \bar { \boldsymbol { O } } } ^ { f }$ and $H$ from the previous   
+178 encoder through a fusing layer.
+
+$$
+\begin{array} { r l } & { E _ { 1 } = \mathrm { R e L U } ( \mathrm { F C } ( [ h ^ { C } , H , h ^ { C } - H , h ^ { C } \circ H ] ) ) , } \\ & { E _ { 2 } = \mathrm { R e L U } ( \mathrm { F C } ( [ h ^ { C } , H , h ^ { C } - O ^ { f } , h ^ { C } \circ O ^ { f } ] ) ) , } \\ & { P = \sigma ( \mathrm { F C } ( [ E _ { 1 } , E _ { 2 } ] ) ) , } \\ & { C = P \circ H + ( 1 - P ) \circ O ^ { f } \in \mathbb { R } ^ { 4 \times d } . } \end{array}
+$$
+
+179 Then another linear layer is applied for final prediction as $z = W _ { z } C + b _ { z } \in \mathbb { R } ^ { 4 }$ . We seek to minimize   
+180 the cross entropy loss over the correct decision $l$ by
+
+$$
+\mathcal { L } _ { a n s } = - \log \operatorname { s o f t m a x } ( z ) _ { l } .
+$$
+
+181 Logical Fact Regularization Inspired by [45], the embedding of the tail argument should be close   
+182 to the embedding of the head argument plus a relation-related vector in the hidden representation   
+183 space. Without loss of generality, we assume that in our settings, the summation of the subject vector   
+184 and the relation vector should be close to the object vector as much as possible, i.e.,
+
+$$
+v _ { s u b j e c t } + v _ { r e l a t i o n }  v _ { o b j e c t } .
+$$
+
+185 In order to make the logical facts more of factual correctness, we introduce a regularization for the   
+186 extracted logical facts based on the hidden states of the sequence $h _ { i }$ where $i = 1 , \ldots , L$ and $L$ is the   
+187 total length of the sequence. The regularization is defined as:
+
+$$
+L _ { l f r } = \sum _ { k = 1 } ^ { m } ( 1 - \cos ( h _ { s u b _ { k } } + h _ { r e l _ { k } } , h _ { o b j _ { k } } ) ) ,
+$$
+
+where $m$ is the total number of logical fact triplets extracted from the context as well as the option and $k$ indicates the $k$ -th fact triplet.
+
+Training Objective. During training, the overall loss for answer prediction is:
+
+$$
+\begin{array} { r } { \mathcal { L } = \alpha \mathcal { L } _ { a n s } + \beta \mathcal { L } _ { l f r } , } \end{array}
+$$
+
+where $\alpha$ and $\beta$ are two parameters. In our implementation, we set $\alpha = 1 . 0$ and $\beta = 0 . 5$
+
+# 4 Experiments
+
+# 4.1 Datasets
+
+We conducted the experiments on three datasets. Two for specialized logical reasoning ability testing: ReClor [7] and LogiQA [5] and one for logical reasoning in dialogues: MuTual [46]. For more details, one can refer to Appendix A.
+
+# 4.2 Implementation Details
+
+We fine-tune RoBERTa as the backbone PrLM for FOCAL REASONER. The overall model is end-toend trained and updated by Adam [47] optimizer with an overall learning rate 8e-6 for ReClor and LogiQA, and 4e-6 for MuTual. The weight decay is 0.01. We set the warm-up proportion during training to 0.1. Graph encoders are implemented using DGL, an open-source lib of python. The layer number of the graph encoder is 2 for ReClor and 3 for LogiQA. The maximum sequence length is 256 for LogiQA and MuTual, and 384 for ReClor. The model is trained for 10 epochs with a total batch size 16 and an overall dropout rate 0.1 on 4 NVIDIA Tesla V100 GPUs, which takes around 2 hours for ReClor and 4 hours for LogiQA3.
+
+Table 1: Experimental results of our model compared with baseline models on ReClor and LogiQA dataset. Test-E and Test-H denote Test-Easy and Test-Hard respectively. We performed Pitman’s permutation test [48] and found that our model significantly outperformed the baseline $_ { ( \mathrm { p < 0 . 0 5 } ) }$ .   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">ReClor</td><td colspan="2">LogiQA</td></tr><tr><td>Dev</td><td>Test</td><td>Test-E</td><td>Test-H</td><td>Dev</td><td>Test</td></tr><tr><td>Human [7]</td><td></td><td>63.00</td><td>57.10</td><td>67.20</td><td></td><td>86.00</td></tr><tr><td>BERT-Large[7]</td><td>53.80</td><td>49.80</td><td>72.00</td><td>32.30</td><td>34.10</td><td>31.03</td></tr><tr><td>XLNet-Large [7]</td><td>62.00</td><td>56.00</td><td>75.70</td><td>40.50</td><td>=</td><td>=</td></tr><tr><td>RoBERTa-Large [7]</td><td>62.60</td><td>55.60</td><td>75.50</td><td>40.00</td><td>35.02</td><td>35.33</td></tr><tr><td>DAGN[10]</td><td>65.20</td><td>58.20</td><td>76.14</td><td>44.11</td><td>35.48</td><td>38.71</td></tr><tr><td>DAGN_(Aug)[10]</td><td>65.80</td><td>58.30</td><td>75.91</td><td>44.46</td><td>36.87</td><td>39.32</td></tr><tr><td>FOCALREASONER</td><td>66.80</td><td>58.90</td><td>77.05</td><td>44.64</td><td>41.01</td><td>40.25</td></tr></table>
+
+<table><tr><td rowspan="3">Model</td><td colspan="6">MuTual</td><td colspan="6">MuTualplus</td></tr><tr><td colspan="2">Dev Set</td><td colspan="2"></td><td colspan="2">Test Set</td><td colspan="2">Dev Set</td><td colspan="2"></td><td colspan="2">Test Set</td></tr><tr><td>R4@1</td><td>R4@2</td><td>MRR R4@1</td><td></td><td>R4@2 MRR</td><td></td><td>R4@1</td><td>R4@2</td><td>MRR</td><td>R4@1</td><td>R4@2 MRR</td><td></td></tr><tr><td>RoBERTabase [46]</td><td>69.5</td><td>87.8</td><td>82.4</td><td>71.3</td><td>89.2</td><td>83.6</td><td>62.2</td><td>85.3</td><td>78.2</td><td>62.6</td><td>86.6</td><td>78.7</td></tr><tr><td>-MC[46]</td><td>69.3</td><td>88.7</td><td>82.5</td><td>68.6</td><td>88.7</td><td>82.2</td><td>62.1</td><td>83.0</td><td>77.8</td><td>64.3</td><td>84.5</td><td>79.2</td></tr><tr><td>FOCAL REASONER</td><td>73.4</td><td>1 90.3</td><td>84.9</td><td>72.7</td><td>91.0</td><td>-84.6</td><td>63.7</td><td>86.1</td><td>79.1</td><td>65.5</td><td>84.3</td><td>-79.7</td></tr></table>
+
+Table 2: Experimental results of our model compared with baseline PrLM on MuTual dataset.
+
+# 206 4.3 Results
+
+Tables 1 and 2 show the results on ReClor, LogiQA, and MuTual, respectively. All the best results are shown in bold. Based on our implemented baseline models (basically consistent with public results), we observe dramatic improvements on both of the logical reasoning benchmarks, e.g., on ReClor test set, FOCAL REASONER achieves $+ 4 . 2 \%$ on dev set and $+ 3 . 3 . \%$ on the test set. FOCAL REASONER also outperforms the prior best system $\mathrm { D A G N ^ { 4 } }$ , reaching $7 7 . 0 5 \%$ on the EASY subset, and $4 4 . 6 4 \%$ on the HARD subset. The performance suggests that FOCAL REASONER makes better use of logical structure inherent in the given context to perform reasoning than existing methods. On the dialogue reasoning dataset MuTual, our model achieves quite a jump compared with the RoBERTa-base $\mathrm { L } \mathbf { \bar { M } } ^ { 5 }$ This verifies our model’s generalizability on other downstream reasoning task settings.
+
+In addition, Table 5 lists the accuracy of our model on the dev set of ReClor of different question types. Results show that our model can perform well on most of the question types, especially "Strengthen" and "Weaken". This means that our model can well interpret the question type from the question statement and make the correct choice corresponding to the question.
+
+# 5 Analysis
+
+# 5.1 Ablation Study
+
+To dive into the effectiveness of different components in FOCAL REASONER, we conduct an ablation study which takes RoBERTa as the backbone on the ReClor dev set. Table 3 summarizes the results.
+
+Supergraph reasoning: The first key component is the supergraph reasoning. We ablate the global atom and erase all the edges connected with it. The results suggest that the global atom indeed betters message propagation, leveraging performance from $6 4 . 6 \%$ to $6 6 . 8 \%$ . We also find that replacing the initial QA pair representation of the global atom with only question representation hurts the performance. In addition, without the logical fact regularization, the performance drops from $6 6 . 8 \%$
+
+![](images/07dc959c4d12f76c1a9f1a395a3d23db7b62a9c08bdfdde53d99e7606e5d23ce.jpg)  
+Figure 6: Accuracy of models on number of fact units on dev set of ReClor (left) and LogiQA (right).
+
+229 to $6 4 . 2 \%$ , indicating its usefulness. For edge analysis, when (1) all edges are regarded as a single   
+230 type rather than the original designed 8 types in total and (2) co-reference edges are removed, the   
+231 accuracy drops to $6 3 . 7 \%$ and $6 4 . { \bar { 8 } } \%$ , respectively. It is proved that in our supergraph, edges link the   
+232 fact units in reasonable manners, which properly uncovers the logical structures.   
+233   
+234   
+235   
+236   
+237   
+238   
+239   
+240   
+241   
+242   
+243   
+244   
+245   
+246   
+247   
+248   
+249   
+250   
+251
+
+Fact Units Variants Apart from our syntactically constructed fact units, there are another two ways in different granularities for construction. We replace the fact units with named entities which are used in previous works like [49]. The statistics of fact units and named entities of ReClor and LogiQA are stated in Table 4, from which we can infer that there are indeed more fact units than named entities. Thus using fact units can better incorporate the logical information within the context. When replacing all the fact units with named entities, we can see from Table 3 that it significantly decreases the performance. We also explore the performance using semantic role labeling the similar way as in [50]. We can see that SRL, leveraging a much more complex information as well as computation complexity, fails to achieve a performance as good as our original fact unit.
+
+Table 3: Ablation results on the dev set of ReClor.   
+
+<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>FOCAL REASONER Supergraph Reasoning</td><td>66.8±0.13</td></tr><tr><td>- global node</td><td>64.6±0.32</td></tr><tr><td>- co-reference</td><td>64.8±0.24</td></tr><tr><td>- logical fact regularization</td><td></td></tr><tr><td>- QA context node → Q node</td><td>64.2±0.12 66.4±0.16</td></tr><tr><td>- question reformulation</td><td></td></tr><tr><td>- edge type</td><td>65.2±0.16 63.7±0.19</td></tr><tr><td>Fact Unit Variants</td><td></td></tr><tr><td> - named entity</td><td>62.8±0.26</td></tr><tr><td>- SRL</td><td>62.2±0.32</td></tr><tr><td>Interactions - interactions</td><td>65.5±0.52</td></tr></table>
+
+# Interactions: We further experimented with
+
+the query-option-interactions setting to see how it affects the performance. The results suggest that the features learned from the interaction process enhance the model. Considering that the logical relations between different options are a strong indicator of the right answer, this means that the model learns from a comparative reasoning strategy.
+
+# 5.2 Effects of Fact Units Numbers
+
+To inspect the effects of the number of fact units, we split the original dev set of ReClor and LogiQA into 5 subsets. The statistics of the fact unit distribution on the datasets are shown in Table 6. Numbers of fact units for most contexts in ReClor and LogiQA are in [3, 6) and $[ 0 , 3 )$ respectively.
+
+<table><tr><td rowspan="2">Number</td><td colspan="2">ReClor</td><td colspan="2">LogiQA</td></tr><tr><td>Train</td><td>Dev</td><td>Train</td><td>Dev</td></tr><tr><td>Fact Unit Argument</td><td>14,895</td><td>1,665</td><td>20,676</td><td>1,981</td></tr><tr><td>Named Entity</td><td>9,495</td><td>984</td><td>12,439</td><td>1,515</td></tr></table>
+
+Comparing the accuracies of RoBERTa-large baseline, prior SOTA DAGN and our proposed
+
+Table 4: Statistics for fact unit entities and traditional named entities in datasets.
+
+67 FOCAL REASONER in Figure 6, our model outperforms baseline models on all the divided subsets,   
+68 which demonstrates the effectiveness and robustness of our proposed method. Specifically, for ReClor,   
+69 FOCAL REASONER performers better when there are more fact units in the context, while for LogiQA,   
+70 FOCAL REASONER works better when the number of fact units locates in $[ 0 , 3 )$ and [9, 12). The   
+271 reason may lie in the difference in style of the two datasets. However, all the models include ours   
+272 struggle when the number of fact units is above certain thresholds, i.e., the logical structure is more   
+273 complicated, calling for better mechanisms to cope with.
+
+Table 5: Accuracy on the dev set of ReClor corresponding to several representative question types. S: Strengthen, W: Weaken, I: Implication, CMP: Conclusion/Main Point, ER: Explain or Resolve, D: Dispute, R: Role, IF: Identify a Flaw, MS: Match Structures.   
+
+<table><tr><td>Model</td><td>S</td><td>W</td><td>I</td><td>CMP</td><td>ER</td><td>P</td><td>D</td><td>R</td><td>IF</td><td>MS</td></tr><tr><td>RoBERTalarge [7]</td><td>61.70</td><td>47.79</td><td>39.13</td><td>63.89</td><td>58.33</td><td>50.77</td><td>50.00</td><td>56.25</td><td>61.54</td><td>56.67</td></tr><tr><td>DAGN[10]</td><td>63.83</td><td>46.02</td><td>39.13</td><td>69.44</td><td>57.14</td><td>5385</td><td>46.67</td><td>62.50</td><td>62.39</td><td>56.67</td></tr><tr><td>FOCAL REASONER</td><td>65.96</td><td>51.33</td><td>43.48</td><td>72.22</td><td>67.86</td><td>53.85</td><td>50.00</td><td>62.50</td><td>62.39</td><td>60.0</td></tr></table>
+
+# 5.3 Interpretability: a Case Study
+
+We aim to interpret FOCAL REASONER’s reasoning process by analyzing the nodeto-node attention weights induced in the supergraph in Figure 7. We can see that our FOCAL REASONER can well bridge the reasoning process between context, question and option. Specifically, in the graph, "students rank $30 \%$ " attends strongly to "playing improve performance". Under the guidance of question
+
+<table><tr><td>Dataset</td><td>[0,3)</td><td>[3,6)</td><td>[6,9）</td><td>[9,12)</td><td>）[12,00）</td></tr><tr><td>ReClor</td><td>37.2%</td><td>48.6%</td><td>12.6%</td><td>0.6%</td><td>1.2%</td></tr><tr><td>LogiQA</td><td>47.5%</td><td>37.5%</td><td>10.9%</td><td>3.5%</td><td>0.6%</td></tr></table>
+
+Table 6: Distribution of fact unit number on dev set of the training datasets.
+
+84 to select the option that weakens the statement and option interaction, our model is able to tell that "students rank $30 \%$ can play" mostly undermines the conclusion that "playing improves performance".
+
+A recent survey in a key middle school showed that high school students in this school have a special preference for playing football, and it far surpasses other balls.The survey also found that students who regularly play football are better at academic performance than students who do not often play football.This shows that often playing football can improve students' academic performance.
+
+A. Only high school students who are ranked in the top $30 \%$ of grades can often play football.   
+B. Regular football can exercise and maintain a strong learning energy.   
+C. Often playing football delays the study time.   
+D. Research has not proved that playing football can contribute to intellectual development.
+
+![](images/5dda7fa84f9379e1010c0eb9f4954b34cb7510f7d1998b2e6e804f77d2f8c54b.jpg)  
+Figure 7: An example of how our model reasons to get the final answer.
+
+# 286 6 Conclusion
+
+287 For logical reasoning arising from machine reading comprehension, it is well known that clear and   
+288 accurate forms like global knowledge are crucial. In this work, we make a finding that existing   
+289 studies miss focusing on quite a lot of non-knowledge parts which is also indispensable for better   
+290 reasoning. Thus we propose extracting a general form called "fact unit" to cover both global and   
+291 local logical units, hoping to shed light on the basis of structural modeling for logical reasoning.   
+292 Our proposed FOCAL REASONER not only better uncovers the logical structures within the context,   
+293 which can be a general method for other sophisticated reasoning tasks, but also better captures the   
+294 logical interactions between context and options. The experimental results verify the effectiveness of   
+295 our method.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 5.2.   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/MBH29cOeohr/MBH29cOeohr_content_list.json

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+        "type": "text",
+        "text": "Fact-driven Logical Reasoning ",
+        "text_level": 1,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            200,
+            580,
+            256
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 Logical reasoning deeply relies on accurate, clearly presented clue forms which   \n2 are usually modeled as entity-like knowledge in existing studies. However, in   \n3 real hierarchical reasoning motivated machine reading comprehension (MRC),   \n4 such one-side modeling are insufficient for those indispensable local complete   \n5 facts or events when only \"global\" knowledge is really paid attention to. Thus, in   \n6 view of language being a complete knowledge/clue carrier, we propose a general   \n7 formalism to support representing logic units by extracting backbone constituents   \n8 of the sentence such as the subject-verb-object formed \"facts\", covering both global   \n9 and local knowledge pieces that are necessary as the basis for logical reasoning.   \n10 Beyond building the ad-hoc graphs, we propose a more general and convenient   \n11 fact-driven approach to construct a supergraph on top of our newly defined fact   \n12 units, and enhance the supergraph with further explicit guidance of local question   \n13 and option interactions. Experiments on two challenging logical reasoning MRC   \n14 benchmarks show that our proposed model, FOCAL REASONER, outperforms the   \n15 baseline models dramatically. ",
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+    {
+        "type": "text",
+        "text": "16 1 Introduction ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
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+        "text": "17 Machine reading comprehension (MRC) requires machine to answer question according to given   \n18 passage [1, 2, 2, 3, 4]. Logical reasoning [5] from MRC accounts for human intuition about entailment   \n19 of sentences and reflects the semantic relations between sentential constituents [6]. Recently, there is   \n20 a surging trend of research into logical reasoning ability, among which ReClor [7] and LogiQA [5] are   \n21 two representative datasets introduced to promote the development of logical reasoning, where logical   \n22 reasoning questions are selected from standardized exams such as GMAT1, requiring models to read   \n23 and comprehend the complicated logical relationships. Similar to the standard question-answering   \n24 (QA)-based MRC tasks in form, our concerned logical reasoning QA tasks contain three elements:   \n25 passage, question and the candidate options as examples shown in Figure 1. ",
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+    {
+        "type": "text",
+        "text": "MRC models usually exploit a pre-trained language model (PrLM) as a key encoder for effective contextualized representation. Meanwhile, the major challenge of logical reasoning is to uncover logical structures, and reasoning with the candidate options and questions to predict the correct answer. However, it is difficult for PrLMs to capture the logical structure inherent in the texts since logical supervision is rarely available during pre-training. Existing logical reasoning has shown serious dependence on knowledge-like clues. This is due to the lengthy, noisy text in human language which is though a natural carrier of knowledge but does not provide a clean, exact knowledge form. Thus, an increasing interest is using graph networks to model the entity-aware relationships in the passages [8, 9, 10, 11]. However, all these methods may insufficiently capture indispensable logical units from two perspectives. First, they mostly focus on entity-aware commonsense knowledge, but pay little attention to those non-entity, non-commonsense clues [12]. Second, when existing models ",
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+        "table_caption": [
+            "Figure 1: Two examples from LogiQA and ReClor respectively are illustrated. There are arguments and relations between arguments. Both are emphasized by different colors: arguments, relations. Key words in questions are highlighted in Purple. Key options are highlighted in gray. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Question</td><td>Passage</td><td>Answer</td></tr><tr><td>iExample1i From this we know</td><td>Xiao Wang is taller than Xiao Li, Xiao Zhao is taller than Xiao Qian, Xiao Li isshorter than Xiao Sun,and</td><td>A. Xiao Li is shorter than Xiao Zhao. B. Xiao Wang is taller than Xiao Zhao. C. Xiao Sun is shorter than Xiao Wang.</td></tr><tr><td>iExample 21</td><td>Xiao Sun is shorter than Xiao Qian. .... .A large enough comet colliding</td><td>D. Xiao Sun is taller than Xiao Zhao. A. Many other animal species from same era did not become extinct at the same time the dinosaurs did.</td></tr><tr><td>Which one of the follow- ing statements,most seriously weakens theargument?</td><td>with Earth could have caused a cloud of dust that enshrouded the planet and cooled the climate long enough to result in the dinosaurs&#x27; demise.</td><td>B. It cannot be determined from dinosaur skeletons whether the animals died from the effects of a dust cloud. C.The consequences for vegetation and animals of a comet colliding with Earth are not fully understood. D.Various species of animals from the same era and similar to them in habitat and physiology did not become extinct.</td></tr></table>",
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+        "text": "37 extract predicate logic inside language into knowledge, they only exploit quite limited predicates like   \n38 hasA and isA but ignore a broad range of predicates in real language. From either of the perspectives,   \n39 the existing methods actually only concern about those \"global\" knowledge that keeps valid across   \n40 the entire data, without sufficient \"local\" perception of complete facts or events in the given specific   \n41 part of MRC task. We argue such insufficient modeling on logic units roots from the ignorance of   \n42 language itself being the complete knowledge/clue carrier. Thus, we propose extracting a kind of   \n43 broad facts according to backbone constituents of a sentence to effectively cover such indispensable   \n44 logic reasoning basis, filling the gap of local, non-commonsense, non-entity, or even non-knowledge   \n45 clues in existing methods as shown in Figure 2. For example, these units may reflect the facts of who   \n46 did what to whom, or who is what in Figure 3. Such groups can be defined as \"fact unit\" following   \n47 [13] in Definition 1. The fact units are further organized into a supergraph following Definition 2.   \n48 Definition 1 (Fact Unit) Given an triplet $T ~ = ~ \\{ E _ { 1 } , P , E _ { 2 } \\}$ , where $E _ { 1 }$ and $E _ { 2 }$ are arguments   \n49 (including entity and non-entity), $P$ is the predicate between them, a fact unit $F$ is the set of all   \n50 entities in $T$ and their corresponding relations. ",
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+        "type": "text",
+        "text": "Definition 2 (Supergraph) A supergraph is a structure made of fact units (regarded as subgraphs) 52 as the vertices, and the relations between fact units as undirected edges. ",
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+        "text": "53 As shown in Figure 2, we regard the defined   \n54 fact as the results of syntactic processing, rather   \n55 than those from semantic role labeling (SRL) as   \n56 in previous study, thus the proposed fact also   \n57 extends the processing means in existing work.   \n58 Correspondingly, in this work, we propose   \n59 a fact-driven logical reasoning model, called   \n60 FOCAL REASONER, which builds supergraphs   \n61 on top of fact units as the basis for logical   \n62 reasoning, to capture both global connections   \n63 between facts and the local concepts or actions   \n64 inside the fact. In addition, we strengthen our   \n65 model by the question-option-aware interaction.   \n66 Specifically, we explicitly reformulate questions   \n67 with negation expressions to compensate for the   \n68 insensitiveness of PrLMs, all of which are interacted in our supergraph. Such resulted FOCAL   \n69 REASONER is evaluated on two challenging logical reasoning benchmarks including ReClor, LogiQA,   \n70 and one dialogue reasoning dataset Mutual for generalizability, achieving new state-of-the-art results. ",
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+        "image_caption": [
+            "Figure 2: Our \"fact\" V.S. existing approaches. "
+        ],
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+        "type": "text",
+        "text": "71 2 Related Work ",
+        "text_level": 1,
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+        "text": "72 Machine Reading Comprehension Recent years have witnessed massive researches on Machine   \n73 Reading Comprehension, which has become one of the most important areas of NLP [14, 15, 16,   \n74 17, 18, 19, 20, 21, 22]. Despite the success of MRC models on various datasets such as CNN/Daily   \n75 Mail [1], SQuAD [2], RACE [3] and so on, researchers began to rethink to what extent does the   \n76 problem been solved. Nowadays, there are massive researches into the reasoning ability of machines.   \n77 According to [23, 24, 25], reasoning abilities can be broadly categorized into (1) commonsense   \n78 reasoning [26, 27, 28, 29]; (2) numerical reasoning [30]; (3) multi-hop reasoning [31] and (4) logical   \n79 reasoning [5, 7], among which logical reasoning is essential in human intelligence but has merely   \n80 been delved into. Natural Language Inference (NLI) [32, 33, 34] is a task closely related to logical   \n81 reasoning. However, it has two obvious drawbacks in measuring logical reasoning abilities. One is   \n82 that it only has three logical types which are entailment, contradiction and neutral. The other is its   \n83 limitation on sentence-level reasoning. Hence, it is important to research more comprehensive and   \n84 deeper logical reasoning abilities.   \n85 Logical Reasoning in MRC There are two   \n86 main kinds of features in language data that   \n87 would be the necessary basis for logical   \n88 reasoning: 1) knowledge: global facts that   \n89 keep consistency regardless of the context,   \n90 such as commonsense, mostly derived from   \n91 named entities; 2) non-knowledge: local facts   \n92 or events that may be sensitive to the context,   \n93 mostly derived from detailed language. Existing   \n94 works have made progress in improving logical   \n95 reasoning ability [8, 9, 10, 11, 12, 38]. However,   \n96 these approaches are barely satisfactory as they   \n97 mostly focus on the global facts such as typical   \n98 entity or sentence-level relations, which are   \n99 obviously not sufficient. In this work, we   \n100 strengthen the basis for logical reasoning by   \n101 unifying both types of the features as \"facts\".   \n102 Different from previous studies that focus on   \n103 the knowledge components, we propose a fact  \n104 driven logical reasoning framework that builds   \n105 supergraphs on top of fact units to capture both ",
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+        "img_path": "images/a9e163928e4ba3e276dd99d23f68db19edfdb59b8eb6053a7b621407621dc4a4.jpg",
+        "image_caption": [
+            "Figure 3: An example of constructed supergraph. In contrast, the dotted vertices and edges are focused in most existing studies [35, 36, 37]. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "06 global connections between entity-aware facts and the local concepts or events inside the fact. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "107 3 Approaches ",
+        "text_level": 1,
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+        "text": "108 In this section, we will describe our method in detail. The overall architecture of the model is shown   \n109 in Figure 4 . We first construct a supergraph from the raw text based on the fact units extracted.   \n110 Then we conduct reasoning over the supergraph with question-option guided approaches to learn and   \n111 update the features, which are further incorporated in answer prediction. ",
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+    {
+        "type": "text",
+        "text": "112 3.1 Supergraph Construction ",
+        "text_level": 1,
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+        "text": "13 Figure 5 illustrates our method for constructing a supergraph from raw text inputs. The first step is   \n14 to obtain triplets that constitute a fact unit. To keep the framework generic, we use a fairly simple   \n15 fact unit extractor based on the syntactic relations. Given a context consisting multiple sentences, we   \n116 first conduct dependency parsing of each sentence. After that, we extract the subject, the predicate,   \n117 and the object tokens to get the \"Argument-Predicate-Argument\" triplets corresponding to   \n18 each sentence in the context.   \n119 With the obtained triplets, the fact units are organized in the form of Levi graph [39], which turns   \n120 arguments and predicates all into nodes. An original fact unit is in the form of $F = ( V , E , R )$ ,   \n121 where $V$ is the set of the arguments, $E$ is the set of edges connected between arguments, and $R$ is   \n122 the relations of each edge which are predicates here. The corresponding Levi graph is denoted as   \n123 $F _ { l } = \\left( V _ { L } , E _ { L } , R _ { L } \\right)$ where $V _ { L } = V \\cup R$ , which makes the originally directly connected arguments   \n124 be intermediately connected via relations. As for $R _ { L }$ , previous works such as [40, 41] designed three   \n125 types of edges $\\dot { R } _ { L } = \\{ d e f a u l t , r e v e r s e , s e l f \\}$ to enhance information flow. Here in our settings,   \n126 we extend it into five types: default-in, default-out, reverse-in, reverse-out, self, corresponding to the   \n127 directions of edges towards the predicates.   \n128 We construct the supergraph by making connections between fact units $F _ { l }$ . In particular, we take   \n129 three strategies according to question-option, identical concept and co-reference information. (1) For   \n130 question-option pair, We initialize a global node $V _ { g }$ with its representation and connect it to all the fact   \n131 unit nodes. The edge type are set as global. The global node ensures that all fact units are connected   \n132 so that information can be exchanged during graph encoding. (2) There can be identical mentions   \n133 in different sentences, resulting in repeated nodes in fact units. We connect nodes corresponding   \n134 to the same non-pronoun arguments by edges with edge type same. (3) We conduct co-reference   \n135 resolution on context using an off-to-shelf model2 in order to identify arguments in fact units that   \n136 refer to the same one. We add edges with type coref between them. The final supergraph is denoted   \n137 as $S = ( F _ { l } \\cup V _ { g } , E )$ where $E$ is the set of edges added with the previous three strategies. ",
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+        "image_caption": [
+            "Figure 4: The framework or our model. For supergraph reasoning, in each iteration, each node selectively receives the message from the neighboring nodes to update its representation. The dashed circle means zero vector. "
+        ],
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+        "type": "text",
+        "text": "3.2 Encoder ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.2.1 Context Encoder ",
+        "text_level": 1,
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+        "text": "140 Our context encoder $F _ { C } ( . )$ is initialized with a pre-trained language model, i.e., RoBERTa-large   \n141 [42]. Question, context and option are concatenated and then fed into the encoder. If the question is   \n142 detected to contain negative meanings, we add a special token <pos> before the question, else we add   \n143 <neg>. In a whole, we get the hidden representation as following: ",
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+    {
+        "type": "text",
+        "text": "$\\{ h _ { c , 0 } , . . . , h _ { c , l _ { c } + 1 } , h _ { q , 1 } , . . . , h _ { o , 1 } , . . . , h _ { o , l _ { o } + 1 } \\} = F _ { C } ( \\{ x _ { c , 0 } , . . . , x _ { c , l _ { c } + 1 } , x _ { q , 0 } , . . . , x _ { o , 1 } , . . . , x _ { o , l _ { o } + 1 } \\} ) ,$ (1) where 44 $x _ { c , 0 } = < s >$ , $x _ { c , l _ { c } + 1 } = x _ { o , l _ { o } + 1 } = < / \\mathrm { s } >$ , $x _ { q , 0 } = < \\mathrm { p o s } > / < \\mathrm { n e g } >$ and $h _ { i } \\in \\mathbb { R } ^ { d }$ , $d$ is the hidden size. ",
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+    {
+        "type": "text",
+        "text": "3.2.2 Supegraph Encoder ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Graph Initialization $F _ { C } ( . )$ encodes each token in nodes $V _ { L }$ , and then the averaged hidden state is used as the initial representation of the original word of each node, because PrLMs like RoBERTa take subwords as input while our triplets extraction performs in word-level. For the global QA-context node, we averaged the embeddings of tokens in question and option for initialization. We also use a one-hot embedding layer to encode the relations between two nodes. ",
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+    {
+        "type": "image",
+        "img_path": "images/97f35831f1809843a8867e5d46260e7f36caa3817533d4bde6170c05f3d14c45.jpg",
+        "image_caption": [
+            "Which one of the following ......, most seriously weakens the argument? Various species of animals from the same era as dinosaurs and similar to them ... did not become extinct when the dinosaurs did. ",
+            "Figure 5: The process of constructing the fact chain and its corresponding Levi graph form of an example in Figure 1. Entities and relations are illustrated in its corresponding color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "151 Graph Attention Network Based on the relational graph convolutional network [43] and given   \n152 the initial representation $h _ { i } ^ { 0 }$ for every node $v _ { i }$ , the feed-forward or the message-passing process with   \n153 information control can be written as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0a435cc11986a10b288ed98e4158b4026cca57eab70a2c520f67b1baa4b616c8.jpg",
+        "text": "$$\nh _ { i } ^ { ( l + 1 ) } = \\mathrm { R e L U } ( \\sum _ { r \\in R _ { L } } \\sum _ { v _ { j } \\in \\mathcal { N } _ { r } ( v _ { i } ) } g _ { q } ^ { ( l ) } \\frac { 1 } { c _ { i , r } } w _ { r } ^ { ( l ) } h _ { j } ^ { ( l ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "154 where $\\mathcal { N } _ { r } ( v _ { i } )$ denotes the neighbors of node $v _ { i }$ under relation $r$ and $c _ { i , r }$ is the number of those nodes.   \n155 $w _ { r } ^ { ( l ) }$ is the learnable parameters of layer $l$ . $g _ { q } ^ { ( l ) }$ is a gated value between 0 and 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Through the graph encoder $F _ { G } ( . )$ , we then obtain the hidden representations of nodes in fact units as: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a27cebb4060e2018d78f516218fc2f9e28b1d9691bf8b723b6d65393b91fe527.jpg",
+        "text": "$$\n\\{ h _ { 0 } ^ { F } , . . . h _ { m } ^ { F } \\} = F _ { G } ( \\{ v _ { L , 0 } , . . . v _ { L , m } \\} , E _ { L } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "158 These features are further concatenated to get the final node representation of the supergraph: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/803ee5fbc30f76fbc1e377d4e44ff238ca02c101a8be8c88ca7754202b52537a.jpg",
+        "text": "$$\n\\{ h _ { 0 } ^ { S } , . . . h _ { m } ^ { S } \\} = F _ { G } ( \\{ h _ { 0 } ^ { F } , . . . h _ { m } ^ { F } \\} , E _ { C } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "159 For node features on the supergraph, it is fused via the attention and gating mechanisms with the   \n160 original representations of the context encoder. Specifically, denoting the original whole sequence   \n161 representation after context encoder as $H ^ { C }$ , we apply attention mechanism to append the supergraph   \n162 representation to the original one: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/59b9ecfcc1f73293cfbc70631079e7bd27779c5dc10a4eed1f149ac1c5f6dfe4.jpg",
+        "text": "$$\n\\tilde { H } = \\mathrm { A t t n } ( H ^ { c } , K _ { f } , V _ { f } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "163 where $\\{ K _ { f } , V _ { f } \\}$ are packed from the learned representations of the supergraph. We compute   \n164 $\\lambda \\in [ 0 , \\bar { 1 } ]$ to weigh the expected importance of supergraph representation of each source word: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/992b9fd53eb5225105f0279728ca8ad3ad5b1dad9ccb777770eb211d270467f8.jpg",
+        "text": "$$\n\\lambda _ { 1 } = \\sigma ( W _ { \\lambda } \\tilde { H } + U _ { \\lambda } H ^ { C } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $W _ { \\lambda }$ and $U _ { \\lambda }$ are learnable parameters. $H ^ { C }$ and $\\tilde { H }$ are then fused for an effective representation: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6a6214c6010547d33147ed11094a812dca0ccd45a1c6da103ae08e4c93ce817.jpg",
+        "text": "$$\nH = H ^ { C } + \\lambda \\tilde { H } \\in \\mathbb { R } ^ { 4 \\times d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.2.3 Question-Option-aware Interaction ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Options have their inherent logical relations, which can be leveraged to aid answer prediction. Inspired by [44], we use an attention-based mechanism to gather option correlation information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically for an option 0 $O _ { i }$ , the information it get by interaction with option $O _ { j }$ is calculated as: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/98e319f0bdbe9ec73dd05fe7b12f4d6b5c8a50bb3bfda31dc8bc26867f846ffa.jpg",
+        "text": "$$\nO _ { i } ^ { ( j ) } = [ O _ { i } ^ { q } - O _ { i } ^ { q } \\mathrm { A t t n } ( O _ { i } ^ { q } , O _ { j } ^ { q } ; v ) ; O _ { i } ^ { q } \\circ O _ { i } ^ { q } \\mathrm { A t t n } ( O _ { i } ^ { q } , O _ { j } ^ { q } ; v ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "171 where $O _ { i } ^ { q }$ is the representation of the concatenation for the $i$ -th option and question after the context   \n172 encoder. Then the option-wise information are gathered to fuse the option correlation information: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2e76594a3a62f0a454bdd86149a11058e70a928a66f8dd068ac1e5130bea7bf.jpg",
+        "text": "$$\n\\hat { O } _ { i } = \\operatorname { t a n h } ( W _ { c } [ O _ { i } ^ { q } ; \\{ O _ { i } ^ { ( j ) } \\} _ { i \\neq j } ] + b _ { c } ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "where 173 $\\mathbf { W } _ { c } \\in \\mathbb { R } ^ { d \\times 7 d }$ and $b _ { c } \\in \\mathbb { R } ^ { d }$ . Finally, a gating mechanism is used to fuse the option features: ",
+        "bbox": [
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+            90,
+            808,
+            107
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae8dbedd3c3142195dc09ff0281574dc926b2a5f8352dec568acc97dd75ae688.jpg",
+        "text": "$$\nO _ { i , : k } ^ { q } = g _ { i , : k } \\circ O _ { i , : k } ^ { q } + ( 1 - g _ { i , : k } ) \\circ \\hat { O } _ { i , : k } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            107,
+            635,
+            127
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the 174 $g _ { i , : k } = \\sigma ( W _ { g } [ O _ { i , : k } ; O _ { i , : k } ^ { \\widehat { q } } ; \\widetilde { Q } ] + b _ { g } ) \\in \\mathbb { R } ^ { d }$ is the $i$ -th column of gate $g$ ",
+        "bbox": [
+            153,
+            131,
+            699,
+            150
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 Hierarchical Decoder ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            164,
+            362,
+            179
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "176 To better incorporate the information obtained above, apart from getting the original pooled context  \n177 attended representation $h ^ { C } \\in \\mathbb { R } ^ { 4 \\times d }$ , we combine the attended vectors ${ \\bf \\bar { \\boldsymbol { O } } } ^ { f }$ and $H$ from the previous   \n178 encoder through a fusing layer. ",
+        "bbox": [
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+            189,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ad9c0996faaa314aa95e0ecc8a68b155c76e74d1da0e5656e1f83d1e614db6e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { E _ { 1 } = \\mathrm { R e L U } ( \\mathrm { F C } ( [ h ^ { C } , H , h ^ { C } - H , h ^ { C } \\circ H ] ) ) , } \\\\ & { E _ { 2 } = \\mathrm { R e L U } ( \\mathrm { F C } ( [ h ^ { C } , H , h ^ { C } - O ^ { f } , h ^ { C } \\circ O ^ { f } ] ) ) , } \\\\ & { P = \\sigma ( \\mathrm { F C } ( [ E _ { 1 } , E _ { 2 } ] ) ) , } \\\\ & { C = P \\circ H + ( 1 - P ) \\circ O ^ { f } \\in \\mathbb { R } ^ { 4 \\times d } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "179 Then another linear layer is applied for final prediction as $z = W _ { z } C + b _ { z } \\in \\mathbb { R } ^ { 4 }$ . We seek to minimize   \n180 the cross entropy loss over the correct decision $l$ by ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9409a91f722aba95edba5ed25d596b8e83a479f4c1578a2695cfd79e170abb1.jpg",
+        "text": "$$\n\\mathcal { L } _ { a n s } = - \\log \\operatorname { s o f t m a x } ( z ) _ { l } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "181 Logical Fact Regularization Inspired by [45], the embedding of the tail argument should be close   \n182 to the embedding of the head argument plus a relation-related vector in the hidden representation   \n183 space. Without loss of generality, we assume that in our settings, the summation of the subject vector   \n184 and the relation vector should be close to the object vector as much as possible, i.e., ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afa0fb444d18c609ab931bdcfaadb1cd6c765621add02ea50dfe38a595fcfe49.jpg",
+        "text": "$$\nv _ { s u b j e c t } + v _ { r e l a t i o n }  v _ { o b j e c t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "185 In order to make the logical facts more of factual correctness, we introduce a regularization for the   \n186 extracted logical facts based on the hidden states of the sequence $h _ { i }$ where $i = 1 , \\ldots , L$ and $L$ is the   \n187 total length of the sequence. The regularization is defined as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cdeff7d491204c26ca907e7b986347428f9fe2f7900436e7a68357cdd0cf95b8.jpg",
+        "text": "$$\nL _ { l f r } = \\sum _ { k = 1 } ^ { m } ( 1 - \\cos ( h _ { s u b _ { k } } + h _ { r e l _ { k } } , h _ { o b j _ { k } } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $m$ is the total number of logical fact triplets extracted from the context as well as the option and $k$ indicates the $k$ -th fact triplet. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Training Objective. During training, the overall loss for answer prediction is: ",
+        "bbox": [
+            178,
+            574,
+            702,
+            589
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4eb8961ac9906c3bca6fb94c1f1a797d0faabe2265b157c7011cf7874396ed8d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } = \\alpha \\mathcal { L } _ { a n s } + \\beta \\mathcal { L } _ { l f r } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha$ and $\\beta$ are two parameters. In our implementation, we set $\\alpha = 1 . 0$ and $\\beta = 0 . 5$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We conducted the experiments on three datasets. Two for specialized logical reasoning ability testing: ReClor [7] and LogiQA [5] and one for logical reasoning in dialogues: MuTual [46]. For more details, one can refer to Appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We fine-tune RoBERTa as the backbone PrLM for FOCAL REASONER. The overall model is end-toend trained and updated by Adam [47] optimizer with an overall learning rate 8e-6 for ReClor and LogiQA, and 4e-6 for MuTual. The weight decay is 0.01. We set the warm-up proportion during training to 0.1. Graph encoders are implemented using DGL, an open-source lib of python. The layer number of the graph encoder is 2 for ReClor and 3 for LogiQA. The maximum sequence length is 256 for LogiQA and MuTual, and 384 for ReClor. The model is trained for 10 epochs with a total batch size 16 and an overall dropout rate 0.1 on 4 NVIDIA Tesla V100 GPUs, which takes around 2 hours for ReClor and 4 hours for LogiQA3. ",
+        "bbox": [
+            173,
+            777,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/96bdb6697c63122be6bf3d9a5c8978e59e9f32d39a6aa07fb5f9294a08bce4b1.jpg",
+        "table_caption": [
+            "Table 1: Experimental results of our model compared with baseline models on ReClor and LogiQA dataset. Test-E and Test-H denote Test-Easy and Test-Hard respectively. We performed Pitman’s permutation test [48] and found that our model significantly outperformed the baseline $_ { ( \\mathrm { p < 0 . 0 5 } ) }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">ReClor</td><td colspan=\"2\">LogiQA</td></tr><tr><td>Dev</td><td>Test</td><td>Test-E</td><td>Test-H</td><td>Dev</td><td>Test</td></tr><tr><td>Human [7]</td><td></td><td>63.00</td><td>57.10</td><td>67.20</td><td></td><td>86.00</td></tr><tr><td>BERT-Large[7]</td><td>53.80</td><td>49.80</td><td>72.00</td><td>32.30</td><td>34.10</td><td>31.03</td></tr><tr><td>XLNet-Large [7]</td><td>62.00</td><td>56.00</td><td>75.70</td><td>40.50</td><td>=</td><td>=</td></tr><tr><td>RoBERTa-Large [7]</td><td>62.60</td><td>55.60</td><td>75.50</td><td>40.00</td><td>35.02</td><td>35.33</td></tr><tr><td>DAGN[10]</td><td>65.20</td><td>58.20</td><td>76.14</td><td>44.11</td><td>35.48</td><td>38.71</td></tr><tr><td>DAGN_(Aug)[10]</td><td>65.80</td><td>58.30</td><td>75.91</td><td>44.46</td><td>36.87</td><td>39.32</td></tr><tr><td>FOCALREASONER</td><td>66.80</td><td>58.90</td><td>77.05</td><td>44.64</td><td>41.01</td><td>40.25</td></tr></table>",
+        "bbox": [
+            176,
+            88,
+            818,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/126f16681ca95b5b7de3e198cebcca24ec6747f79ecddad4783e3c5c0c653520.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Experimental results of our model compared with baseline PrLM on MuTual dataset. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"3\">Model</td><td colspan=\"6\">MuTual</td><td colspan=\"6\">MuTualplus</td></tr><tr><td colspan=\"2\">Dev Set</td><td colspan=\"2\"></td><td colspan=\"2\">Test Set</td><td colspan=\"2\">Dev Set</td><td colspan=\"2\"></td><td colspan=\"2\">Test Set</td></tr><tr><td>R4@1</td><td>R4@2</td><td>MRR R4@1</td><td></td><td>R4@2 MRR</td><td></td><td>R4@1</td><td>R4@2</td><td>MRR</td><td>R4@1</td><td>R4@2 MRR</td><td></td></tr><tr><td>RoBERTabase [46]</td><td>69.5</td><td>87.8</td><td>82.4</td><td>71.3</td><td>89.2</td><td>83.6</td><td>62.2</td><td>85.3</td><td>78.2</td><td>62.6</td><td>86.6</td><td>78.7</td></tr><tr><td>-MC[46]</td><td>69.3</td><td>88.7</td><td>82.5</td><td>68.6</td><td>88.7</td><td>82.2</td><td>62.1</td><td>83.0</td><td>77.8</td><td>64.3</td><td>84.5</td><td>79.2</td></tr><tr><td>FOCAL REASONER</td><td>73.4</td><td>1 90.3</td><td>84.9</td><td>72.7</td><td>91.0</td><td>-84.6</td><td>63.7</td><td>86.1</td><td>79.1</td><td>65.5</td><td>84.3</td><td>-79.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "206 4.3 Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Tables 1 and 2 show the results on ReClor, LogiQA, and MuTual, respectively. All the best results are shown in bold. Based on our implemented baseline models (basically consistent with public results), we observe dramatic improvements on both of the logical reasoning benchmarks, e.g., on ReClor test set, FOCAL REASONER achieves $+ 4 . 2 \\%$ on dev set and $+ 3 . 3 . \\%$ on the test set. FOCAL REASONER also outperforms the prior best system $\\mathrm { D A G N ^ { 4 } }$ , reaching $7 7 . 0 5 \\%$ on the EASY subset, and $4 4 . 6 4 \\%$ on the HARD subset. The performance suggests that FOCAL REASONER makes better use of logical structure inherent in the given context to perform reasoning than existing methods. On the dialogue reasoning dataset MuTual, our model achieves quite a jump compared with the RoBERTa-base $\\mathrm { L } \\mathbf { \\bar { M } } ^ { 5 }$ This verifies our model’s generalizability on other downstream reasoning task settings. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In addition, Table 5 lists the accuracy of our model on the dev set of ReClor of different question types. Results show that our model can perform well on most of the question types, especially \"Strengthen\" and \"Weaken\". This means that our model can well interpret the question type from the question statement and make the correct choice corresponding to the question. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To dive into the effectiveness of different components in FOCAL REASONER, we conduct an ablation study which takes RoBERTa as the backbone on the ReClor dev set. Table 3 summarizes the results. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Supergraph reasoning: The first key component is the supergraph reasoning. We ablate the global atom and erase all the edges connected with it. The results suggest that the global atom indeed betters message propagation, leveraging performance from $6 4 . 6 \\%$ to $6 6 . 8 \\%$ . We also find that replacing the initial QA pair representation of the global atom with only question representation hurts the performance. In addition, without the logical fact regularization, the performance drops from $6 6 . 8 \\%$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/07dc959c4d12f76c1a9f1a395a3d23db7b62a9c08bdfdde53d99e7606e5d23ce.jpg",
+        "image_caption": [
+            "Figure 6: Accuracy of models on number of fact units on dev set of ReClor (left) and LogiQA (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "229 to $6 4 . 2 \\%$ , indicating its usefulness. For edge analysis, when (1) all edges are regarded as a single   \n230 type rather than the original designed 8 types in total and (2) co-reference edges are removed, the   \n231 accuracy drops to $6 3 . 7 \\%$ and $6 4 . { \\bar { 8 } } \\%$ , respectively. It is proved that in our supergraph, edges link the   \n232 fact units in reasonable manners, which properly uncovers the logical structures.   \n233   \n234   \n235   \n236   \n237   \n238   \n239   \n240   \n241   \n242   \n243   \n244   \n245   \n246   \n247   \n248   \n249   \n250   \n251 ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Fact Units Variants Apart from our syntactically constructed fact units, there are another two ways in different granularities for construction. We replace the fact units with named entities which are used in previous works like [49]. The statistics of fact units and named entities of ReClor and LogiQA are stated in Table 4, from which we can infer that there are indeed more fact units than named entities. Thus using fact units can better incorporate the logical information within the context. When replacing all the fact units with named entities, we can see from Table 3 that it significantly decreases the performance. We also explore the performance using semantic role labeling the similar way as in [50]. We can see that SRL, leveraging a much more complex information as well as computation complexity, fails to achieve a performance as good as our original fact unit. ",
+        "bbox": [
+            163,
+            335,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/fca35c809edeb66a2e0050166a2f51043819ad480f0a1c2aa56115d872639efa.jpg",
+        "table_caption": [
+            "Table 3: Ablation results on the dev set of ReClor. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>FOCAL REASONER Supergraph Reasoning</td><td>66.8±0.13</td></tr><tr><td>- global node</td><td>64.6±0.32</td></tr><tr><td>- co-reference</td><td>64.8±0.24</td></tr><tr><td>- logical fact regularization</td><td></td></tr><tr><td>- QA context node → Q node</td><td>64.2±0.12 66.4±0.16</td></tr><tr><td>- question reformulation</td><td></td></tr><tr><td>- edge type</td><td>65.2±0.16 63.7±0.19</td></tr><tr><td>Fact Unit Variants</td><td></td></tr><tr><td> - named entity</td><td>62.8±0.26</td></tr><tr><td>- SRL</td><td>62.2±0.32</td></tr><tr><td>Interactions - interactions</td><td>65.5±0.52</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Interactions: We further experimented with ",
+        "text_level": 1,
+        "bbox": [
+            165,
+            613,
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+            627
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "the query-option-interactions setting to see how it affects the performance. The results suggest that the features learned from the interaction process enhance the model. Considering that the logical relations between different options are a strong indicator of the right answer, this means that the model learns from a comparative reasoning strategy. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Effects of Fact Units Numbers ",
+        "text_level": 1,
+        "bbox": [
+            168,
+            699,
+            423,
+            713
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To inspect the effects of the number of fact units, we split the original dev set of ReClor and LogiQA into 5 subsets. The statistics of the fact unit distribution on the datasets are shown in Table 6. Numbers of fact units for most contexts in ReClor and LogiQA are in [3, 6) and $[ 0 , 3 )$ respectively. ",
+        "bbox": [
+            174,
+            724,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b1811ae8e3d8bacfe39b489a66140f2cb1a4ee39f43988d4ffc83fc82bd9245e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Number</td><td colspan=\"2\">ReClor</td><td colspan=\"2\">LogiQA</td></tr><tr><td>Train</td><td>Dev</td><td>Train</td><td>Dev</td></tr><tr><td>Fact Unit Argument</td><td>14,895</td><td>1,665</td><td>20,676</td><td>1,981</td></tr><tr><td>Named Entity</td><td>9,495</td><td>984</td><td>12,439</td><td>1,515</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparing the accuracies of RoBERTa-large baseline, prior SOTA DAGN and our proposed ",
+        "bbox": [
+            168,
+            828,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Statistics for fact unit entities and traditional named entities in datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "67 FOCAL REASONER in Figure 6, our model outperforms baseline models on all the divided subsets,   \n68 which demonstrates the effectiveness and robustness of our proposed method. Specifically, for ReClor,   \n69 FOCAL REASONER performers better when there are more fact units in the context, while for LogiQA,   \n70 FOCAL REASONER works better when the number of fact units locates in $[ 0 , 3 )$ and [9, 12). The   \n271 reason may lie in the difference in style of the two datasets. However, all the models include ours   \n272 struggle when the number of fact units is above certain thresholds, i.e., the logical structure is more   \n273 complicated, calling for better mechanisms to cope with. ",
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+            "Table 5: Accuracy on the dev set of ReClor corresponding to several representative question types. S: Strengthen, W: Weaken, I: Implication, CMP: Conclusion/Main Point, ER: Explain or Resolve, D: Dispute, R: Role, IF: Identify a Flaw, MS: Match Structures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>S</td><td>W</td><td>I</td><td>CMP</td><td>ER</td><td>P</td><td>D</td><td>R</td><td>IF</td><td>MS</td></tr><tr><td>RoBERTalarge [7]</td><td>61.70</td><td>47.79</td><td>39.13</td><td>63.89</td><td>58.33</td><td>50.77</td><td>50.00</td><td>56.25</td><td>61.54</td><td>56.67</td></tr><tr><td>DAGN[10]</td><td>63.83</td><td>46.02</td><td>39.13</td><td>69.44</td><td>57.14</td><td>5385</td><td>46.67</td><td>62.50</td><td>62.39</td><td>56.67</td></tr><tr><td>FOCAL REASONER</td><td>65.96</td><td>51.33</td><td>43.48</td><td>72.22</td><td>67.86</td><td>53.85</td><td>50.00</td><td>62.50</td><td>62.39</td><td>60.0</td></tr></table>",
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+        "text": "5.3 Interpretability: a Case Study ",
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+        "text": "We aim to interpret FOCAL REASONER’s reasoning process by analyzing the nodeto-node attention weights induced in the supergraph in Figure 7. We can see that our FOCAL REASONER can well bridge the reasoning process between context, question and option. Specifically, in the graph, \"students rank $30 \\%$ \" attends strongly to \"playing improve performance\". Under the guidance of question ",
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+        "table_body": "<table><tr><td>Dataset</td><td>[0,3)</td><td>[3,6)</td><td>[6,9）</td><td>[9,12)</td><td>）[12,00）</td></tr><tr><td>ReClor</td><td>37.2%</td><td>48.6%</td><td>12.6%</td><td>0.6%</td><td>1.2%</td></tr><tr><td>LogiQA</td><td>47.5%</td><td>37.5%</td><td>10.9%</td><td>3.5%</td><td>0.6%</td></tr></table>",
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+        "text": "Table 6: Distribution of fact unit number on dev set of the training datasets. ",
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+        "type": "text",
+        "text": "84 to select the option that weakens the statement and option interaction, our model is able to tell that \"students rank $30 \\%$ can play\" mostly undermines the conclusion that \"playing improves performance\". ",
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+        "text": "A recent survey in a key middle school showed that high school students in this school have a special preference for playing football, and it far surpasses other balls.The survey also found that students who regularly play football are better at academic performance than students who do not often play football.This shows that often playing football can improve students' academic performance. ",
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+        "text": "A. Only high school students who are ranked in the top $30 \\%$ of grades can often play football.   \nB. Regular football can exercise and maintain a strong learning energy.   \nC. Often playing football delays the study time.   \nD. Research has not proved that playing football can contribute to intellectual development. ",
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+        "image_caption": [
+            "Figure 7: An example of how our model reasons to get the final answer. "
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+        "text": "286 6 Conclusion ",
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+        "text": "287 For logical reasoning arising from machine reading comprehension, it is well known that clear and   \n288 accurate forms like global knowledge are crucial. In this work, we make a finding that existing   \n289 studies miss focusing on quite a lot of non-knowledge parts which is also indispensable for better   \n290 reasoning. Thus we propose extracting a general form called \"fact unit\" to cover both global and   \n291 local logical units, hoping to shed light on the basis of structural modeling for logical reasoning.   \n292 Our proposed FOCAL REASONER not only better uncovers the logical structures within the context,   \n293 which can be a general method for other sophisticated reasoning tasks, but also better captures the   \n294 logical interactions between context and options. The experimental results verify the effectiveness of   \n295 our method. ",
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+        "text": "References   \n[1] Karl Moritz Hermann, Tomáš Kocisk ˇ y, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa \\` Suleyman, and Phil Blunsom. Teaching machines to read and comprehend. In Proceedings of the 28th International Conference on Neural Information Processing Systems-Volume 1, pages 1693–1701, 2015.   \n[2] Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: $^ { 1 0 0 , 0 0 0 + }$ questions for machine comprehension of text. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 2383–2392, Austin, Texas, November 2016. Association for Computational Linguistics.   \n[3] Guokun Lai, Qizhe Xie, Hanxiao Liu, Yiming Yang, and Eduard Hovy. RACE: Large-scale ReAding comprehension dataset from examinations. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 785–794, Copenhagen, Denmark, September 2017. Association for Computational Linguistics.   \n[4] Zhuosheng Zhang, Hai Zhao, and Rui Wang. Machine reading comprehension: The role of contextualized language models and beyond. arXiv preprint arXiv:2005.06249, 2020.   \n[5] Jian Liu, Leyang Cui, Hanmeng Liu, Dandan Huang, Yile Wang, and Yue Zhang. Logiqa: A challenge dataset for machine reading comprehension with logical reasoning. In Christian Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 3622–3628. International Joint Conferences on Artificial Intelligence Organization, 7 2020. Main track.   \n[6] Lucja Iwanska. Logical reasoning in natural language: It is all about knowledge. ´ Minds and Machines, 3(4):475–510, 1993.   \n[7] Weihao Yu, Zihang Jiang, Yanfei Dong, and Jiashi Feng. Reclor: A reading comprehension dataset requiring logical reasoning. In International Conference on Learning Representations (ICLR), April 2020.   \n[8] Michihiro Yasunaga, Hongyu Ren, Antoine Bosselut, Percy Liang, and Jure Leskovec. Qa-gnn: Reasoning with language models and knowledge graphs for question answering. In North American Chapter of the Association for Computational Linguistics (NAACL), 2021.   \n[9] Hongyu Ren and Jure Leskovec. Beta embeddings for multi-hop logical reasoning in knowledge graphs. Advances in Neural Information Processing Systems, 33, 2020.   \n[10] Yinya Huang, Meng Fang, Yu Cao, Liwei Wang, and Xiaodan Liang. DAGN: Discourse-aware graph network for logical reasoning. In NAACL, 2021.   \n[11] Siddharth Krishna, Alexander J Summers, and Thomas Wies. Local reasoning for global graph properties. In European Symposium on Programming, pages 308–335. Springer, Cham, 2020.   \n[12] Wanjun Zhong, Siyuan Wang, Duyu Tang, Zenan Xu, Daya Guo, Jiahai Wang, Jian Yin, Ming Zhou, and Nan Duan. AR-LSAT: Investigating Analytical Reasoning of Text. arXiv e-prints, page arXiv:2104.06598, April 2021.   \n[13] Ndapandula Nakashole and Tom Mitchell. Language-aware truth assessment of fact candidates. In Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1009–1019, 2014.   \n[14] Danqi Chen, Jason Bolton, and Christopher D Manning. A thorough examination of the cnn/daily mail reading comprehension task. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2358–2367, 2016.   \n[15] Minjoon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. Bidirectional attention flow for machine comprehension. In ICLR 2017, 2017.   \n[16] Bhuwan Dhingra, Hanxiao Liu, Zhilin Yang, William Cohen, and Ruslan Salakhutdinov. Gatedattention readers for text comprehension. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1832–1846, 2017. ",
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parse/train/Mh1Abj33qI/Mh1Abj33qI.md

@@ -0,0 +1,424 @@
+# DATA-DRIVEN LEARNING OF GEOMETRIC SCATTERING NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Many popular graph neural network (GNN) architectures, which are often considered as the current state of the art, rely on encoding graph structure via smoothness or similarity between neighbors. While this approach performs well on a surprising number of standard benchmarks, the efficacy of such models does not translate consistently to more complex domains, such as graph data in the biochemistry domain. We argue that these more complex domains require priors that encourage learning of longer range features rather than oversmoothed signals of standard GNN architectures. Here, we propose an alternative GNN architecture, based on a relaxation of recently proposed geometric scattering transforms, which consists of a cascade of graph wavelet filters. Our learned geometric scattering (LEGS) architecture adaptively tunes these wavelets and their scales to encourage band-pass features to emerge in learned representations. This results in a simplified GNN with significantly fewer learned parameters compared to competing methods. We demonstrate the predictive performance of our method on several biochemistry graph classification benchmarks, as well as the descriptive quality of its learned features in biochemical graph data exploration tasks. Our results show that the proposed LEGS network matches or outperforms popular GNNs, as well as the original geometric scattering construction, while retaining certain mathematical properties of its handcrafted (nonlearned) design.
+
+# 1 INTRODUCTION
+
+Geometric deep learning has recently emerged as an increasingly prominent branch of machine learning in general, and deep learning in particular (Bronstein et al., 2017). It is based on the observation that many of the impressive achievements of neural networks come in applications where the data has an intrinsic geometric structure which can be used to inform network design and training procedures. For example, in computer vision, convolutional neural networks use the spatial organization of pixels to define convolutional filters that hierarchically aggregate local information at multiple scales that in turn encode shape and texture information in data and task-driven representations. Similarly, in time-series analysis, recurrent neural networks leverage memory mechanisms based on the temporal organization of input data to collect multiresolution information from local subsequences, which can be interpreted geometrically via tools from dynamical systems and spectral analysis. While these examples only leverage Euclidean spatiotemporal structure in data, they exemplify the potential benefits of incorporating information about intrinsic data geometry in neural network design and processing. Indeed, recent advances have further generalized the utilization of geometric information in neural networks design to consider non-Euclidean structures, with particular interest in graphs that represent data geometry, either directly given as input or constructed as an approximation of a data manifold.
+
+At the core of geometric deep learning is the use of graph neural networks (GNNs) in general, and graph convolutional networks (GCNs) in particular, which ensure neuron activations follow the geometric organization of input data by propagating information across graph neighborhoods (Bruna et al., 2014; Defferrard et al., 2016; Kipf & Welling, 2016; Hamilton et al., 2017; Xu et al., 2019; Abu-El-Haija et al., 2019). However, recent work has shown the difficulty in generalizing these methods to more complex structures, identifying common problems and phrasing them in terms of oversmoothing (Li et al., 2018), oversquashing (Alon & Yahav, 2020) or under-reaching (Barcelo´ et al., 2020). Using graph signal processing terminology from Kipf & Welling (2016), these issues can be partly attributed to the limited construction of convolutional filters in many commonly used GCN architectures. Inspired by the filters learned in convolutional neural networks, GCNs consider node features as graph signals and aim to aggregate information from neighboring nodes. For example, Kipf & Welling (2016) presented a typical implementation of a GCN with a cascade of averaging (essentially low pass) filters. We note that more general variations of GCN architectures exist (Defferrard et al., 2016; Hamilton et al., 2017; Xu et al., 2019), which are capable of representing other filters, but as investigated in Alon & Yahav (2020), they too often have difficulty in learning long range connections.
+
+Recently, an alternative approach was presented to provide deep geometric representation learning by generalizing Mallat’s scattering transform (Mallat, 2012), originally proposed to provide a mathematical framework for understanding convolutional neural networks, to graphs (Gao et al., 2019; Gama et al., 2019a; Zou & Lerman, 2019) and manifolds (Perlmutter et al., 2018). Similar to traditional scattering, which can be seen as a convolutional network with nonlearned wavelet filters, geometric scattering is defined as a GNN with handcrafted graph filters, typically constructed as diffusion wavelets over the input graph (Coifman & Maggioni, 2006), which are then cascaded with pointwise absolute-value nonlinearities. This wavelet cascade results in permutation equivariant node features that are typically aggregated via statistical moments over the graph nodes, as explained in detail in Sec. 2, to provide a permutation invariant graph-level representation. The efficacy of geometric scattering features in graph processing tasks was demonstrated in Gao et al. (2019), with both supervised learning and data exploration applications. Moreover, their handcrafted design enables rigorous study of their properties, such as stability to deformations and perturbations, and provides a clear understanding of the information extracted by them, which by design (e.g., the cascaded band-pass filters) goes beyond low frequencies to consider richer notions of regularity (Gama et al., 2019b; Perlmutter et al., 2019).
+
+However, while graph scattering transforms provide effective universal feature extractors, their rigid handcrafted design does not allow for the automatic task-driven representation learning that naturally arises in traditional GNNs. To address this deficiency, recent work has proposed a hybrid scattering-GCN (Min et al., 2020) model for obtaining node-level representations, which ensembles a GCN model with a fixed scattering feature extractor. In Min et al. (2020), integrating channels from both architectures alleviates the well-known oversmoothing problem and outperforms popular GNNs on node classification tasks. Here, we focus on improving the geometric scattering transform by learning, in particular its scales. We focus on whole-graph representations with an emphasis on biochemical molecular graphs, where relatively large diameters and non-planar structures usually limit the effectiveness of traditional GNNs. Instead of the ensemble approach of Min et al. (2020), we propose a native neural network architecture for learned geometric scattering (LEGS), which directly modifies the scattering architecture from Gao et al. (2019); Perlmutter et al. (2019), via relaxations described in Sec. 3, to allow a task-driven adaptation of its wavelet configuration via backpropagation implemented in Sec. 4. We note that other recent graph spectrum-based methods approach the learning of long range connections by approximating the spectrum of the graph with the Lancoz algorithm Liao et al. (2019), or learning in block Krylov subspaces Luan et al. (2019). Such methods are complementary to the work presented here, in that their spectral approximation can also be applied in the computation of geometric scattering when considering very long range scales (e.g., via spectral formulation of graph wavelet filters). However, we find that such approximations are not necessary in the datasets considered here and in other recent work focusing on whole-graph tasks, where direct computation of polynomials of the Laplacian is sufficient.
+
+The resulting learnable geometric scattering network balances the mathematical properties inherited from the scattering transform (as shown in Sec. 3) with the flexibility enabled by adaptive representation learning. The benefits of our construction over standard GNNs, as well as pure geometric scattering, are discussed and demonstrated on graph classification and regression tasks in Sec. 5. In particular, we find that our network maintains the robustness to small training sets present in graph scattering while improving classification on biological graph classification and regression tasks, and we show that in tasks where the graphs have a large diameter relative to their size, learnable scattering features improve performance over competing methods.
+
+# 2 PRELIMINARIES: GEOMETRIC SCATTERING FEATURES
+
+Let $\mathcal { G } = ( V , E , w )$ be a weighted graph with $V : = \{ v _ { 1 } , \ldots , v _ { n } \}$ the set of nodes, $E \subset \{ \{ v _ { i } , v _ { j } \} \in$ $V \times V , i \ne j \}$ the set of (undirected) edges and $w : E \to ( 0 , \infty )$ assigning (positive) edge weights to the graph edges. Note that $w$ can equivalently be considered as a function of $V \times V$ , where we set the weights of non-adjacent node pairs to zero. We define a graph signal as a function $x : V \to \mathbb { R }$ on the nodes of $\mathcal { G }$ and aggregate them in a signal vector $\pmb { x } \in \mathbb { R } ^ { n }$ with the $i ^ { t h }$ entry being $x [ v _ { i } ]$ .
+
+We define the weighted adjacency matrix $W \in \mathbb { R } ^ { n \times n }$ of the graph $\mathcal { G }$ as
+
+$$
+W [ v _ { i } , v _ { j } ] : = { \left\{ \begin{array} { l l } { w ( v _ { i } , v _ { j } ) } & { { \mathrm { i f ~ } } \{ v _ { i } , v _ { j } \} \in E } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } ,
+$$
+
+and the degree matrix $\ b { D } \in \mathbb { R } ^ { n \times n }$ of $\mathcal { G }$ as $D : = \mathrm { d i a g } ( d _ { 1 } , . . . , d _ { n } )$ with $d _ { i } : = \deg ( v _ { i } ) : =$ $\textstyle \sum _ { j = 1 } ^ { n } W [ v _ { i } , v _ { j } ]$ being the degree of the node $v _ { i }$ .
+
+The geometric scattering transform (Gao et al., 2019) relies on a cascade of graph filters constructed from a row stochastic diffusion matrix $P : = { \textstyle { \frac { 1 } { 2 } } } \big ( I _ { n } + W D ^ { - 1 } \big )$ , which corresponds to transition probabilities of a lazy random walk Markov process. The laziness of the process signifies that at each step it has equal probability of either staying at the current node or transitioning to a neighbor, where transition probabilities in the latter case are determined by (normalized) edge weights. Scattering filters are then defined via the graph-wavelet matrices $\Psi _ { j } \in \mathbb { R } ^ { n \times n }$ of scale $j \in  { \mathbb { N } } _ { 0 }$ , as
+
+$$
+\begin{array} { r l } & { \Psi _ { 0 } : = I _ { n } - P , } \\ & { \Psi _ { j } : = { P ^ { 2 ^ { j - 1 } } } - { P ^ { 2 ^ { j } } } = { P ^ { 2 ^ { j - 1 } } } \big ( I _ { n } - { P ^ { 2 ^ { j - 1 } } } \big ) , \quad j \geq 1 . } \end{array}
+$$
+
+These diffusion wavelet operators partition the frequency spectrum into dyadic frequency bands, which are then organized into a full wavelet filter bank $\mathcal { W } _ { J } : = \{ \Psi _ { j } , \Phi _ { J } \} _ { 0 \leq j \leq J }$ , where $\Phi _ { J } : = P ^ { 2 ^ { J } }$ is a pure low-pass filter, similar to the one used in GCNs. It is easy to verify that the resulting wavelet transform is invertible, since a simple sum of filter matrices in $\mathcal { W } _ { J }$ yields the identity. Moreover, as discussed in Perlmutter et al. (2019), this filter bank forms a nonexpansive frame, which provides energy preservation guarantees as well as stability to perturbations, and can be generalized to a wider family of constructions that encompasses the variations of scattering transforms on graphs from Gama et al. $( 2 0 1 9 \mathrm { a } ; \mathrm { b } )$ and Zou & Lerman (2019).
+
+Given the wavelet filter bank $\mathcal { W } _ { J }$ , node-level scattering features are computed by stacking cascades of bandpass filters and element-wise absolute value nonlinearities to form
+
+$$
+U _ { p } \pmb { x } : = \Psi _ { j _ { m } } | \Psi _ { j _ { m - 1 } } \cdot . . . | \Psi _ { j _ { 2 } } | \Psi _ { j _ { 1 } } \pmb { x } | | . . . | ,
+$$
+
+indexed (or parametrized) by the scattering path $p : = ( j _ { 1 } , \ldots , j _ { m } ) \in \cup _ { m \in \mathbb { N } } \mathbb { N } _ { 0 } ^ { m }$ that determines the filter scales captured by each scattering coefficient. Then, a whole-graph scattering representation is obtained by aggregating together node-level features via statistical moments over the nodes of the graph (Gao et al., 2019). This construction yields the geometric scattering features
+
+$$
+S _ { p , q } { \pmb x } : = \sum _ { i = 1 } ^ { n } | U _ { p } { \pmb x } [ v _ { i } ] | ^ { q } .
+$$
+
+indexed by the scattering path $p$ and moment order $q$ . Finally, we note that it can be shown that the graph-level scattering transform $S _ { p , q }$ guarantees node-permutation invariance, while $U _ { p }$ is permutation equivariant (Perlmutter et al., 2019; Gao et al., 2019).
+
+# 3 RELAXED GEOMETRIC SCATTERING CONSTRUCTION TO ALLOW TRAINING
+
+The geometric scattering construction, described in Sec. 2, can be seen as a particular GNN with handcrafted layers, rather than learned ones. This provides a solid mathematical framework for understanding the encoding of geometric information in GNNs, as shown in Perlmutter et al. (2019), while also providing effective unsupervised graph representation learning for data exploration, which also has some advantages even in supervised learning task, as shown in Gao et al. (2019). While the handcrafted design in Perlmutter et al. (2019); Gao et al. (2019) is not a priori amenable to task-driven tuning provided by end-to-end GNN training, we note that the cascade in Eq. 3 does conform to a neural network architecture suitable for backpropagation. Therefore, in this section, we show how and under what conditions a relaxation of the laziness of the random walk and the selection of the scales preserves some of the useful mathematical properties established in Perlmutter et al. (2019). We then establish in section 5 the empirical benefits of learning the diffusion scales over a purely handcrafted design.
+
+We first note that the construction of the diffusion matrix $_ { P }$ that forms the lowpass filter used in the fixed scattering construction can be relaxed to encode adaptive laziness by setting $P _ { \alpha } : =$ $\alpha pmb { I } _ { n } + ( 1 - \alpha ) \pmb { W } \pmb { D } ^ { - \top }$ . Where $\alpha \in [ 1 / 2 , 1 )$ controls the reluctance of the random walk to transition from one node to another. $\alpha = 1 / 2$ gives an equal probability to stay in the same node as to transition to one of its neighbors. At this point, we note that one difference between the diffusion lowpass filter here and the one typically used in GCN and its variation is the symmetrization applied in Kipf & Welling (2016). However, Perlmutter et al. (2019) established that for the original construction, this is only a technical difference since $_ { P }$ can be regarded as self-adjoint under an appropriate measure which encodes degree variations in the graph. This is then used to generate a Hilbert space $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ of graph signals with inner product $\langle \pmb { x } , \pmb { y } \rangle _ { D ^ { - 1 / 2 } } : = \langle D ^ { - 1 / 2 } \pmb { x } , D ^ { - 1 / 2 } \pmb { y } \rangle$ . The following lemma shows that a similar property is retained for our adaptive lowpass filter $P _ { \alpha }$ .
+
+Lemma 1. The matrix $P _ { \alpha }$ is self-adjoint on the Hilbert space $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ from Perlmutter et al.   
+(2019).
+
+We note that the self-adjointness shown here is interesting, as it links models that use symmetric and asymmetric versions of the Laplacian or adjacency matrix. Namely, Lemma 1 shows that the diffusion matrix $_ { r }$ (which is column normalized but not row normalized) is self-adjoint, as an operator, and can thus be considered as “symmetric” in a suitable inner product space, thus establishing a theoretical link between these design choices.
+
+As a second relaxation, we propose to replace the handcrafted dyadic scales in Eq. 1 with an adaptive monotonic sequence of integer diffusion time scales tuned via training. Then, an adaptive filter bank is co $0 < t _ { 1 } < \cdots < t _ { J }$ $\mathcal { W } _ { J } ^ { \prime } : = \{ \boldsymbol { \Psi } _ { j } ^ { \prime } , \boldsymbol { \Phi } _ { J } ^ { \prime } \} _ { j = 0 } ^ { J - 1 }$ selected or, with
+
+$$
+\begin{array} { r l } & { \Phi _ { J } ^ { \prime } : = P _ { \alpha } ^ { t _ { J } } , } \\ & { \Psi _ { 0 } ^ { \prime } : = I _ { n } - P _ { \alpha } ^ { t _ { 1 } } , } \\ & { \Psi _ { j } ^ { \prime } : = P _ { \alpha } ^ { t _ { j } } - P _ { \alpha } ^ { t _ { j + 1 } } , \quad 1 \leq j \leq J - 1 . } \end{array}
+$$
+
+The following theorem shows that for any selection of scales, the relaxed construction of $\mathcal { W } _ { J } ^ { \prime }$ constructs a nonexpansive frame, similar to the result from Perlmutter et al. (2019) shown for the original handcrafted construction.
+
+Theorem 1. There exist a constant $C > 0$ that only depends on $t _ { 1 }$ and $t _ { J }$ such that for all ${ \textbf { \em x } } \in$ $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ ,
+
+$$
+C \| \pmb { x } \| _ { { D ^ { - 1 / 2 } } } ^ { 2 } \leqslant \| \pmb { \Phi } _ { J } ^ { \prime } \pmb { x } \| _ { { D ^ { - 1 / 2 } } } ^ { 2 } + \sum _ { j = 0 } ^ { J } \| \pmb { \Psi } _ { j } ^ { \prime } \pmb { x } \| _ { { D ^ { - 1 / 2 } } } ^ { 2 } \leqslant \| \pmb { x } \| _ { { D ^ { - 1 / 2 } } } ^ { 2 } ,
+$$
+
+where the norm considered here is the one induced by the space $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$
+
+Intuitively, the upper (i.e., nonexpansive) frame bound implies stability in the sense that small perturbations in the input graph signal will only result in small perturbations in the representation extracted by the constructed filter bank. Further, the lower frame bound ensures certain energy preservation by the constructed filter bank, thus indicating the nonexpansiveness is not implemented in a trivial fashion (e.g., by constant features independent of input signal).
+
+In the next section we leverage the two relaxations described here to design a neural network architecture for learning the configuration $\alpha , t _ { 1 } , \ldots , t _ { J }$ of this relaxed construction via backpropagation through the resulting scattering filter cascade. The following theorem establishes that for any such configuration, extracted from $\mathcal { W } _ { J } ^ { \prime }$ via Eqs. 2-3, is permutation equivariant at the node-level and permutation invariant at the graph level. This guarantees that the extracted (in this case learned) features indeed encode intrinsic graph geometry rather than a priori indexation.
+
+Theorem 2. Let $U _ { p } ^ { \prime }$ and $S _ { p , q } ^ { \prime }$ be defined as in Eq. 2 and 3 (correspondingly), with the filters from $\mathcal { W } _ { J } ^ { \prime }$ with an arbitrary configuration $0 < \alpha < 1$ , $0 < t _ { 1 } < \cdots < t _ { J }$ . Then, for any permutation Π over the nodes of $\mathcal { G }$ , and any graph signal $\pmb { x } \in L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$
+
+$$
+\begin{array} { r } { U _ { p } ^ { \prime } \Pi \boldsymbol { x } = \Pi U _ { p } ^ { \prime } \boldsymbol { x } \quad a n d \quad S _ { p , q } ^ { \prime } \Pi \boldsymbol { x } = S _ { p , q } ^ { \prime } \boldsymbol { x } \qquad p \in \cup _ { m \in \mathbb { N } } \mathbb { N } _ { 0 } ^ { m } , q \in \mathbb { N } } \end{array}
+$$
+
+where geometric scattering implicitly considers here the node ordering supporting its input signal.
+
+We note that the results in Lemma 1 and Theorems 1-2, as well as their proofs, closely follow the theoretical framework proposed by Perlmutter et al. (2019). We carefully account here for the relaxed learned configuration, which replaces the originally handcrafted configuration there. For completeness, the adjusted proofs appear in Sec. A of the Appendix.
+
+![](images/52268c36b1ad9e135d2563831d93b42d80f0700bf196ad580baa8267255d7a8d.jpg)  
+Figure 1: LEGSNet learns to select the appropriate scattering scales from the data.
+
+# 4 LEARNABLE GEOMETRIC SCATTERING NETWORK ARCHITECTURE
+
+In order to implement the relaxed geometric scattering construction (Sec. 3) via a trainable neural network, throughout this section, we consider an input graph signal $\ b { x } \in \mathbb { R } ^ { n }$ or, equivalently, a collection of graph signals $\pmb { X } \in \mathbb { R } ^ { n \times N _ { \ell - 1 } }$ . The propagation of these signals can be divided into three major modules. First, a diffusion module implements the Markov process that forms the basis of the filter bank and transform, while allowing learning of the laziness parameter $\alpha$ . Then, a scattering module implements the filters and the corresponding cascade, while allowing the learning of the scales $t _ { 1 } , \ldots , t _ { J }$ . Finally, the aggregation module collects the extracted features to provide a graph and produces the task-dependent output.
+
+Building a diffusion process. We build a set of $m \in \mathbb { N }$ subsequent diffusion steps of the signal $_ { \textbf { \em x } }$ by iteratively multiplying the diffusion matrix $P _ { \alpha }$ to the left of the signal, resulting in
+
+$$
+\left[ P _ { \alpha } x , P _ { \alpha } ^ { 2 } x , P _ { \alpha } ^ { 3 } x , . . . , P _ { \alpha } ^ { m } x \right] ,
+$$
+
+Since $P _ { \alpha }$ is often sparse, for efficiency reasons these filter responses are implemented via an RNN structure consisting of $m$ RNN modules. Each module propagates the incoming hidden state $h _ { t - 1 } , t = 1 , \ldots , m$ with $P _ { \alpha }$ with the readout $\mathbf { } _ { o _ { t } }$ equal to the produced hidden state,
+
+$$
+h _ { t } : = P _ { \alpha } h _ { t - 1 } , \quad o _ { t } : = h _ { t } .
+$$
+
+Our architecture and theory enable the implementation of either trainable or nontrainable $\alpha$ , which we believe will be useful for future work as indicated, for example, in Gao & Ji (2019). However, in the applications considered here (see Sec. 5), we find that training $\alpha$ made training unstable and did not improve performance. Therefore, for simplicity, we leave it fixed as $\alpha = 1 / 2$ for the remainder of this work. In this case, the RNN portion of the network contains no trainable parameters, thus speeding up the computation, but still enables a convenient gradient flow back to the model input.
+
+Learning diffusion filter bank. Next, we consider the selection of $J \le m$ diffusion scales for the relaxed filter bank construction with the wavelets defined according to Eq. 5. We found this was the most influential part of the architecture. We experimented with methods of increasing flexibility:
+
+1. Selection of $\{ t _ { j } \} _ { j = 1 } ^ { J - 1 }$ as dyadic scales (as in Sec. 2 and Eq. 1), fixed for all datasets (LEGSFIXED),   
+2. Selection of each $t _ { j }$ using softmax and sorting by $j$ , learnable per model (LEGS-FCN and LEGS-RBF, depending on output layer explained below).
+
+For the softmax selection, we use a selection matrix $\pmb { F } \in \mathbb { R } ^ { J \times m }$ , where each row $F _ { ( j , \cdot ) } , j \ =$ $1 , \ldots , J$ is dedicated to identifying the diffusion scale of the wavelet $P _ { \alpha } ^ { t _ { j } }$ via a one-hot encoding. This is achieved by setting
+
+$$
+\boldsymbol F : = \mathrm { s o f t m a x } ( \boldsymbol \Theta ) = [ \mathrm { s o f t m a x } ( \pmb \theta _ { 1 } ) , \mathrm { s o f t m a x } ( \pmb \theta _ { 2 } ) , \dots , \mathrm { s o f t m a x } ( \pmb \theta _ { J } ) ] ^ { T }
+$$
+
+where $\pmb { \theta } _ { j } \in \mathbb { R } ^ { m }$ constitute the rows of the trainable weight matrix $\Theta$ . While this construction may not strictly guarantee an exact one-hot encoding, we assume that the softmax activations yield a sufficient approximation. Further, without loss of generality, we assume that the rows of $\pmb { F }$ are ordered according to the position of the leading “one” activated in every row. In practice, this can be easily enforced by reordering the rows. We now construct the filter bank with the filters $\widetilde { \mathcal { W } } _ { F } : = \{ \widetilde { \Psi } _ { j } , \widetilde { \Phi } _ { J } \} _ { j = 0 } ^ { J - 1 }$
+
+$$
+\begin{array} { r l r l } & { \widetilde { \Phi } _ { J } \pmb { x } = \sum _ { t = 1 } ^ { m } F _ { ( J , t ) } { \pmb { P } } _ { \alpha } ^ { t } \pmb { x } , } \\ & { \widetilde { \Psi } _ { 0 } \pmb { x } = { \pmb { I } } _ { n } - \sum _ { t = 1 } ^ { m } F _ { ( 1 , t ) } { \pmb { P } } _ { \alpha } ^ { t } \pmb { x } } \\ & { \widetilde { \Psi } _ { j } \pmb { x } = \sum _ { t = 1 } ^ { m } \left[ F _ { ( j , t ) } { \pmb { P } } _ { \alpha } ^ { t } \pmb { x } - F _ { j + 1 , t } { \pmb { P } } _ { \alpha } ^ { t } \pmb { x } \right] } & & { 1 \leq j \leq J - 1 } \end{array}
+$$
+
+matching and implementing the construction of $\mathcal { W } _ { J } ^ { \prime }$ from Eq. 4.
+
+Aggregating and classifying scattering features. While many approaches may be applied to aggregate node-level features into graph-level features such as max, mean, sum pooling, and the more powerful TopK (Gao & Ji, 2019) or attention pooling (Velickovi ˇ c et al., 2018), we follow the ´ statistical-moment aggregation explained in Secs. 2-3 (motivated by Gao et al., 2019; Perlmutter et al., 2019) and leave exploration of other pooling methods to future work. As shown in Gao et al. (2019) on graph classification, this aggregation works particularly well in conjunction with support vector machines (SVMs) based on the radial basis function (RBF) kernel.
+
+Here, we consider two configurations for the task-dependent output layer of the network, either using a small neural network with two fully connected layers, which we denote LEGS-FCN, or using a modified RBF network (Broomhead & Lowe, 1988), which we denote LEGS-RBF, to produce the final classification. The latter configuration more accurately processes scattering features as shown in Table 2. Our RBF network works by first initializing a fixed number of movable anchor points. Then, for every point, new features are calculated based on the radial distances to these anchor points. In previous work on radial basis networks these anchor points were initialized independent of the data. We found that this led to training issues if the range of the data was not similar to the initialization of the centers. Instead, we first use a batch normalization layer to constrain the scale of the features and then pick anchors randomly from the initial features of the first pass through our data. This gives an RBF-kernel network with anchors that are always in the range of the data. Our RBF layer is then $\mathrm { R B F } ( { \pmb x } ) = \phi ( \| \mathrm { B a t c h N o r m } ( { \pmb x } ) - { \pmb c } \| )$ with $\phi ( \pmb { x } ) = e ^ { - \| \pmb { x } \| ^ { 2 } }$ .
+
+# 5 EMPIRICAL RESULTS
+
+Here we show results of LEGSNet on whole graph classification and graph regression tasks, that arise in a variety of contexts, with emphasis on the more complex biochemic datasets. We use biochemical graph datasets as they represent a new challenge in the field of graph learning. Unlike
+
+Table 1: Dataset statistics, diameter, nodes, edges, and clustering coefficient averaged over graphs.   
+
+<table><tr><td></td><td># Graphs</td><td># Classes</td><td>Diameter</td><td>Nodes</td><td>Edges</td><td>Clust. Coeff</td></tr><tr><td>DD</td><td>1178</td><td>2</td><td>19.81</td><td>284.32</td><td>715.66</td><td>0.48</td></tr><tr><td>ENZYMES</td><td>600</td><td>6</td><td>10.92</td><td>32.63</td><td>62.14</td><td>0.45</td></tr><tr><td>MUTAG</td><td>188</td><td>2</td><td>8.22</td><td>17.93</td><td>19.79</td><td>0.00</td></tr><tr><td>NCI1</td><td>4110</td><td>2</td><td>13.33</td><td>29.87</td><td>32.30</td><td>0.00</td></tr><tr><td>NCI109</td><td>4127</td><td></td><td>13.14</td><td>29.68</td><td>32.13</td><td>0.00</td></tr><tr><td>PROTEINS</td><td>1113</td><td>２2</td><td>11.62</td><td>39.06</td><td>72.82</td><td>0.51</td></tr><tr><td>PTC</td><td>344</td><td>2</td><td>7.52</td><td>14.29</td><td>14.69</td><td>0.01</td></tr></table>
+
+other types of data, these datasets do not exhibit the small-world structure of social datasets and may have large graph diameters for their size. Further, the connectivity patterns of biomolecules are very irregular due to 3D folding and long range connections, and thus ordinary local node aggregation methods may miss such connectivity differences.
+
+# 5.1 WHOLE GRAPH CLASSIFICATION
+
+We perform whole graph classification by using eccentricity and clustering coefficient as node features as is done in Gao et al. (2019). We compare against graph convolutional networks (GCN) (Kipf & Welling, 2016), GraphSAGE (Hamilton et al., 2017), graph attention network (GAT) (Velickovi ˇ c´ et al., 2018), graph isomorphism network (GIN) (Xu et al., 2019), Snowball network (Luan et al., 2019), and fixed geometric scattering with a support vector machine classifier (GS-SVM) as in Gao et al. (2019), and a baseline which is a 2-layer neural network on the features averaged across nodes (disregarding graph structure). These comparisons are meant to inform when including learnable graph scattering features are helpful in extracting whole graph features. Specifically, we are interested in the types of graph datasets where existing graph neural network performance can be improved upon with scattering features. We evaluate these methods across 7 benchmark biochemical datasets: DD, ENZYMES, MUTAG, NCI1, NCI109, PROTEINS, and PTC where the goal is to classify between two or more classes of compounds with hundreds to thousands of graphs and tens to hundreds of nodes (See Table 1). For completeness we also show results on six social network datasets in Table S2. For more specific information on individual datasets see Appendix B. We use 10-fold cross validation on all models which is elaborated on in Appendix C. For an ensembling comparison to Scattering-GCN (Min et al., 2020) see Appendix D.
+
+Table 2: Mean $\pm$ standard deviation test set accuracy on biochemical datasets. Time limit expired (TLE) individual denotes models that did not finish in 10 hours.   
+
+<table><tr><td></td><td>DD</td><td>ENZYMES</td><td>MUTAG</td><td>NCI1</td><td>NCI109</td><td>PROTEINS</td><td>PTC</td></tr><tr><td>LEGS-RBF</td><td>72.58 ± 3.35</td><td>36.33 ± 4.50</td><td>33.51 ± 4.34</td><td>74.26 ± 1.53</td><td>72.47 ± 2.11</td><td>70.89 ± 3.91</td><td>57.26± 5.54</td></tr><tr><td>LEGS-FCN</td><td>72.07 ± 2.37</td><td>38.50 ± 8.18</td><td>82.98 ± 9.85</td><td>70.83 ± 2.65</td><td>70.17 ± 1.46</td><td>71.06 ± 3.17</td><td>56.92 ± 9.36</td></tr><tr><td>LEGS-FIXED</td><td>69.09 ± 4.82</td><td>32.33 ±5.04</td><td>81.84 ±11.24</td><td>71.24 ± 1.63</td><td>69.25 ± 1.75</td><td>67.30 ± 2.94</td><td>54.31 ±6.92</td></tr><tr><td>GCN</td><td>67.82 ± 3.81</td><td>31.33 ± 6.89</td><td>79.30 ± 9.66</td><td>60.80 ± 4.26</td><td>61.30 ± 2.99</td><td>74.03 ± 3.20</td><td>56.34 ± 10.29</td></tr><tr><td>GraphSAGE</td><td>66.37 ± 4.45</td><td>15.83 ± 9.10</td><td>81.43 ± 11.64</td><td>57.54 ± 3.33</td><td>55.15 ± 2.58</td><td>71.87 ± 3.50</td><td>55.22 ±9.13</td></tr><tr><td>GAT</td><td>68.50 ± 3.62</td><td>25.83 ± 4.73</td><td>79.85 ± 9.44</td><td>62.19 ± 2.18</td><td>61.28 ± 2.24</td><td>73.22 ± 3.55</td><td>55.50 ± 6.90</td></tr><tr><td>GIN</td><td>42.37 ± 4.32</td><td>36.83 ± 4.81</td><td>83.57 ± 9.68</td><td>66.67 ± 2.90</td><td>65.23 ± 1.82</td><td>75.02 ± 4.55</td><td>55.82 ± 8.07</td></tr><tr><td>Snowball</td><td>TLE</td><td>18.00 ± 1.89</td><td>50.56 ± 20.87</td><td>48.56 ± 2.92</td><td>50.86 ± 2.65</td><td>39.36± 4.29</td><td>50.84 ±9.32</td></tr><tr><td>GS-SVM</td><td>72.66± 4.94</td><td>27.33 ±5.10</td><td>85.09 ± 7.44</td><td>69.68 ± 2.38</td><td>68.55 ± 2.06</td><td>70.98 ± 2.67</td><td>56.96 ± 7.09</td></tr><tr><td>Baseline</td><td>75.98 ± 2.81</td><td>20.50± 5.99</td><td>79.80 ± 9.92</td><td>56.69 ± 3.07</td><td>57.38 ± 2.20</td><td>73.22 ± 3.76</td><td>56.71 ± 5.54</td></tr></table>
+
+LEGS outperforms on biological datasets. A somewhat less explored domain for GNNs is in biochemical graphs that represent molecules and tend to be overall smaller and less connected (see Tables 1 and S1) than social networks. In particular we find that LEGSNet outperforms other methods by a significant margin on biochemical datasets with relatively small but high diameter graphs (NCI1, NCI109, ENZYMES, PTC), as shown in Table 2. On extremely small graphs we find that GS-SVM performs best, which is expected as other methods with more parameters can easily overfit the data. We reason that the performance increases exhibited by LEGSNet, and to a lesser extent GS-SVM, on these chemical and biological benchmarks is due the ability of geometric scattering to compute complex connectivity features via its multiscale diffusion wavelets. Thus, methods that rely on a scattering construction would in general perform better, with the flexibility and trainability LEGSNet giving it an edge on most tasks.
+
+LEGS performs consistently on social network datasets. On the social network datasets LEGSNet performs consistently well, although its benefits here are not as clear as in the biochemical datasets. Ignoring the fixed scattering transform GS-SVM, which was tuned in Gao et al. (2019) with a focus on these particular social network datasets, a version of LEGSNet is best on three out of the six social datasets and second best on the other three. Since the advantages are clearer in the biochemical domain, we focus on this in the remainder of this section. However, for completeness, we provide results on social network datasets in Table S2, and leave further discussion to Appendix B.1.
+
+LEGS preserves enzyme exchange preferences while increasing performance. One advantage of geometric scattering over other graph embedding techniques lies in the rich information present within the scattering feature space. This was demonstrated in Gao et al. (2019) where it was shown that the embeddings created through fixed geometric scattering can be used to accurately infer inter-graph relationships. Scattering features of enzyme graphs within the ENZYMES dataset (Borgwardt et al., 2005) possessed sufficient
+
+global information to recreate the enzyme class exchange preferences observed empirically by Cuesta et al. (2015), using only linear methods of analysis, and despite working with a much smaller and artificially balanced dataset. We demonstrate here that LEGSNet retains similar descriptive capabilities, as shown in Figure 2 via chord diagrams where each exchange preference between enzyme classes (estimated as suggested in Gao et al., 2019) is represented as a ribbon of the corresponding size. Our results here (and in Table S5, which provides complementary quantitative comparison) show that, with relaxations on the scattering parameters, LEGS-FCN achieves better classification accuracy than both LEGS-FIXED and GCN (see Table 1) while also retaining a more descriptive embedding that maintains the global structure of relations between enzyme classes. We ran two varieties of LEGSNet on the ENZYMES dataset: LEGS-FIXED and LEGSFCN, which allows the diffusion scales to be learned. For comparison, we also ran a standard GCN whose graph embeddings were obtained via mean pooling. To infer enzyme ex
+
+![](images/1fad2b87854fbb8733ddca9964b1b221db129b1e5d15dc972a607f72661e76b7.jpg)  
+Figure 2: Enzyme class exchange preferences empirically observed in Cuesta et al. (2015), and estimated from LEGS and GCN embeddings.
+
+change preferences from their embeddings, we followed Gao et al. (2019) in defining the distance from an enzyme $e$ to the enzyme class $\mathrm { E C } _ { j }$ as $\mathrm { d i s t } ( e , \mathrm { E C } _ { j } ) : = \| v _ { e } - \mathrm { p r o j } _ { C _ { j } } ( v _ { e } ) \|$ , where $v _ { i }$ is the embedding of $e$ , and $C _ { j }$ is the PCA subspace of the enzyme feature vectors within $\mathrm { E C } _ { j }$ . The distance between the enzyme classes $\operatorname { E C } _ { i }$ and $\mathrm { E C } _ { j }$ is the average of the individual distances, mean $\{ \mathrm { d i s t } ( e , \mathbf { E C } _ { j } ) : e \in \mathbf { E C } _ { i } \}$ . From here, the affinity between two enzyme classes is computed as pref $\begin{array} { r } { ( \mathrm { E C } _ { i } , \mathrm { E C } _ { j } ) = w _ { i } / \operatorname* { m i n } ( \frac { D _ { i , i } } { D _ { i , j } } , \frac { D _ { j , j } } { D _ { j , i } } ) , } \end{array}$ , where $w _ { i }$ is the percentage of enzymes in class $i$ which are closer to another class than their own, and $D _ { i , j }$ is the distance between $\mathrm { E C } _ { i }$ and $\mathrm { E C } _ { j }$ .
+
+Robustness to reduced training set size. We remark that similar to the robustness shown in (Gao et al., 2019) for handcrafted scattering, LEGSNet is able to maintain accuracy even when the training set size is shrunk to as low as $20 \%$ of the dataset, with a median decrease of $4 . 7 \%$ accuracy as when $80 \%$ of the data is used for training, as discussed in the supplement (see Table S3).
+
+# 5.2 GRAPH REGRESSION
+
+We next evaluate learnable scattering on two graph regression tasks, the QM9 (Gilmer et al., 2017; Wu et al., 2018) graph regression dataset, and a new task from the critical assessment of structure prediction (CASP) challenge (Moult et al., 2018). On the CASP task, the main objective is to score protein structure prediction/simulation models in terms of the discrepancy between their predicted structure and the actual structure of the protein (which is known a priori). The accuracy of such 3D structure predictions are evaluated using a variety of met
+
+Table 3: Train and test set mean squared error on CASP GDT regression task over three seeds.   
+
+<table><tr><td>(μ±σ)</td><td>Train MSE</td><td>Test MSE</td></tr><tr><td>LEGS-FCN</td><td>134.34 ± 8.62</td><td>144.14 ± 15.48</td></tr><tr><td>LEGS-RBF</td><td>140.46 ± 9.76</td><td>152.59 ± 14.56</td></tr><tr><td>LEGS-FIXED</td><td>136.84 ± 15.57</td><td>160.03 ± 1.81</td></tr><tr><td>GCN</td><td>289.33 ± 15.75</td><td>303.52 ± 18.90</td></tr><tr><td>GraphSAGE</td><td>221.14 ± 42.56</td><td>219.44 ± 34.84</td></tr><tr><td>GIN</td><td>221.14 ± 42.56</td><td>219.44 ± 34.84</td></tr><tr><td>Baseline</td><td>393.78 ± 4.02</td><td>402.21 ± 21.45</td></tr></table>
+
+rics, but we focus on the global distance test (GDT) score (Modi et al., 2016). The GDT score measures the similarity between tertiary structures of two proteins with amino-acid correspondence. A higher score means two structures are more similar. For a set of predicted 3D structures for a protein, we would like to score their quality as quantified by the GDT score.
+
+For this task we use the CASP12 dataset (Moult et al., 2018) and preprocess the data similarly to Ingraham et al. (2019), creating a KNN graph between proteins based on the 3D coordinates of each amino acid. From this KNN graph we regress against the GDT score. We evaluate on 12 proteins from the CASP12 dataset and choose random (but consistent) splits with $80 \%$ train, $10 \%$ validation, and $10 \%$ test data out of 4000 total structures. We are only concerned with structure similarity so use no non-structural node features.
+
+LEGSNet outperforms on all CASP targets Across all CASP targets we find that LEGSNet significantly outperforms GNN and baseline methods (See Table S4). This performance improvement is particularly stark on the easiest structures (measured by average GDT) but is consistent across all structures. In Figure 3 we show the relationship between percent improvement of LEGSNet over the GCN model and the average GDT score across the target structures. We draw attention to target t0879, where LEGSNet shows the greatest improvement over other methods. This target has long range dependencies (Ovchinnikov et al., 2018) as it exhibits metal coupling (Li et al., 2015)
+
+![](images/58bb713364518dbaba815b43a47322e0d1242e135b7d1231ccb7146066330250.jpg)  
+Figure 3: CASP dataset LEGS-FCN $\%$ improvement over GCN in MSE of GDT prediction vs. Average GDT score.
+
+creating long range connections over the sequence. Since other methods are unable to model these long range connections LEGSNet is particularly important on these more difficult to model targets.
+
+LEGSNet outperforms on the QM9 dataset We evaluate the performance of LEGSNet on the quantum chemistry dataset QM9 (Gilmer et al., 2017; Wu et al., 2018), which consists of 130,000 molecules with ${ \sim } 1 8$ nodes per molecule. We use the node features from Gilmer et al. (2017), with the addition of eccentricity and clustering coefficient features, and ignore the edge features. We whiten all targets to have zero mean and unit standard deviation. We train each network against all 19 targets and evaluate the mean squared error on the test set with mean and std. over four runs. We find that learning the scales improves the overall MSE, and particularly improves the results over difficult targets (see Table 4 for overall results and Table S7 for results by target). Indeed, on more difficult targets (i.e., those with large test error) LEGS-FCN is able to perform better, where on easy targets GIN is the best. Overall, scattering features offer a robust signal over many targets, and while perhaps less flexible (by construction), they achieve good average performance with significantly fewer parameters.
+
+Table 4: Mean $\pm$ std. over four runs of mean squared error over 19 targets for the QM9 dataset, lower is better.   
+
+<table><tr><td>(μ±σ)</td><td>Test MSE</td></tr><tr><td>LEGS-FCN</td><td>0.216 ± 0.009</td></tr><tr><td>LEGS-FIXED</td><td>0.228 ± 0.019</td></tr><tr><td>GraphSAGE</td><td>0.524 ± 0.224</td></tr><tr><td>GCN GIN</td><td>0.417 ± 0.061</td></tr><tr><td>Baseline</td><td>0.247 ± 0.037 0.533 ± 0.041</td></tr></table>
+
+# 6 CONCLUSION
+
+In this work we have established a relaxation from fixed geometric scattering with strong guarantees to a more flexible network with better performance by learning data dependent scales. Allowing the network to choose data-driven diffusion scales leads to improved performance particularly on biochemical datasets, while keeping strong guarantees on extracted features. This parameterization has advantages in representing long range connections with a small number of weights, which are necessary in complex biochemical data. This also opens the possibility to provide additional relaxation to enable node-specific or graph-specific tuning via attention mechanisms, which we regard as an exciting future direction, but out of scope for the current work.
+
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+
+# APPENDIX
+
+# A PROOFS FOR SECTION 3
+
+A.1 PROOF OF LEMMA 1
+
+Let $M _ { \alpha } = D ^ { - 1 / 2 } P _ { \alpha } D ^ { 1 / 2 }$ then it can be verified that $M _ { \alpha }$ is a symmetric conjugate of $P _ { \alpha }$ , and by construction is self-adjoint with respect to the standard inner product of $L ^ { \bar { 2 } } ( \bar { \mathcal { G } } )$ . Let $\mathbf { { } _ { \pmb { x } } , { \pmb y } _ { \mathbf { \mu } } \in }$ $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ then we have
+
+$$
+\begin{array} { r l } & { \langle P _ { \alpha } x , y \rangle _ { D ^ { - 1 / 2 } } = \langle D ^ { - 1 / 2 } P _ { \alpha } x , D ^ { - 1 / 2 } y \rangle } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \end{array}
+$$
+
+which gives the result of the lemma.
+
+# A.2 PROOF OF THEOREM 1
+
+As shown in the previous proof (Sec. A.1), $P _ { \alpha }$ has a symmetric conjugate $M _ { \alpha }$ . Given the eigendecomposition $M _ { \alpha } = Q \Lambda Q ^ { T }$ , we can write $P _ { \alpha } ^ { t } = D ^ { 1 / \bar { 2 } } Q \Lambda ^ { t } Q ^ { T } D ^ { - \bar { 1 } / 2 }$ , giving the eigendecomposition of the propagated diffusion matrices. Furthermore, it can be verified that the eigenvalues on the diagonal of $\Lambda$ are nonnegative. Briefly, this results from graph Laplacian eigenvalues being within the range $[ 0 , 1 ]$ , which means those of $W D ^ { - 1 }$ are in $[ - 1 , 1 ]$ , which combined with $1 / 2 \bar { \le } \alpha \le 1$ result in $\lambda _ { i } : = [ \Lambda ] _ { i i } \in [ 0 , 1 ]$ for every $j$ . Next, given this decomposition we can write:
+
+$$
+\begin{array} { r l } & { \Phi _ { J } ^ { \prime } = D ^ { 1 / 2 } Q \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } , } \\ & { \Psi _ { j } ^ { \prime } = D ^ { 1 / 2 } Q ( \Lambda ^ { t _ { j } } - \Lambda ^ { t _ { j + 1 } } ) Q ^ { T } D ^ { - 1 / 2 } , \quad 0 \leq j \leq J - 1 . } \end{array}
+$$
+
+where we set $t _ { 0 } = 0$ to simplify notations. Then, we have:
+
+$$
+\begin{array} { l } { \| \Phi _ { J } ^ { \prime } x \| _ { D ^ { - 1 / 2 } } ^ { 2 } = \langle \Phi _ { J } ^ { \prime } x , \Phi _ { J } ^ { \prime } x \rangle _ { D ^ { - 1 / 2 } } } \\ { = \langle D ^ { - 1 / 2 } D ^ { 1 / 2 } Q \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } x , D ^ { - 1 / 2 } D ^ { 1 / 2 } Q \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } x \rangle } \\ { = x ^ { T } D ^ { - 1 / 2 } Q \Lambda ^ { t _ { J } } Q ^ { T } Q \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } x = ( x ^ { T } D ^ { - 1 / 2 } Q \Lambda ^ { t _ { J } } ) ( \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } x ) } \\ { = \| \Lambda ^ { t _ { J } } Q ^ { T } D ^ { - 1 / 2 } x \| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+Further, since $Q$ is orthogonal (as it is constructed from an eigenbasis of a symmetric matrix), if we consider a change of variable to $\pmb { y } = \pmb { Q } ^ { T } \pmb { D } ^ { - 1 / 2 } \pmb { x }$ , we have $\| \pmb { x } \| _ { D ^ { - 1 / 2 } } ^ { 2 } = \| \pmb { D } ^ { - 1 / 2 } \pmb { x } \| _ { 2 } ^ { 2 } = \| \pmb { y } \| _ { 2 } ^ { 2 }$ while $\| \Phi _ { J } ^ { \prime } \pmb { x } \| _ { D ^ { - 1 / 2 } } ^ { 2 } = \| \Lambda ^ { t _ { J } } \pmb { y } \| _ { 2 } ^ { 2 }$ . Similarly, we can also reformulate the operation of other filters in terms of diagonal matrices applied to $\textbf {  { y } }$ as $\mathcal { W } _ { J } ^ { \prime }$ as $\| \Psi _ { j } ^ { \prime } \pmb { x } \| _ { D ^ { - 1 / 2 } } ^ { 2 } = \| ( \Lambda ^ { t _ { j } } - \Lambda ^ { t _ { j + 1 } } ) \pmb { y } \| _ { 2 } ^ { 2 }$ .
+
+Given the reformulation in terms of $\textbf {  { y } }$ and standard $L ^ { 2 } ( { \mathcal { G } } )$ , we can now write
+
+$$
+\| \Lambda ^ { t , \ j } y \| _ { 2 } ^ { 2 } + \sum _ { j = 0 } ^ { J - 1 } \| ( \Lambda ^ { t _ { j } } - \Lambda ^ { t _ { j + 1 } } ) y \| _ { 2 } ^ { 2 } = \sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } \cdot \left( \lambda ^ { 2 t _ { J } } + \sum _ { j = 0 } ^ { J - 1 } ( \lambda _ { i } ^ { t _ { j } } - \lambda _ { i } ^ { t _ { j + 1 } } ) ^ { 2 } \right) .
+$$
+
+Then, since $0 \leq \lambda _ { i } \leq 1$ and $0 = t _ { 0 } < t _ { 1 } < \cdot \cdot \cdot < t _ { J }$ we have
+
+$$
+\lambda ^ { 2 t , t } + \sum _ { j = 0 } ^ { J - 1 } ( \lambda _ { i } ^ { t _ { j } } - \lambda _ { i } ^ { t _ { j + 1 } } ) ^ { 2 } \leq \left( \lambda ^ { t , t } + \sum _ { j = 0 } ^ { J - 1 } \lambda _ { i } ^ { t _ { j } } - \lambda _ { i } ^ { t _ { j + 1 } } \right) ^ { 2 } = \left( \lambda ^ { t , t } + \lambda _ { i } ^ { t _ { 0 } } - \lambda _ { i } ^ { t _ { j } } \right) ^ { 2 } = 1 ,
+$$
+
+which yields the upper bound $\begin{array} { r } { \| \boldsymbol { \Lambda } ^ { t _ { J } } \pmb { y } \| _ { 2 } ^ { 2 } + \sum _ { j = 0 } ^ { J - 1 } \| \big ( \boldsymbol { \Lambda } ^ { t _ { j } } - \boldsymbol { \Lambda } ^ { t _ { j + 1 } } \big ) \pmb { y } \| _ { 2 } ^ { 2 } \leq \| \pmb { y } \| _ { 2 } ^ { 2 } } \end{array}$ . On the other hand, since $t _ { 1 } > 0 = t _ { 0 }$ , then we also have
+
+$$
+\lambda ^ { 2 t _ { J } } + \sum _ { j = 0 } ^ { J - 1 } ( \lambda _ { i } ^ { t _ { j } } - \lambda _ { i } ^ { t _ { j + 1 } } ) ^ { 2 } \geq \lambda ^ { 2 t _ { J } } + ( 1 - \lambda _ { i } ^ { t _ { 1 } } ) ^ { 2 }
+$$
+
+and therefore, by setting $C : = \mathrm { m i n } _ { 0 \le \xi \le 1 } ( \xi ^ { 2 t _ { J } } + ( 1 - \xi ^ { t _ { 1 } } ) ^ { 2 } ) > 0$ , whose positivity is not difficult to verify, we get the lower bound $\begin{array} { r } { \| \boldsymbol { \Lambda } ^ { t _ { J } } \pmb { y } \| _ { 2 } ^ { 2 } + \sum _ { j = 0 } ^ { J - 1 } \| \big ( \boldsymbol { \Lambda } ^ { t _ { j } } - \boldsymbol { \Lambda } ^ { t _ { j + 1 } } \big ) \pmb { y } \| _ { 2 } ^ { 2 } \geq C \| \pmb { y } \| _ { 2 } ^ { 2 } } \end{array}$ . Finally, applying the reverse change of variable to $_ { \textbf { \em x } }$ and $L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ yields the result of the theorem. □
+
+# A.3 PROOF OF THEOREM 2
+
+Denote the permutation group on $n$ elements as $S _ { n }$ , then for a permutation $\Pi \in S _ { n }$ we let ${ \overline { { \mathcal { G } } } } = \Pi ( { \mathcal { G } } )$ be the graph obtained by permuting the vertices of $\mathcal { G }$ with $\Pi$ . The corresponding permutation operation on a graph signal $\pmb { x } \in L ^ { 2 } ( \mathcal { G } , D ^ { - 1 / 2 } )$ gives a signal $\Pi { \pmb x } \in L ^ { 2 } ( \overline { { \mathcal { G } } } , \bar { D ^ { - 1 / 2 } } )$ , which we implicitly considered in the statement of the theorem, without specifying these notations for simplicity. Rewriting the statement of the theorem more rigorously with the introduced notations, we aim to show that $\overline { { U } } _ { p } ^ { \prime } \Pi \pmb { x } = \Pi U _ { p } ^ { \prime } \pmb { x }$ and $\overline { { S } } _ { p , q } ^ { \prime } \Pi \pmb { x } = S _ { p , q } ^ { \prime } \pmb { x }$ under suitable conditions, where the operation $U _ { p } ^ { \prime }$ from $\mathcal { G }$ on the permuted graph $\overline { { \mathcal { G } } }$ is denoted here by $\overline { { U } } _ { p } ^ { \prime }$ and likewise for $S _ { p , q } ^ { \prime }$ we have $\overline { { S } } _ { p , q } ^ { \prime }$ .
+
+We start by showing $U _ { p } ^ { \prime }$ is permutation equivariant. First, we notice that for any $\Psi _ { j }$ , $0 < j < J$ we have that $\overline { { \Psi } } _ { j } \Pi \pmb { x } = \Pi \bar { \Psi _ { j } } \pmb { x }$ , as for $1 \leq j \leq J - 1$
+
+$$
+\begin{array} { r l } & { \overline { { \Psi } } _ { j } \Pi \pmb { x } = ( \Pi P ^ { t _ { j } } \Pi ^ { T } - \Pi P ^ { t _ { j + 1 } } \Pi ^ { T } ) \Pi \pmb { x } } \\ & { \qquad = \Pi ( \pmb { P } ^ { t _ { j } } - \pmb { P } ^ { t _ { j + 1 } } ) \pmb { x } } \\ & { \qquad = \Pi \Psi _ { j } \pmb { x } . } \end{array}
+$$
+
+Similar reasoning also holds for $j \in \{ 0 , J \}$ . Further, notice that for the element-wise nature of the absolute value nonlinearity yields $| \Pi { \pmb x } | = \Pi | { \pmb x } |$ for any permutation matrix $\Pi$ . Using these two observations, it follows inductively that
+
+$$
+\begin{array} { r l } & { \overline { { U } } _ { p } ^ { \prime } \Pi x : = \Psi _ { j _ { m } } ^ { \prime } | \Psi _ { j _ { m - 1 } } ^ { \prime } \cdot . . . | \Psi _ { j _ { 2 } } ^ { \prime } | \Psi _ { j _ { 1 } } ^ { \prime } \Pi x | | \cdot . . . | } \\ & { \qquad = \Psi _ { j _ { m } } ^ { \prime } | \Psi _ { j _ { m - 1 } } ^ { \prime } \cdot . . . | \Psi _ { j _ { 2 } } ^ { \prime } \Pi | \Psi _ { j _ { 1 } } ^ { \prime } x | | \cdot . . . | } \\ & { \qquad \vdots } \\ & { \qquad = \Pi \Psi _ { j _ { m } } ^ { \prime } | \Psi _ { j _ { m - 1 } } ^ { \prime } \cdot . . . | \Psi _ { j _ { 2 } } ^ { \prime } | \Psi _ { j _ { 1 } } ^ { \prime } x | | \cdot . . . | } \\ & { \qquad = \Pi U _ { p } ^ { \prime } x . } \end{array}
+$$
+
+To show $S _ { p , q } ^ { \prime }$ is permutation invariant, first notice that for any statistical moment $q > 0$ , we have $| \Pi { \pmb x } | ^ { q } = \Pi | { \pmb x } | ^ { q }$ and further as sums are commutative, $\begin{array} { r } { \sum _ { j } ( \Pi \pmb { x } ) _ { j } = \sum _ { j } \pmb { x } _ { j } } \end{array}$ . We then have
+
+$$
+\overline { { S } } _ { p , q } ^ { ' } \Pi \boldsymbol { x } = \sum _ { i = 1 } ^ { n } | \overline { { U } } _ { p } ^ { ' } \Pi \boldsymbol { x } [ v _ { i } ] | ^ { q } = \sum _ { i = 1 } ^ { n } | \Pi U _ { p } ^ { \prime } \boldsymbol { x } [ v _ { i } ] | ^ { q } = \sum _ { i = 1 } ^ { n } | U _ { p } ^ { \prime } \boldsymbol { x } [ v _ { i } ] | ^ { q } = S _ { p , q } ^ { ' } \boldsymbol { x } ,
+$$
+
+which, together with the previous result, completes the proof of the theorem.
+
+# B DATASETS
+
+In this section we further analyze individual datasets. Relating composition of the dataset as shown in Table S1 to the relative performance of our models as shown in Table S2.
+
+DD Dobson & Doig (2003): Is a dataset extracted from the protein data bank (PDB) of 1178 high resolution proteins. The task is to distinguish between enzymes and non-enzymes. Since these are high resolution structures, these graphs are significantly larger than those found in our other biochemical datasets with a mean graph size of 284 nodes with the next largest biochemical dataset with a mean size of 39 nodes.
+
+ENZYMES Borgwardt et al. (2005): Is a dataset of 600 enzymes divided into 6 balanced classes of 100 enzymes each. As we analyzed in the main text, scattering features are better able to preserve the structure between classes. LEGS-FCN slightly relaxes this structure but improves accuracy from 32 to $39 \%$ over LEGS-FIXED.
+
+NCI1, NCI109 Wale et al. (2008): Contains slight variants of 4100 chemical compounds encoded as graphs. Each compound is separated into one of two classes based on its activity against nonsmall cell lung cancer and ovarian cancer cell lines. Graphs in this dataset are 30 nodes with a similar number of edges. This makes for long graphs with high diameter.
+
+PROTEINS Borgwardt et al. (2005): Contains 1178 protein structures with the goal of classifying enzymes vs. non enzymes. GCN outperforms all other models on this dataset, however the Baseline model, where no structure is used also performs very similarly. This suggests that the graph structure within this dataset does not add much information over the structure encoded in the eccentricity and clustering coefficient.
+
+PTC Toivonen et al. (2003): Contains 344 chemical compound graphs divided into two classes based on whether or not they cause cancer in rats. This dataset is very difficult to classify without features however LEGS-RBF and LEGS-FCN are able to capture the long range connections slightly better than other methods.
+
+COLLAB Yanardag & Vishwanathan (2015): 5000 ego-networks of different researchers from high energy physics, condensed matter physics or astrophysics. The goal is to determine which field the research belongs to. The GraphSAGE model performs best on this dataset although the LEGS-RBF network performs nearly as well. Ego graphs have a very small average diameter. Thus shallow networks can perform quite well on them as is the case here.
+
+IMDB Yanardag & Vishwanathan (2015): For each graph nodes represent actresses/actors and there is an edge between them if they are in the same move. These graphs are also ego graphs around specific actors. IMDB-BINARY classifies between action and romance genres. IMDB-MULTI classifies between 3 classes. Somewhat surprisingly GS-SVM performs the best with other LEGS networks close behind. This could be due to oversmoothing on the part of GCN and GraphSAGE when the graphs are so small.
+
+REDDIT Yanardag & Vishwanathan (2015): Graphs in REDDIT-BINARY/MULTI-5K/MULTI12K datasets each graph represents a discussion thread where nodes correspond to users and there is an edge between two nodes if one replied to the other’s comment. The task is to identify which subreddit a given graph came from. On these datasets GCN outperforms other models.
+
+QM9 Gilmer et al. (2017); Wu et al. (2018): Graphs in the QM9 dataset each represent chemicals with 18 atoms. Regression targets represent chemical properties of the molecules.
+
+# B.1 PERFORMANCE OF LEGSNET ON SOCIAL NETWORK DATASETS
+
+Table S2 shows that our model outperforms other GNNs on some biomedical benchmarks and that it performs comparably on social network datasets. Out of the six social network datasets, ignoring the fixed scattering model GS-SVM, which has been hand tuned with these datasets in mind, our model outperforms both GNN models on three of them, and is second best on the other three. This is at least comparable if not slightly superior performance. GraphSAGE does a bit better on Collab, but much worse on IMDB-Binary and Reddit-Binary. GCN does a bit better on Reddit-Multi, but worse on Collab, IMDB-Binary, and Reddit-Binary.
+
+LEGSNet has significantly fewer parameters and achieves comparable or superior accuracy on common benchmarks. Even when our method shows comparable results, and definitely when it outperforms other GNNs, we believe that its smaller number of parameters could be useful in applications with limited compute or limited training examples.
+
+Table S1: Dataset statistics, diameter, nodes, edges, clustering coefficient averaged over all graphs. Split into bio-chemical and social network types.   
+
+<table><tr><td></td><td># Graphs</td><td># Classes</td><td>Diameter</td><td>Nodes</td><td>Edges</td><td>Clust. Coeff</td></tr><tr><td>DD</td><td>1178</td><td>2</td><td>19.81</td><td>284.32</td><td>715.66</td><td>0.48</td></tr><tr><td>ENZYMES</td><td>600</td><td>6</td><td>10.92</td><td>32.63</td><td>62.14</td><td>0.45</td></tr><tr><td>MUTAG</td><td>188</td><td>2</td><td>8.22</td><td>17.93</td><td>19.79</td><td>0.00</td></tr><tr><td>NCI1</td><td>4110</td><td>2</td><td>13.33</td><td>29.87</td><td>32.30</td><td>0.00</td></tr><tr><td>NCI109</td><td>4127</td><td>2</td><td>13.14</td><td>29.68</td><td>32.13</td><td>0.00</td></tr><tr><td>PROTEINS</td><td>1113</td><td>2</td><td>11.62</td><td>39.06</td><td>72.82</td><td>0.51</td></tr><tr><td>PTC</td><td>344</td><td>2</td><td>7.52</td><td>14.29</td><td>14.69</td><td>0.01</td></tr><tr><td>COLLAB</td><td>5000</td><td>3</td><td>1.86</td><td>74.49</td><td>2457.22</td><td>0.89</td></tr><tr><td>IMDB-BINARY</td><td>1000</td><td>2</td><td>1.86</td><td>19.77</td><td>96.53</td><td>0.95</td></tr><tr><td>IMDB-MULTI</td><td>1500</td><td>3</td><td>1.47</td><td>13.00</td><td>65.94</td><td>0.97</td></tr><tr><td>REDDIT-BINARY</td><td>2000</td><td>2</td><td>8.59</td><td>429.63</td><td>497.75</td><td>0.05</td></tr><tr><td>REDDIT-MULTI-12K</td><td>11929</td><td>11</td><td>9.53</td><td>391.41</td><td>456.89</td><td>0.03</td></tr><tr><td>REDDIT-MULTI-5K</td><td>4999</td><td>5</td><td>10.57</td><td>508.52</td><td>594.87</td><td>0.03</td></tr></table>
+
+Table S2: Mean $\pm$ std. over 10 test sets on bio-chemical and social datasets.   
+
+<table><tr><td></td><td>LEGS-RBF</td><td>LEGS-FCN</td><td>LEGS-FIXED</td><td>GCN</td><td>GraphSAGE</td><td>GAT</td><td>GIN</td><td>GS-SVM</td><td>Baseline</td></tr><tr><td>DD</td><td>72.58 ± 3.35</td><td>72.07 ± 2.37</td><td>69.09 ± 4.82</td><td>67.82 ± 3.81</td><td>66.37 ± 4.45</td><td>68.50± 3.62</td><td>42.37 ± 4.32</td><td>72.66 ± 4.94</td><td>75.98 ± 2.81</td></tr><tr><td>ENZYMES</td><td>36.33 ± 4.50</td><td>38.50 ± 8.18</td><td>32.33 ± 5.04</td><td>31.33 ± 6.89</td><td>15.83 ± 9.10</td><td>25.83 ± 4.73</td><td>36.83 ± 4.81</td><td>27.33 ± 5.10</td><td>20.50 ± 5.99</td></tr><tr><td>MUTAG</td><td>33.51 ± 4.34</td><td>82.98 ± 9.85</td><td>81.84 ± 11.24</td><td>79.30 ± 9.66</td><td>81.43 ± 11.64</td><td>79.85 ± 9.44</td><td>83.57 ± 9.68</td><td>85.09 ± 7.44</td><td>79.80 ± 9.92</td></tr><tr><td>NCI1</td><td>74.26± 1.53</td><td>70.83 ± 2.65</td><td>71.24 ± 1.63</td><td>60.80 ± 4.26</td><td>57.54 ± 3.33</td><td>62.19 ± 2.18</td><td>66.67 ± 2.90</td><td>69.68 ± 2.38</td><td>56.69 ± 3.07</td></tr><tr><td>NCI109</td><td>72.47 ± 2.11</td><td>70.17 ± 1.46</td><td>69.25 ± 1.75</td><td>61.30 ± 2.99</td><td>55.15 ± 2.58</td><td>61.28 ± 2.24</td><td>65.23 ± 1.82</td><td>68.55 ± 2.06</td><td>57.38 ± 2.20</td></tr><tr><td>PROTEINS</td><td>70.89 ± 3.91</td><td>71.06 ± 3.17</td><td>67.30 ± 2.94</td><td>74.03 ± 3.20</td><td>71.87 ± 3.50</td><td>73.22 ± 3.55</td><td>75.02 ± 4.55</td><td>70.98 ± 2.67</td><td>73.22 ± 3.76</td></tr><tr><td>PTC</td><td>57.26± 5.54</td><td>56.92 ± 9.36</td><td>54.31 ± 6.92</td><td>56.34 ± 10.29</td><td>55.22 ± 9.13</td><td>55.50 ± 6.90</td><td>55.82 ± 8.07</td><td>56.96 ± 7.09</td><td>56.71 ± 5.54</td></tr><tr><td>COLLAB</td><td>75.78 ± 1.95</td><td>75.40 ± 1.80</td><td>72.94 ± 1.70</td><td>73.80 ±1.73</td><td>76.12 ± 1.58</td><td>72.88 ± 2.06</td><td>62.98 ± 3.92</td><td>74.54 ± 2.32</td><td>64.76 ± 2.63</td></tr><tr><td>IMDB-BINARY</td><td>64.90 ± 3.48</td><td>64.50± 3.50</td><td>64.30 ± 3.68</td><td>47.40 ± 6.24</td><td>46.40 ± 4.03</td><td>45.50 ± 3.14</td><td>64.20 ± 5.77</td><td>66.70 ± 3.53</td><td>47.20 ± 5.67</td></tr><tr><td>IMDB-MULTI</td><td>41.93 ± 3.01</td><td>40.13 ± 2.77</td><td>41.67 ± 3.19</td><td>39.33 ± 3.13</td><td>39.73 ± 3.45</td><td>39.73 ± 3.61</td><td>38.67± 3.93</td><td>42.13 ± 2.53</td><td>39.53 ± 3.63</td></tr><tr><td>REDDIT-BINARY</td><td>86.10 ± 2.92</td><td>78.15 ± 5.42</td><td>85.00 ± 1.93</td><td>81.60 ± 2.32</td><td>73.40 ± 4.38</td><td>73.35 ± 2.27</td><td>71.40 ± 6.98</td><td>85.15 ± 2.78</td><td>69.30 ± 5.08</td></tr><tr><td>REDDIT-MULTI-12K</td><td>38.47 ± 1.07</td><td>38.46 ± 1.31</td><td>39.74 ± 1.31</td><td>42.57 ± 0.90</td><td>32.17 ± 2.04</td><td>32.74 ± 0.75</td><td>24.45 ± 5.52</td><td>39.79 ± 1.11</td><td>22.07 ± 0.98</td></tr><tr><td>REDDIT-MULTI-5K</td><td>47.83 ± 2.61</td><td>46.97 ± 3.06</td><td>47.17 ± 2.93</td><td>52.79 ± 2.11</td><td>45.71 ± 2.88</td><td>44.03 ± 2.57</td><td>35.73 ± 8.35</td><td>48.79 ± 2.95</td><td>36.41 ± 1.80</td></tr></table>
+
+# C TRAINING DETAILS
+
+We train all models for a maximum of 1000 epochs with an initial learning rate of $1 e ^ { - 4 }$ using the ADAM optimizer (Kingma & Ba, 2015). We terminate training if validation loss does not improve for 100 epochs testing every 10 epochs. Our models are implemented with Pytorch Paszke et al. (2019) and Pytorch geometric. Models were run on a variety of hardware resources. For all models we use $q \ : = \ : 4$ normalized statistical moments for the node to graph level feature extraction and $m = 1 6$ diffusion scales in line with choices in Gao et al. (2019).
+
+# C.1 CROSS VALIDATION PROCEDURE
+
+For all datasets we use 10-fold cross validation with $80 \%$ training data $10 \%$ validation data and $10 \%$ test data for each model. We first split the data into 10 (roughly) equal partitions. For each model we take exactly one of the partitions to be the test set and one of the remaining nine to be the validation set. We then train the model on the remaining eight partitions using the cross-entropy loss on the validation for early stopping checking every ten epochs. For each test set, we use majority voting of the nine models trained with that test set. We then take the mean and standard deviation across these test set scores to average out any variability in the particular split chosen. This results in 900 models trained on every dataset. With mean and standard deviation over 10 ensembled models each with a separate test set.
+
+# D ENSEMBLING EVALUATION
+
+Recent work by Min et al. (2020) combines the features from a fixed scattering transform with a GCN network, showing that this has empirical advantages in semi-supervised node classification, and theoretical representation advantages over a standard Kipf & Welling (2016) style GCN. We ensemble the learned features from a learnable scattering network (LEGS-FCN) with those of GCN and compare this to ensembling fixed scattering features with GCN as in Min et al. (2020), as well as the solo features. Our setting is slightly different in that we use the GCN features from pretrained networks, only training a small 2-layer ensembling network on the combined graph level features. This network consists of a batch norm layer, a 128 width fully connected layer, a leakyReLU activation, and a final classification layer down to the number of classes. In Table S6 we see that combining GCN features with fixed scattering features in LEGS-FIXED or learned scattering features in LEGS-FCN always helps classification. Learnable scattering features help more than fixed scattering features overall and particularly in the biochemical domain.
+
+Table S3: Mean $\pm$ std. over test set selection on cross-validated LEGS-RBF Net with reduced training set size.   
+
+<table><tr><td>Train, Val, Test %</td><td>80%,10%,10%</td><td>70%, 10%,20%</td><td>40%, 10%,50%</td><td>20%,10%,70%</td></tr><tr><td>COLLAB</td><td>75.78 ± 1.95</td><td>75.00 ± 1.83</td><td>74.00 ± 0.51</td><td>72.73 ± 0.59</td></tr><tr><td>DD</td><td>72.58 ± 3.35</td><td>70.88 ± 2.83</td><td>69.95 ± 1.85</td><td>69.43 ± 1.24</td></tr><tr><td>ENZYMES</td><td>36.33 ± 4.50</td><td>34.17 ± 3.77</td><td>29.83 ± 3.54</td><td>23.98 ± 3.32</td></tr><tr><td>IMDB-BINARY</td><td>64.90 ± 3.48</td><td>63.00 ± 2.03</td><td>63.30 ±1.27</td><td>57.67 ± 6.04</td></tr><tr><td>IMDB-MULTI</td><td>41.93 ± 3.01</td><td>40.80 ± 1.79</td><td>41.80 ± 1.23</td><td>36.83 ± 3.31</td></tr><tr><td>MUTAG</td><td>33.51 ± 4.34</td><td>33.51 ± 1.14</td><td>33.52 ± 1.26</td><td>33.51 ± 0.77</td></tr><tr><td>NCI1</td><td>74.26 ± 1.53</td><td>74.38 ± 1.38</td><td>72.07 ± 0.28</td><td>70.30 ± 0.72</td></tr><tr><td>NCI109</td><td>72.47 ± 2.11</td><td>72.21 ± 0.92</td><td>70.44 ± 0.78</td><td>68.46 ± 0.96</td></tr><tr><td>PROTIENS</td><td>70.89 ± 3.91</td><td>69.27 ± 1.95</td><td>69.72 ± 0.27</td><td>68.96 ± 1.63</td></tr><tr><td>PTC</td><td>57.26 ± 5.54</td><td>57.83 ± 4.39</td><td>54.62 ± 3.21</td><td>55.45 ± 2.35</td></tr><tr><td>REDDIT-BINARY</td><td>86.10 ± 2.92</td><td>86.05 ± 2.51</td><td>85.15 ± 1.77</td><td>83.71 ± 0.97</td></tr><tr><td>REDDIT-MULTI-12K</td><td>38.47 ± 1.07</td><td>38.60 ± 0.52</td><td>37.55 ± 0.05</td><td>36.65 ± 0.50</td></tr><tr><td>REDDIT-MULTI-5K</td><td>47.83 ± 2.61</td><td>47.81 ± 1.32</td><td>46.73 ± 1.46</td><td>44.59 ± 1.02</td></tr></table>
+
+Table S4: Test set mean squared error on CASP GDT regression task across targets over 3 nonoverlapping test sets.   
+
+<table><tr><td></td><td>LEGS-RBF</td><td>LEGS-FCN</td><td>LEGS-FIXED</td><td>GCN</td><td>GraphSAGE</td><td>GIN</td><td>Baseline</td></tr><tr><td>t0860</td><td>197.68 ± 34.29</td><td>164.22 ±10.28</td><td>206.20 ± 28.46</td><td>314.90 ± 29.66</td><td>230.45 ± 79.72</td><td>262.35 ± 66.88</td><td>414.41 ± 26.96</td></tr><tr><td>t0868</td><td>131.42 ± 8.12</td><td>127.71 ± 14.26</td><td>178.45 ± 5.64</td><td>272.14± 26.34</td><td>191.08 ± 21.96</td><td>170.05 ± 27.26</td><td>411.98 ± 57.39</td></tr><tr><td>t0869</td><td>106.69 ± 9.97</td><td>132.12 ± 31.37</td><td>104.47 ±14.16</td><td>317.22 ± 12.75</td><td>244.38 ± 40.58</td><td>217.02 ± 57.01</td><td>393.12 ± 48.70</td></tr><tr><td>t0872</td><td>144.11 ± 24.88</td><td>148.20 ± 23.63</td><td>134.48 ± 8.25</td><td>293.96 ± 19.00</td><td>221.13 ± 28.74</td><td>240.89 ± 24.17</td><td>374.48 ± 33.70</td></tr><tr><td>t0879</td><td>89.00± 44.94</td><td>80.14 ± 16.21</td><td>64.63±15.92</td><td>309.23 ± 69.40</td><td>172.41 ± 73.07</td><td>147.77 ± 15.72</td><td>364.79±144.32</td></tr><tr><td>t0900</td><td>193.74± 10.78</td><td>171.05 ± 25.41</td><td>158.56 ± 9.87</td><td>254.11 ± 18.63</td><td>209.07 ±11.90</td><td>265.77 ± 79.99</td><td>399.16 ± 83.48</td></tr><tr><td>t0912</td><td>113.00 ± 22.31</td><td>169.55 ± 27.35</td><td>150.70 ± 8.53</td><td>227.17 ± 22.11</td><td>192.28 ± 39.45</td><td>271.30 ± 28.89</td><td>406.25 ± 31.42</td></tr><tr><td>t0920</td><td>80.46 ± 14.98</td><td>136.94 ± 36.43</td><td>84.83 ± 19.70</td><td>361.19 ± 71.25</td><td>261.72 ± 59.67</td><td>191.86 ± 37.85</td><td>398.22 ± 25.60</td></tr><tr><td>t0921</td><td>187.89 ± 46.15</td><td>165.97 ± 42.39</td><td>142.97 ± 27.09</td><td>382.69 ± 20.27</td><td>260.49 ±16.09</td><td>207.19 ± 24.84</td><td>363.92 ± 35.79</td></tr><tr><td>t0922</td><td>254.83 ± 91.28</td><td>110.54 ± 43.99</td><td>227.73 ± 26.41</td><td>366.72 ± 8.10</td><td>290.71 ± 7.22</td><td>130.46 ± 11.64</td><td>419.14 ± 45.49</td></tr><tr><td>t0942</td><td>188.55 ± 11.10</td><td>167.53 ± 22.01</td><td>137.21 ± 7.43</td><td>371.31 ± 9.90</td><td>233.78± 84.95</td><td>254.38 ± 47.21</td><td>393.03± 24.93</td></tr><tr><td>t0944</td><td>146.59 ± 8.41</td><td>138.67 ± 50.36</td><td>245.79 ± 58.16</td><td>263.03 ±9.43</td><td>199.40 ± 51.11</td><td>157.90 ± 2.57</td><td>404.12 ± 40.82</td></tr></table>
+
+Table S5: Quantified distance between the empirically observed enzyme class exchange preferences of Cuesta et al. (2015) and the class exchange preferences inferred from LEGS-FIXED, LEGS-FCN, and a GCN. We measure the cosine distance between the graphs represented by the chord diagrams in Figure 2. As before, the self-affinities were discarded. LEGS-Fixed reproduces the exchange preferences the best, but LEGS-FCN still reproduces well and has significantly better classification accuracy.   
+
+<table><tr><td>LEGS-FIXED</td><td>LEGS-FCN</td><td>GCN</td></tr><tr><td>0.132</td><td>0.146</td><td>0.155</td></tr></table>
+
+Table S6: Mean $\pm$ standard deviation test set accuracy on biochemical and social network datasets.   
+
+<table><tr><td></td><td>GCN</td><td>GCN-LEGS-FIXED</td><td>GCN-LEGS-FCN</td></tr><tr><td>DD</td><td>67.82 ± 3.81</td><td>74.02 ± 2.79</td><td>73.34 ± 3.57</td></tr><tr><td>ENZYMES</td><td>31.33 ± 6.89</td><td>31.83 ± 6.78</td><td>35.83 ± 5.57</td></tr><tr><td>MUTAG</td><td>79.30 ± 9.66</td><td>82.46 ± 7.88</td><td>83.54 ± 9.39</td></tr><tr><td>NCI1</td><td>60.80 ± 4.26</td><td>70.80 ± 2.27</td><td>72.21 ± 2.32</td></tr><tr><td>NCI109</td><td>61.30 ± 2.99</td><td>68.82 ± 1.80</td><td>69.52 ± 1.99</td></tr><tr><td>PROTEINS</td><td>74.03 ± 3.20</td><td>73.94 ± 3.88</td><td>74.30 ± 3.41</td></tr><tr><td>PTC</td><td>56.34 ± 10.29</td><td>58.11 ± 6.06</td><td>56.64 ± 7.34</td></tr><tr><td>COLLAB</td><td>73.80 ± 1.73</td><td>76.60 ± 1.75</td><td>75.76 ± 1.83</td></tr><tr><td>IMDB-BINARY</td><td>47.40 ± 6.24</td><td>65.10 ± 3.75</td><td>65.90 ± 4.33</td></tr><tr><td>IMDB-MULTI</td><td>39.33 ± 3.13</td><td>39.93 ± 2.69</td><td>39.87 ± 2.24</td></tr><tr><td>REDDIT-BINARY</td><td>81.60 ± 2.32</td><td>86.90 ± 1.90</td><td>87.00 ± 2.36</td></tr><tr><td>REDDIT-MULTI-12K</td><td>42.57 ± 0.90</td><td>45.41 ± 1.24</td><td>45.55 ± 1.00</td></tr><tr><td>REDDIT-MULTI-5K</td><td>52.79 ± 2.11</td><td>53.87 ± 2.75</td><td>53.41 ± 3.07</td></tr></table>
+
+Table S7: Mean $\pm$ std. over four runs of mean squared error over 19 targets for the QM9 dataset, lower is better.   
+
+<table><tr><td></td><td>LEGS-FCN</td><td>LEGS-FIXED</td><td>GCN</td><td>GraphSAGE</td><td>GIN</td><td>Baseline</td></tr><tr><td>Target 0</td><td>0.749 ± 0.025</td><td>0.761 ± 0.026</td><td>0.776 ± 0.021</td><td>0.876 ± 0.083</td><td>0.786 ± 0.032</td><td>0.985 ± 0.020</td></tr><tr><td>Target 1</td><td>0.158 ± 0.014</td><td>0.164 ± 0.024</td><td>0.448 ± 0.007</td><td>0.555 ± 0.295</td><td>0.191 ± 0.060</td><td>0.593 ± 0.013</td></tr><tr><td>Target 2</td><td>0.830 ± 0.016</td><td>0.856 ± 0.026</td><td>0.899 ± 0.051</td><td>0.961 ± 0.057</td><td>0.903 ±0.033</td><td>0.982 ± 0.027</td></tr><tr><td>Target 3</td><td>0.511 ± 0.012</td><td>0.508 ± 0.005</td><td>0.549 ± 0.010</td><td>0.688 ± 0.216</td><td>0.555 ± 0.006</td><td>0.805 ± 0.025</td></tr><tr><td>Target 4</td><td>0.587 ± 0.007</td><td>0.587 ± 0.006</td><td>0.609 ± 0.009</td><td>0.755 ± 0.177</td><td>0.613 ± 0.013</td><td>0.792 ± 0.010</td></tr><tr><td>Target 5</td><td>0.646 ± 0.013</td><td>0.674 ± 0.047</td><td>0.889 ± 0.014</td><td>0.882 ± 0.118</td><td>0.699 ± 0.033</td><td>0.833 ± 0.026</td></tr><tr><td>Target 6</td><td>0.018 ±0.012</td><td>0.020 ± 0.011</td><td>0.099 ± 0.011</td><td>0.321 ± 0.454</td><td>0.012 ± 0.006</td><td>0.468 ± 0.005</td></tr><tr><td>Target7</td><td>0.017 ± 0.005</td><td>0.024 ± 0.008</td><td>0.368 ± 0.015</td><td>0.532 ± 0.405</td><td>0.015 ± 0.005</td><td>0.379 ± 0.013</td></tr><tr><td>Target 8</td><td>0.017 ± 0.005</td><td>0.024 ± 0.008</td><td>0.368 ± 0.015</td><td>0.532 ± 0.404</td><td>0.015 ± 0.005</td><td>0.378 ± 0.013</td></tr><tr><td>Target 9</td><td>0.017 ± 0.005</td><td>0.024 ± 0.008</td><td>0.368 ± 0.015</td><td>0.532 ± 0.404</td><td>0.015 ± 0.005</td><td>0.378 ± 0.013</td></tr><tr><td>Target 10</td><td>0.017 ± 0.005</td><td>0.024 ± 0.008</td><td>0.368 ± 0.015</td><td>0.533 ± 0.404</td><td>0.015 ± 0.005</td><td>0.380 ± 0.014</td></tr><tr><td>Target 11</td><td>0.254 ± 0.013</td><td>0.279 ± 0.023</td><td>0.548 ± 0.023</td><td>0.617 ± 0.282</td><td>0.294 ± 0.003</td><td>0.631± 0.013</td></tr><tr><td>Target 12</td><td>0.034 ± 0.014</td><td>0.033 ± 0.010</td><td>0.215 ± 0.009</td><td>0.356 ± 0.437</td><td>0.020 ± 0.002</td><td>0.478 ± 0.014</td></tr><tr><td>Target 13</td><td>0.033 ± 0.014</td><td>0.033 ± 0.010</td><td>0.214 ± 0.009</td><td>0.356 ± 0.438</td><td>0.020 ±0.002</td><td>0.478 ± 0.014</td></tr><tr><td>Target 14</td><td>0.033 ± 0.014</td><td>0.033 ± 0.010</td><td>0.213 ± 0.009</td><td>0.355 ± 0.438</td><td>0.020 ±0.002</td><td>0.478 ± 0.014</td></tr><tr><td>Target 15</td><td>0.036 ± 0.014</td><td>0.036 ± 0.011</td><td>0.219 ± 0.009</td><td>0.359 ± 0.436</td><td>0.023 ± 0.002</td><td>0.479 ± 0.014</td></tr><tr><td>Target 16</td><td>0.002 ± 0.002</td><td>0.001 ± 0.001</td><td>0.017 ± 0.034</td><td>0.012 ± 0.022</td><td>0.000 ± 0.000</td><td>0.033 ± 0.013</td></tr><tr><td>Target 17</td><td>0.083 ± 0.047</td><td>0.079 ± 0.033</td><td>0.280 ± 0.354</td><td>0.264 ± 0.347</td><td>0.169 ± 0.206</td><td>0.205 ± 0.220</td></tr><tr><td>Target 18</td><td>0.062 ± 0.005</td><td>0.176 ± 0.231</td><td>0.482 ± 0.753</td><td>0.470± 0.740</td><td>0.321 ± 0.507</td><td>0.368 ± 0.525</td></tr></table>

parse/train/S1sRrN-CW/S1sRrN-CW.md

@@ -0,0 +1,262 @@
+# REVISITING KNOWLEDGE BASE EMBEDDING AS TEN-SOR DECOMPOSITION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We study the problem of knowledge base (KB) embedding, which is usually addressed through two frameworks—neural KB embedding and tensor decomposition. In this work, we theoretically analyze the neural embedding framework and subsequently connect it with tensor based embedding. Specifically, we show that in neural KB embedding the two commonly adopted optimization solutions— margin-based and negative sampling losses—are closely related to each other. We also reach the closed-form tensor that is implicitly approximated by popular neural KB approaches, revealing the underlying connection between neural and tensor based KB embedding models. Grounded in the theoretical results, we further present a tensor decomposition based framework KBTD to directly approximate the derived closed form tensor. Under this framework, the neural KB embedding models, such as NTN (Socher et al., 2013), TransE (Bordes et al., 2013), Bilinear (Jenatton et al., 2012), and DISTMULT (Yang et al., 2015), are unified into a general tensor optimization architecture. Finally, we conduct experiments on the link prediction task in WordNet and Freebase, empirically demonstrating the effectiveness of the KBTD framework.
+
+# 1 INTRODUCTION
+
+Knowledge bases (KBs) power many of semantic-oriented techniques and applications, such as question answering and intelligent personal assistant. A classical example is the automatic answer to the query “Who is Barack Obama’s wife” by KB-supported search engines. Most if not all of KBs achieve this by storing the facts about the world in the form of RDF triplets (W3C, 1999), wherein a triplet (subject, predicate, object), in short $( s , r , o )$ , records a piece of fact about the relation between the two entities—the subject and object. To automatically construct Web-scale KBs with billions of facts (triplets), a significant line of effort has been devoted to knowledge base embedding—the technique of encoding entities and their relational information into latent representations (Bordes et al., 2011; 2013; Socher et al., 2013; Yang et al., 2015).
+
+In particular, the direction of neural embedding has been extensively explored for learning representations for KBs, offering state-of-the-art performance for validating and completing unseen facts (Yang et al., 2015). Briefly, given a KB represented by triplets $T = \{ ( s , r , o ) \}$ , neural embedding models take these known triplets as positive instances and corrupted triplets $\{ ( \boldsymbol { s } ^ { \prime } , \boldsymbol { r } , o ^ { \prime } ) \}$ as negative ones. For each triplet, a scoring function $f ( s , r , o )$ — parameterized with a neural network—is designed to project the associated entities $s , o$ and their relational information $r$ into a scalar. Most of the existing models are then trained through two popular choices of loss functions, including the margin-based ranking loss by NTN (Socher et al., 2013), TransE (Bordes et al., 2013), and DISTMULT (Yang et al., 2015), as well as several practices of the Negative Sampling loss (Mikolov et al., 2013) by Bilinear (or LFM) (Jenatton et al., 2012) and CONV (Toutanova et al., 2015). In addition, another line of KB embedding is focused on tensor decomposition based frameworks, such as RESCAL (Nickel et al., 2011; 2012).
+
+Notwithstanding the rapid development and progress of KB embedding techniques, insights concerning their underlying mechanisms are to date sorely lacking. For example, a natural question that arises is what are the relationships or differences between the margin-based ranking loss and the negative sampling loss for neural KB embedding. Moreover, what is the quantity that is optimized by conventional neural KB embedding models? With an eye toward comprehensively understanding
+
+Table 1: Knowledge Base Embedding.   
+
+<table><tr><td rowspan=1 colspan=1>local margin-based loss (ReLU)</td><td rowspan=1 colspan=4>max(0,f(s,r,o&#x27;)- f(s,r,o)+ γ)</td></tr><tr><td rowspan=1 colspan=1>local margin-based loss (Softplus)</td><td rowspan=1 colspan=1>log (</td><td rowspan=1 colspan=2>log(1+ ef(s&#x27;,r,o&#x27;)-f(s,r,)+γ)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>local negative sampling loss</td><td rowspan=1 colspan=1>log(</td><td rowspan=1 colspan=3>log(1+ef(s&#x27;ro&#x27;)-f(s,r）+ef(s&#x27;r,0&#x27;）+e-f(s,r,o))</td></tr><tr><td rowspan=1 colspan=1>closed-form tensor</td><td rowspan=1 colspan=1>log</td><td rowspan=1 colspan=1>2|E|Xs,r.o（b（do+d，)</td><td rowspan=1 colspan=2></td></tr></table>
+
+See detailed notations in Sections 3 and 4. A brief introduction is listed below:   
+• $( s , r , o )$ & $( s ^ { \prime } , r , o ^ { \prime } )$ : the known and corrupted triplets, respectively; • $E$ & $R$ : the entity and relation sets of a given knowledge base, with s, $o \in E$ and $r \in R$ ; • $\mathcal { X } \in \{ 0 , 1 \} ^ { E \times R \times E }$ : a three-way binary tensor, with $\mathcal { X } _ { s , r , o } = 1$ indicating $( s , r , o ) \in T$ and otherwise 0; • $\begin{array} { r } { d _ { s , r } ^ { \mathrm { o u t } } = \sum _ { o } \mathcal { X } _ { s , r , o } } \end{array}$ & $\begin{array} { r } { d _ { o , r } ^ { \mathrm { i n } } = \sum _ { s } \mathcal { X } _ { s , r , o } } \end{array}$ : the out- $/ \mathrm { i n } \cdot$ - degree of entity $s / o$ under relation $r$ , respectively; • $b$ : the number of negative samples.
+
+KB embedding, we investigate (1) the connection between margin-based ranking loss and negative sampling loss in neural KB models, (2) the relationship between neural KB models and classical tensor-based KB models, and (3) the universal framework for KB learning.
+
+Contributions. In this work, we unveil two fundamentals of KB embedding, according to which we further present a tensor decomposition based KB embedding framework—KBTD, yielding significant outperformance over neural KB embedding models in most cases.
+
+First, with softplus’ smooth approximation to ReLU in the margin-based loss (Dugas et al., 2001), we show that the margin-based loss is closely connected to the negative sampling loss (See rows 2 & 3 in Table 1). In specific, both losses aim to encourage positive triplets $( s , r , o )$ and penalize corrupted ones $( s ^ { \prime } , \bar { r } , o ^ { \prime } )$ , and the slight difference lies in the extra reward or penalization to corresponding triplets in the negative sampling loss.
+
+Second, we derive the closed form tensor (See row 4 in Table 1) whose entry is implicitly fitted (approximated) by scoring function $f ( s , r , o )$ , when optimizing neural KB embedding models through the negative sampling loss. This closed form generalizes the ultimate objective of previous attempts on designing various scoring functions, such as NTN, TransE, Bilinear, and DISTMULT. This finding also links the neural KB embedding framework with the tensor-based KB embedding approach.
+
+Third, building upon the discoveries, we propose a tensor decomposition based KB embedding framework, KBTD, to directly fit the closed form tensor with by leveraging the scoring functions proposed in several popular neural models. Our extensive experiments on WordNet and FreeBase demonstrate the outstanding performance of KBTD over the conventional margin-based neural framework. In addition, we point out the limitation of dissimilarity/distance based scoring function design, which is wildly adopted by the TransE/H/R/D models (Bordes et al., 2013; Wang et al., 2014; Lin et al., 2015; Ji et al., 2015).
+
+The rest of this paper is organized as follows. Section 2 discusses related work. Section 3 unveils the connection between the margin-based ranking loss and the negative sampling loss. Section 4 performs the theoretical analysis and subsequently presents our KBTD framework. Section 5 introduces the detailed experiments on the link prediction task for KBs. Section 6 concludes this paper.
+
+# 2 RELATED WORK
+
+Knowledge base embedding is being extensively explored and developed over the last few years, during which the major breakthroughs are resulted from the neural embedding and tensor factorization models (Bordes et al., 2013; Nickel et al., 2012). Our work focuses on understanding the fundamentals of neural KB embedding, as well as its connection with tensor models.
+
+Neural KB Embedding: Loss Functions. Neural knowledge base embedding is usually formulated as an optimization problem with different loss functions. The majority of existing KB models employ the margin-based ranking loss, which was first proposed (Collobert et al., 2011) for addressing the efficiency issue of softmax (Bengio et al., 2003) in the field of natural language models. A brief collection of recent margin-based KB embedding methods include the SE, Unstructured, and SME models (Bordes et al., 2011; 2012; 2014), SLM and NTN (Socher et al., 2013), DISTMULT (Yang et al., 2015), TransE (Bordes et al., 2013), TransH (Wang et al., 2014), TransR (Lin et al., 2015), TransD (Ji et al., 2015), and ProjE (Shi & Weninger, 2017). In addition, there are several models— Bilinear model (Jenatton et al., 2012) and CONV (Toutanova et al., 2015)—that adopt the negative sampling (NE) loss. Our work contributes to this line of research by providing the relationship between margin-based and negative sampling based KB embedding models.
+
+Neural KB Embedding: Scoring Functions. As summarized in (Yang et al., 2015), given a KB represented by a list of triplets, neural models learn embeddings by utilizing a neural network, wherein the first layer projects the two entities of each triplet into latent low-dimensional vectors, and the second layer leverages a scoring function to operate on each pair of entity vectors with relationspecific parameters. The major difference between neural KB models lie in the various ways that they design the scoring functions — TransE, NTN, Bilinear, and DISTMULT (See details in Table 3 of Section 5). To date, few attempts have been conducted to understand any common grounds behind these models. Our work furthers this direction by proposing a general tensor decomposition framework (KBTD) which unifies existing neural KB models.
+
+Tensor Decomposition for KB Embedding. Tensor decomposition has seen successes in structural and relational learning over decades (Kolda & Bader, 2009; Sun et al., 2006). Recent years also witness the natural application of this technique on learning KB embeddings, including the BCTF model (Sutskever et al., 2009), and RESCAL (Nickel et al., 2011). In this work, we show the closed relationship between tensor decomposition based KB embedding and neural KB embedding models.
+
+# 3 CONNECTING MARGIN-BASED LOSS WITH NEGATIVE SAMPLING LOSS
+
+Given a KB with entity set $E$ and relation set $R$ , represented by a set of triplets $T = \{ ( s , r , o ) \}$ with $s , o \in E$ and $r \in R$ , the goal of neural KB models is to learn a scoring function $f ( s , r , o )$ which evaluates an arbitrary triplet and outputs a scalar to measure the acceptability of this triplet, where high/low score indicates that the input triplet tends to be correct/wrong. As summarized by (Yang et al., 2015), most existing scoring functions can be unified by a neural network, where the first layer projects the two entities of each triplet into latent low-dimensional vectors, and the second layer applies either linear or bilinear transformation (or both) on entity vectors with relation-specific parameters.
+
+The scoring function is then fitted to a loss function to learn the representations of both entities and relations. The majority of neural KB embedding models adopt either a margin-based ranking loss or a negative sampling loss. Both loss functions leverage the known triplets $T$ as positive samples and the corrupted triplets $T ^ { \prime }$ as the negative ones. Following the literature, given a known triplet $( s , r , o ) \in T$ , its corrupted triplets $\mathbf { \Phi } _ { T _ { ( s , r , o ) } ^ { \prime } } ^ { - }$ are constructed by replacing either the subject entity $s$ or the object entity $t$ with an arbitrary entity from $E$ , i.e., $T _ { ( s , r , o ) } ^ { \prime } = \{ ( s ^ { \prime } , r , o ) | s ^ { \prime } \in E \} \cup$ $\{ ( s , r , o ^ { \prime } ) | o ^ { \prime } \in E \}$ . Given both positive and corrupted triplets, the objective of the margin-based ranking loss is to minimize:
+
+$$
+\mathcal { L } _ { \mathrm { M A R G I N } } = \sum _ { ( s , r , o ) \in T } \sum _ { ( s ^ { \prime } , r , o ^ { \prime } ) \in T _ { ( s , r , o ) } ^ { \prime } } \operatorname* { m a x } \left( 0 , \gamma + f ( s ^ { \prime } , r , o ^ { \prime } ) - f ( s , r , o ) \right) .
+$$
+
+Its challenge lies in, however, the second summation, which takes $O ( | E | )$ complexity to enumerate the whole entity set $E$ and is extremely time demanding. Therefore, in practice, the second summation is commonly approximated with sampling and the sample size is usually set to one (Bordes et al., 2013). Considering the local loss function $\ell _ { \mathrm { M A R G I N } } ( s , \dot { r } , o )$ for each positive triplet $( s , r , o )$ associated with one (sampled) corrupted triplet $( s ^ { \prime } , r , o ^ { \prime } )$ , the $\operatorname* { m a x } ( 0 , \cdot )$ loss can be smoothly approximated by the softplus function (Dugas et al., 2001), that is:
+
+$$
+\begin{array} { r l } & { \ell _ { \mathrm { M A R G I N } } ( s , r , o ) = \operatorname* { m a x } \left( 0 , \gamma + f ( s ^ { \prime } , r , o ^ { \prime } ) - f ( s , r , o ) \right) } \\ & { \qquad \approx \log \left( 1 + e ^ { \gamma + f ( s ^ { \prime } , r , o ^ { \prime } ) - f ( s , r , o ) } \right) . } \end{array}
+$$
+
+On the other hand, the negative sampling loss aims to optimize the following objective:
+
+$$
+\mathcal { L } _ { \mathrm { N E G } } = - \sum _ { ( s , r , o ) \in T } \left( \log \sigma ( f ( s , r , o ) ) + b \mathbb { E } _ { ( s ^ { \prime } , r , o ^ { \prime } ) \sim T _ { ( s , r , o ) } ^ { \prime } } \left[ \log \sigma ( - f ( s ^ { \prime } , r , o ^ { \prime } ) ) \right] \right) ,
+$$
+
+where $b$ is the number of negative samples and $\sigma ( \cdot )$ is the sigmoid function. Similarly, the expectation term can be replaced with its Monte Carlo approximation. To align with our previous
+
+discussion on the margin-based ranking loss, we also set negative sample size $b = 1$ and derive the local objective for a certain positive triplet $( s , r , o )$ to be:
+
+$$
+\begin{array} { r l } & { \ell _ { \mathrm { N E G } } ( s , r , o ) = - \log \sigma ( f ( s , r , o ) ) - \log \sigma ( - f ( s ^ { \prime } , r , o ^ { \prime } ) ) } \\ & { \phantom { \ell _ { \mathrm { N E G } } ( s , r , o ) = - \log \sigma ( f ( s , r , o ) ) - \log \sigma ( - f ( s ^ { \prime } , r , o ^ { \prime } ) ) } = \log \left( 1 + e ^ { f ( s ^ { \prime } , r , o ^ { \prime } ) - f ( s , r , o ) } + e ^ { f ( s ^ { \prime } , r , o ^ { \prime } ) } + e ^ { - f ( s , r , o ) } \right) . } \end{array}
+$$
+
+Eq. 1 and Eq. 3 reveal the close relationship between margin-based ranking loss and negative sampling loss in neural KB embedding. First, observed from the similar term $e ^ { f ( s ^ { \prime } , r , o ^ { \prime } ) - f ( s , r , o ) }$ , both loss functions implicitly encourage positive triplets to have relatively higher scores than the corrupted ones. Second, the extra term $e ^ { f ( s ^ { \prime } , r , o ^ { \prime } ) } \bar { + } e ^ { - f ( s , r , o ) }$ in the negative sampling loss suggests that in addition to the implicit comparison, it explicitly rewards the positive triplets to have high scores, and also encourages the corrupted ones to have low scores.
+
+# 4 UNIFYING NEURAL KB EMBEDDING AS TENSOR DECOMPOSITION
+
+From the above section, we observe that the margin-based loss and negative sampling loss share very similar form. In this section, we unify existing neural KB embedding models by assuming a general scoring function $f : E \times R \times E \to \mathbb { R }$ and the use of negative sampling loss. We present a theoretical analysis in Section 4.1, followed by our KBTD framework which formally defines KB embedding problem as a tensor decomposition problem in Section 4.2. Additionally, the connection between KBTD and classical tensor decomposition models (Nickel et al., 2011; 2012) is discussed in Section 4.2.
+
+# 4.1 THEORETICAL ANALYSIS OF NEURAL KB EMBEDDING
+
+To facilitate our analysis, we represent the KB triplets as a three-way binary tensor $ { \mathcal { X } } \_ { } \in$ $\{ 0 , 1 \} ^ { E \times R \times E }$ , where $\mathcal { X } _ { s , r , o } ~ = ~ 1$ indicates $( s , r , o ) \in T$ , while $\mathcal { X } _ { s , r , o } ~ = ~ 0$ for non-existing or unknown triplets. The loss function $\mathcal { L } _ { \mathrm { N E G } }$ in Eq. 2 can be re-formatted with $\mathcal { X } _ { s , r , o }$ as
+
+$$
+- \sum _ { s , r , o } \chi _ { s , r , o } \left\{ \log \sigma \left( f ( s , r , o ) \right) + \frac { b } { 2 } \mathbb { E } _ { o ^ { \prime } \sim P _ { N } } \left[ \log \sigma \left( - f ( s , r , o ^ { \prime } ) \right) \right] + \frac { b } { 2 } \mathbb { E } _ { s ^ { \prime } \sim P _ { N } } \left[ \log \sigma \left( - f ( s ^ { \prime } , r , o ) \right) \right] \right\} ,
+$$
+
+where $P _ { N }$ is a uniform distribution over all entities, i.e., $\begin{array} { r } { { P _ { N } } ( \cdot ) = \frac { 1 } { | E | } } \end{array}$ . We further break down the summation and arrive at the following form:
+
+$$
+\begin{array} { r l } {  { \mathcal { L } _ { \mathrm { N E G } } = - \sum _ { s , r , o } \chi _ { s , r , o } \log \sigma ( f ( s , r , o ) ) } } \\ & { \quad - \frac { b } { 2 } ( \sum _ { s , r } d _ { o , r } ^ { \mathrm { o u t } } \mathbb { E } _ { o ^ { \prime } \sim P _ { N } } [ \log \sigma ( - f ( s , r , o ^ { \prime } ) ) ] + \sum _ { r , o } d _ { o , r } ^ { \mathrm { i n } } \mathbb { E } _ { s ^ { \prime } \sim P _ { N } } [ \log \sigma ( - f ( s ^ { \prime } , r , o ) ) ] ) , } \end{array}
+$$
+
+where $d _ { s , r } ^ { \mathrm { o u t } } = \textstyle \sum _ { o } \mathcal { X } _ { s , r , o }$ is the out-degree of entity $s$ under relation $r$ , and $\begin{array} { r } { d _ { o , r } ^ { \mathrm { i n } } = \sum _ { s } \mathcal { X } _ { s , r , o } } \end{array}$ is the in-degree of entity $o$ under relation $r$ . Then we explicitly express the two expectation terms:
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \sigma ^ { \prime } \sim P _ { N } } \left[ \log \sigma \left( - f ( s , r , \sigma ^ { \prime } ) \right) \right] = \displaystyle \sum _ { \sigma ^ { \prime } } \frac { 1 } { | E | } \log \sigma \left( - f ( s , r , \sigma ^ { \prime } ) \right) } \\ & { \qquad = \displaystyle \frac { 1 } { | E | } \log \sigma \left( - f ( s , r , \sigma ) \right) + \displaystyle \sum _ { \sigma ^ { \prime } \neq \sigma } \frac { 1 } { | E | } \log \sigma \left( - f ( s , r , \sigma ^ { \prime } ) \right) } \\ & { \mathbb { E } _ { s ^ { \prime } \sim P _ { N } } \left[ \log \sigma \left( - f ( s ^ { \prime } , r , \sigma ) \right) \right] = \displaystyle \frac { 1 } { | E | } \log \sigma \left( - f ( s , r , \sigma ) \right) + \displaystyle \sum _ { s ^ { \prime } \neq s } \frac { 1 } { | E | } \log \sigma \left( - f ( s ^ { \prime } , r , \sigma ) \right) . } \end{array}
+$$
+
+Then, by utilizing the above expectation terms, the local loss function for each specific triplet $( s , r , o )$ in Eq. 4 can be defined as
+
+$$
+\ell ( s , r , o ) = - \mathcal { X } _ { s , r , o } \log \sigma \left( f ( s , r , o ) \right) - \frac { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } { 2 \left. E \right. } \log \sigma \left( - f ( s , r , o ) \right) .
+$$
+
+Table 2: The comparison between RESCAL and our KBTD framework with parameters $\Theta$ initialized in the bilinear way, that is, $\Theta = \{ \pmb { a } _ { 1 } , \cdot \cdot \cdot , \pmb { a } _ { | E | } , \pmb { W } _ { 1 } , \cdot \cdot \cdot , \pmb { W } _ { | R | } \}$ .   
+
+<table><tr><td>RESCAL</td><td> mine∑s,ro (xs,ro-asTWrat)²</td></tr><tr><td>KBTD</td><td> mine ∑s,roWs,r, (Us,ro - asT Wrat)²</td></tr></table>
+
+The work in (Levy & Goldberg, 2014) suggested that for sufficient large embedding dimensionality, each individual $f ( s , r , o )$ can assume a value independence1. Following this assumption enables us to treat the objective $\mathcal { L }$ as a function of independent $f ( s , r , o )$ terms. The partial derivative with respect to $f ( s , \overline { { r } } , o )$ can be taken as:
+
+$$
+\frac { \partial \mathcal { L } } { \partial f ( s , r , o ) } = \frac { \partial \ell ( s , r , o ) } { \partial f ( s , r , o ) } = - \mathcal { X } _ { s , r , o } \sigma \left( - f ( s , r , o ) \right) + \frac { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } { \left| E \right| } \sigma \left( f ( s , r , o ) \right) .
+$$
+
+By setting the derivative to zero, we have
+
+$$
+e ^ { 2 f ( s , r , o ) } - \left( \frac { 2 \left| E \right| \mathcal { X } _ { s , r , o } } { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } - 1 \right) e ^ { f ( s , r , o ) } - \frac { 2 \left| E \right| \mathcal { X } _ { s , r , o } } { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } = 0 ,
+$$
+
+which implies
+
+$$
+f ( s , r , o ) = \log \left( \frac { 2 \left. E \right. \mathcal { X } _ { s , r , o } } { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } \right) .
+$$
+
+The RHS of Eq. 5 defines a “transformed” tensor based on $\mathcal { X }$ , and the LHS of Eq. 5 implies a regression problem, i.e., try to fit the $( s , r , o )$ -entry of the transformed tensor with the scoring function ${ \bar { f } } ( s , r , o )$ . In next section, we formally define this problem and then propose our KBTD framework.
+
+# 4.2 KBTD: NEURAL KB EMBEDDING AS TENSOR DECOMPOSITION
+
+In this section, we formalize the KB embedding problem analyzed in Section 4.1 as a tensor decomposition problem. We further present our framework—KBTD—to learn latent embedding for KB entities and relations. Its connection with classical tensor decomposition methods is also discussed.
+
+First, as mentioned at the end of Section 4.1, RHS of Eq. 5 defines a transformed tensor based on $\mathcal { X }$ . Here we denote it to be tensor $\mathcal { V } \in \mathbb { R } ^ { | E | \times | R | \times | E | }$ , with the $( s , r , o )$ -entry defined to be
+
+$$
+\mathcal { V } _ { s , r , o } = \log \left( \frac { 2 \left| E \right| \mathcal { X } _ { s , r , o } } { b \cdot \left( d _ { s , r } ^ { \mathrm { o u t } } + d _ { o , r } ^ { \mathrm { i n } } \right) } \right) .
+$$
+
+Second, our discussion in Section 4.1 actually implies a weighted tensor decomposition problem. This is mainly due to the negative sampling mechanism — we only care about positive triplets and corrupted triplets. This mechanism can be characterized by a binary tensor $\mathcal { W } \in \{ 0 , 1 \} ^ { | E | \times | R | \times | E | }$ , wherein $\bar { \mathcal { W } _ { s , r , o } } = 1$ if and only if $( s , r , o )$ is either a positive triplet or a corrupted triplet. Given the definition of tensor $\mathcal { V }$ and tensor $\mathcal { W }$ , we can formalize the following tensor decomposition problem:
+
+$$
+\operatorname* { m i n } _ { \Theta } \sum _ { s , r , o } \mathcal { W } _ { s , r , o } \left( \mathcal { V } _ { s , r , o } - f _ { \Theta } ( s , r , o ) \right) ^ { 2 } ,
+$$
+
+where $f _ { \Theta }$ is the scoring function parameterized by $\Theta$ .
+
+Revisiting RESCAL (Nickel et al., 2011; 2012). Before introducing how we optimize Eq. 7, we would like to discuss the connection between KBTD and RESCAL—a classical tensor decomposition model for KB embedding. Table 2 lists the optimization problems solved by RESCAL and our framework, in which the scoring function in Eq. 7 is initialized as a bilinear function, i.e., $f ( s , r , o ) = \pmb { a } _ { s } ^ { \top } \pmb { W } _ { r } \pmb { a } _ { o }$ , where $\mathbf { a } _ { s } , \pmb { a } _ { o } \in \bar { \mathbb { R } } ^ { d }$ and $W _ { r } \in \mathbb { R } ^ { \bar { d } \times d }$ . We observe the following connections and differences between them. First, both models explain a RDF triplet $( s , r , o )$ through the latent
+
+# Algorithm 1: The KBTD Framework
+
+input: Training set $T = \{ ( s , r , o ) \}$ , entity and relation set $E$ and $R$ , corrupted triplets multiplier $\lambda$ , mini-batch size $B$ output: Models parameters $\Theta$ , including entity and relation embeddings 1 Initialize model parameters $\Theta$ ; 2 while do $/ \star$ Sample a mini-batch of size $B$ \*/ 3 $T _ { b a t c h } \gets \mathrm { s a m p l e } ( T , B )$ ; $/ \star \delta { \sf a m p 1 } \mathrm { e }$ corrupted triplets for this mini-batch \*/ 4 $T _ { b a t c h } ^ { \prime } \gets \emptyset$ ; 5 for $( s , r , o ) \in T _ { b a t c h }$ do 6 for $i = 1$ to $\lambda$ do 7 s0 ← sample $( E )$ ; 8 o0 ← sample(E); 9 $T _ { b a t c h } ^ { \prime } \gets T _ { b a t c h } ^ { \prime } \cup ( s ^ { \prime } , r , o ) \cup ( s , r , o ^ { \prime } ) ;$ 10 Update parameter Θ w.r.t. P(s,r,o)∈Tbatch∪T 0 $\begin{array} { r } { \sum _ { ( s , r , o ) \in T _ { b a t c h } \cup T _ { b a t c h } ^ { \prime } } \left( \mathcal { V } _ { s , r , o } - f _ { \Theta } ( s , r , o ) \right) ^ { 2 } ; } \end{array}$
+
+representations $\mathbf { \delta } _ { a _ { s } , a _ { o } }$ and $W _ { r }$ . To learn the representations, however, RESCAL directly factorizes the binary tensor $\mathcal { X }$ , while our model decomposes a transformed real-value tensor $\mathcal { V }$ . Second, KBTD also differs with RESCAL in the way they treat the unobserved triplets. Notice that given a KB of observed (positive) triplets, the unobserved triples includes both positive and negative ones. This issue is known as the one-class problem (Moya & Hush, 1996; Pan et al., 2008). Two common solutions to this problem are AMAN (all missing as negative) and AMAU (all missing as unknown). The RESCAL model simply adopts the AMAN strategy by assuming all unobserved triplets as negative ones. However, our model is able to implicitly compromise between AMAN and AMAU by only treating corrupted triplets as negative.
+
+KBTD Learning. The detailed optimization procedure for KBTD is described in Algorithm 1. We optimize the objective function in Eq. 7 using mini-batch stochastic gradient descent with AdaGrad (Duchi et al., 2011). At each main iteration (Line 3-10), we first sample a mini-batch of positive triplets (Line 4) and then sample their corrupted triplets whose size is controlled by a multiplier $\lambda$ (Line 6-10). The parameters are updated with respect to the sampled positive triplets as well as corresponding corrupted ones. In this setting, we avoid generating the dense tensors $\mathcal { V }$ and $\mathcal { W }$ , which may in practice result in memory issues.
+
+There is a computational issue that comes from the log operator. For an unobserved triplet $( s , r , o )$ (i.e., $\mathcal { X } _ { s , r , o } = 0 \mathrm { , }$ ), $\mathcal { V } _ { s , r , o } = \log 0 = - \infty$ . Previously, two approaches have been proposed for addressing it (Levy & Goldberg, 2014). One is to smooth the logarithm by adding a small constant to tensor $\mathcal { X }$ , generating a dense tensor. The other one is to apply an additional shifted-truncated operator, that is, $\operatorname* { m a x } ( \mathcal { V } _ { s , r , o } - c , 0 )$ , generating a sparse tensor with the loss of certain information. Due to the obvious drawbacks, we instead propose to use a simple and effective solution, wherein the operation $\log x$ is replaced with $\log ( \epsilon + \bar { x } )$ with $\epsilon$ as a tunable parameter.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate the proposed KBTD framework on the canonical link prediction task against several popular KB embedding methods on two datasets extracted from WordNet and FreeBase. In this task, we are given a KB with a certain fraction of triplets removed, and our target is to predict these missing triplets. We first introduce our experimental setup in Section 5.1, followed by detailed discussion on experimental results in Section 5.2.
+
+# 5.1 EXPERIMENTAL SETUP
+
+Datasets. We use WordNet (WN18) and FreeBase (FB15k) datasets as introduced in (Bordes et al., 2013) where WN18 consists of 151, 442 triplets with 40,943 entities and 18 relations, and FB15k contains 592,213 triplets with 14,951 entities and 1,345 relations. We use the same training/validation/test split as in (Bordes et al., 2013; Yang et al., 2015).
+
+Table 3: Scoring Functions and Parameters.   
+
+<table><tr><td rowspan=1 colspan=1>Model Name</td><td rowspan=1 colspan=3>ParametersΘ</td><td rowspan=1 colspan=3>Scoring Function fe(s,r,o)</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=3>{1,,|R1，W[:.k]V1,.,|R|，,V1,.,|R|,a1,.,|Ei}</td><td rowspan=1 colspan=1>ur tanh(asW1:k]lao+Vr</td><td rowspan=1 colspan=1>asao</td><td rowspan=1 colspan=1>+br</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{w1,..,R,a.,E}</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>-llas+wr-all</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{W1,..,|R,a1.,E</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>aWrao</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{w1...,|,a1...,E</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>adiag(wr)ao</td></tr></table>
+
+Table 4: Experimental Results on the WN18 Dataset.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Results from Neural KB Embedding</td><td rowspan=1 colspan=2>Results from our KBTD framework</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS @10</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=1>0.53</td><td rowspan=1 colspan=1>66.10</td><td rowspan=1 colspan=1>0.85</td><td rowspan=1 colspan=1>90.50</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1>0.38</td><td rowspan=1 colspan=1>90.90</td><td rowspan=1 colspan=1>0.39</td><td rowspan=1 colspan=1>82.18</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>92.80</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>94.62</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>94.20</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>94.62</td></tr></table>
+
+Table 5: Experimental Results on the FB15k Dataset.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Results from Neural KB Embedding</td><td rowspan=1 colspan=2>Results fromourKBTD framework</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>41.40</td><td rowspan=1 colspan=1>0.37</td><td rowspan=1 colspan=1>59.06</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>53.90</td><td rowspan=1 colspan=1>0.30</td><td rowspan=1 colspan=1>49.94</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>51.90</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>54.95</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>57.70</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>59.91</td></tr></table>
+
+Baselines. We compare our proposed framework with TransE (Bordes et al., 2013), NTN (Socher et al., 2013), Bilinear (Jenatton et al., 2012) and DISTMULT (Yang et al., 2015). The original TransE model is based on dissimilarity/distance function. To fit our framework, we define the scoring function for TransE to be negative dissimilarity/distance function, i.e., $f ( s , r , o ) ~ =$ $- \left\| \pmb { a } _ { s } + \pmb { a } _ { r } - \pmb { a } _ { o } \right\| _ { 2 } ^ { 2 }$ . For NTN, Bilinear and DISTMULT, we inherit the scoring functions from their paper. The detailed scoring functions as well as their parameters are listed in Table 3. For the meaning of parameters and the intuition behind scoring functions, readers can refer to the original papers.
+
+Evaluation Protocol. We exactly follow the experimental procedure and treatment used in TransE (Bordes et al., 2013) and DISTMULT (Yang et al., 2015). For each triplet $( s , r , o )$ in the test set, the subject entity $s$ is replaced with each of entities from entity set $E$ in turn. We apply corresponding scoring function $f$ on those corrupted triplets and then sort them in non-increasing order to get the rank of the correct triplet. This procedure is then repeated for the object entity $o$ . For evaluation metrics, we consider Mean Reciprocal Rank (MRR) which is defined to be an average of the reciprocal rank of the correct triplets over all test triplets, and $H I T S @ I O$ (top-10 accuracy). If possible, we list the experimental results reported in (Yang et al., 2015) directly. In addition, we apply the filtered setting from (Bordes et al., 2013; Yang et al., 2015) in evaluation. In this setting, for one certain test triplet $( s , r , o )$ , we removed from the list of corrupted triplets all the triplets which appear in training, validation, or test set, except $( s , r , o )$ itself. This setting, for example, can avoid cases where lots of triplets in training set rank above the one of interest.
+
+Implementation Details. All the models in our framework were implemented using PyTorch in a machine with one 12GB GPU. Since the complexities of the aforementioned approaches vary a lot, in order to achieve the best accuracy for all the models, we cross-validate using the validation set to find the best hyperparameters. We found that, except for TransE on WN18, all the methods on both datasets can share the same hyper-parameters: (1) dimensionality $d = 1 0 0$ ; (2) smoothing parameter $\epsilon = 0 . 0 1$ ; (3) multiplier $\lambda$ mentioned in Algorithm 1 was set to 2; (4) the learning rate of AdaGrad algorithm was set to 0.1 (0.01 for TransE on WN18); (5) $\ell _ { 2 }$ -regularization applied to all the parameters using the weight 0.0001; (6) the mini-batch size is set to 2,048 (4,800 for TransE on WN18); (6) $b = 1$ in Eq. 6 (200 for TransE on WN18). For the additional hyper-parameter in NTN method, i.e., the number of slices $k$ , was set to 2. We allow all the algorithms to run at most 10,000 epochs over the training data, and the best model was selected by early stopping using $\mathrm { H I T S } @ 1 0$ score on the validation sets. By taking advantages of GPU computation, every training experiment can be finished within 4 hours.
+
+Table 6: Model Complexity in terms of #Parameters. $d$ is the embedding dimension, and $k$ in NTN is the number of slices.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3># Parameters</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=3>O(R|d²k +|E|d)</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=3>O(Rd+Ed)</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=3>O(R|d² +E|d)</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>O(R</td><td rowspan=1 colspan=1>d+</td><td rowspan=1 colspan=1>Ed)</td></tr></table>
+
+# 5.2 EXPERIMENTAL RESULTS
+
+Table 4 and Table 5 list the overall results on the WN18 and FB15k datasets for several popular models under both our KBTD framework and the neural KB embedding framework, respectively. In general, we have the following key observations and insights:
+
+(1) On the WN18 dataset, the KBTD framework achieves the best performance among most cases. In terms of MRR, KBTD outperforms all baselines except DISTMULT with an impressive improvement up to $6 0 . 4 \%$ (0.85 v.s. 0.53) on NTN. In terms of $\mathrm { H I T S } @ 1 0$ , KBTD outperforms baselines except TransE with an improvement up to $3 6 . 9 \%$ (90.50 v.s. 66.10) on NTN. Similar results can also be observed on the FB15k dataset in Table 5. In terms of both MRR and $\mathrm { H I T S } @ 1 0 .$ , KBTD outperforms all baselines except TransE with an improvement greater than $4 2 . 7 \%$ on NTN.
+
+(2) It is notable that KBTD outperforms NTN by large margins on both datasets. We conjecture that this comes from the very high model complexity of NTN, as suggested by Table 6. In our KBTD framework, we reduce the previously considered margin-based ranking problem in the original NTN to a simple regression problem. As a result, KBTD enables the efficient training procedure to significantly boost up NTN with respect to both the MRR and $\mathrm { H I T S } @ 1 0$ metrics.
+
+(3) It is also worth noting that under the KBTD framework, most models generate comparable or better results than their neural KB embedding versions. In terms of $\mathrm { H I T S } @ 1 0$ , KBTD underperforms TransE by $9 . 6 \%$ (82.18 v.s. 90.90) on WN18 and $7 . 3 \%$ (49.94 v.s. 53.90) on $\mathrm { F B } 1 5 \mathrm { k }$ . We attribute this underperformance to the constraint on TransE’s scoring function. As showed in Table 3, when instantiating the KBTD framework with TransE, the scoring function is set to be the negative dissimilarity function— $\begin{array} { r } { \mathbf { \sigma } _ { - } f ( s , r , o ) = - \left. \mathbf { a } _ { s } + \mathbf { w } _ { r } - \mathbf { a } _ { o } \right. _ { 2 } ^ { 2 } } \end{array}$ , which is for sure non-positive. However, the tensor $\mathcal { V }$ in Eq. 7 that KBTD aims to fit allows both positive and negative entries. In specific, for the observed triplets and a moderate $b$ , tensor entries are usually positive; for the corrupted triplets and a small $\epsilon$ , tensor entries reach negative values. On the contrary, the scoring functions of NTN, Bilinear, and DISTMULT are able to model both positive and negative tensor entries. That said, the non-positive constraint of TransE’s scoring function limits its ability to learn better latent KB representations in this link prediction task.
+
+# 6 CONCLUSION
+
+In this work, we provide a theoretical analysis of conventional neural KB embedding models and unveil the link between them and tensor-based KB embedding models. We show that the existing neural KB models can be unified into one tensor decomposition framework. We further propose the KBTD framework to directly fit the derived closed-form tensor. Our extensive experiments suggest that KBTD achieves consistent performance improvements over NTN, Bilinear, and DISTMULT under the neural KB embedding framework.
+
+For further work, one interesting direction is to exploit efficient and scalable algorithms that extend KBTD to web-scale KBs. Another direction is to leverage the effective techniques from the matrix factorization community to enhance our tensor framework, such as the usage of bias terms and rich contextual information.
+
+# REFERENCES
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+Antoine Bordes, Jason Weston, Ronan Collobert, Yoshua Bengio, et al. Learning structured embeddings of knowledge bases. In AAAI ’11, volume 6, pp. 6, 2011.
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+Richard Socher, Danqi Chen, Christopher D Manning, and Andrew Ng. Reasoning with neural tensor networks for knowledge base completion. In NIPS ’13, pp. 926–934, 2013.
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parse/train/S1sRrN-CW/S1sRrN-CW_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "REVISITING KNOWLEDGE BASE EMBEDDING AS TEN-SOR DECOMPOSITION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            234,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We study the problem of knowledge base (KB) embedding, which is usually addressed through two frameworks—neural KB embedding and tensor decomposition. In this work, we theoretically analyze the neural embedding framework and subsequently connect it with tensor based embedding. Specifically, we show that in neural KB embedding the two commonly adopted optimization solutions— margin-based and negative sampling losses—are closely related to each other. We also reach the closed-form tensor that is implicitly approximated by popular neural KB approaches, revealing the underlying connection between neural and tensor based KB embedding models. Grounded in the theoretical results, we further present a tensor decomposition based framework KBTD to directly approximate the derived closed form tensor. Under this framework, the neural KB embedding models, such as NTN (Socher et al., 2013), TransE (Bordes et al., 2013), Bilinear (Jenatton et al., 2012), and DISTMULT (Yang et al., 2015), are unified into a general tensor optimization architecture. Finally, we conduct experiments on the link prediction task in WordNet and Freebase, empirically demonstrating the effectiveness of the KBTD framework. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Knowledge bases (KBs) power many of semantic-oriented techniques and applications, such as question answering and intelligent personal assistant. A classical example is the automatic answer to the query “Who is Barack Obama’s wife” by KB-supported search engines. Most if not all of KBs achieve this by storing the facts about the world in the form of RDF triplets (W3C, 1999), wherein a triplet (subject, predicate, object), in short $( s , r , o )$ , records a piece of fact about the relation between the two entities—the subject and object. To automatically construct Web-scale KBs with billions of facts (triplets), a significant line of effort has been devoted to knowledge base embedding—the technique of encoding entities and their relational information into latent representations (Bordes et al., 2011; 2013; Socher et al., 2013; Yang et al., 2015). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In particular, the direction of neural embedding has been extensively explored for learning representations for KBs, offering state-of-the-art performance for validating and completing unseen facts (Yang et al., 2015). Briefly, given a KB represented by triplets $T = \\{ ( s , r , o ) \\}$ , neural embedding models take these known triplets as positive instances and corrupted triplets $\\{ ( \\boldsymbol { s } ^ { \\prime } , \\boldsymbol { r } , o ^ { \\prime } ) \\}$ as negative ones. For each triplet, a scoring function $f ( s , r , o )$ — parameterized with a neural network—is designed to project the associated entities $s , o$ and their relational information $r$ into a scalar. Most of the existing models are then trained through two popular choices of loss functions, including the margin-based ranking loss by NTN (Socher et al., 2013), TransE (Bordes et al., 2013), and DISTMULT (Yang et al., 2015), as well as several practices of the Negative Sampling loss (Mikolov et al., 2013) by Bilinear (or LFM) (Jenatton et al., 2012) and CONV (Toutanova et al., 2015). In addition, another line of KB embedding is focused on tensor decomposition based frameworks, such as RESCAL (Nickel et al., 2011; 2012). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Notwithstanding the rapid development and progress of KB embedding techniques, insights concerning their underlying mechanisms are to date sorely lacking. For example, a natural question that arises is what are the relationships or differences between the margin-based ranking loss and the negative sampling loss for neural KB embedding. Moreover, what is the quantity that is optimized by conventional neural KB embedding models? With an eye toward comprehensively understanding ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/9b694e2614285efaed6c6e8014128e9f087c6c202fefa20c1ba1ef86d145ce24.jpg",
+        "table_caption": [
+            "Table 1: Knowledge Base Embedding. "
+        ],
+        "table_footnote": [
+            "See detailed notations in Sections 3 and 4. A brief introduction is listed below: ",
+            "• $( s , r , o )$ & $( s ^ { \\prime } , r , o ^ { \\prime } )$ : the known and corrupted triplets, respectively; • $E$ & $R$ : the entity and relation sets of a given knowledge base, with s, $o \\in E$ and $r \\in R$ ; • $\\mathcal { X } \\in \\{ 0 , 1 \\} ^ { E \\times R \\times E }$ : a three-way binary tensor, with $\\mathcal { X } _ { s , r , o } = 1$ indicating $( s , r , o ) \\in T$ and otherwise 0; • $\\begin{array} { r } { d _ { s , r } ^ { \\mathrm { o u t } } = \\sum _ { o } \\mathcal { X } _ { s , r , o } } \\end{array}$ & $\\begin{array} { r } { d _ { o , r } ^ { \\mathrm { i n } } = \\sum _ { s } \\mathcal { X } _ { s , r , o } } \\end{array}$ : the out- $/ \\mathrm { i n } \\cdot$ - degree of entity $s / o$ under relation $r$ , respectively; • $b$ : the number of negative samples. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>local margin-based loss (ReLU)</td><td rowspan=1 colspan=4>max(0,f(s,r,o&#x27;)- f(s,r,o)+ γ)</td></tr><tr><td rowspan=1 colspan=1>local margin-based loss (Softplus)</td><td rowspan=1 colspan=1>log (</td><td rowspan=1 colspan=2>log(1+ ef(s&#x27;,r,o&#x27;)-f(s,r,)+γ)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>local negative sampling loss</td><td rowspan=1 colspan=1>log(</td><td rowspan=1 colspan=3>log(1+ef(s&#x27;ro&#x27;)-f(s,r）+ef(s&#x27;r,0&#x27;）+e-f(s,r,o))</td></tr><tr><td rowspan=1 colspan=1>closed-form tensor</td><td rowspan=1 colspan=1>log</td><td rowspan=1 colspan=1>2|E|Xs,r.o（b（do+d，)</td><td rowspan=1 colspan=2></td></tr></table>",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "KB embedding, we investigate (1) the connection between margin-based ranking loss and negative sampling loss in neural KB models, (2) the relationship between neural KB models and classical tensor-based KB models, and (3) the universal framework for KB learning. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions. In this work, we unveil two fundamentals of KB embedding, according to which we further present a tensor decomposition based KB embedding framework—KBTD, yielding significant outperformance over neural KB embedding models in most cases. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "First, with softplus’ smooth approximation to ReLU in the margin-based loss (Dugas et al., 2001), we show that the margin-based loss is closely connected to the negative sampling loss (See rows 2 & 3 in Table 1). In specific, both losses aim to encourage positive triplets $( s , r , o )$ and penalize corrupted ones $( s ^ { \\prime } , \\bar { r } , o ^ { \\prime } )$ , and the slight difference lies in the extra reward or penalization to corresponding triplets in the negative sampling loss. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Second, we derive the closed form tensor (See row 4 in Table 1) whose entry is implicitly fitted (approximated) by scoring function $f ( s , r , o )$ , when optimizing neural KB embedding models through the negative sampling loss. This closed form generalizes the ultimate objective of previous attempts on designing various scoring functions, such as NTN, TransE, Bilinear, and DISTMULT. This finding also links the neural KB embedding framework with the tensor-based KB embedding approach. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Third, building upon the discoveries, we propose a tensor decomposition based KB embedding framework, KBTD, to directly fit the closed form tensor with by leveraging the scoring functions proposed in several popular neural models. Our extensive experiments on WordNet and FreeBase demonstrate the outstanding performance of KBTD over the conventional margin-based neural framework. In addition, we point out the limitation of dissimilarity/distance based scoring function design, which is wildly adopted by the TransE/H/R/D models (Bordes et al., 2013; Wang et al., 2014; Lin et al., 2015; Ji et al., 2015). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The rest of this paper is organized as follows. Section 2 discusses related work. Section 3 unveils the connection between the margin-based ranking loss and the negative sampling loss. Section 4 performs the theoretical analysis and subsequently presents our KBTD framework. Section 5 introduces the detailed experiments on the link prediction task for KBs. Section 6 concludes this paper. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge base embedding is being extensively explored and developed over the last few years, during which the major breakthroughs are resulted from the neural embedding and tensor factorization models (Bordes et al., 2013; Nickel et al., 2012). Our work focuses on understanding the fundamentals of neural KB embedding, as well as its connection with tensor models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Neural KB Embedding: Loss Functions. Neural knowledge base embedding is usually formulated as an optimization problem with different loss functions. The majority of existing KB models employ the margin-based ranking loss, which was first proposed (Collobert et al., 2011) for addressing the efficiency issue of softmax (Bengio et al., 2003) in the field of natural language models. A brief collection of recent margin-based KB embedding methods include the SE, Unstructured, and SME models (Bordes et al., 2011; 2012; 2014), SLM and NTN (Socher et al., 2013), DISTMULT (Yang et al., 2015), TransE (Bordes et al., 2013), TransH (Wang et al., 2014), TransR (Lin et al., 2015), TransD (Ji et al., 2015), and ProjE (Shi & Weninger, 2017). In addition, there are several models— Bilinear model (Jenatton et al., 2012) and CONV (Toutanova et al., 2015)—that adopt the negative sampling (NE) loss. Our work contributes to this line of research by providing the relationship between margin-based and negative sampling based KB embedding models. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural KB Embedding: Scoring Functions. As summarized in (Yang et al., 2015), given a KB represented by a list of triplets, neural models learn embeddings by utilizing a neural network, wherein the first layer projects the two entities of each triplet into latent low-dimensional vectors, and the second layer leverages a scoring function to operate on each pair of entity vectors with relationspecific parameters. The major difference between neural KB models lie in the various ways that they design the scoring functions — TransE, NTN, Bilinear, and DISTMULT (See details in Table 3 of Section 5). To date, few attempts have been conducted to understand any common grounds behind these models. Our work furthers this direction by proposing a general tensor decomposition framework (KBTD) which unifies existing neural KB models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Tensor Decomposition for KB Embedding. Tensor decomposition has seen successes in structural and relational learning over decades (Kolda & Bader, 2009; Sun et al., 2006). Recent years also witness the natural application of this technique on learning KB embeddings, including the BCTF model (Sutskever et al., 2009), and RESCAL (Nickel et al., 2011). In this work, we show the closed relationship between tensor decomposition based KB embedding and neural KB embedding models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 CONNECTING MARGIN-BASED LOSS WITH NEGATIVE SAMPLING LOSS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a KB with entity set $E$ and relation set $R$ , represented by a set of triplets $T = \\{ ( s , r , o ) \\}$ with $s , o \\in E$ and $r \\in R$ , the goal of neural KB models is to learn a scoring function $f ( s , r , o )$ which evaluates an arbitrary triplet and outputs a scalar to measure the acceptability of this triplet, where high/low score indicates that the input triplet tends to be correct/wrong. As summarized by (Yang et al., 2015), most existing scoring functions can be unified by a neural network, where the first layer projects the two entities of each triplet into latent low-dimensional vectors, and the second layer applies either linear or bilinear transformation (or both) on entity vectors with relation-specific parameters. ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The scoring function is then fitted to a loss function to learn the representations of both entities and relations. The majority of neural KB embedding models adopt either a margin-based ranking loss or a negative sampling loss. Both loss functions leverage the known triplets $T$ as positive samples and the corrupted triplets $T ^ { \\prime }$ as the negative ones. Following the literature, given a known triplet $( s , r , o ) \\in T$ , its corrupted triplets $\\mathbf { \\Phi } _ { T _ { ( s , r , o ) } ^ { \\prime } } ^ { - }$ are constructed by replacing either the subject entity $s$ or the object entity $t$ with an arbitrary entity from $E$ , i.e., $T _ { ( s , r , o ) } ^ { \\prime } = \\{ ( s ^ { \\prime } , r , o ) | s ^ { \\prime } \\in E \\} \\cup$ $\\{ ( s , r , o ^ { \\prime } ) | o ^ { \\prime } \\in E \\}$ . Given both positive and corrupted triplets, the objective of the margin-based ranking loss is to minimize: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2cef3da948bf8347010b04c8bce1ebe8a676f26076ea3b6d712ca39106191c6.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { M A R G I N } } = \\sum _ { ( s , r , o ) \\in T } \\sum _ { ( s ^ { \\prime } , r , o ^ { \\prime } ) \\in T _ { ( s , r , o ) } ^ { \\prime } } \\operatorname* { m a x } \\left( 0 , \\gamma + f ( s ^ { \\prime } , r , o ^ { \\prime } ) - f ( s , r , o ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Its challenge lies in, however, the second summation, which takes $O ( | E | )$ complexity to enumerate the whole entity set $E$ and is extremely time demanding. Therefore, in practice, the second summation is commonly approximated with sampling and the sample size is usually set to one (Bordes et al., 2013). Considering the local loss function $\\ell _ { \\mathrm { M A R G I N } } ( s , \\dot { r } , o )$ for each positive triplet $( s , r , o )$ associated with one (sampled) corrupted triplet $( s ^ { \\prime } , r , o ^ { \\prime } )$ , the $\\operatorname* { m a x } ( 0 , \\cdot )$ loss can be smoothly approximated by the softplus function (Dugas et al., 2001), that is: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8204bba0972c099c36fe7df784e8e829c532a753b6b19569fd1225ba6b920794.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\ell _ { \\mathrm { M A R G I N } } ( s , r , o ) = \\operatorname* { m a x } \\left( 0 , \\gamma + f ( s ^ { \\prime } , r , o ^ { \\prime } ) - f ( s , r , o ) \\right) } \\\\ & { \\qquad \\approx \\log \\left( 1 + e ^ { \\gamma + f ( s ^ { \\prime } , r , o ^ { \\prime } ) - f ( s , r , o ) } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, the negative sampling loss aims to optimize the following objective: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0f35141ee3545949a8f68de290191541623376ed995b4ffe4e17b5b18cceb11.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { N E G } } = - \\sum _ { ( s , r , o ) \\in T } \\left( \\log \\sigma ( f ( s , r , o ) ) + b \\mathbb { E } _ { ( s ^ { \\prime } , r , o ^ { \\prime } ) \\sim T _ { ( s , r , o ) } ^ { \\prime } } \\left[ \\log \\sigma ( - f ( s ^ { \\prime } , r , o ^ { \\prime } ) ) \\right] \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $b$ is the number of negative samples and $\\sigma ( \\cdot )$ is the sigmoid function. Similarly, the expectation term can be replaced with its Monte Carlo approximation. To align with our previous ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "discussion on the margin-based ranking loss, we also set negative sample size $b = 1$ and derive the local objective for a certain positive triplet $( s , r , o )$ to be: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77385169041a171f7328ab47fd62432b4a5c74c49142e76c3502196a897cd72a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\ell _ { \\mathrm { N E G } } ( s , r , o ) = - \\log \\sigma ( f ( s , r , o ) ) - \\log \\sigma ( - f ( s ^ { \\prime } , r , o ^ { \\prime } ) ) } \\\\ & { \\phantom { \\ell _ { \\mathrm { N E G } } ( s , r , o ) = - \\log \\sigma ( f ( s , r , o ) ) - \\log \\sigma ( - f ( s ^ { \\prime } , r , o ^ { \\prime } ) ) } = \\log \\left( 1 + e ^ { f ( s ^ { \\prime } , r , o ^ { \\prime } ) - f ( s , r , o ) } + e ^ { f ( s ^ { \\prime } , r , o ^ { \\prime } ) } + e ^ { - f ( s , r , o ) } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Eq. 1 and Eq. 3 reveal the close relationship between margin-based ranking loss and negative sampling loss in neural KB embedding. First, observed from the similar term $e ^ { f ( s ^ { \\prime } , r , o ^ { \\prime } ) - f ( s , r , o ) }$ , both loss functions implicitly encourage positive triplets to have relatively higher scores than the corrupted ones. Second, the extra term $e ^ { f ( s ^ { \\prime } , r , o ^ { \\prime } ) } \\bar { + } e ^ { - f ( s , r , o ) }$ in the negative sampling loss suggests that in addition to the implicit comparison, it explicitly rewards the positive triplets to have high scores, and also encourages the corrupted ones to have low scores. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 UNIFYING NEURAL KB EMBEDDING AS TENSOR DECOMPOSITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From the above section, we observe that the margin-based loss and negative sampling loss share very similar form. In this section, we unify existing neural KB embedding models by assuming a general scoring function $f : E \\times R \\times E \\to \\mathbb { R }$ and the use of negative sampling loss. We present a theoretical analysis in Section 4.1, followed by our KBTD framework which formally defines KB embedding problem as a tensor decomposition problem in Section 4.2. Additionally, the connection between KBTD and classical tensor decomposition models (Nickel et al., 2011; 2012) is discussed in Section 4.2. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 THEORETICAL ANALYSIS OF NEURAL KB EMBEDDING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To facilitate our analysis, we represent the KB triplets as a three-way binary tensor $ { \\mathcal { X } } \\_ { } \\in$ $\\{ 0 , 1 \\} ^ { E \\times R \\times E }$ , where $\\mathcal { X } _ { s , r , o } ~ = ~ 1$ indicates $( s , r , o ) \\in T$ , while $\\mathcal { X } _ { s , r , o } ~ = ~ 0$ for non-existing or unknown triplets. The loss function $\\mathcal { L } _ { \\mathrm { N E G } }$ in Eq. 2 can be re-formatted with $\\mathcal { X } _ { s , r , o }$ as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bc747f5d5c85e57f1b00f685bc9886a6fc8342d2cfa37b75eaf7bd9e8af9528.jpg",
+        "text": "$$\n- \\sum _ { s , r , o } \\chi _ { s , r , o } \\left\\{ \\log \\sigma \\left( f ( s , r , o ) \\right) + \\frac { b } { 2 } \\mathbb { E } _ { o ^ { \\prime } \\sim P _ { N } } \\left[ \\log \\sigma \\left( - f ( s , r , o ^ { \\prime } ) \\right) \\right] + \\frac { b } { 2 } \\mathbb { E } _ { s ^ { \\prime } \\sim P _ { N } } \\left[ \\log \\sigma \\left( - f ( s ^ { \\prime } , r , o ) \\right) \\right] \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $P _ { N }$ is a uniform distribution over all entities, i.e., $\\begin{array} { r } { { P _ { N } } ( \\cdot ) = \\frac { 1 } { | E | } } \\end{array}$ . We further break down the summation and arrive at the following form: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac6d391b23ef76d4565f27eba3cb5b94865a7e20c20b0e8485d9fcfc02bebf4d.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\mathcal { L } _ { \\mathrm { N E G } } = - \\sum _ { s , r , o } \\chi _ { s , r , o } \\log \\sigma ( f ( s , r , o ) ) } } \\\\ & { \\quad - \\frac { b } { 2 } ( \\sum _ { s , r } d _ { o , r } ^ { \\mathrm { o u t } } \\mathbb { E } _ { o ^ { \\prime } \\sim P _ { N } } [ \\log \\sigma ( - f ( s , r , o ^ { \\prime } ) ) ] + \\sum _ { r , o } d _ { o , r } ^ { \\mathrm { i n } } \\mathbb { E } _ { s ^ { \\prime } \\sim P _ { N } } [ \\log \\sigma ( - f ( s ^ { \\prime } , r , o ) ) ] ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $d _ { s , r } ^ { \\mathrm { o u t } } = \\textstyle \\sum _ { o } \\mathcal { X } _ { s , r , o }$ is the out-degree of entity $s$ under relation $r$ , and $\\begin{array} { r } { d _ { o , r } ^ { \\mathrm { i n } } = \\sum _ { s } \\mathcal { X } _ { s , r , o } } \\end{array}$ is the in-degree of entity $o$ under relation $r$ . Then we explicitly express the two expectation terms: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02edc2392057520a33bc415e05bf09a67a8f419018b236fa412f990f2a639679.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { \\sigma ^ { \\prime } \\sim P _ { N } } \\left[ \\log \\sigma \\left( - f ( s , r , \\sigma ^ { \\prime } ) \\right) \\right] = \\displaystyle \\sum _ { \\sigma ^ { \\prime } } \\frac { 1 } { | E | } \\log \\sigma \\left( - f ( s , r , \\sigma ^ { \\prime } ) \\right) } \\\\ & { \\qquad = \\displaystyle \\frac { 1 } { | E | } \\log \\sigma \\left( - f ( s , r , \\sigma ) \\right) + \\displaystyle \\sum _ { \\sigma ^ { \\prime } \\neq \\sigma } \\frac { 1 } { | E | } \\log \\sigma \\left( - f ( s , r , \\sigma ^ { \\prime } ) \\right) } \\\\ & { \\mathbb { E } _ { s ^ { \\prime } \\sim P _ { N } } \\left[ \\log \\sigma \\left( - f ( s ^ { \\prime } , r , \\sigma ) \\right) \\right] = \\displaystyle \\frac { 1 } { | E | } \\log \\sigma \\left( - f ( s , r , \\sigma ) \\right) + \\displaystyle \\sum _ { s ^ { \\prime } \\neq s } \\frac { 1 } { | E | } \\log \\sigma \\left( - f ( s ^ { \\prime } , r , \\sigma ) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, by utilizing the above expectation terms, the local loss function for each specific triplet $( s , r , o )$ in Eq. 4 can be defined as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81f93ccfa446cb9eb43faf5dd3f0c84e6d18900bcaa2f29912b4971d3995cd94.jpg",
+        "text": "$$\n\\ell ( s , r , o ) = - \\mathcal { X } _ { s , r , o } \\log \\sigma \\left( f ( s , r , o ) \\right) - \\frac { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } { 2 \\left. E \\right. } \\log \\sigma \\left( - f ( s , r , o ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/49bd52e029a2312c243236e557302f2486ed23fd880bfca3f80b7ceaed01e3fe.jpg",
+        "table_caption": [
+            "Table 2: The comparison between RESCAL and our KBTD framework with parameters $\\Theta$ initialized in the bilinear way, that is, $\\Theta = \\{ \\pmb { a } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { a } _ { | E | } , \\pmb { W } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { W } _ { | R | } \\}$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>RESCAL</td><td> mine∑s,ro (xs,ro-asTWrat)²</td></tr><tr><td>KBTD</td><td> mine ∑s,roWs,r, (Us,ro - asT Wrat)²</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The work in (Levy & Goldberg, 2014) suggested that for sufficient large embedding dimensionality, each individual $f ( s , r , o )$ can assume a value independence1. Following this assumption enables us to treat the objective $\\mathcal { L }$ as a function of independent $f ( s , r , o )$ terms. The partial derivative with respect to $f ( s , \\overline { { r } } , o )$ can be taken as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5586aafb08d11d4d956623dfe59f8aedc1ce97010de159441c1829c81fb6f35.jpg",
+        "text": "$$\n\\frac { \\partial \\mathcal { L } } { \\partial f ( s , r , o ) } = \\frac { \\partial \\ell ( s , r , o ) } { \\partial f ( s , r , o ) } = - \\mathcal { X } _ { s , r , o } \\sigma \\left( - f ( s , r , o ) \\right) + \\frac { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } { \\left| E \\right| } \\sigma \\left( f ( s , r , o ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By setting the derivative to zero, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/937ef1820e43784dbc9fe8ee2a36be49a57457c327837557884ed27b5aab93bd.jpg",
+        "text": "$$\ne ^ { 2 f ( s , r , o ) } - \\left( \\frac { 2 \\left| E \\right| \\mathcal { X } _ { s , r , o } } { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } - 1 \\right) e ^ { f ( s , r , o ) } - \\frac { 2 \\left| E \\right| \\mathcal { X } _ { s , r , o } } { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which implies ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60a6e63813805ac9307f60ba0b8b6ca0979ba7619f12b2d09102c7c0a6db848e.jpg",
+        "text": "$$\nf ( s , r , o ) = \\log \\left( \\frac { 2 \\left. E \\right. \\mathcal { X } _ { s , r , o } } { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The RHS of Eq. 5 defines a “transformed” tensor based on $\\mathcal { X }$ , and the LHS of Eq. 5 implies a regression problem, i.e., try to fit the $( s , r , o )$ -entry of the transformed tensor with the scoring function ${ \\bar { f } } ( s , r , o )$ . In next section, we formally define this problem and then propose our KBTD framework. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 KBTD: NEURAL KB EMBEDDING AS TENSOR DECOMPOSITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we formalize the KB embedding problem analyzed in Section 4.1 as a tensor decomposition problem. We further present our framework—KBTD—to learn latent embedding for KB entities and relations. Its connection with classical tensor decomposition methods is also discussed. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First, as mentioned at the end of Section 4.1, RHS of Eq. 5 defines a transformed tensor based on $\\mathcal { X }$ . Here we denote it to be tensor $\\mathcal { V } \\in \\mathbb { R } ^ { | E | \\times | R | \\times | E | }$ , with the $( s , r , o )$ -entry defined to be ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c6f7cd7343a28f22e281855198e41c8adcadf173ef702aec470ad6ae81fa535b.jpg",
+        "text": "$$\n\\mathcal { V } _ { s , r , o } = \\log \\left( \\frac { 2 \\left| E \\right| \\mathcal { X } _ { s , r , o } } { b \\cdot \\left( d _ { s , r } ^ { \\mathrm { o u t } } + d _ { o , r } ^ { \\mathrm { i n } } \\right) } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Second, our discussion in Section 4.1 actually implies a weighted tensor decomposition problem. This is mainly due to the negative sampling mechanism — we only care about positive triplets and corrupted triplets. This mechanism can be characterized by a binary tensor $\\mathcal { W } \\in \\{ 0 , 1 \\} ^ { | E | \\times | R | \\times | E | }$ , wherein $\\bar { \\mathcal { W } _ { s , r , o } } = 1$ if and only if $( s , r , o )$ is either a positive triplet or a corrupted triplet. Given the definition of tensor $\\mathcal { V }$ and tensor $\\mathcal { W }$ , we can formalize the following tensor decomposition problem: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f549cd22023099190757e14de283db89b99e47b0841487391ca83f9aeb7bfe13.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\sum _ { s , r , o } \\mathcal { W } _ { s , r , o } \\left( \\mathcal { V } _ { s , r , o } - f _ { \\Theta } ( s , r , o ) \\right) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { \\Theta }$ is the scoring function parameterized by $\\Theta$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Revisiting RESCAL (Nickel et al., 2011; 2012). Before introducing how we optimize Eq. 7, we would like to discuss the connection between KBTD and RESCAL—a classical tensor decomposition model for KB embedding. Table 2 lists the optimization problems solved by RESCAL and our framework, in which the scoring function in Eq. 7 is initialized as a bilinear function, i.e., $f ( s , r , o ) = \\pmb { a } _ { s } ^ { \\top } \\pmb { W } _ { r } \\pmb { a } _ { o }$ , where $\\mathbf { a } _ { s } , \\pmb { a } _ { o } \\in \\bar { \\mathbb { R } } ^ { d }$ and $W _ { r } \\in \\mathbb { R } ^ { \\bar { d } \\times d }$ . We observe the following connections and differences between them. First, both models explain a RDF triplet $( s , r , o )$ through the latent ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: The KBTD Framework ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "input: Training set $T = \\{ ( s , r , o ) \\}$ , entity and relation set $E$ and $R$ , corrupted triplets multiplier $\\lambda$ , mini-batch size $B$ output: Models parameters $\\Theta$ , including entity and relation embeddings 1 Initialize model parameters $\\Theta$ ; 2 while do $/ \\star$ Sample a mini-batch of size $B$ \\*/ 3 $T _ { b a t c h } \\gets \\mathrm { s a m p l e } ( T , B )$ ; $/ \\star \\delta { \\sf a m p 1 } \\mathrm { e }$ corrupted triplets for this mini-batch \\*/ 4 $T _ { b a t c h } ^ { \\prime } \\gets \\emptyset$ ; 5 for $( s , r , o ) \\in T _ { b a t c h }$ do 6 for $i = 1$ to $\\lambda$ do 7 s0 ← sample $( E )$ ; 8 o0 ← sample(E); 9 $T _ { b a t c h } ^ { \\prime } \\gets T _ { b a t c h } ^ { \\prime } \\cup ( s ^ { \\prime } , r , o ) \\cup ( s , r , o ^ { \\prime } ) ;$ 10 Update parameter Θ w.r.t. P(s,r,o)∈Tbatch∪T 0 $\\begin{array} { r } { \\sum _ { ( s , r , o ) \\in T _ { b a t c h } \\cup T _ { b a t c h } ^ { \\prime } } \\left( \\mathcal { V } _ { s , r , o } - f _ { \\Theta } ( s , r , o ) \\right) ^ { 2 } ; } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "representations $\\mathbf { \\delta } _ { a _ { s } , a _ { o } }$ and $W _ { r }$ . To learn the representations, however, RESCAL directly factorizes the binary tensor $\\mathcal { X }$ , while our model decomposes a transformed real-value tensor $\\mathcal { V }$ . Second, KBTD also differs with RESCAL in the way they treat the unobserved triplets. Notice that given a KB of observed (positive) triplets, the unobserved triples includes both positive and negative ones. This issue is known as the one-class problem (Moya & Hush, 1996; Pan et al., 2008). Two common solutions to this problem are AMAN (all missing as negative) and AMAU (all missing as unknown). The RESCAL model simply adopts the AMAN strategy by assuming all unobserved triplets as negative ones. However, our model is able to implicitly compromise between AMAN and AMAU by only treating corrupted triplets as negative. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "KBTD Learning. The detailed optimization procedure for KBTD is described in Algorithm 1. We optimize the objective function in Eq. 7 using mini-batch stochastic gradient descent with AdaGrad (Duchi et al., 2011). At each main iteration (Line 3-10), we first sample a mini-batch of positive triplets (Line 4) and then sample their corrupted triplets whose size is controlled by a multiplier $\\lambda$ (Line 6-10). The parameters are updated with respect to the sampled positive triplets as well as corresponding corrupted ones. In this setting, we avoid generating the dense tensors $\\mathcal { V }$ and $\\mathcal { W }$ , which may in practice result in memory issues. ",
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+    {
+        "type": "text",
+        "text": "There is a computational issue that comes from the log operator. For an unobserved triplet $( s , r , o )$ (i.e., $\\mathcal { X } _ { s , r , o } = 0 \\mathrm { , }$ ), $\\mathcal { V } _ { s , r , o } = \\log 0 = - \\infty$ . Previously, two approaches have been proposed for addressing it (Levy & Goldberg, 2014). One is to smooth the logarithm by adding a small constant to tensor $\\mathcal { X }$ , generating a dense tensor. The other one is to apply an additional shifted-truncated operator, that is, $\\operatorname* { m a x } ( \\mathcal { V } _ { s , r , o } - c , 0 )$ , generating a sparse tensor with the loss of certain information. Due to the obvious drawbacks, we instead propose to use a simple and effective solution, wherein the operation $\\log x$ is replaced with $\\log ( \\epsilon + \\bar { x } )$ with $\\epsilon$ as a tunable parameter. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we evaluate the proposed KBTD framework on the canonical link prediction task against several popular KB embedding methods on two datasets extracted from WordNet and FreeBase. In this task, we are given a KB with a certain fraction of triplets removed, and our target is to predict these missing triplets. We first introduce our experimental setup in Section 5.1, followed by detailed discussion on experimental results in Section 5.2. ",
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+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Datasets. We use WordNet (WN18) and FreeBase (FB15k) datasets as introduced in (Bordes et al., 2013) where WN18 consists of 151, 442 triplets with 40,943 entities and 18 relations, and FB15k contains 592,213 triplets with 14,951 entities and 1,345 relations. We use the same training/validation/test split as in (Bordes et al., 2013; Yang et al., 2015). ",
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+    {
+        "type": "table",
+        "img_path": "images/d3d13a613d2696b89aca99f0747f6e84df4c06fabd07c195ccbc60e43d6baca6.jpg",
+        "table_caption": [
+            "Table 3: Scoring Functions and Parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model Name</td><td rowspan=1 colspan=3>ParametersΘ</td><td rowspan=1 colspan=3>Scoring Function fe(s,r,o)</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=3>{1,,|R1，W[:.k]V1,.,|R|，,V1,.,|R|,a1,.,|Ei}</td><td rowspan=1 colspan=1>ur tanh(asW1:k]lao+Vr</td><td rowspan=1 colspan=1>asao</td><td rowspan=1 colspan=1>+br</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{w1,..,R,a.,E}</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>-llas+wr-all</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{W1,..,|R,a1.,E</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>aWrao</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{w1...,|,a1...,E</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>adiag(wr)ao</td></tr></table>",
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+    },
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+        "type": "table",
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+        "table_caption": [
+            "Table 4: Experimental Results on the WN18 Dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Results from Neural KB Embedding</td><td rowspan=1 colspan=2>Results from our KBTD framework</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS @10</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=1>0.53</td><td rowspan=1 colspan=1>66.10</td><td rowspan=1 colspan=1>0.85</td><td rowspan=1 colspan=1>90.50</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1>0.38</td><td rowspan=1 colspan=1>90.90</td><td rowspan=1 colspan=1>0.39</td><td rowspan=1 colspan=1>82.18</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>92.80</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>94.62</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>94.20</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>94.62</td></tr></table>",
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+    },
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+        "type": "table",
+        "img_path": "images/aa4deee07f5d86d621efe86f02b8ae84afdaf40489f28213ab29e78ec1b2631b.jpg",
+        "table_caption": [
+            "Table 5: Experimental Results on the FB15k Dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Results from Neural KB Embedding</td><td rowspan=1 colspan=2>Results fromourKBTD framework</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td><td rowspan=1 colspan=1>MRR</td><td rowspan=1 colspan=1>HITS@10</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>41.40</td><td rowspan=1 colspan=1>0.37</td><td rowspan=1 colspan=1>59.06</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>53.90</td><td rowspan=1 colspan=1>0.30</td><td rowspan=1 colspan=1>49.94</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>51.90</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>54.95</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>57.70</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>59.91</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Baselines. We compare our proposed framework with TransE (Bordes et al., 2013), NTN (Socher et al., 2013), Bilinear (Jenatton et al., 2012) and DISTMULT (Yang et al., 2015). The original TransE model is based on dissimilarity/distance function. To fit our framework, we define the scoring function for TransE to be negative dissimilarity/distance function, i.e., $f ( s , r , o ) ~ =$ $- \\left\\| \\pmb { a } _ { s } + \\pmb { a } _ { r } - \\pmb { a } _ { o } \\right\\| _ { 2 } ^ { 2 }$ . For NTN, Bilinear and DISTMULT, we inherit the scoring functions from their paper. The detailed scoring functions as well as their parameters are listed in Table 3. For the meaning of parameters and the intuition behind scoring functions, readers can refer to the original papers. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation Protocol. We exactly follow the experimental procedure and treatment used in TransE (Bordes et al., 2013) and DISTMULT (Yang et al., 2015). For each triplet $( s , r , o )$ in the test set, the subject entity $s$ is replaced with each of entities from entity set $E$ in turn. We apply corresponding scoring function $f$ on those corrupted triplets and then sort them in non-increasing order to get the rank of the correct triplet. This procedure is then repeated for the object entity $o$ . For evaluation metrics, we consider Mean Reciprocal Rank (MRR) which is defined to be an average of the reciprocal rank of the correct triplets over all test triplets, and $H I T S @ I O$ (top-10 accuracy). If possible, we list the experimental results reported in (Yang et al., 2015) directly. In addition, we apply the filtered setting from (Bordes et al., 2013; Yang et al., 2015) in evaluation. In this setting, for one certain test triplet $( s , r , o )$ , we removed from the list of corrupted triplets all the triplets which appear in training, validation, or test set, except $( s , r , o )$ itself. This setting, for example, can avoid cases where lots of triplets in training set rank above the one of interest. ",
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+        "type": "text",
+        "text": "Implementation Details. All the models in our framework were implemented using PyTorch in a machine with one 12GB GPU. Since the complexities of the aforementioned approaches vary a lot, in order to achieve the best accuracy for all the models, we cross-validate using the validation set to find the best hyperparameters. We found that, except for TransE on WN18, all the methods on both datasets can share the same hyper-parameters: (1) dimensionality $d = 1 0 0$ ; (2) smoothing parameter $\\epsilon = 0 . 0 1$ ; (3) multiplier $\\lambda$ mentioned in Algorithm 1 was set to 2; (4) the learning rate of AdaGrad algorithm was set to 0.1 (0.01 for TransE on WN18); (5) $\\ell _ { 2 }$ -regularization applied to all the parameters using the weight 0.0001; (6) the mini-batch size is set to 2,048 (4,800 for TransE on WN18); (6) $b = 1$ in Eq. 6 (200 for TransE on WN18). For the additional hyper-parameter in NTN method, i.e., the number of slices $k$ , was set to 2. We allow all the algorithms to run at most 10,000 epochs over the training data, and the best model was selected by early stopping using $\\mathrm { H I T S } @ 1 0$ score on the validation sets. By taking advantages of GPU computation, every training experiment can be finished within 4 hours. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/02c31f164feeb29ca4fcd52048e54a9ff3b494740fc241e7cd8826c4c72f73c6.jpg",
+        "table_caption": [
+            "Table 6: Model Complexity in terms of #Parameters. $d$ is the embedding dimension, and $k$ in NTN is the number of slices. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=3># Parameters</td></tr><tr><td rowspan=1 colspan=1>NTN</td><td rowspan=1 colspan=3>O(R|d²k +|E|d)</td></tr><tr><td rowspan=1 colspan=1>TransE</td><td rowspan=1 colspan=3>O(Rd+Ed)</td></tr><tr><td rowspan=1 colspan=1>Bilinear</td><td rowspan=1 colspan=3>O(R|d² +E|d)</td></tr><tr><td rowspan=1 colspan=1>DISTMULT</td><td rowspan=1 colspan=1>O(R</td><td rowspan=1 colspan=1>d+</td><td rowspan=1 colspan=1>Ed)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 4 and Table 5 list the overall results on the WN18 and FB15k datasets for several popular models under both our KBTD framework and the neural KB embedding framework, respectively. In general, we have the following key observations and insights: ",
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+    {
+        "type": "text",
+        "text": "(1) On the WN18 dataset, the KBTD framework achieves the best performance among most cases. In terms of MRR, KBTD outperforms all baselines except DISTMULT with an impressive improvement up to $6 0 . 4 \\%$ (0.85 v.s. 0.53) on NTN. In terms of $\\mathrm { H I T S } @ 1 0$ , KBTD outperforms baselines except TransE with an improvement up to $3 6 . 9 \\%$ (90.50 v.s. 66.10) on NTN. Similar results can also be observed on the FB15k dataset in Table 5. In terms of both MRR and $\\mathrm { H I T S } @ 1 0 .$ , KBTD outperforms all baselines except TransE with an improvement greater than $4 2 . 7 \\%$ on NTN. ",
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+    {
+        "type": "text",
+        "text": "(2) It is notable that KBTD outperforms NTN by large margins on both datasets. We conjecture that this comes from the very high model complexity of NTN, as suggested by Table 6. In our KBTD framework, we reduce the previously considered margin-based ranking problem in the original NTN to a simple regression problem. As a result, KBTD enables the efficient training procedure to significantly boost up NTN with respect to both the MRR and $\\mathrm { H I T S } @ 1 0$ metrics. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(3) It is also worth noting that under the KBTD framework, most models generate comparable or better results than their neural KB embedding versions. In terms of $\\mathrm { H I T S } @ 1 0$ , KBTD underperforms TransE by $9 . 6 \\%$ (82.18 v.s. 90.90) on WN18 and $7 . 3 \\%$ (49.94 v.s. 53.90) on $\\mathrm { F B } 1 5 \\mathrm { k }$ . We attribute this underperformance to the constraint on TransE’s scoring function. As showed in Table 3, when instantiating the KBTD framework with TransE, the scoring function is set to be the negative dissimilarity function— $\\begin{array} { r } { \\mathbf { \\sigma } _ { - } f ( s , r , o ) = - \\left. \\mathbf { a } _ { s } + \\mathbf { w } _ { r } - \\mathbf { a } _ { o } \\right. _ { 2 } ^ { 2 } } \\end{array}$ , which is for sure non-positive. However, the tensor $\\mathcal { V }$ in Eq. 7 that KBTD aims to fit allows both positive and negative entries. In specific, for the observed triplets and a moderate $b$ , tensor entries are usually positive; for the corrupted triplets and a small $\\epsilon$ , tensor entries reach negative values. On the contrary, the scoring functions of NTN, Bilinear, and DISTMULT are able to model both positive and negative tensor entries. That said, the non-positive constraint of TransE’s scoring function limits its ability to learn better latent KB representations in this link prediction task. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we provide a theoretical analysis of conventional neural KB embedding models and unveil the link between them and tensor-based KB embedding models. We show that the existing neural KB models can be unified into one tensor decomposition framework. We further propose the KBTD framework to directly fit the derived closed-form tensor. Our extensive experiments suggest that KBTD achieves consistent performance improvements over NTN, Bilinear, and DISTMULT under the neural KB embedding framework. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For further work, one interesting direction is to exploit efficient and scalable algorithms that extend KBTD to web-scale KBs. Another direction is to leverage the effective techniques from the matrix factorization community to enhance our tensor framework, such as the usage of bias terms and rich contextual information. ",
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+    {
+        "type": "text",
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+# A NEURAL DIRICHLET PROCESS MIXTURE MODEL FOR TASK-FREE CONTINUAL LEARNING
+
+Soochan Lee, Junsoo Ha, Dongsu Zhang & Gunhee Kim   
+Department of Computer Science, Seoul National University, Seoul, Republic of Korea {soochan.lee,junsoo.ha}@vision.snu.ac.kr,{96lives,gunhee}@snu.ac.kr http://vision.snu.ac.kr/projects/cn-dpm
+
+# ABSTRACT
+
+Despite the growing interest in continual learning, most of its contemporary works have been studied in a rather restricted setting where tasks are clearly distinguishable, and task boundaries are known during training. However, if our goal is to develop an algorithm that learns as humans do, this setting is far from realistic, and it is essential to develop a methodology that works in a task-free manner. Meanwhile, among several branches of continual learning, expansion-based methods have the advantage of eliminating catastrophic forgetting by allocating new resources to learn new data. In this work, we propose an expansion-based approach for task-free continual learning. Our model, named Continual Neural Dirichlet Process Mixture (CN-DPM), consists of a set of neural network experts that are in charge of a subset of the data. CN-DPM expands the number of experts in a principled way under the Bayesian nonparametric framework. With extensive experiments, we show that our model successfully performs task-free continual learning for both discriminative and generative tasks such as image classification and image generation.
+
+# 1 INTRODUCTION
+
+Humans consistently encounter new information throughout their lifetime. The way the information is provided, however, is vastly different from that of conventional deep learning where each minibatch is iid-sampled from the whole dataset. Data points adjacent in time can be highly correlated, and the overall distribution of the data can shift drastically as the training progresses. Continual learning (CL) aims at imitating incredible human’s ability to learn from a non-iid stream of data without catastrophically forgetting the previously learned knowledge.
+
+Most CL approaches (Aljundi et al., 2018; 2017; Lopez-Paz & Ranzato, 2017; Kirkpatrick et al., 2017; Rusu et al., 2016; Shin et al., 2017; Yoon et al., 2018) assume that the data stream is explicitly divided into a sequence of tasks that are known at training time. Since this assumption is far from realistic, task-free CL is more practical and demanding but has been largely understudied with only a few exceptions of (Aljundi et al., 2019a;b). In this general CL, not only is explicit task definition unavailable but also the data distribution gradually shifts without a clear task boundary.
+
+Meanwhile, existing CL methods can be classified into three different categories (Parisi et al., 2019): regularization, replay, and expansion methods. Regularization and replay approaches address the catastrophic forgetting by regularizing the update of a specific set of weights or replaying the previously seen data, respectively. On the other hand, the expansion methods are different from the two approaches in that it can expand the model architecture to accommodate new data instead of fixing it beforehand. Therefore, the expansion methods can bypass catastrophic forgetting by preventing pre-existing components from being overwritten by the new information. The critical limitation of prior expansion methods, however, is that the decisions of when to expand and which resource to use heavily rely on explicitly given task definition and heuristics.
+
+In this work, our goal is to propose a novel expansion-based approach for task-free CL. Inspired by the Mixture of Experts (MoE) (Jacobs et al., 1991), our model consists of a set of experts, each of which is in charge of a subset of the data in a stream. The model expansion (i.e., adding more experts) is governed by the Bayesian nonparametric framework, which determines the model complexity by the data, as opposed to the parametric methods that fix the model complexity before training. We formulate the task-free CL as an online variational inference of Dirichlet process mixture models consisting of a set of neural experts; thus, we name our approach as the Continual Neural Dirichlet Process Mixture (CN-DPM) model.
+
+We highlight the key contributions of this work as follows.
+
+• We are one of the first to propose an expansion-based approach for task-free CL. Hence, our model not only prevents catastrophic forgetting but also applies to the setting where no task definition and boundaries are given at both training and test time. Our model named CN-DPM consists of a set of neural network experts, which are expanded in a principled way built upon the Bayesian nonparametrics that have not been adopted in general CL research.   
+• Our model can deal with both generative and discriminative tasks of CL. With several benchmark experiments of CL literature on MNIST, SVHN, and CIFAR 10/100, we show that our model successfully performs multiple types of CL tasks, including image classification and generation.
+
+# 2 BACKGROUND AND RELATED WORK
+
+# 2.1 CONTINUAL LEARNING
+
+Parisi et al. (2019) classify CL approaches into three branches: regularization (Kirkpatrick et al., 2017; Aljundi et al., 2018), replay (Shin et al., 2017) and expansion (Aljundi et al., 2017; Rusu et al., 2016; Yoon et al., 2018) methods. Regularization and replay approaches fix the model architecture before training and prevent catastrophic forgetting by regularizing the change of a specific set of weights or replaying previously learned data. Hybrids of replay and regularization also exist, such as Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2019a). On the other hand, methods based on expansion add new network components to learn new data. Conceptually, such direction has the following advantages compared to the first two: (i) catastrophic forgetting can be eliminated since new information is not overwritten on pre-existing components and (ii) the model capacity is determined adaptively depending on the data.
+
+Task-Free Continual Learning. All the works mentioned above heavily rely on explicit task definition. However, in real-world scenarios, task definition is rarely given at training time. Moreover, the data domain may gradually shift without any clear task boundary. Despite its importance, taskfree CL has been largely understudied; to the best of our knowledge, there are only a few works (Aljundi et al., 2019a;b; Rao et al., 2019), each of which is respectively based on regularization, replay, and a hybrid of replay and expansion. Specifically, Aljundi et al. (2019a) extend MAS (Aljundi et al., 2018) by adding heuristics to determine when to update the importance weights with no task definition. In their following work (Aljundi et al., 2019b), they improve the memory management algorithm of GEM (Lopez-Paz & Ranzato, 2017) such that the memory elements are carefully selected to minimize catastrophic forgetting. While focused on unsupervised learning, Rao et al. (2019) is a parallel work that shares several similarities with our method, e.g., model expansion and short-term memory. However, due to their model architecture, expansion is not enough to stop catastrophic forgetting; consequently, generative replay plays a crucial role in Rao et al. (2019). As such, it can be categorized as a hybrid of replay and expansion. More detailed comparison between our method and Rao et al. (2019) is deferred to Appendix M.
+
+# 2.2 DIRICHLET PROCESS MIXTURE MODELS
+
+We briefly review the Dirichlet process mixture (DPM) model (Antoniak, 1974; Ferguson, 1983), and a variational method to approximate the posterior of DPM models in an online setting: Sequential Variational Approximation (SVA) (Lin, 2013). For a more detailed review, refer to Appendix A.
+
+Dirichlet Process Mixture (DPM). The DPM model is often applied to clustering problems where the number of clusters is not known in advance. The generative process of a DPM model is
+
+$$
+x _ { n } \sim p ( x ; \theta _ { n } ) , \theta _ { n } \sim G , G \sim \mathrm { D P } ( \alpha , G _ { 0 } ) ,
+$$
+
+where $x _ { n }$ is the $n$ -th data, and $\theta _ { n }$ is the $n$ -th latent variable sampled from $G$ , which itself is a distribution sampled from a Dirichlet process (DP). The DP is parameterized by a concentration parameter $\alpha$ and a base distribution $G _ { 0 }$ . The expected number of clusters is proportional to $\alpha$ , and $G _ { 0 }$ is the marginal distribution of $\theta$ when $G$ is marginalized out. Since $G$ is discrete with probability 1 (Teh, 2010), same values can be sampled multiple times for $\theta$ . If $\theta _ { n } = \theta _ { m }$ , the two data points $x _ { n }$ and $x _ { m }$ belong to the same cluster. An alternative formulation uses the variable $z _ { n }$ that indicates to which cluster the $n$ -th data belongs such that $\theta _ { n } = \phi _ { z _ { n } }$ where $\phi _ { k }$ is the parameter of the $k$ -th cluster. In the context of this paper, $\phi _ { k }$ refers to the parameters of the $k$ -th expert.
+
+Approximation of the Posterior of DPM Models. Since the exact inference of the posterior of DPM models is infeasible, approximate inference methods are applied. Among many approximation methods, we adopt the Sequential Variational Approximation (SVA) (Lin, 2013). While the data is given one by one, SVA sequentially determines $\rho _ { n }$ and $\nu _ { k }$ , which are the variational approximation for the distribution of $z _ { n }$ and $\phi _ { k }$ respectively. Since $\rho _ { n }$ satisfies $\textstyle \sum _ { k } \rho _ { n , k } = 1$ and $\rho _ { n , k } \ > = 0$ , $\rho _ { n , k }$ can be interpreted as the probability of $n$ -th data belonging to $k$ -th cluster and is often called responsibility. $\rho _ { n + 1 }$ and $\nu ^ { ( n + 1 ) }$ at step $n + 1$ are computed as:
+
+$$
+\begin{array} { r l } & { \rho _ { n + 1 , k } \propto \left\{ ( \sum _ { i = 1 } ^ { n } \rho _ { i , k } ) \int _ { \phi } p ( x _ { n + 1 } | \phi ) \nu _ { k } ^ { ( n ) } ( d \phi ) \right. \mathrm { ~ i f ~ } 1 \leq k \leq K } \\ & { \qquad \quad \alpha \int _ { \phi } p ( x _ { n + 1 } | \phi ) G _ { 0 } ( d \phi ) \qquad \mathrm { i f ~ } k = K + 1 } \\ & { \qquad \quad \nu _ { k } ^ { ( n + 1 ) } ( d \phi ) \propto \left\{ G _ { 0 } ( d \phi ) \prod _ { i = 1 } ^ { n + 1 } p ( x _ { i } | \phi ) ^ { \rho _ { i , k } } \right. \quad \mathrm { i f ~ } 1 \leq k \leq K } \\ & { \qquad \quad \left. G _ { 0 } ( d \phi ) p ( x _ { n + 1 } | \phi ) ^ { \rho _ { n + 1 , k } } \right. \quad \mathrm { i f ~ } k = K + 1 } \end{array} .
+$$
+
+In practice, SVA adds a new component only when $\rho _ { K + 1 }$ is greater than a certain threshold $\epsilon$ . If $G _ { 0 }$ and $p ( x _ { i } | \phi )$ are not a conjugate pair, stochastic gradient descent (SGD) is used to find the MAP estimation $\hat { \phi }$ with a learning rate of $\lambda$ instead of calculating the whole distribution $\nu _ { k }$ :
+
+$$
+\begin{array} { r } { \hat { \phi } _ { k } ^ { ( n + 1 ) } \gets \hat { \phi } _ { k } ^ { ( n ) } + \lambda ( \nabla _ { \hat { \phi } _ { k } ^ { ( n ) } } \log G _ { 0 } ( \hat { \phi } _ { k } ^ { ( n ) } ) + \nabla _ { \hat { \phi } _ { k } ^ { ( n ) } } \log p ( x | \hat { \phi } _ { k } ^ { ( n ) } ) ) . } \end{array}
+$$
+
+DPM for Discriminative Tasks. DPM can be extended to discriminative tasks where each data point is an input-output pair $( x , y )$ , and the goal is to learn the conditional distribution $p ( y | x )$ . To use DPM, which is a generative model, for discriminative tasks, we first learn the joint distribution $p ( x , y )$ and induce the conditional distribution from it: $\begin{array} { r } { p ( y | x ) = p ( x , y ) / \int _ { y } p ( x , y ) } \end{array}$ . The joint distribution modeled by each component can be decomposed as $p ( x , y | z ) = \bar { p ( y | x , z ) } p ( x | z )$ (Rasmussen & Ghahramani, 2002; Shahbaba & Neal, 2009).
+
+DPM in Related Fields. Recent works of Nagabandi et al. (2019) and Jerfel et al. (2019) exploit the DPM framework to add new components without supervision in the meta-learning context. Nagabandi et al. (2019) apply DPM to the model-based reinforcement learning to predict the next state from a given state-action pair. When a new task appears, they add a component under the DPM framework to handle predictions in the new task. Jerfel et al. (2019) apply DPM to online metalearning. Extending MAML (Finn et al., 2017), they assume that similar tasks can be grouped into a super-task in which the parameter initialization is shared among tasks. DPM is exploited to find the super-tasks and the parameter initialization for each super-task. Therefore, it can be regarded as a meta-level CL method. These works, however, lack generative components, which are often essential to infer the responsible component at test time, as will be described in the next section. As a consequence, it is not straightforward to extend their algorithms to other CL settings beyond modelbased RL or meta-learning. In contrast, our method implements a DPM model that is applicable to general task-free CL.
+
+# 3 APPROACH
+
+We aim at general task-free CL, where the number of tasks and task descriptions are not available at both training and test time. We even consider the case where the data stream cannot be split into separate tasks in Appendix F. All of the existing expansion methods are not task-free since they require task definition at training (Aljundi et al., 2017) or even at test time (Rusu et al., 2016; Xu & Zhu, 2018; Li et al., 2019). We propose a novel expansion method that automatically determines when to expand and which component to use. We first deal with generative tasks and generalize them into discriminative ones.
+
+# 3.1 CONTINUAL LEARNING AS MODELING OF THE MIXTURE DISTRIBUTION
+
+We can formulate a CL scenario as a stream of data involving different tasks $\mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , . . .$ where each task $\mathcal { D } _ { k }$ is a set of data sampled from a (possibly) distinct distribution $p ( x | z = k )$ . If $K$ tasks are given so far, the overall distribution is expressed as the mixture distribution:
+
+$$
+p ( x ) = \sum _ { k = 1 } ^ { K } p ( x | z = k ) p ( z = k ) ,
+$$
+
+where $p ( z = k )$ can be approximated by $N _ { k } / N$ where $N _ { k } = | \mathcal { D } _ { k } |$ and $\begin{array} { r } { N = \sum _ { k } N _ { k } } \end{array}$ . The goal of CL is to learn the mixture distribution in an online manner. Regularization and replay methods directly model the approximate distribution $p ( x ; \phi )$ parameterized by a single component $\phi$ and update it to fit the overall distribution $p ( x )$ . When updating $\phi$ , however, they do not have full access to all the previous data, and thus the information of previous tasks is at risk of being lost as more tasks are learned. Another way to solve $\mathrm { C L }$ is to use a mixture model: approximating each $p ( x | z = k )$ with $p ( x ; \phi _ { k } )$ . If we learn a new task distribution $p ( x | z = K + \bar { 1 } )$ with new parameter $\phi _ { K + 1 }$ and leave the existing parameters intact, we can preserve the knowledge of the previous tasks. The expansion-based CL methods follow this idea.
+
+Similarly, in the discriminative task, the goal of CL is to model the overall conditional distribution, which is a mixture of task-wise conditional distribution $p ( y | x , z = k )$ :
+
+$$
+p ( \boldsymbol { y } | \boldsymbol { x } ) = \sum _ { k = 1 } ^ { K } p ( \boldsymbol { y } | \boldsymbol { x } , z = k ) p ( z = k | \boldsymbol { x } ) .
+$$
+
+Prior expansion methods use expert networks each of which models a task-wise conditional distribution $p ( \boldsymbol { y } | \boldsymbol { x } ; \phi _ { k } ) ^ { 1 }$ . However, a new problem arises in expansion methods: choosing the right expert given $x$ , i.e., $p ( z | x )$ in Eq.(6). Existing methods assume that explicit task descriptor $z$ is given, which is generally not true in human-like learning scenarios. That is, we need a gating mechanism that can infer $p ( z | x )$ only from $x$ (i.e., which expert should process $x$ ). With the gating, the model prediction naturally reduces to the sum of expert outputs weighted by the gate values, which is the mixture of experts (MoE) (Jacobs et al., 1991) formulation: $\begin{array} { r } { \bar { p ( \boldsymbol { y } | \boldsymbol { x } ) } \approx \sum _ { k } \bar { p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \phi } _ { k } ) } p ( \boldsymbol { z } = k | \boldsymbol { x } ) } \end{array}$ .
+
+However, it is not possible to use a single gate network as in Shazeer et al. (2017) to model $p ( z | x )$ in CL; since the gate network is a classifier that finds the correct expert for a given data, training it in an online setting causes catastrophic forgetting. Thus, one possible solution to replace a gating network is to couple each expert $k$ with a generative model that represents $p ( x | z = \bar { k } )$ as in Rasmussen & Ghahramani (2002) and Shahbaba & Neal (2009). As a result, we can build a gating mechanism without catastrophic forgetting as
+
+$$
+p ( y | x ) \approx \sum _ { k } p ( y | x ; \phi _ { k } ^ { D } ) p ( z = k | x ) \approx \sum _ { k } p ( y | x ; \phi _ { k } ^ { D } ) \frac { p ( x ; \phi _ { k } ^ { G } ) p ( z = k ) } { \sum _ { k ^ { \prime } } p ( x ; \phi _ { k ^ { \prime } } ^ { G } ) p ( z = k ^ { \prime } ) } ,
+$$
+
+where $p ( z = k ) \approx N _ { k } / N$ . We also differentiate the notation for the parameters of discriminative models for classification and generative models for gating by the superscript $D$ and $G$ .
+
+If we know the true assignment of $z$ , which is the case of task-based $\mathrm { C L }$ , we can independently train a discriminative model (i.e., $p ( \boldsymbol { y } | \boldsymbol { x } ; \phi _ { k } ^ { D } ) )$ and a generative model (i.e., $p ( x ; \phi _ { k } ^ { G } ) )$ for each task $k$ . In task-free CL, however, $z$ is unknown, so the model needs to infer the posterior $p ( z | x , y )$ . Even worse, the total number of experts is unknown beforehand. Therefore, we propose to employ a Bayesian nonparametric framework, specifically the Dirichlet process mixture (DPM) model, which can fit a mixture distribution with no prefixed number of components. We use SVA described in section 2.2 to approximate the posterior in an online setting. Although SVA is originally designed for the generative tasks, it is easily applicable to discriminative tasks by making each component $k$ to model $p ( x , y | z ) = p ( y | x , z ) p ( \bar { x } | z )$ .
+
+![](images/0b43b0ae0ab3e245f2d29a0d0000bb530c6f405c227da1e08a26089018c69448.jpg)  
+Figure 1: Overview of our CN-DPM model. Each expert $k$ (blue boxes) contains a discriminative component for modeling $p ( \boldsymbol { y } | \boldsymbol { x } ; \phi _ { k } ^ { D } )$ and a generative component for modeling $p ( x ; \phi _ { k } ^ { G } )$ , jointly representing $p ( x , y ; \phi _ { k } )$ . We also keep the assigned data count $N _ { k }$ per expert. (a) During training, each sample $( x , y )$ coming in a sequence is evaluated by every expert to calculate the responsibility $\rho _ { k }$ of each expert. If $\rho _ { K + 1 }$ is high enough, i.e., none of the existing experts is responsible, the data is stored into short-term memory (STM). Otherwise, it is learned by the corresponding expert. When STM is full, a new expert is created from the data in STM. (b) Since CN-DPM is a generative model, we first compute the joint distribution $p ( x , y )$ for a given $x$ , from which it is trivial to infer $p ( y | x )$ .
+
+# 3.2 THE CONTINUAL NEURAL DIRICHLET PROCESS MIXTURE (CN-DPM) MODEL
+
+The proposed approach for task-free CL, named Continual Neural Dirichlet Process Mixture (CNDPM) model, consists of a set of experts, each of which is associated with a discriminative model (classifier) and a generative model (density estimator). More specifically, the classifier models $p ( y | x , z \ = \ k )$ , for which we can adopt any classifier or regressor using deep neural networks, while the density estimator describes the marginal likelihood $p ( x | z = k \bar { ) }$ , for which we can use any explicit density model such as VAEs (Kingma & Welling, 2014) and PixelRNN (Oord et al., 2016). We respectively denote the classifier and the density estimator of expert $k$ as $p ( y | x ; \phi _ { k } ^ { D } )$ and $p ( x ; \phi _ { k } ^ { G } )$ , where $\phi _ { k } ^ { D }$ and $\phi _ { k } ^ { G }$ are the parameters of the models. Finally, the prediction $p ( y | x )$ can be obtained from Eq.(7) by plugging in the output of the classifier and the density estimator. Note that the number of experts is not prefixed but expanded via the DPM framework. Figure 1 illustrates the overall training and inference process of our model.
+
+Training. We assume that samples sequentially arrive one at a time during training. For a new sample, we first decide whether the sample should be assigned to an existing expert or a new expert should be created for it. Suppose that samples up to $( x _ { n } , y _ { n } )$ are sequentially processed and $K$ experts are already created when a new sample $( x _ { n + 1 } , y _ { n + 1 } )$ arrives. We compute the responsibility ρn+1,k as follows:
+
+$$
+\rho _ { n + 1 , k } \propto \left\{ \begin{array} { l l } { ( \sum _ { i = 1 } ^ { n } \rho _ { i , k } ) p ( y _ { n + 1 } | x _ { n + 1 } ; \hat { \phi } _ { k } ^ { D } ) p ( x _ { n + 1 } ; \hat { \phi } _ { k } ^ { G } ) } & { \mathrm { i f ~ } 1 \leq k \leq K } \\ { \alpha p ( y _ { n + 1 } | x _ { n + 1 } ; \hat { \phi } _ { 0 } ^ { D } ) p ( x _ { n + 1 } ; \hat { \phi } _ { 0 } ^ { G } ) \mathrm { ~ w h e r e ~ } \hat { \phi } _ { 0 } \sim G _ { 0 } ( \phi ) } & { \mathrm { i f ~ } k = K + 1 } \end{array} \right.
+$$
+
+where $G _ { 0 }$ is a distribution corresponding to the weight initialization. If arg $\textstyle \operatorname* { m a x } _ { k } \rho _ { n + 1 , k } \neq K + 1$ , the sample is assigned to the existing experts proportional to $\rho _ { n + 1 , k }$ , and the parameters of the experts are updated with the new sample by Eq.(4) such that $\hat { \phi } _ { k }$ is the MAP approximation given the data assigned up to the current time step. Otherwise, we create a new expert.
+
+Short-Term Memory. However, it is not a good idea to create a new expert immediately and initialize it to be the MAP estimation given $x _ { n + 1 }$ . Since both the classifier and density estimator of an expert are neural networks, training the new expert with only a single example leads to severe overfitting. To mitigate this issue, we employ short-term memory (STM) to collect sufficient data before creating a new expert. When a data point is classified as new, we store it to the STM. Once the STM reaches its maximum capacity $M$ , we stop the data inflow for a while and train a new expert with the data in the STM for multiple epochs until convergence. We call this procedure sleep phase. After sleep, the STM is emptied, and the newly trained expert is added to the expert pool. During the subsequent wake phase, the expert is learned from the data assigned to it. This STM trick assumes that the data in the STM belong to the same expert. We empirically find that this assumption is acceptable in many CL settings where adjacent data are highly correlated. The overall training procedure is described in Algorithm 1. Note that we use $\rho _ { n , 0 }$ instead of $\rho _ { n , K + 1 }$ in the algorithm for brevity.
+
+Inference. At test time, we infer $p ( y | x )$ from the collaboration of the learned experts as in Eq.(7).
+
+Techniques for Practicality. Naively adding a new expert has two major problems: (i) the number of parameters grows unnecessarily large as the experts redundantly learn common features and (ii) there is no positive transfer of knowledge between experts. Therefore, we propose a simple method to share parameters between experts. When creating a new expert, we add lateral connections to the features of the previous experts similar to Rusu et al. (2016). To prevent catastrophic forgetting in the existing experts, we block the gradient from the new expert. In this way, we can greatly reduce the number of parameters while allowing positive knowledge transfer. More techniques such as sparse regularization in Yoon et al. (2018) can be employed to reduce redundant parameters further. As they are orthogonal to our approach, we do not use such techniques in our experiments. Another effective technique that we use in the classification experiments is adding a temperature parameter to the classifier. Since the range of $\log p ( x | z )$ is far broader than $\log p ( y | x , z )$ , the classifier has almost no effect without proper scaling. Thus, we can increase overall accuracy by adjusting the relative importance of images and labels. We also introduce an algorithm to prune redundant experts in Appendix D, and discuss further practical issues of CN-DPM in Appendix B.
+
+<table><tr><td colspan="3">Algorithm1 Training of the Continual Neural Dirichlet Process Mixture (CD-NDP) Model</td></tr><tr><td>Require: Data (x1,y1),.,(xN,yN)， concen- 12: tration α，base measure Go， short-term memory capacity M,learning rate 入</td><td>13: 14:</td><td>M←{xn}UM if |M| ≥ M then {Add new expert} 𝜙K+1 ←FindMAP(M,Go)</td></tr><tr><td>1: M ← 0 {Short-term memory} 2:K←O {Number of experts}</td><td></td><td>Nk+1←|M|;M←0</td></tr><tr><td></td><td></td><td>K←K+1</td></tr><tr><td>3:No ← α; Φo ← Sample(Go)</td><td></td><td>end if</td></tr><tr><td></td><td></td><td></td></tr><tr><td>4:for n=1 to N do</td><td></td><td>else {Update existing experts}</td></tr><tr><td>5:</td><td>for k= O to K do</td><td></td></tr><tr><td>6:</td><td>lk ←p(ynlxn;R）p(xn;E）</td><td>for k=1 to K do</td></tr><tr><td>7:</td><td>Pn,k←Nklk</td><td>Nk←Nk +Pn,k</td></tr><tr><td>8:</td><td>end for</td><td>k←k+pn,k入Vloglk</td></tr><tr><td>9:</td><td>Pn,0:K ← Pn,0:k/∑k=0 Pnk K</td><td>end for</td></tr><tr><td>10:</td><td>if arg maxk Pn,k = O then</td><td>23:</td></tr><tr><td>11:</td><td></td><td>end if</td></tr><tr><td></td><td>{Save xn to short-term memory}</td><td>24: 25: end for</td></tr></table>
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed CN-DPM model in task-free CL with four benchmark datasets. Appendices include more detailed model architecture, additional experiments, and analyses.
+
+# 4.1 CONTINUAL LEARNING SCENARIOS
+
+A CL scenario defines a sequence of tasks where the data distribution for each task is assumed to be different from others. Below we describe the task-free CL scenarios used in the experiments. At both train and test time, the model cannot access the task information. Unless stated otherwise, each task is presented for a single epoch (i.e., a completely online setting) with a batch size of 10.
+
+Split-MNIST (Zenke et al., 2017). The MNIST dataset (LeCun et al., 1998) is split into five tasks, each containing approximately 12K images of two classes, namely (0/1, 2/3, 4/5, 6/7, 8/9). We conduct both classification and generation experiments in this scenario.
+
+Table 1: Test scores and the numbers of parameters in task-free CL on Split-MNIST, MNIST-SVHN, and Split-CIFAR100 scenarios. Note that iid- $^ { \ast }$ baselines are not CL methods.   
+
+<table><tr><td>Method</td><td colspan="2">Split-MNIST</td><td colspan="2">Split-MNIST(Gen.)</td><td colspan="2">MNIST-SVHN</td><td colspan="2">Split-CIFAR100</td></tr><tr><td></td><td>Acc. (%)</td><td>Param.</td><td>bits/dim</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td></tr><tr><td>iid-offline</td><td>98.63</td><td>478K</td><td>0.1806</td><td>988K</td><td>96.69</td><td>11.2M</td><td>73.80</td><td>11.2M</td></tr><tr><td>iid-online</td><td>96.18</td><td>478K</td><td>0.2156</td><td>988K</td><td>95.24</td><td>11.2M</td><td>20.46</td><td>11.2M</td></tr><tr><td>Fine-tune</td><td>19.43</td><td>478K</td><td>0.2817</td><td>988K</td><td>83.35</td><td>11.2M</td><td>2.43</td><td>11.2M</td></tr><tr><td>Reservoir</td><td>85.69</td><td>478K</td><td>0.2234</td><td>988K</td><td>94.12</td><td>11.2M</td><td>10.01</td><td>11.2M</td></tr><tr><td>CN-DPM</td><td>93.23</td><td>524K</td><td>0.2110</td><td>970K</td><td>94.46</td><td>7.80M</td><td>20.10</td><td>19.2M</td></tr></table>
+
+Table 2: Performance comparison on Split-CIFAR10 with various scenario length.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Split-CIFAR10 Acc. (%)</td><td rowspan="2">Param.</td></tr><tr><td>0.2 Epoch</td><td>1 Epoch</td><td>10 Epochs</td></tr><tr><td>iid-offline</td><td>93.17</td><td>93.17</td><td>93.17</td><td>11.2M</td></tr><tr><td>iid-online</td><td>36.65</td><td>62.79</td><td>83.19</td><td>11.2M</td></tr><tr><td>Fine-tune</td><td>12.68</td><td>18.08</td><td>19.31</td><td>11.2M</td></tr><tr><td>Reservoir</td><td>37.09</td><td>44.00</td><td>43.82</td><td>11.2M</td></tr><tr><td>GSS</td><td>33.56</td><td>1</td><td>1</td><td>11.2M</td></tr><tr><td>CN-DPM</td><td>41.78</td><td>45.21</td><td>46.98</td><td>4.60M</td></tr></table>
+
+Table 3: Dissecting the performance of CN-DPM.   
+
+<table><tr><td>Acc. Type</td><td>Split-CIFAR10</td><td>Split-CIFAR100</td></tr><tr><td>Classifier (init)</td><td>88.20</td><td>55.42</td></tr><tr><td>Classifier (final)</td><td>88.20</td><td>55.24</td></tr><tr><td>Gating (VAEs)</td><td>48.18</td><td>31.14</td></tr></table>
+
+![](images/38bb2aad5539de663fd064e383e1b02fc8358d48d65347d57f3445c49b185851.jpg)
+
+![](images/9fe1b9fb451f1464de3a6b1b859b3dbb83c604844492552fc16c81ba0f28a57f.jpg)  
+Figure 2: Split-CIFAR10 (0.2 Epoch).   
+Figure 3: Split-CIFAR100.
+
+MNIST-SVHN (Shin et al., 2017). It is a two-stage scenario where the first consists of MNIST, and the second contains SVHN (Netzer et al., 2011). This scenario is different from Split-MNIST; in Split-MNIST, new classes are introduced when transitioning into a new task, whereas the two stages in MNIST-SVHN share the same set of class labels and have different input domains.
+
+Split-CIFAR10 and Split-CIFAR100. In Split-CIFAR10, we split CIFAR10 (Krizhevsky & Hinton, 2009) into five tasks in the same manner as Split-MNIST. For Split-CIFAR100, we build 20 tasks, each containing five classes according to the pre-defined superclasses in CIFAR100. The training sets of CIFAR10 and CIFAR100 consist of 50K examples each. Note that most of the previous works (Rebuffi et al., 2017; Zenke et al., 2017; Lopez-Paz & Ranzato, 2017; Aljundi et al., 2019c; Chaudhry et al., 2019a), except Maltoni & Lomonaco (2019), use task information at test time in Split-CIFAR100 experiments. They assign distinct output heads for each task and utilize the task identity to choose the responsible output head at both training and test time. Knowing the right output head, however, the task reduces to 5-way classification. Therefore, our setting is far more difficult than the prior works since the model has to perform 100-way classification only from the given input.
+
+# 4.2 COMPARED METHODS
+
+All the following baselines use the same base network that will be discussed in section 4.3.
+
+iid-offline and iid-online. iid-offline shows the maximum performance achieved by combining standard training techniques such as data augmentation, learning rate decay, multiple iterations (up to 100 epochs), and larger batch size. iid-online is the model trained with the same number of epoch and batch size with other CL baselines.
+
+Fine-tune. As a popular baseline in the previous works, the base model is naively trained as data enters.
+
+Reservoir. As Chaudhry et al. (2019b) show that simple experience replay (ER) can outperform most CL methods, we test the ER with reservoir sampling as a strong baseline. Reservoir sampling randomly chooses a fixed number of samples with a uniform probability from an indefinitely long stream of data, and thus, it is suitable for managing the replay memory in task-free CL. At each training step, the model is trained using a mini-batch from the data stream and another one of the same sizes from the memory.
+
+Gradient-Based Sample Selection (GSS). Aljundi et al. (2019b) propose a sampling method called GSS that diversifies the gradients of the samples in the replay memory. Since it is designed to work in task-free settings, we report the scores in their paper for comparison.
+
+# 4.3 MODEL ARCHITECTURE
+
+Split-MNIST. Following Hsu et al. (2018), we use a simple two-hidden-layer MLP classifier with ReLU activation as the base model for classification. The dimension of each layer is 400. For generation experiments, we use VAE, whose encoder and decoder have the same hidden layer configuration with the classifier. Each expert in CN-DPM has a similar classifier and VAE with smaller hidden dimensions. The first expert starts with 64 hidden units per layer and adds 16 units when a new expert is added. For classification, we adjust hyperparameter $\alpha$ such that five experts are created. For generation, we set $\alpha$ to produce 12 experts since more experts produce a better score. We set the memory size in both Reservoir and CN-DPM to 500 for classification and 1000 for generation.
+
+MNIST-SVHN and Split-CIFAR10/100. We use ResNet-18 (He et al., 2016) as the base model. In CN-DPM, we use a 10-layer ResNet for the classifier and a CNN-based VAE. The encoder and the decoder of VAE have two CONV layers and two FC layers. We set $\alpha$ such that 2, 5, and 20 experts are created for each scenario. The memory sizes in Reservoir, GSS, and CN-DPM are set to 500 for MNIST-SVHN and 1000 for Split-CIFAR10/100. More details can be found in Appendix C.
+
+# 4.4 RESULTS OF TASK-FREE CONTINUAL LEARNING
+
+All reported numbers in our experiments are the average of 10 runs. Table 1 and 2 show our main experimental results. In every setting, CN-DPM outperforms the baselines by significant margins with reasonable parameter usage. Table 2 and Figure 2 shows the results of Split-CIFAR10 experiments. Since Aljundi et al. (2019b) test GSS using only 10K examples of CIFAR10, which is 1/5 of the whole train set, we follow their setting (denoted by $0 . 2 ~ E p o c h )$ for a fair comparison. We also test a Split-CIFAR10 variant where each task is presented for 10 epochs. The accuracy and the training graph of GSS are excerpted from the original paper, where the accuracy is the average of three runs, and the graph is from one of the runs. In Figure 2, the bold line represents the average of 10 runs (except GSS, which is a single run), and the faint lines are the individual runs. Surprisingly, Reservoir even surpasses the accuracy of GSS and proves to be a simple but powerful CL method.
+
+One interesting observation in Table 2 is that the performance of Reservoir degrades as each task is extended up to 10 epochs. This is due to the nature of replay methods; since the same samples are replayed repeatedly as representatives of the previous tasks, the model tends to be overfitted to the replay memory as training continues. This degradation is more severe when the memory size is small, as presented in Appendix I. Our CN-DPM, on the other hand, uses the memory to buffer recent examples temporarily, so there is no such overfitting problem. This is also confirmed by the CN-DPM’s accuracy consistently increasing as learning progresses.
+
+In addition, CN-DPM is particularly strong compared to other baselines when the number of tasks increases. For example, Reservoir, which performs reasonably well in other tasks, scores poorly in Split-CIFAR100, which involves 20 tasks and 100 classes. Even with the large replay memory of size 1000, the Reservoir suffers from the shortage of memory (e.g., only 50 slots per task). In contrast, CN-DPM’s accuracy is more than double of Reservoir and comparable to that of iid-online.
+
+Table 3 analyzes the accuracy of CN-DPM in Split-CIFAR10/100. We assess the performance and forgetting of individual components. At the end of each task, we measure the test accuracy of the responsible classifier and report the average of such task-wise classifier accuracies as Classifier (init).
+
+We report the average of the task-wise accuracies after learning all tasks as Classifier (final). With little difference between the two scores, we confirm that forgetting barely occurs in the classifiers. In addition, we report the gating accuracy measured after training as Gating $( V A E s )$ , which is the accuracy of the task identification performed jointly by the VAEs. The relatively low gating accuracy suggests that CN-DPM has much room for improvement through better density estimates.
+
+Overall, CN-DPM does not suffer from catastrophic forgetting, which is a major problem in regularization and replay methods. As a trade-off, however, choosing the right expert arises as another problem in CN-DPM. Nonetheless, the results show that this new direction is especially promising when the number of tasks is very large.
+
+# 5 CONCLUSION
+
+In this work, we formulated expansion-based task-free CL as learning of a Dirichlet process mixture model with neural experts. We demonstrated that the proposed CN-DPM model achieves great performance in multiple task-free settings, better than the existing methods. We believe there are several interesting research directions beyond this work: (i) improving the accuracy of expert selection, which is the main bottleneck of our method, and (ii) applying our method to different domains such as natural language processing and reinforcement learning.
+
+# ACKNOWLEDGMENTS
+
+We thank Chris Dongjoo Kim and Yookoon Park for helpful discussion and advice. This work was supported by Video Analytics Center of Excellence in AIX center of SK telecom, Institute of Information & communications Technology Planning & Evaluation (IITP) grant (No.2019-0-01082, SW StarLab) and Basic Science Research Program through National Research Foundation of Korea (NRF) funded by the Korea government (MSIT) (2017R1E1A1A01077431). Gunhee Kim is the corresponding author.
+
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+
+# A REVIEW OF DIRICHLET PROCESS MIXTURE MODEL
+
+We review the Dirichlet process mixture (DPM) model and a variational method to approximate the posterior of DPM models in an online setting: Sequential Variational Approximation (SVA) (Lin, 2013).
+
+Dirichlet Process. Dirichlet process (DP) is a distribution over distributions that are defined over infinitely many dimensions. DP is parameterized by a concentration parameter $\alpha \in \mathbb { R } ^ { + }$ and a base distribution $G _ { 0 }$ . For a distribution $G$ sampled from $\mathrm { D P } ( \alpha , G _ { 0 } )$ , the following holds for any finite measurable partition $\{ A _ { 1 } , A _ { 2 } , . . . , A _ { K } \}$ of probability space $\Theta$ (Teh, 2010):
+
+$$
+( G ( A _ { 1 } ) , . . . , G ( A _ { K } ) ) \sim \mathrm { D i r } ( \alpha G _ { 0 } ( A _ { 1 } ) , . . . , \alpha G _ { 0 } ( A _ { K } ) ) .
+$$
+
+The stick-breaking process is often used as a more intuitive construction of DP:
+
+$$
+G = \sum _ { k = 1 } ^ { \infty } \left( v _ { k } \prod _ { l = 1 } ^ { k - 1 } ( 1 - v _ { l } ) \right) \delta _ { \phi _ { k } , } v _ { k } \sim \mathrm { B e t a } ( 1 , \alpha ) , \phi _ { k } \sim G _ { 0 } .
+$$
+
+Initially, we start with a stick of length one, which represents the total probability. At each step $k$ , we cut a proportion $v _ { k }$ off from the remaining stick (probability) and assign it to the atom $\phi _ { k }$ sampled from the base distribution $G _ { 0 }$ . This formulation shows DP is discrete with probability 1 (Teh, 2010). In our problem setting, $G$ is a distribution over expert’s parameter space and has positive probability only at the countably many $\phi _ { k }$ , which are independently sampled from the base distribution.
+
+Dirichlet Process Mixture (DPM) Model. The DPM model is often applied to clustering problems where the number of clusters is not known in advance. The generative process of DPM model is
+
+$$
+x _ { n } \sim p ( \theta _ { n } ) , \theta _ { n } \sim G , G \sim \mathrm { D P } ( \alpha , G _ { 0 } ) ,
+$$
+
+where $x _ { n }$ is the $n$ -th data, and $\theta _ { n }$ is the $n$ -th latent variable sampled from $G$ , which itself is a distribution sampled from a Dirichlet process (DP). Since $G$ is discrete with probability 1, the same values can be sampled multiple times for $\theta$ . If $\theta _ { n } = \theta _ { m }$ , the two data points $x _ { n }$ and $x _ { m }$ belong to the same cluster. An alternative formulation uses the indicator variable $z _ { n }$ that indicates to which cluster the $n$ -th data belongs such that $\theta _ { n } = \phi _ { z _ { n } }$ where $\phi _ { k }$ is the parameter of $k$ -th cluster. The data $x _ { n }$ is sampled from a distribution parameterized by $\theta _ { n }$ . For a DP Gaussian mixture model as an example, each $\theta = \{ \mu , \sigma ^ { 2 } \}$ parameterizes a Gaussian distribution.
+
+The Posterior of DPM Models. The posterior of a DPM model for given $\theta _ { 1 } , . . . , \theta _ { n }$ is also a DP (Teh, 2010):
+
+$$
+G | \theta _ { 1 } , . . . , \theta _ { n } \sim \mathrm { D P } \left( \alpha + n , \frac { \alpha } { \alpha + n } G _ { 0 } + \frac { 1 } { \alpha + n } \sum _ { i = 1 } ^ { n } \delta ( \theta _ { i } ) \right) .
+$$
+
+tion The base distribution of the posterior, which is a weighted average of $\textstyle { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \delta ( \theta _ { i } )$ , is in fact the predictive distribution of $\theta _ { n + 1 }$ given $G _ { 0 }$ $\theta _ { 1 : n }$ and the empirical distribu- (Teh, 2010):
+
+$$
+\theta _ { n + 1 } | \theta _ { 1 } , . . . , \theta _ { n } \sim \frac { \alpha } { \alpha + n } G _ { 0 } + \frac { 1 } { \alpha + n } \sum _ { i = 1 } ^ { n } \delta ( \theta _ { i } ) .
+$$
+
+If we additionally condition $x _ { n }$ and reflect the likelihood, we obtain (Neal, 2000):
+
+$$
+\theta _ { n + 1 } | \theta _ { 1 } , . . . , \theta _ { n } , x _ { n + 1 } \sim { \frac { 1 } { Z } } \left( { \frac { \alpha } { \alpha + n } } \int p ( x _ { n + 1 } | \theta ) d G _ { 0 } ( \theta ) + { \frac { 1 } { \alpha + n } } \sum _ { i = 1 } ^ { n } p ( x _ { n + 1 } | \theta _ { i } ) \delta ( \theta _ { i } ) \right) ,
+$$
+
+where $Z$ is the normalizing constant. Note that $\theta _ { n + 1 }$ is independent from $x _ { 1 : n }$ given $\theta _ { 1 : n }$
+
+Approximation of the Posterior of DPM Models. Since the exact inference of the posterior of DPM models is infeasible, approximate inference methods are adopted such as Markov chain Monte Carlo (MCMC) (Maceachern, 1994; Escobar & West, 1995; Neal, 2000) or variational inference (Blei & Jordan, 2006; Wang & Dunson, 2011; Lin, 2013). Among many variational methods, the Sequential Variational Approximation (SVA) (Lin, 2013) approximates the posterior as
+
+$$
+p ( G | x _ { 1 : n } ) = \sum _ { z _ { 1 : n } } p ( z _ { 1 : n } | x _ { 1 : n } ) p ( G | x _ { 1 : n } , z _ { 1 : n } ) \approx q ( G | \rho , \nu ) = \sum _ { z _ { 1 : n } } \Big ( \prod _ { i = 1 } ^ { n } \rho _ { i , z _ { i } } \Big ) q _ { \nu } ^ { ( z ) } ( G | z _ { 1 : n } ) ,
+$$
+
+where $p ( z _ { 1 : n } | x _ { 1 : n } )$ is represented by the product of individual variational probabilities $\rho _ { z _ { i } }$ for $z _ { i }$ , which greatly simplifies the distribution. Moreover, $p ( G | x _ { 1 : n } , z _ { 1 : n } )$ is approximated by a stochastic process $q _ { \nu } ^ { ( z ) } ( G | z _ { 1 : n } )$ . Sampling from $q _ { \nu } ^ { ( z ) } ( G | z _ { 1 : n } )$ is equivalent to constructing a distribution as
+
+$$
+\beta _ { 0 } D ^ { \prime } + \sum _ { k = 1 } ^ { K } \beta _ { k } \delta _ { \phi _ { k } } , D ^ { \prime } \sim \mathrm { D P } ( \alpha G _ { 0 } ) , ( \beta _ { 0 } , \dots , \beta _ { K } ) \sim \mathrm { D i r } ( \alpha , | C _ { 1 } ^ { ( z ) } | , \dots , | C _ { K } ^ { ( z ) } | ) , \phi _ { k } \sim \nu _ { k } ,
+$$
+
+$\{ C _ { 1 } ^ { ( z ) } , C _ { 2 } ^ { ( z ) } , . . . , C _ { K } ^ { ( z ) } \}$ is the partition of $x _ { 1 : n }$ characterized by $z$
+
+The approximation yields the following tractable predictive distribution:
+
+$$
+q ( \theta ^ { \prime } | \rho , \nu ) = \mathbb { E } _ { q ( G | \rho , \nu ) } [ p ( \theta ^ { \prime } | G ) ] = \frac { \alpha } { \alpha + n } G _ { 0 } ( \theta ^ { \prime } ) + \sum _ { k = 1 } ^ { K } \frac { \sum _ { i = 1 } ^ { n } \rho _ { i , k } } { \alpha + n } \nu _ { k } ( \theta ^ { \prime } ) .
+$$
+
+SVA uses this predictive distribution for sequential approximation of the posterior of $z$ and $\phi$ .
+
+$$
+\begin{array} { r l } & { p ( z _ { n + 1 } , \phi ^ { ( n + 1 ) } | x _ { 1 : n + 1 } ) \propto p ( x _ { n + 1 } | z _ { n + 1 } , \phi ^ { ( n + 1 ) } ) p ( z _ { n + 1 } , \phi ^ { ( n + 1 ) } | x _ { 1 : n } ) } \\ & { \qquad \approx p ( x _ { n + 1 } | z _ { n + 1 } , \phi ^ { ( n + 1 ) } ) q ( z _ { n + 1 } , \phi ^ { ( n + 1 ) } | \rho _ { 1 : n } , \nu ^ { ( n ) } ) . } \end{array}
+$$
+
+While the data is given one by one, SVA sequentially updates the variational parameters; the following $\rho _ { n + 1 }$ and $\nu ^ { ( n + 1 ) }$ at step $n + 1$ minimizes the $\mathrm { K L }$ divergence between $q ( z _ { n + 1 } , \phi ^ { ( n + 1 ) } | \rho _ { 1 : n + 1 } , \nu ^ { ( n + 1 ) } )$ and the posterior:
+
+$$
+\begin{array} { r l } & { \rho _ { n + 1 , k } \propto \left\{ ( \sum _ { i = 1 } ^ { n } \rho _ { i , k } ) \int _ { \theta } p ( x _ { n + 1 } | \theta ) \nu _ { k } ^ { ( n ) } ( d \theta ) \right. \mathrm { ~ i f ~ } 1 \leq k \leq K } \\ & { \left. \alpha \int _ { \theta } p ( x _ { n + 1 } | \theta ) G _ { 0 } ( d \theta ) \right. \mathrm { ~ i f ~ } k = K + 1 } \\ & { \nu _ { k } ^ { ( n + 1 ) } ( d \theta ) \propto \left\{ G _ { 0 } ( d \theta ) \prod _ { i = 1 } ^ { n + 1 } p ( x _ { i } | \theta ) ^ { \rho _ { i , k } } \right. \mathrm { ~ i f ~ } 1 \leq k \leq K } \\ & { \left. G _ { 0 } ( d \theta ) p ( x _ { n + 1 } | \theta ) ^ { \rho _ { n + 1 , k } } \right. \mathrm { ~ i f ~ } k = K + 1 } \end{array} .
+$$
+
+In practice, SVA adds a new component only when $\rho _ { n + 1 , K + 1 }$ is greater than a threshold $\epsilon$ . It uses stochastic gradient descent to find and maintain the MAP estimation of parameters instead of calculating the whole distribution $\nu _ { k }$ :
+
+$$
+\begin{array} { r } { \hat { \phi } _ { k } ^ { ( n + 1 ) } \gets \hat { \phi } _ { k } ^ { ( n ) } + \lambda _ { n } ( \nabla _ { \hat { \phi } _ { k } ^ { ( n ) } } \log G _ { 0 } ( \hat { \phi } _ { k } ^ { ( n ) } ) + \nabla _ { \hat { \phi } _ { k } ^ { ( n ) } } \log p ( x | \hat { \phi } _ { k } ^ { ( n ) } ) ) , } \end{array}
+$$
+
+where $\lambda _ { k } ^ { ( n ) }$ is a learning rate of component $k$ at step $n$ , which decreases as in the Robbins-Monro algorithm.
+
+# B PRACTICAL ISSUES OF CN-DPM
+
+CN-DPN is designed based on strong theoretical foundations, including the nonparametric Bayesian framework. In this section, we further discuss some practical issues of CN-DPM with intuitive explanations.
+
+Bounded expansion of CN-DPM. The number of components in the DPM model is determined by the data distribution and the concentration parameter. If the true distribution consists of $K$ clusters, the number of effective components converges to $K$ under an appropriate concentration parameter $\alpha$ . Typically, the number of components is bounded by $O ( \alpha \log N )$ (Teh, 2010). Experiments in Appendix H empirically show that CN-DPM does not blindly increase the number of experts.
+
+The continued increase in model capacity. Our model capacity keeps increasing as it learns new tasks. However, we believe this is one of the strengths of our method, since it may not make sense to use a fixed-capacity neural network to learn an indefinitely long sequence of tasks. The underlying assumption of using a fixed-capacity model is that the pre-set model capacity is adequate (at least not insufficient) to learn the incoming tasks. On the other hand, CN-DPM approaches the problem in a different direction: start small and add more as needed. This property is essential in task-free settings where the total number of tasks is not known. If there are too many tasks than expected, a fixed-capacity model would not be able to learn them successfully. Conversely, if there are fewer tasks than expected, resources would be wasted. We argue that expansion is a promising direction since it does not need to fix the model capacity beforehand. Moreover, we also introduce an algorithm to prune redundant experts in Appendix D,
+
+Generality of the concentration parameter. The concentration parameter controls how sensitive the model is to new data. In other words, it determines the level of discrepancy between tasks, that makes the tasks modeled by distinct components. As an example, suppose that we are designing a hand-written alphabet classifier that continually learns in the real world. In the development, we only have the character images for half of the alphabets, i.e., from $\mathbf { \dot { a } } _ { } ^ { \dagger }$ to $\cdot _ { \mathrm { m } } \cdot$ . If we can find a good concentration parameter $\alpha$ for the data from $\mathbf { \dot { a } } _ { } ^ { \dagger }$ to $\cdot _ { \mathrm { m } } \cdot$ , the same $\alpha$ can work well with novel alphabets (i.e., from $\cdot _ { \mathrm { n } } \cdot$ to $\cdot _ { z } ,$ ) because the alphabets would have a similar level of discrepancies between tasks. Therefore, we do not need to access the whole data to determine $\alpha$ if the discrepancy between tasks is steady.
+
+# C MODEL ARCHITECTURES AND EXPERIMENTAL DETAILS
+
+# C.1 BASE MODELS
+
+# C.1.1 SPLIT-MNIST
+
+Following Hsu et al. (2018), we use two-hidden-layer MLP classifier with 400 hidden units per layer. For generation tasks, we use a simple VAE with the two-hidden-layer MLP encoder and decoder, where each layer contains 400 units. The dimension of the latent code is set to 32. We use ReLU for all intermediate activation functions.
+
+# C.1.2 MNIST-SVHN AND SPLIT-CIFAR10/100
+
+We use ResNet-18 (He et al., 2016). The input images are transformed to $3 2 \times 3 2$ RGB images.
+
+# C.2 CN-DPM
+
+# C.2.1 SPLIT-MNIST
+
+For the classifiers in experts, we use a smaller version of the base MLP classifier. In the first expert, we set the number of hidden units per layer to 64. In the second or later experts, we introduce 16 new units per layer which are connected to the lower layers of the existing experts. For the encoder and decoder of VAEs, we use a two-layer MLP. The encoder is expanded in the same manner as the classifier. However, we do not share the parameters beyond the encoders; with a latent code of dimension 16, we use the two-hidden-layer MLP decoder as done in the classifier. For generation tasks, we double the size; for example, we set the size of initial and additional hidden units to 128 and 32, respectively.
+
+# C.2.2 SPLIT-CIFAR10/100
+
+The ResNet-18 base network has eight residual blocks. After passing through 2 residual blocks, the width and height of the feature are halved, and the number of channels is doubled. The initial number of channels is set to 64.
+
+For the classifiers in CN-DPM, we use a smaller version of ResNet that has only four residual blocks and resizes the feature every block. The initial number of channels is set to 20 in the first expert, and four initial channels are added with a new expert. Thus, 4, 8, 16, and 32 channels are added for the four blocks. The first layer of each block is connected to the last layer of the previous block of prior experts.
+
+For the VAEs, we use a simple CNN-based VAEs. The encoder has two $3 \times 3$ convolutions followed by two fully connected layers. Each convolution is followed by $2 \times 2$ max-pool and ReLU activation. The numbers of channels and hidden units are doubled after each layer. In the first expert, the first convolution outputs 32 channels, while four new channels are added with each new expert.
+
+As done for the VAE in Split-MNIST, each expert’s VAE has an unshared decoder with a 64- dimensional latent code. The decoder is the mirrored encoder where $3 \times 3$ convolution is replaced by $4 \times 4$ transposed convolution with a stride of 2.
+
+# C.2.3 MNIST-SVHN
+
+For the classifier, we use ResNet-18 with 32 channels for the first expert and additional 32 channels for each new expert. We use the same VAE as in Split-CIFAR10.
+
+# C.3 EXPERIMENTAL DETAILS
+
+We use the classifier temperature parameter of 0.01 for Split-MNIST, Split-CIFAR10/100, and no temperature parameter on MNIST-SVHN. Weight decay 0.00001 has been used for every model in the paper. Gradients are clipped by value with a threshold of 0.5. All the CN-DPM models are trained by Adam optimizer. During the sleep phase, we train the new expert for multiple epochs with a batch size of 50. In classification tasks, we improve the density estimation of VAEs by sampling 16 latent codes and averaging the ELBOs, following Burda et al. (2015).
+
+# C.3.1 SPLIT-MNIST
+
+The learning rate of 0.0001 and 0.0004 has been used for the classifier and VAE of each expert in the classification task. We use learning rate 0.003 for the VAE of each expert in generation task. In the generation task, we decay the learning rate of the expert by 0.003 before it enters the wake phase. Following the existing works in VAE literature, we use binarized MNIST for the generation experiments. VAEs are trained to maximize Bernoulli log-likelihood in the generation task, while Gaussian log-likelihood is used for the classification task.
+
+# C.3.2 SPLIT-CIFAR10
+
+The learning rate of 0.005 and 0.0002 has been used for the classifier and VAE of each expert in CIFAR10. We decay the learning rate of the expert by 0.1 before it enters the wake phase. VAEs are trained to maximize Gaussian log-likelihood.
+
+# C.3.3 SPLIT-CIFAR100
+
+The learning rate of 0.0002 and 0.0001 has been used for the classifier and VAE of each expert in CIFAR10. We decay the learning rate of the expert by 0.2 before it enters the wake phase. VAEs are trained to maximize Gaussian log-likelihood.
+
+# C.3.4 MNIST-SVHN
+
+The learning rate of 0.0001 and 0.0003 has been used for the classifier and VAE of each expert in CIFAR10. We decay the learning rates of classifier and VAE of each expert by 0.5 and 0.1 before it enters the wake phase. VAEs are trained to maximize Gaussian log-likelihood.
+
+# D PRUNING REDUNDANT EXPERTS
+
+Lin (2013) propose a simple algorithm to prune and merge redundant components in DPM models. Following the basic principle of the algorithm, we provide a pruning algorithm for CN-DPM. First, we need to measure the similarities between experts to choose which expert to prune. We compute the log-likelihood $l _ { n k } = p ( x _ { n } , y _ { n } | \hat { \phi } _ { k } )$ of each expert $k$ for data $\left( x _ { 1 : N } , y _ { 1 : N } \right)$ . As a result, we can obtain $K$ vectors with $N$ dimensions. We define the similarity $s ( \boldsymbol { k } , \boldsymbol { k } ^ { \prime } )$ between two experts $k$ and $k ^ { \prime }$ as the cosine similarity between the two corresponding vectors $l . \boldsymbol { k }$ and ${ \mathit { l } } . _ { k ^ { \prime } }$ , i.e., $\begin{array} { r } { s ( k , k ^ { \prime } ) = \frac { l \cdot k \cdot l _ { \cdot k ^ { \prime } } } { | l _ { \cdot k } | | l _ { \cdot k ^ { \prime } } | } } \end{array}$ . If the similarity is greater than a certain threshold $\epsilon$ , we remove one of the experts with smaller $\begin{array} { r } { N _ { k } = \sum _ { n } \rho _ { n , k } . } \end{array}$ . The $N _ { k }$ data of the removed expert are added to the remaining experts.
+
+Figure 4 shows an example of an expert pruning. We test CN-DPM on Split-MNIST with an $\alpha$ higher than the optimal value such that more than five experts are created. In this case, seven experts are created. If we build a similarity matrix as shown in Figure 4b, we can see which pair of experts are similar. We then threshold the matrix at 0.9 in Figure 4c and choose expert pairs (2/3) and $( 5 / 6 )$ for pruning. Comparing $N _ { k }$ within each pair, we can finally choose to prune expert 3 and 6. After pruning, the test accuracy marginally drops from $8 7 . 0 7 \%$ to $8 6 . 0 1 \%$ .
+
+![](images/818c2d43fb266324792f8469be351a664a19418170881f82b4384a2adf4b4125.jpg)  
+Figure 4: An example of the expert pruning in the Split-MNIST scenario.
+
+# E COMPARISON WITH TASK-BASED METHODS ON SPLIT-MNIST
+
+Table 4 compares our method with task-based methods for Split-MNIST classification. All the numbers except for our CN-DPM are excerpted from Hsu et al. (2018), in which all methods are trained for four epochs per task with a batch size of 128. Our method is trained for four epochs per task with a batch size of 10. The model architecture used in compared methods is the same as our baselines: a two-hidden-layer MLP with 400 hidden units per layer. All compared methods use a single output head, and the task information is given at training time but not at test time. For CN-DPM, we test two training settings where the first one uses task information to select experts, while the second one infers the responsible expert by the DPM principle. Task information is not given at test time in both cases.
+
+Notice that regularization methods often suffer from catastrophic forgetting while replay methods yield decent accuracies. Even though the task-free condition is a far more difficult setting, the performance of our method is significantly better than regularization and replay methods that exploit the task description. If task information is available at train time, we can utilize it to improve the performance even more.
+
+Table 4: Comparison with task-based methods on Split-MNIST classification. We report the average of 10 runs with $\pm$ standard error of the mean. The numbers except ours are from Hsu et al. (2018).   
+
+<table><tr><td>Type</td><td>Method</td><td>Task labels</td><td>Accuracy (%)</td></tr><tr><td rowspan="5">Regularization</td><td>EWC (Kirkpatrick et al.,2017)</td><td>&lt;&lt;&gt;</td><td>19.80 ± 0.05</td></tr><tr><td>Online EWC (Schwarz et al.,2018)</td><td></td><td>19.77 ± 0.04</td></tr><tr><td>SI (Zenke et al., 2017)</td><td></td><td>19.67 ± 0.09</td></tr><tr><td>MAS (Aljundi et al., 2018)</td><td>&lt;</td><td>19.52 ± 0.04</td></tr><tr><td>LwF (Li &amp; Hoiem, 2017)</td><td>√</td><td>24.17 ± 0.33</td></tr><tr><td rowspan="3">Replay</td><td>GEM (Lopez-Paz &amp; Ranzato,2017)</td><td>&lt;</td><td>92.20 ± 0.12</td></tr><tr><td>DGR (Shin et al.,2017)</td><td>√</td><td>91.24 ± 0.33</td></tr><tr><td>RtF (van de Ven &amp; Tolias,2018)</td><td>√</td><td>92.56 ± 0.21</td></tr><tr><td rowspan="2">Expansion</td><td>CN-DPM</td><td>√</td><td>93.81 ± 0.07</td></tr><tr><td>CN-DPM</td><td>×</td><td>93.70 ± 0.07</td></tr><tr><td></td><td>Upper bound (iid)</td><td></td><td>97.53 ± 0.30</td></tr></table>
+
+Table 5: Fuzzy Split-MNIST   
+
+<table><tr><td>Method</td><td>Acc. (%)</td><td>Param.</td></tr><tr><td>Fine-tune</td><td>28.41± 0.52</td><td>478K</td></tr><tr><td>Reservoir</td><td>88.64±0.48</td><td>478K</td></tr><tr><td>CN-DPM</td><td>93.22 ± 0.07</td><td>524K</td></tr></table>
+
+![](images/c24e00a633ff9c1799260f8d62c11ab4a7e17bf175aa03c227406faa41021be1.jpg)  
+Figure 5: Scenario configuration of Fuzzy Split-MNIST
+
+![](images/fa54a2521b527f3055721706e126414092a975fb4df125e58254fa1d0970461c.jpg)  
+Figure 6: Examples of generation samples by CN-DPM trained on Split-MNIST.
+
+# F FUZZY SPLIT-MNIST
+
+In addition, we experiment with the case where the task boundaries are not clearly defined, which we call Fuzzy-Split-MNIST. Instead of discrete task boundaries, we have transition stages between tasks where the data of existing and new tasks are mixed, but the proportion of the new task linearly increases. This condition adds another level of difficulty since it makes the methods unable to rely on clear task boundaries. The scenario is visualized in Figure 5. As shown in Table 5, CN-DPM can perform continual learning without task boundaries.
+
+# G GENERATION OF SAMPLES
+
+Even in discriminative tasks where the goal is to model $p ( y | x )$ , CN-DPM learns the joint distribution $p ( x , y )$ . Since CN-DPM is a complete generative model, it can generate $( x , y )$ pairs. To e first, i.e., ample oose $z$ from  exper $p ( z )$ whiven led by the c first sample egorical distribution from the generator $\begin{array} { r } { \mathrm { C a t } ( \frac { N _ { 1 } } { N } , \frac { N _ { 2 } } { N } , . . . , \frac { N _ { K } } { N } ) } \end{array}$ $z = k$ $x$ $p ( x ; \phi _ { k } ^ { G } )$ , and then sample $y$ from the discriminator $p ( \boldsymbol { y } | \boldsymbol { x } ; \phi _ { k } ^ { D } )$ . Figure 6 presents 50 sample examples generated from a CN-DPM trained on Split-MNIST for a single epoch. We observe that CN-DPM successfully generates examples of all tasks with no catastrophic forgetting.
+
+# H EXPERIMENTS WITH LONGER CONTINUAL LEARNING SCENARIOS
+
+We present experiments with much longer continual learning scenarios on Split-MNIST, SplitCIFAR10 and Split-CIFAR100 in Table 6, 7 and 8, respectively. We report the average of 10 runs with $\pm$ standard error of the mean. To compare with the default 1-epoch scenario, we carry out experiments that repeat each task 10 times, which are denoted 10 Epochs. In addition, we also present the results of repeating the whole scenario 10 times, which are denoted 1 Epoch $\times I 0$ . For example, in Split-MNIST, the 10 Epochs scenario consists of 10-epoch 0/1, 10-epoch 2/3, ..., 10-epoch 8/9 tasks. On the other hand, the 1 Epoch $\times I 0$ scenario revisits each task multiple times, i.e., 1-epoch 0/1, 1-epoch 2/3, ..., 1-epoch 8/9, 1-epoch 0/1, ..., 1-epoch $8 / 9$ . We use the same hyperparameters tuned for the 1-epoch scenario.
+
+We find that the accuracy of Reservoir drops as the length of each task increases. As mentioned in the main text, this phenomenon seems to be caused by overfitting on the samples in the replay memory. Since only a small number of examples in the memory represent each task, replaying them for a long period degrades the performance. On the other hand, the performance of our CN-DPM improves as the learning process is extended.
+
+In the 1 Epoch $\times I O$ setting, CN-DPM shows similar performance with 10 Epoch since the model sees each data point 10 times in both scenarios. On the other hand, Reservoir’s scores in the 1 Epoch $\times I 0$ largely increase compared to both 1 Epoch and 10 Epoch This difference can be explained by how the replay memory changes while training progresses. In the 10 Epoch setting, if a task is finished, it is not visited again. Therefore, the examples of the task in the replay memory monotonically decreases, and the remaining examples are replayed repeatedly. As the training progresses, the model is overfitted to the old examples in the memory and fails to generalize in the old tasks. In contrast, in $I { E p o c h } \times I O$ setting, each task is revisited multiple times, and each time a task is revisited, the replay memory is also updated with the new examples of the task. Therefore, the overfitting problem in the old tasks is greatly relieved.
+
+Another important remark is that CN-DPM does not blindly increase the number of experts. If we add a new expert at every constant steps, we would have 10 times more experts in the longer scenarios. However, this is not the case. CN-DPM determines whether it needs a new expert on a data-by-data basis such that the number of experts is determined by the task distribution, not by the length of training.
+
+Table 6: Experiments with longer training episodes on Split-MNIST   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">1Epoch</td><td colspan="2">10 Epochs</td><td colspan="2">1 Epoch ×10</td></tr><tr><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td></tr><tr><td>iid-offline</td><td>98.63 ± 0.01</td><td>478K</td><td>98.63 ± 0.01</td><td>478K</td><td>98.63 ± 0.01</td><td>478K</td></tr><tr><td>iid-online</td><td>96.18 ± 0.19</td><td>478K</td><td>97.67 ± 0.05</td><td>478K</td><td>97.67 ± 0.05</td><td>478K</td></tr><tr><td>Fine-tune</td><td>19.43 ± 0.02</td><td>478K</td><td>19.68 ± 0.01</td><td>478K</td><td>20.27± 0.26</td><td>478K</td></tr><tr><td>Reservoir</td><td>85.69 ± 0.48</td><td>478K</td><td>78.82 ± 0.71</td><td>478K</td><td>92.06 ± 0.11</td><td>478K</td></tr><tr><td>CN-DPM</td><td>93.23±0.09</td><td>524K</td><td>94.39 ± 0.04</td><td>524K</td><td>94.15 ± 0.04</td><td>616K</td></tr></table>
+
+Table 7: Experiments with longer training episodes on Split-CIFAR10   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">1Epoch</td><td colspan="2">10Epochs</td><td colspan="2">1 Epoch ×10</td></tr><tr><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td></tr><tr><td>iid-offline</td><td>93.17± 0.03</td><td>11.2M</td><td>93.17± 0.03</td><td>11.2M</td><td>93.17± 0.03</td><td>11.2M</td></tr><tr><td>iid-online</td><td>62.79 ± 1.30</td><td>11.2M</td><td>83.19 ±0.27</td><td>11.2M</td><td>83.19 ±0.27</td><td>11.2M</td></tr><tr><td>Fine-tune</td><td>18.08 ± 0.13</td><td>11.2M</td><td>19.31 ± 0.03</td><td>11.2M</td><td>19.33 ± 0.03</td><td>11.2M</td></tr><tr><td>Reservoir</td><td>44.00 ± 0.92</td><td>11.2M</td><td>43.82 ± 0.53</td><td>11.2M</td><td>51.44 ± 0.42</td><td>11.2M</td></tr><tr><td>CN-DPM</td><td>45.21 ± 0.18</td><td>4.60M</td><td>46.98 ± 0.18</td><td>4.60M</td><td>47.10 ± 0.16</td><td>4.60M</td></tr></table>
+
+# I EXPERIMENTS WITH DIFFERENT MEMORY SIZES
+
+In Split-CIFAR10/100 experiments in the main text, we set the memory size of Reservoir and CNDPM to 1000, following Aljundi et al. (2019b). Table 9 compares the experimental results with different memory sizes of 500 and 1000 on Split-CIFAR10/100. Compared to Reservoir, whose performance drops significantly with smaller memory, CN-DPM’s accuracy drop is relatively marginal.
+
+Table 8: Experiments with longer training episodes on Split-CIFAR100   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">1 Epoch</td><td colspan="2">10 Epochs</td><td colspan="2">1Epoch ×10</td></tr><tr><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td><td>Acc. (%)</td><td>Param.</td></tr><tr><td>iid-offline</td><td>73.80 ± 0.11</td><td>11.2M</td><td>73.80 ± 0.11</td><td>11.2M</td><td>73.80 ± 0.11</td><td>11.2M</td></tr><tr><td>iid-online</td><td>20.46± 0.30</td><td>11.2M</td><td>54.58±0.27</td><td>11.2M</td><td>54.58±0.27</td><td>11.2M</td></tr><tr><td>Fine-tune</td><td>2.43±0.05</td><td>11.2M</td><td>3.99 ± 0.03</td><td>11.2M</td><td>4.30±0.02</td><td>11.2M</td></tr><tr><td>Reservoir</td><td>10.01 ± 0.35</td><td>11.2M</td><td>6.61± 0.20</td><td>11.2M</td><td>14.53 ± 0.35</td><td>11.2M</td></tr><tr><td>CN-DPM</td><td>20.10± 0.12</td><td>19.2M</td><td>20.95 ±0.09</td><td>19.2M</td><td>20.67± 0.13</td><td>19.2M</td></tr></table>
+
+Table 9: Experiments with different memory sizes.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Memory</td><td colspan="2">Split-CIFAR10 Acc. (%)</td><td colspan="2">Split-CIFAR100 Acc. (%)</td></tr><tr><td>1Epoch</td><td>10 Epoch</td><td>1 Epoch</td><td>10 Epoch</td></tr><tr><td>Reservoir</td><td>500</td><td>33.53 ± 1.03</td><td>34.46 ± 0.49</td><td>6.24±0.25</td><td>4.99 ± 0.09</td></tr><tr><td>CN-DPM</td><td>500</td><td>43.07 ± 0.16</td><td>47.01± 0.22</td><td>19.17 ± 0.13</td><td>20.77 ± 0.11</td></tr><tr><td>Reservoir</td><td>1000</td><td>44.00 ±0.92</td><td>43.82 ± 0.53</td><td>10.01± 0.35</td><td>6.61± 0.20</td></tr><tr><td>CN-DPM</td><td>1000</td><td>45.21± 0.18</td><td>46.98 ± 0.18</td><td>20.10 ± 0.12</td><td>20.95 ± 0.09</td></tr></table>
+
+# J THE EFFECT OF CONCENTRATION PARAMETER
+
+Table 10 shows the results of CN-DPM on Split-MNIST classification according to the concentration parameter $\alpha$ , which defines the prior of how sensitive CN-DPM is to new data. With a higher $\alpha$ , an expert tends to be created more easily. In the experiment reported in the prior sections, we set $\log \alpha = - 4 0 0$ . At $\log \alpha = - 6 0 0$ , too few experts are created, and the accuracy is rather low. As $\alpha$ increases, the number of experts grows along with the accuracy. Although the CN-DPM model is task-free and automatically decides the task assignments to experts, we still need to tune the concentration parameter to find the best balance point between performance and model capacity, as all Bayesian nonparametric models require.
+
+Table 10: The effects of concentration parameter $\alpha$ .   
+
+<table><tr><td>loga</td><td>Acc. (%)</td><td>Experts</td><td>Param.</td></tr><tr><td>-600</td><td>54.04 ± 2.22</td><td>3.20 ± 0.13</td><td>362K</td></tr><tr><td>-400</td><td>93.23 ± 0.09</td><td>5.00± 0.00</td><td>524K</td></tr><tr><td>80</td><td>93.54 ± 0.21</td><td>14.4 ± 1.35</td><td>1.44M</td></tr></table>
+
+![](images/09823615acba6ae26835c9d330c7d4287b5fcb2e823f21eeef1b6fe1e7680d25.jpg)  
+Figure 7: The effects of concentration parameter $\alpha$ .
+
+# K THE EFFECT OF PARAMETER SHARING
+
+Table 11 compares when the parameters are shared between experts and when they are not shared. By sharing the parameters, we could reduce the number of parameters by approximately $38 \%$ without sacrificing accuracy.
+
+Table 11: The effects of parameter sharing.   
+
+<table><tr><td>Model</td><td>Acc. (%)</td><td>Experts</td><td>Param.</td></tr><tr><td>CN-DPM</td><td>93.23± 0.09</td><td>5</td><td>524K</td></tr><tr><td>CN-DPM w/o PS</td><td>93.30 ± 0.24</td><td>5</td><td>839K</td></tr></table>
+
+# L TRAINING GRAPHS
+
+Figure 8 shows the training graphs of our experiments. In addition to the performance metrics, we present the number of experts in CN-DPM and compare the total number of parameters with the baselines. The bold lines represent the average of the 10 runs while the faint lines represent individual runs.
+
+Figure 9 and Figure 10 show how the accuracy of each task changes during training. We also present the average accuracy of learned tasks at the bottom right.
+
+# M COMPARISON WITH THE CURL
+
+Continual Unsupervised Representation Learning (CURL) (Rao et al., 2019) is a parallel work that shares some characteristics with our CN-DPM in terms of model expansion and short-term memory. However, there are several key differences that distinguish our method from CURL, which will be elaborated in this section. Following the notations of Rao et al. (2019), here $y$ denotes the cluster assignment, and $z$ denotes the latent variable.
+
+1. The Generative Process. The primary goal of CURL is to continually learn a unified latent representation $z$ , which is shared across all tasks. Therefore, the generative model of CURL explicitly consists of the latent variable $z$ as summarized as follows:
+
+$$
+\begin{array} { r } { \iota ( x , y , z ) = p ( y ) p ( z | y ) p ( x | z ) \mathrm { ~ w h e r e ~ } y \sim \mathrm { C a t } ( \pi ) , z \sim { \mathcal N } ( \mu _ { z } ( y ) , \sigma _ { z } ^ { 2 } ( y ) ) , x \sim \mathrm { B e r n o u l l i } ( \mu _ { x } ( z ) ) . } \end{array}
+$$
+
+The overall distribution of $z$ is the mixture of Gaussians, and $z$ includes the information of $y$ such that $x$ and $y$ are conditionally independent given $z$ . Then, $z$ is fed into a single decoder network $\mu _ { x }$ to generate the mean of $x$ , which is modeled by a Bernoulli distribution. On the other hand, the generative version of CN-DPM, which does not include classifiers, has a simpler generative process:
+
+$$
+p ( x , y ) = p ( y ) p ( x | y ) { \mathrm { ~ w h e r e ~ } } y \sim \operatorname { C a t } ( \pi ) , x \sim p ( x | y ) .
+$$
+
+The choice of $p ( x | y )$ here is not necessarily restricted to VAEs (Kingma & Welling, 2014); one may use other kinds of explicit density models such as PixelRNN (Oord et al., 2016). Even if we use VAEs to model $p ( x | y )$ , the generative process is different from CURL:
+
+$$
+p ( x , y , z ) = p ( y ) p ( z ) p ( x | y , z ) { \mathrm { ~ w h e r e ~ } } y \sim \operatorname { C a t } ( \pi ) , z \sim \mathcal { N } ( 0 , I ) , x \sim { \mathrm { B e r n o u l l i } } ( \mu _ { x } ^ { y } ( z ) ) .
+$$
+
+Unlike CURL, CN-DPM generates $y$ and $z$ independently and maintains a separate decoder $\mu _ { x } ^ { y }$ for each cluster $y$ .
+
+2. The Necessity for Generative Replay in CURL. CURL periodically saves a copy of its parameters and use it to generate samples of learned distribution. The generated samples are played together with new data such that the main model does not forget previously learned knowledge. This process is called generative replay. The generative replay is an essential element in CURL, unlike our CN-DPM. CURL assumes a factorized variational posterior $q ( y , z | x ) = q ( y | x ) q ( z | x , y )$ where $q ( y | x )$ and $q ( z | x , y )$ are modeled by separate output heads of the encoder neural network. However, the output head for $\dot { \mathbf { \zeta } } q ( y | x )$ is basically a gating network that could be vulnerable to catastrophic forgetting, as mentioned in Section 3.1. Moreover, CURL shares a single decoder $\mu _ { x }$ across all tasks. As a consequence, expansion alone is not enough to stop catastrophic forgetting, and CURL needs another CL method to prevent catastrophic forgetting in the shared components. This is the main reason why the generative replay is crucial in CURL. As shown in the ablation test of Rao et al. (2019), the performance of CURL drops without the generative replay. In contrast, the components of CN-DPM are separated for each task (although they may share low-level representations) such that no additional treatment is needed.
+
+![](images/11d3a920dc579a2c056e292bc9de50e1c3b3eed41efcd9ba908d4488836a5ab9.jpg)  
+Figure 8: Full training graphs.
+
+![](images/cec9d0022828c582d28702fc64dcec282d27d97323c3f4f817554dbc388b5197.jpg)  
+Figure 9: Accuracy for each task in Split-CIFAR10.
+
+![](images/bb765766a41e1ef31e9bfa87e14ed1a8247a778abf5b5bc71830f05916d127db.jpg)  
+Figure 10: Accuracy for each task in Split-CIFAR100.