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+# On the Generalization of Neural Networks Trained with SGD: Information-Theoretical Bounds and Implications
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Understanding the generalization behaviour of deep neural networks is an important   
+2 theme of modern research in machine learning. In this paper, we follow up on   
+3 a recent work of Neu [49] and present new information-theoretic upper bounds   
+4 for the generalization error of neural networks trained with SGD. Our bounds and   
+5 experimental study provide new insights on the SGD training of neural networks.   
+6 They also point to a new and simple regularization scheme which we show performs   
+7 comparably to the current state of the art.
+
+# 8 1 Introduction
+
+9 The outstanding performance of deep learning has brought to the surface some intriguing properties   
+10 of deep neural networks, one of which is the observation that despite their high capacity, deep neural   
+11 networks tend to generalize well [80]. This contradicts classical wisdom in statistical learning theory   
+12 (e.g., [71] ) and has stimulated intense research interest in understanding the generalization behaviour   
+13 of modern neural networks.   
+14 One theme of research focuses on the study of over-parameterized neural networks, where generaliza  
+15 tion bounds are obtained [22, 1, 6, 50, 52, 53, 2, 3] and a curious “double descent” phenomenon is   
+16 observed and analyzed [9, 46, 78]. New bounding techniques for analyzing generalization have also   
+17 been developed, utilizing information theoretic quantities [64, 65, 77, 5, 13, 69, 23, 7, 4, 30, 84]. The   
+18 bounds provided by these techniques have the advantages of accounting for both the data structure   
+19 and the learning algorithm.   
+20 The generalization ability of neural networks trained with mini-batched stochastic gradient descent   
+21 [61], simply referred to SGD in this paper, has also been widely studied. Specifically, built on a   
+22 connection between stability and generalization [12], a stability-based bound is first presented in [32],   
+23 followed by a surge of research effort exploiting similar approaches [44, 17, 26, 43, 8]. Information  
+24 theoretic bounding techniques have also demonstrated great power in analyzing SGD-like algorithms.   
+25 For example, [55] is the first to utilize information-theoretical bound in analyzing the generalization   
+26 ability of SGLD [28, 74]. The bound was subsquently improved by [47, 31, 62, 72]. Inspired by the   
+27 work of [55], [49] presents an information- theoretic analysis of the models trained with SGD. The   
+28 analysis of [49] constructs an auxiliary weight process parallel to SGD training and upper-bounds the   
+29 generalization error through this auxiliary process.   
+30 Another line of research connects the generalization of neural networks with the flatness of loss   
+31 minima [35] found by SGD or its variant [40, 20, 24, 51, 16, 37, 38, 83, 27]. This understanding   
+32 has led to the discovery of new SGD-based training algorithms for improved generalization. For   
+33 example, in a concurrent development by [83] and [27], a local “max-pooling” operation is applied to   
+34 the loss landscape prior to the SGD updates. This approach, referred to as AMP[83] or SAM[27], is   
+35 shown to make SGD favor flatter minima and achieve the state-of-the-art performance among various   
+36 competitive regularization schemes [83].   
+37 In this paper, we focus on investigating the generalization of neural networks trained with SGD.   
+38 We build upon the work of [49]. Following the same construction of the auxiliary weight process   
+39 in [49], we present upper bounds of generalization error that improve upon [49] in two ways. The   
+40 first improvement is via removing an unnecessary term in the bounds of [49] by invoking the HWI   
+41 inequality [60]. The second improvement is via replacing a sample-level mutual information term in   
+42 [49] with an instance-level mutual information term, exploiting a recent result of [13]. The bounds we   
+43 obtain decompose into two terms, one measuring the impact of training trajectories (“the trajectory   
+44 term”) and the other measuring the impact of the flatness of the found solution (“the flatness term”).   
+45 We empirically validate the derived bounds. Various insights are also obtained experimentally   
+46 concerning the generalization of neural networks under SGD training. For example, the batch size of   
+47 SGD appears to impact the trajectory term and the flatness term in opposite ways, which complicates   
+48 the overall dependency of generalization error on batch sizes. A particular interesting observation   
+49 from our experiments is that a key quantity arising in the trajectory term of the bounds, which we   
+50 refer to as gradient dispersion1, reveals a double descent phenomenon with respect to training epochs.   
+51 Most intriguingly, the valley in the double descent curve appears to mark the great divide between the   
+52 “generalization regime” and the “memorization regime” of training. Furthering from this observation,   
+53 we also show that it is possible to reduce the memorization effect by dynamically clipping the gradient   
+54 and reducing its dispersion.   
+55 Our bounds also inspire a natural and simple solution to alleviate generalization error. Specifically,   
+56 we propose a new training scheme, referred to as Gaussian model perturbation (GMP), aiming at   
+57 reducing the flatness term of the bounds. This scheme effectively applies a local “average pooling” to   
+58 the empirical risk surface prior to SGD, greatly resembling the “max-pooling” approach adopted in   
+59 AMP[83]. We demonstrate experimentally that GMP achieves a competitive performance with the   
+60 current art of regularization schemes.
+
+Length constraints precludes elaboration at places. The reader is referred to supplementary materials for proofs and additional information.
+
+63 Other Related Literature Gradient dispersion is mostly studied from optimization perspectives[11,   
+64 63, 39, 75, 25]. Prior to this work, only a few works relate gradient dispersion with the generalization   
+65 behaviour of the networks. In [49, 72], gradient dispersion also appears in the generalization bounds.   
+66 In [38], gradient dispersion is argued to capture a notion of “flatness” of the local minima of the loss   
+67 landscape, thereby correlating with generalization.   
+68 Injecting noise in the training process has been proposed in various regularization schemes, for   
+69 example, [10, 14, 15, 68, 73]. But unlike the Gaussian model perturbation scheme derived in this   
+70 paper, where noise is injected to the model parameters, noise in those schemes is injected either to   
+71 the training data or to the network activation.   
+72 Gradient clipping is a common technique for preventing gradient exploding (see, e.g., [45, 56]).   
+73 This technique is also used in [82] to accelerate training. In this paper, gradient clipping is used to   
+74 investigate and control the impact of gradient dispersion on generalization error.
+
+# 75 2 Preliminaries
+
+Population Risk, Empirical Risk and Generalization Error Unless otherwise noted, a random variable will be denoted by a capitalized letter (e.g., $Z$ ), and its realization denoted by the corresponding lower-case letter (e.g. $z$ ). Let $\mathcal { Z }$ be the instance space of interest and $\mu$ be an unknown distribution on $\mathcal { Z }$ , specifying random variable $Z$ . Let $\mathcal { W } \subseteq \bar { \mathbb { R } } ^ { d }$ be the space of hypotheses. Suppose that a training sample ${ \bar { S } } = ( Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n } )$ is drawn i.i.d. from $\mu$ and that a stochastic learning algorithm $\mathcal { A }$ takes $S$ as its input and outputs a hypothesis $W \in { \mathcal { W } }$ according to some conditional distribution $P _ { W | S }$ mapping ${ \mathcal { Z } } ^ { n }$ to $\mathcal { W }$ . Let $\ell : \mathcal { W } \times \mathcal { Z } \to \mathbb { R } ^ { + }$ be a loss function, where $\ell ( w , z )$ measures the “unfitness” or “error” of any $z \in { \mathcal { Z } }$ with respect to a hypothesis $w \in \mathcal W$ . The population risk, for any
+
+$$
+L _ { \mu } ( w ) \triangleq \mathbb { E } _ { Z \sim \mu } [ \ell ( w , Z ) ] .
+$$
+
+85 The goal of learning is to find a hypothesis $w$ that minimizes the population risk. But since $\mu$ is only   
+6 partially accessible via the sample $S$ , in practice, we instead turn to the empirical risk, defined as
+
+$$
+L _ { S } ( w ) \triangleq { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \ell ( w , Z _ { i } ) .
+$$
+
+87 The expected generalization error of the learning algorithm $\mathcal { A }$ is then defined as
+
+$$
+\mathrm { g e n } ( \mu , P _ { W | S } ) \triangleq \mathbb { E } _ { W , S } [ L _ { \mu } ( W ) - L _ { S } ( W ) ] ,
+$$
+
+where the expectation is taken over the joint distribution of 88 $( S , W )$ (i.e., $\mu ^ { n } \otimes P _ { W | S } )$ .
+
+89 Throughout this paper, we take $\ell$ as a continuous function (adopting the usual notion “surrogate loss”   
+90 [66].). Additionally, we assume that $\ell$ is differentiable almost everywhere with respect to both $w$ and   
+91 $z$ . Furthermore we assume that $\ell ( w , Z )$ is $R$ -subgaussian2 for any $w \in \mathcal { W }$ . Note that a bounded loss   
+92 is guaranteed to be subgaussian for all $\mu$ and all $w \in \mathcal { W }$ . Let $I ( X ; Y )$ denote the mutual information   
+93 [18] between any pair of random variables $( X , Y )$ . The following results are known.
+
+94 Lemma 1 ([77, Theorem 1.]). The expected generalization error of algorithm $\mathcal { A }$ is bounded by
+
+$$
+| \mathrm { g e n } ( \mu , P _ { W | S } ) | \leq \sqrt { \frac { 2 R ^ { 2 } } { n } I ( W ; S ) } ,
+$$
+
+95
+
+96 Lemma 2 ([13, Proposition 1.]). The expected generalization error of algorithm $\mathcal { A }$ is bounded by
+
+$$
+| \mathrm { g e n } ( \mu , P _ { W | S } ) | \leq \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sqrt { 2 R ^ { 2 } I ( W ; Z _ { i } ) } ,
+$$
+
+97
+
+98 Stochastic Gradient Descent We now restrict the learning algorithm $\mathcal { A }$ to be the mini-batched   
+99 stochastic gradient descent (SGD) algorithm for empirical risk minimization. For each training epoch,   
+100 the dataset $S$ is randomly split into $m$ disjoint mini-batches, each having size $b$ , namely, $n = m b$ .   
+101 Based on each batch, one parameter update is performed. Specifically, let $B _ { t }$ denote the batch used   
+102 for the $t ^ { \mathrm { t h } }$ update. Define
+
+$$
+g ( w , B _ { t } ) \triangleq \frac { 1 } { b } \sum _ { z \in B _ { t } } \nabla _ { w } \ell ( w , z ) ,
+$$
+
+103 namely, $g ( w , B _ { t } )$ is the average gradient computed for the batch $B _ { t }$ with respect to parameter $w$ . The rule for the 104 $t ^ { \mathrm { t h } }$ parameter update is then
+
+$$
+W _ { t } \triangleq W _ { t - 1 } - \lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) ,
+$$
+
+105 where $\lambda _ { t }$ is the learning rate at the step $t$ . The initial parameter setting $W _ { 0 }$ is assumed to be drawn   
+106 from the zero-mean spherical Gaussian $\mathcal { N } ( 0 , \sigma _ { 0 } ^ { 2 } \mathbf { I } _ { d } )$ with variance $\sigma _ { 0 } ^ { 2 }$ in each dimension. We will   
+107 assume that the SGD algorithm stops after $T$ updates and outputs $W _ { T }$ as the learned model parameter.   
+108 Given the training sample $S$ , let $\xi$ govern the randomness in the sequence $( B _ { 1 } , B _ { 2 } , \ldots , B _ { T } )$ of   
+109 batches. For the simplicity of notion, we will fix the configuration of $\xi$ . That is, we will assume a   
+110 fixed “batching trajectory”, or a fixed way to shuffle the example indices $\{ 1 , \ldots , n \}$ and divide them   
+111 into $m$ batches in each epoch. The presented generalization bounds of this paper can be extended   
+112 to the case where the batching trajectory is uniformly random (as we set up above). This merely   
+113 involves averaging over all batching trajectories or taking expectation over $\xi$ .   
+114 Auxiliary Weight Process We now associate with the SGD algorithm an auxiliary weight process   
+115 $\{ \widetilde { W _ { t } } \}$ . Let $\sigma ^ { 2 }$ be given, and let $\sigma _ { 1 } , \sigma _ { 2 } , \ldots , \sigma _ { T }$ be a sequence of positive real numbers. Define
+
+$$
+\widetilde { W } _ { 0 } \triangleq W _ { 0 } , \quad \mathrm { a n d } \widetilde { W } _ { t } \triangleq \widetilde { W } _ { t - 1 } - \lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) + N _ { t } , \mathrm { f o r } t > 0 ,
+$$
+
+where 116 $N _ { t } \sim \mathcal N ( 0 , \sigma _ { t } ^ { 2 } \mathbf I _ { d } )$ is a Gaussian noise. The relationship between this auxiliary weight process 117 $\{ \widetilde { W _ { t } } \}$ and the weight process $\{ W _ { t } \}$ in SGD is shown in the Bayesian network below.
+
+118
+
+$$
+\begin{array} { r l r l r l r l r l r l r l r } & { } & & { N _ { 1 } } & & { N _ { 2 } } & & { \cdots } & & { N _ { T - 1 } } & & { N _ { T } } \\ & { } & & { \downarrow } & & { \downarrow } & & { } & & { \downarrow } & & { \downarrow } & & { \downarrow } \\ { \widetilde { W } _ { 0 } } & {  } & { \widetilde { W } _ { 1 } } & {  } & { \widetilde { W } _ { 2 } } & {  } & { \cdots } & {  } & { \widetilde { W } _ { T - 1 } } & {  } & { \widetilde { W } _ { T } } \\ { \parallel } & { \nearrow } & { } & { \chi } & { \chi } & { \nearrow } & { \dots } & {  } & { \dots } & {  } & { W _ { T } } \\ { W _ { 0 } } & {  } & { W _ { 1 } } & {  } & { W _ { 2 } } & {  } & { \cdots } & {  } & { W _ { T - 1 } } & {  } & { W _ { T } } \end{array}
+$$
+
+119 Let $\begin{array} { r } { \Delta _ { t } = \sum _ { \tau = 1 } ^ { t } N _ { \tau } } \end{array}$ . Noting that the weight updates in $\{ \widetilde { W } _ { t } \}$ uses the same gradient signal as that   
+120 used in $\{ W _ { t } \}$ (which depends on $W _ { t - 1 }$ not $\widetilde { W } _ { t - 1 } )$ ), it is immediate that $\widetilde { W } _ { t } = W _ { t } + \Delta _ { t }$ . Note that this   
+121 auxiliary process follows the same construction as [49], which we will use to study the generalization   
+122 error of SGD.
+
+123 To that end, define gradient dispersion at parameter $w$ by
+
+$$
+\mathbb { V } ( w ) \triangleq \mathbb { E } \left[ | | \nabla _ { w } \ell ( w , Z ) - \mathbb { E } \left[ \nabla _ { w } \ell ( w , Z ) \right] | | _ { 2 } ^ { 2 } \right] ,
+$$
+
+124 where the expectation is taken over $Z \sim \mu$ .
+
+For a given sample 125 $s \in \mathcal { Z } ^ { n }$ , define
+
+$$
+\gamma ( w , s ) \triangleq \mathbb { E } \left[ L _ { s } ( w + \Delta _ { T } ) - L _ { s } ( w ) \right] ,
+$$
+
+26 where the expectation is taken over $\Delta _ { T }$ and $L _ { s } ( w )$ is the empirical risk of $s$ at parameter $w$ .
+
+In the remainder of the paper, let 127 $S ^ { \prime }$ denote another sample drawn from $\mu ^ { n }$ , independent of all other 128 random variables. The main generalization bound in [49] is re-stated below.
+
+129 Lemma 3 ([49, Theorem 1.]). The generalization error of SGD is upper bounded by
+
+$$
+\left| \mathrm { g e n } ( \mu , P _ { W _ { T } | S } ) \right| \leq \sqrt { \frac { 2 R ^ { 2 } } { n } \sum _ { t = 1 } ^ { T } \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \Psi ( W _ { t - 1 } ) + \frac { \Psi ( W _ { t - 1 } ) } { b } \right] } + \left| \mathbb { E } \left[ \gamma ( W _ { T } , S ) - \gamma ( W _ { T } , S ^ { \prime } ) \right] \right| ,
+$$
+
+$$
+\begin{array} { r } { \Psi ( w _ { t - 1 } ) \triangleq \mathbb { E } \left[ | | \nabla _ { w } \ell ( w _ { t - 1 } , Z ) - \nabla _ { w } \ell ( w _ { t - 1 } + \zeta , Z ) | | _ { 2 } ^ { 2 } \right] a n d \zeta \sim \mathcal { N } ( 0 , 2 \sum _ { i = 1 } ^ { t - 1 } \sigma _ { i } ^ { 2 } \mathrm { I } _ { d } ) . } \end{array}
+$$
+
+The term $\Psi ( w _ { t - 1 } )$ in the bound is referred to as “local gradient sensitivity” in [49].
+
+# 3 New Generalization Bounds for SGD
+
+133 We first prove that the generalization bound in Lemma 3 can be tightened by removing the local   
+134 gradient sensitivity term $\Psi ( w _ { t - 1 } )$ . The key observation is that an independence condition used for   
+135 establishing Lemma 3 in [49] is unnecessary (see Lemma 4 in [49]). This requires invoking a vector   
+136 version of the HWI inequality [60, Lemma 3.4.2], which we prove in this paper.
+
+Lemma 4. Let37 $X$ and $Y$ be two random vectors in $\mathbb { R } ^ { d }$ , and let $\boldsymbol { N } \sim \mathcal { N } ( 0 , \mathbf { I } _ { d } )$ be independent of 38 $( X , Y )$ . Then, for every $t > 0$ , $\begin{array} { r } { \mathrm { D } _ { \mathrm { K L } } \big ( P _ { X + \sqrt { t } N } | | P _ { Y + \sqrt { t } N } \big ) \leq \frac { 1 } { 2 t } \mathbb { E } \left[ | | X - Y | | ^ { 2 } \right] } \end{array}$ .
+
+39 Here $\mathrm { D } _ { \mathrm { K L } }$ is the KL divergence. Note that the bound in Lemma 3 relies on a similar result which   
+40 however requires the independence of $X$ and $Y$ . Using Lemma 4, we obtain the following theorem.
+
+141 Theorem 1. The generalization error of SGD is upper bounded by
+
+$$
+\left| \mathrm { g e n } ( \mu , P _ { W _ { T } | S } ) \right| \le \sqrt { \frac { 2 R ^ { 2 } } { n b } \sum _ { t = 1 } ^ { T } \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } + \left| \mathbb { E } \left[ \gamma ( W _ { T } , S ) - \gamma ( W _ { T } , S ^ { \prime } ) \right] \right| .
+$$
+
+143 The proof of this theorem, as the bounds in [49], relies on Lemma 1 and the sample-level mutual   
+144 information bound therein. This theorem can be further tightened by exploiting the fact that the   
+145 instance-level mutual information bound in Lemma 2 is in fact tighter than the sample-level mutual   
+146 information bound in Lemma 1, as shown in [13]. The main ingredient to proceed in this direction is   
+147 the following lemma.   
+Lemma 5. Le148 $\begin{array} { r } { \ d _ { t } G _ { t } = - \lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) . \ d H _ { t } Z _ { i } \in B _ { t } , t h e n I ( G _ { t } + N _ { t } ; Z _ { i } | \widetilde W _ { t - 1 } ) \le \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } b ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] . } \end{array}$   
+49   
+150 In this lemma, the mutual information $I ( G _ { t } + N _ { t } ; Z _ { i } | \widetilde { W } _ { t - 1 } )$ roughly indicates the degree by which   
+151 the SGD’s updating signal $G _ { t }$ (smoothed with noise) depends on an individual training instance   
+152 $Z _ { i }$ , when $Z _ { i }$ is used for computing the gradient. When this dependency is strong (giving rise to a   
+153 high value of the mutual information), the model conceivably tends to overfit the individual training   
+154 instances. This lemma suggests that the strength of this dependency can be upper-bounded by the   
+155 expected gradient dispersion at the current weight configuration. In our experiments, we will estimate   
+156 the expected gradient dispersion and validate this intuition.   
+157 It is remarkable that the noise $\{ N _ { t } \}$ plays an important role for the bound to hold. To see this,   
+158 consider $b = 1$ and $\mathcal { Z }$ is countable and large. Then $\bar { I } ( G _ { t } ; Z _ { t } | W _ { t - 1 } )$ is merely the conditional entropy   
+159 $H ( Z _ { i } | W _ { t - 1 } )$ , which would grow with sample size $n$ at least as $\log n$ . Upper-bounding it with a   
+160 quantity independent of $n$ would be impossible – This justifies the construction of the auxiliary   
+161 weight process.   
+162 We now state our main theorem. Unlike Theorem 1, which considers a random batching trajectory,   
+163 this theorem considers a fixed batching trajectory to keep the expression less cluttered. For that   
+164 batching trajectory, we will use $\mathcal { T } _ { i }$ to denote the set of indices of batches $B _ { t }$ containing instance $Z _ { i }$ .
+
+165 Theorem 2. The expected generalization error of SGD is bounded by
+
+$$
+\left| \mathrm { g e n } ( \mu , P _ { W _ { T } | S } ) \right| \leq \frac { R } { n b } \sum _ { i = 1 } ^ { n } \sqrt { \sum _ { t \in { \mathcal T } _ { i } } \frac { 2 \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } + \left| \mathbb { E } \left[ \gamma ( W _ { T } , S ) - \gamma ( W _ { T } , S ^ { \prime } ) \right] \right| .
+$$
+
+167 With an additional assumption, the second term in the bound can be re-expressed, as shown in the   
+168 following corollary.
+
+169 Corollary 1. Assume $L _ { \mu } ( w _ { T } ) \leq \mathbb { E } _ { \Delta } \left[ L _ { \mu } ( w _ { T } + \Delta _ { T } ) \right]$ , then the following holds,
+
+$$
+\mathrm { g e n } ( \mu , P _ { W r | S } ) \leq \frac { R } { n b } \sum _ { i = 1 } ^ { n } \sqrt { \sum _ { t \in \mathcal { T } _ { i } } \frac { 2 \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } + \frac { \sum _ { t = 1 } ^ { T } \sigma _ { t } ^ { 2 } } { 2 } \mathrm { T r } \left( \mathbb { E } \left[ \mathrm { H } _ { W _ { T } } ( Z ) \right] \right) ,
+$$
+
+where 170 $\mathrm { H } _ { W _ { T } }$ is the Hessian matrix of the loss with respect to $W _ { T }$ and $\operatorname { T r } ( \cdot )$ denotes trace.
+
+171 Corollary 1 follows directly from the second order Taylor expansion of the second term in the bound   
+172 of Theorem 2. The condition $L _ { \mu } ( w _ { T } ) \leq \mathbb { E } _ { \Delta } \left[ L _ { \mu } ( \hat { w _ { T } } + \bar { \Delta _ { T } } ) \right]$ indicates that the perturbation does   
+173 not decrease the population risk. This is also assumed in [27] in the derivation of a PAC-Bayesian   
+174 generalization bound.   
+175 Notably, in the bound of Theorem 2, the first term captures the impact of the training trajectory   
+176 (“trajectory term”), and the second term captures the impact of the final solution. As seen in Corollary   
+177 1, this term in fact measures the flatness for the loss landscape at the found solution (“flatness term”).   
+178 The previous bound of [49] (Lemma 3) and its tightened version in Theorem 1 also similarly contain a   
+179 trajectory term and a flatness term. Despite that the flatness term there are identical to that in Theorem   
+180 2, we now show the trajectory term in Theorem 2 does improve on its counter-part in Theorem 1.   
+181 Lemma 6. Assume the instances are sampled without replacement in every epoch. Then the trajectory   
+182 term in Theorem 2 is upper-bounded by
+
+$$
+\operatorname* { m i n } \left\{ \frac { R } { n } \sum _ { t = 1 } ^ { T } \sqrt { \frac { 2 \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } , \sqrt { \frac { 2 R ^ { 2 } } { n b } \sum _ { t = 1 } ^ { T } \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } \right\} .
+$$
+
+184 The condition in Lemma 6 is usually satisfied in practice. This lemma then immediately implies that   
+185 the trajectory term in Theorem 2 is no worse than that in Theorem 1. Incorporating this result, if we   
+186 restrict the smoothness of the loss function $\ell$ , we may obtain another version of the generalization   
+187 bound (although the flatness term therein is expected to be looser than that in Corollary 1).   
+188 Corollary 2. If the loss function is differentiable and $\beta$ -smooth with respect to $w$ , then under the   
+189 condition of Lemma ${ \it 6 }$ ,
+
+$$
+\mathrm { g e n } ( \mu , P _ { W _ { T } | S } ) | \leq \operatorname* { m i n } \left\{ \frac { R } { n } \sum _ { t = 1 } ^ { T } \sqrt { \frac { 2 \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } , \sqrt { \frac { 2 R ^ { 2 } } { n b } \sum _ { t = 1 } ^ { T } \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } \right\} + \beta d \sum _ { t = 1 } ^ { T } \sigma _ { t } ^ { 2 } .
+$$
+
+191 To conclude, we remark that these bounds suggest that in order for the model to generalize well, both   
+192 the trajectory term and the flatness term need to be small — the former involves the interaction of the   
+193 learning rate and batch size with the gradient dispersion along the training trajectory, whereas the   
+194 latter depends on the flatness of the empirical risk surface at the found solution.
+
+# 195 4 Experimental Study
+
+196 Bound Verification We first verify our bound in Corollary 1 by training an MLP (with one hidden   
+197 layer) and an AlexNet [42] on MNIST and CIFAR10 [41], respectively. To simplify estimation, we   
+198 fix the weight initialization and set $\sigma _ { t }$ and $\lambda _ { t }$ to be constants $\sigma$ and $\lambda$ , respectively. To compute   
+199 $\textstyle \sum _ { i = 1 } ^ { n } { \sqrt { \sum _ { t \in { \mathcal { T } } _ { i } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] } }$ , we compute the gradient dispersion as its empirical estimate from a   
+200 batch, utilizing a PyTorch [54] library BackPack [19]. To compute $\mathrm { T r } \left( \mathbb { E } \left[ \mathrm { H } _ { W _ { T } } ( Z ) \right] \right)$ , we randomly   
+201 sample $10 \%$ of the training data and use the PyHessian library [79] to compute the Hessian. Since   
+202 every choice of $\sigma$ gives a valid generalization bound in Corollary 1, we need to find the optimal   
+203 $\sigma$ , which gives the tightest bound. This can be done by simply utilizing the fact $A / \sigma + \bar { \sigma ^ { 2 } } B \geq$   
+204 $3 ( A / 2 ) ^ { 2 / 3 } B ^ { 1 / 3 }$ for any positive $A$ and $B$ , where the equality is achieved by the optimal $\sigma$ . We set   
+205 the sub-gaussian parameter $R = 0 . 1$ . The implementation in this paper is on PyTorch, and all the   
+206 experiments are carried out on NVIDIA Tesla V100 GPUs (32 GB).   
+207 We perform experiments with varying network width and varying levels of label noise. Specifically,   
+208 label noise level $\epsilon$ refers to the setting where we replace the labels of $\epsilon$ fraction of the training and   
+209 testing instances with random labels. The estimated bound is compared against the true generalization   
+210 gap, namely, the difference between the training loss and testing loss, and is shown in Figure 1.   
+211 In Figure 1, we see that in all cases the estimated bound follows closely the trend of the true   
+212 generalization gap. The fact that the bound curve consistently tracks the gap curve under various   
+213 label noise levels indicates that our bound very well captures the changes of the data distribution.   
+214 Note that in Figure 1 (a) and (b), our bound decays with the increase of the model size, showing a   
+215 trend as opposite to the bounds obtained in classical learning theory. But such a trend clearly better   
+216 explains the generalization behaviour of modern neural networks.   
+217 Learning Rate and Batch Size. The learning rate and batch size in SGD have explicitly appeared   
+218 in the trajectory term of the bound in Theorem 2. From the way they appear in the bound, one may be   
+219 tempted to assert that a small learning rate or large batch size will improve generalization. This would   
+220 then contradict some previous observations [37, 76, 33], in which increasing the ratio of learning rate   
+221 to batch size will benefit generalization. We now investigate this by performing experiments with   
+222 varying learning rates and batch sizes. In our experiments, the model is continuously updated until   
+223 the average training loss drops below 0.0001. We separate trajectory and flatness terms of the bound   
+224 and plot them in Figure 2.   
+225 A key observation in Figure 2 is that the learning rate impacts the trajectory term and the flatness term   
+226 in opposite ways, as seen, for example, in (a) and (b), where the two set of curves swap their orders in   
+227 the two figures. On the other hand, the batch size also impacts the two terms in opposite ways, as seen   
+228 in (a) and (b) where curves decrease in (a) but increase in (b). This makes the generalization bound,   
+229 i.e., the sum of the two terms, have a rather complex relationship with the settings of learning rate   
+230 and batch size. This relationship is further complicated by the fact that a small learning rate requires   
+231 a longer training time, or a larger number $T$ of training iterations, which increases the number that   
+232 are summed over in the trajectory term. Nonetheless, we do observe that a smaller batch size gives a   
+233 lower value of the flatness term ((b) and (d)), confirming the previous wisdom that small batch sizes   
+234 enable the neural network to find a flat minima [40].   
+235 Double Descent of Gradient Dispersion We experimentally investigate the impact of gradient   
+236 dispersion on the training of the neural networks by fixing the learning rate, batch size and weight   
+237 initialization for the each model (MLP for MNIST, AlexNet for CIFAR10). For each model and   
+238 various label noise levels, we plot in Figure 3 the evolution of the (empirical) gradient dispersion   
+239 $\widehat { \mathbb { V } } ( \boldsymbol { w } _ { t } )$ , training accuracy and testing accuracy across training epochs.
+
+![](images/e2845eede1a36c96ff767cc3b2a3b3b932b49c4abdcf63f4d16e3823ae0705ae.jpg)  
+Figure 1: Estimated bound and empirical generalization gap (“gap”) as functions of network width ((a) and (b)) and label noise level ((c) and (d)). Left Y-axis: gap value; right Y-axis: bound value.
+
+![](images/d7ecfa641ada864e8b7f67d14386df264cd5899c6554078a3f5f55e475c4a3a6.jpg)  
+Figure 2: The impact of learning rate and batch size on the trajectory term and the flatness term.
+
+![](images/0a762b53a3615ccaa00fe726062e74cdfffe11afbda1cbd432bd3e98a7ce576c.jpg)  
+Figure 3: Epoch-wise double descent of gradient dispersion, in relation to training/testing accuracies.
+
+An intriguing epoch-wise “double descent” phenomenon is observed, particularly when the labels are noisy. According to the double descent curve, the training may be split into three phases (e.g., Figure 3 (h)). In the first phase, the gradient dispersion rapidly descends and maintains a very low level. In this phase, both training and test accuracies increase while maintaining a very small generalization gap. This suggests that the network in this phase is extracting useful patterns and generalizes well. In the second phase, the gradient dispersion starts increasing until it reaches a peak value. In this phase, the training and testing accuracies gradually diverge, marking the model entering an overfitting or “memorization” regime – when the data contains the noisy labels, the network mostly tries to memorize the labels in the training set. In the third phase, the gradient dispersion descends again, reaching a low value. In this phase, the model continuously overfits the training data, until the training and testing curves reach their respective maximum and minimum. It appears that the timing of the three phases depends on the dataset and the label noise level. For simpler data (e.g. MNIST) and cleaner datasets (e.g. CIFAR10 with low label noise), the first phase may be shorter. This is arguably because in these datasets, extracting useful patterns is relatively easier. Nonetheless, the valley in the double-descent curve appears to mark a “great divide” between generalization and memorization.
+
+255 Dynamic Gradient Clipping Inspired by our generalization bounds and above observations, one   
+256 way to reduce the generalization error is to control the trajectory term of the bounds by reducing the   
+257 gradient dispersion in each training step. Here we investigate a simple scheme that dynamically clips   
+258 the gradient norm so as to reduce the gradient dispersion. Specifically, whenever the current gradient   
+259 norm is larger than the gradient norm $K$ steps earlier, or $| | g ( W _ { t } , B _ { t } ) | | _ { 2 } > | | g ( W _ { t - K } , B _ { t - K } ) | | _ { 2 }$ (i.e.,   
+260 the model is expected to have entered the “memorization” regime), we reduce the norm of the current   
+261 gradient $g ( W _ { t } , B _ { t } )$ to $\alpha$ fraction of $| | g ( W _ { t - K } , B _ { t - K } ) | | _ { 2 }$ , for some prescribed value $\alpha < 1$ . The   
+262 effectiveness of this scheme is best demonstrated when the labels contain noise. As shown in Figure   
+263 4, dynamic gradient clipping significantly closes the gap between the training accuracy and the testing   
+264 accuracy. The models trained with this scheme maintain a near-optimal testing accuracy (e.g., about   
+265 $80 \%$ when the label noise level of MNIST is 0.2), without suffering from the severe memorization   
+266 effect as seen in models trained without this scheme. Further understanding of the double-descent   
+267 phenomenon of the gradient dispersion may enable more delicate design of such a dynamic clipping   
+268 scheme and potentially lead to novel and powerful regularization techniques.
+
+![](images/3f2a6039b469e6c40083c61a8febb3a68d18c39f1c50dc01325222b7551f9afc.jpg)  
+Figure 4: Dynamic Gradient Clipping.
+
+# 269 5 A Practical Implication: Gaussian Model Perturbation
+
+270 The appearance of the flatness term in our generalization bounds suggests that for an empirical   
+271 risk minimizer $w ^ { * }$ to generalize well, it is necessary that the empirical risk surface at $w ^ { * }$ is flat,   
+272 or insensitive to a small perturbation of $w ^ { * }$ . This naturally motivates a training scheme using the   
+273 following regularized loss:
+
+$$
+\operatorname* { m i n } _ { w } L _ { s } ( w ) + \rho \operatorname* { l g } _ { \Delta \sim \mathcal { N } ( 0 , \sigma ^ { 2 } \mathbf { I } _ { d } ) } [ L _ { s } ( w + \Delta ) - L _ { s } ( w ) ] ,
+$$
+
+274 where $\rho$ is a hyper-parameter. Replacing the expectation above with its stochastic approximation   
+275 using $k$ realizations of $\Delta$ gives rise to the following optimization problem.
+
+$$
+\operatorname* { m i n } _ { w } \frac { 1 } { b } \sum _ { z \in B } \left( ( 1 - \rho ) \ell ( w , z ) + \rho \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \left( \ell ( w + \delta _ { i } , z ) \right) \right) .
+$$
+
+276 We refer to the SGD training scheme using this loss as Gaussian model perturbation or GMP. Notably,   
+277 GMP requires $k + 1$ forward passes for every parameter update. Empirical evidence shows that a   
+278 small $k$ , for example, $k = 3$ , already gives competitive performance. Implementing the $k + 1$ forward   
+279 passes on parallel processors further reduces the computation load.   
+280 We experimentally compare GMP with several major regularization schemes in the current art, includ  
+281 ing Dropout [68], label smoothing [70], Flooding [36], MixUp [81], adversarial training [29], and   
+282 AMP [83]. The compared schemes are evaluated on three popular benchmark image classification   
+283 datasets SVHN [48], CIFAR-10 and CIFAR-100 [41]. Two representative deep architectures PreAc  
+284 tResNet18 [34] and VGG16 [67] are taken as the underlying model. We train the models for 200   
+285 epochs by SGD. The learning rate is initialized as 0.1 and divided by 10 after 100 and 150 epochs.   
+286 For all compared models, the batch size is set to 50 and weight decay is set to $1 0 ^ { - 4 }$ . For GMP, we   
+287 choose $\rho = 0 . 5$ and set the standard deviation of the Gaussian noise $\Delta$ to 0.03. The value of $k$ is   
+288 chosen as 3 and 10 respectively (referred to as $\mathrm { G M P ^ { 3 } }$ and $\mathrm { G M P ^ { 1 0 } }$ ).   
+289 The performances of all compared schemes are given in Table 1. For the compared regularization   
+290 schemes except GMP, we directly report their performances as given in [83]. Performances of vanilla   
+291 ERM without regularization are also included as a reference.   
+292 Table 1 demonstrates the effectiveness of GMP. Overall GMP performs comparably to the current   
+293 art of regularization schemes, although appearing slighly inferior to the most recent record given by   
+294 AMP [83]. Noting that the key ingredient of AMP, “max-pooling” in the parameter space, greatly   
+295 resembles regularization term in GMP, which may be seen as “average-pooling” in the same space.
+
+Table 1: Top-1 classification accuracy on (a) SVHN, (b) CIFAR-10 and (c) CIFAR-100. We run experiments 10 times and report the mean and the standard deviation of the testing accuracy.   
+
+<table><tr><td>PreActResNet18</td><td>Top-1 Acc. (%)</td><td>PreActResNet18</td><td>Top-1 Acc. (%)</td><td>PreActResNet18</td><td>Top-1 Acc. (%)</td></tr><tr><td>ERM</td><td>97.05±0.063</td><td>ERM</td><td>94.98±0.212</td><td>ERM</td><td>75.69±0.303</td></tr><tr><td>Dropout</td><td>97.20±0.065</td><td>Dropout</td><td>95.14±0.148</td><td>Dropout</td><td>75.52±0.351</td></tr><tr><td>Label Smoothing</td><td>97.22±0.087</td><td>Label Smoothing</td><td>95.15±0.115</td><td>Label Smoothing</td><td>77.93±0.256</td></tr><tr><td>Flooding</td><td>97.16±0.047</td><td>Flooding</td><td>95.03±0.082</td><td>Flooding</td><td>75.50±0.234</td></tr><tr><td>MixUp</td><td>97.26±0.044</td><td>MixUp</td><td>95.91±0.117</td><td>MixUp</td><td>78.22±0.210</td></tr><tr><td>Adv.Training</td><td>97.23±0.080</td><td>Adv. Training</td><td>95.01±0.085</td><td>Adv. Training</td><td>74.77±0.229</td></tr><tr><td>AMP</td><td>97.70±0.025</td><td>AMP</td><td>96.03±0.091</td><td>AMP</td><td>78.49±0.308</td></tr><tr><td>GMP3</td><td>97.43±0.037</td><td>GMP3</td><td>95.64±0.053</td><td>GMP3</td><td>78.05±0.208</td></tr><tr><td>GMP10</td><td>97.34±0.058</td><td>GMP10</td><td>95.71±0.073</td><td>GMP10</td><td>78.07±0.170</td></tr><tr><td>VGG16</td><td>Top-1 Acc. (%)</td><td>VGG16</td><td>Top-1 Acc. (%)</td><td>VGG16</td><td>Top-1 Acc. (%)</td></tr><tr><td>ERM</td><td>96.86±0.060</td><td>ERM</td><td>93.68±0.193</td><td>ERM</td><td>72.16±0.297</td></tr><tr><td>Dropout</td><td>97.04±0.049</td><td>Dropout</td><td>93.78±0.147</td><td>Dropout</td><td>72.28±0.337</td></tr><tr><td>Label Smoothing</td><td>96.93±0.070</td><td>Label Smoothing</td><td>93.71±0.158</td><td>Label Smoothing</td><td>72.51±0.179</td></tr><tr><td>Flooding</td><td>96.85±0.085</td><td>Flooding</td><td>93.74±0.145</td><td>Flooding</td><td>72.07±0.271</td></tr><tr><td>MixUp</td><td>96.91±0.057</td><td>MixUp</td><td>94.52±0.112</td><td>MixUp</td><td>73.19±0.254</td></tr><tr><td>Adv.Training</td><td>97.06±0.091</td><td>Adv. Training</td><td>93.51±0.130</td><td>Adv.Training</td><td>70.88±0.145</td></tr><tr><td>AMP</td><td>97.27±0.015</td><td>AMP</td><td>94.35±0.147</td><td>AMP</td><td>74.40±0.168</td></tr><tr><td>GMP3</td><td>97.18±0.057</td><td>GMP3</td><td>94.33±0.094</td><td>GMP3</td><td>74.45±0.256</td></tr><tr><td>GMP10</td><td>97.09±0.068</td><td>GMP10</td><td>94.45±0.158</td><td>GMP10</td><td>75.09±0.285</td></tr></table>
+
+(a) SVHN (b) CIFAR-10 (c) CIFAR-100
+
+# 296 6 Conclusion and Outlook
+
+This paper presents new generalization bounds for neural networks trained with SGD, improving upon the results of [49]. Our bounds naturally point to new and effective regularization schemes. At the same time, they reveal interesting phenomena in the SGD training of neural networks. While these phenomena deserve further investigation in their own right, we here suggest another direction for improving the bounds, namely, via the use of strong data-processing inequalities (DPI) [57–59] (noting that the standard DPI is in fact needed for establishing Theorem 2).
+
+303 For any Markov chain $U \to X \to Y$ , we will denote by $u , x$ , and $\mathcal { V }$ the spaces in which $U$ ,   
+304 $X$ , $Y$ take values, respectively. For any distribution $P$ on $\mathcal { X }$ , we will use $P _ { Y \mid X } \circ P$ to denote the   
+305 distribution on $\mathcal { V }$ induced by the push-forward of the distribution $P$ by $P _ { Y \mid X }$ , namely, for any $y \in \mathcal { V }$ ,   
+306 $\begin{array} { r } { \left( P _ { Y | X } \circ P \right) ( y ) \triangleq \int P _ { Y | X } ( y | x ) P ( x ) d x } \end{array}$ . Let ${ \cal { S } } ( U )$ be the support of $P _ { U }$ and ${ \mathcal { H } } ( U , P _ { X | U } )$ be the   
+307 convex hull of $\{ P _ { X | U = u } : u \in { \mathcal { S } } ( U ) \}$ . Define
+
+$$
+\eta ( U \to X \to Y ) \triangleq \operatorname* { s u p } _ { P , Q \in \mathcal { H } ( U , P _ { X | U } ) } \frac { \mathrm { D } _ { \mathrm { K L } } ( P _ { Y | X } \circ P | | P _ { Y | X } \circ Q ) } { \mathrm { D } _ { \mathrm { K L } } ( P | | Q ) }
+$$
+
+08 Lemma 7. For any Markov chain $U  X  Y$ , $I ( U ; Y ) \leq \eta ( U  X  Y ) I ( U ; X ) .$
+
+309 Here $\eta ( U \to X \to Y )$ serves as the “contraction coefficient” for the stochastic kernel $P _ { Y \mid X }$ ,   
+310 characterizing the greatest extent by which the kernel may bring closer any two distributions on $S ( U )$   
+311 in its output space. It is easy to see that $\eta ( U \to X \to Y ) \leq 1$ , giving rise to a stronger DPI.   
+12 Denote $V _ { t } \triangleq \widetilde { W } _ { t - 1 } + G _ { t }$ . It can be verified that $Z _ { i }  V _ { t }  \widetilde { W } _ { t }$ form a Markov chain. Denote   
+13 $\eta _ { i , t } \triangleq \eta ( Z _ { i }  V _ { t }  \widetilde { W } _ { t } )$ , and $\Gamma _ { i } ^ { t } = \{ t + 1 , t + 2 , \ldots , T \} \setminus \mathcal { T } _ { i }$ . Theorem 2 can be improved to:
+
+314 Theorem 3. The expected generalization error of SGD is bounded by
+
+$$
+\left| \mathrm { g e n } ( \mu , P _ { W _ { T } | S } ) \right| \le \frac { 2 R } { n b } \sum _ { i = 1 } ^ { n } \sqrt { \sum _ { t \in { \mathcal T } _ { i } } \frac { \lambda _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \mathbb { E } \left[ \mathbb { V } ( W _ { t - 1 } ) \right] \cdot \prod _ { \tau \in \Gamma _ { i } ^ { t } } \eta _ { i , \tau } } + \left| \mathbb { E } \left[ \gamma ( W _ { T } , S ) - \gamma ( W _ { T } , S ^ { \prime } ) \right] \right| .
+$$
+
+316 It remains to characterize the contraction coefficient $\eta _ { i , \tau }$ in a computable form. Simply bounding it   
+317 via the Dobrushin’s coefficient [21], as suggested in [72] for analyzing SGLD, is unlikely to make   
+318 the bound in this theorem significantly tighter than that in Theorem 2.
+
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+
+1. For all authors...
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+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
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+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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+(b) Did you include complete proofs of all theoretical results? [Yes] See supplementary materials.
+
+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] See supplementary materials.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5 and supplementary materials.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section 4 and Section 5.   
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+        "text": "On the Generalization of Neural Networks Trained with SGD: Information-Theoretical Bounds and Implications ",
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+        "text": "1 Understanding the generalization behaviour of deep neural networks is an important   \n2 theme of modern research in machine learning. In this paper, we follow up on   \n3 a recent work of Neu [49] and present new information-theoretic upper bounds   \n4 for the generalization error of neural networks trained with SGD. Our bounds and   \n5 experimental study provide new insights on the SGD training of neural networks.   \n6 They also point to a new and simple regularization scheme which we show performs   \n7 comparably to the current state of the art. ",
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+        "text": "8 1 Introduction ",
+        "text_level": 1,
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+        "text": "9 The outstanding performance of deep learning has brought to the surface some intriguing properties   \n10 of deep neural networks, one of which is the observation that despite their high capacity, deep neural   \n11 networks tend to generalize well [80]. This contradicts classical wisdom in statistical learning theory   \n12 (e.g., [71] ) and has stimulated intense research interest in understanding the generalization behaviour   \n13 of modern neural networks.   \n14 One theme of research focuses on the study of over-parameterized neural networks, where generaliza  \n15 tion bounds are obtained [22, 1, 6, 50, 52, 53, 2, 3] and a curious “double descent” phenomenon is   \n16 observed and analyzed [9, 46, 78]. New bounding techniques for analyzing generalization have also   \n17 been developed, utilizing information theoretic quantities [64, 65, 77, 5, 13, 69, 23, 7, 4, 30, 84]. The   \n18 bounds provided by these techniques have the advantages of accounting for both the data structure   \n19 and the learning algorithm.   \n20 The generalization ability of neural networks trained with mini-batched stochastic gradient descent   \n21 [61], simply referred to SGD in this paper, has also been widely studied. Specifically, built on a   \n22 connection between stability and generalization [12], a stability-based bound is first presented in [32],   \n23 followed by a surge of research effort exploiting similar approaches [44, 17, 26, 43, 8]. Information  \n24 theoretic bounding techniques have also demonstrated great power in analyzing SGD-like algorithms.   \n25 For example, [55] is the first to utilize information-theoretical bound in analyzing the generalization   \n26 ability of SGLD [28, 74]. The bound was subsquently improved by [47, 31, 62, 72]. Inspired by the   \n27 work of [55], [49] presents an information- theoretic analysis of the models trained with SGD. The   \n28 analysis of [49] constructs an auxiliary weight process parallel to SGD training and upper-bounds the   \n29 generalization error through this auxiliary process.   \n30 Another line of research connects the generalization of neural networks with the flatness of loss   \n31 minima [35] found by SGD or its variant [40, 20, 24, 51, 16, 37, 38, 83, 27]. This understanding   \n32 has led to the discovery of new SGD-based training algorithms for improved generalization. For   \n33 example, in a concurrent development by [83] and [27], a local “max-pooling” operation is applied to   \n34 the loss landscape prior to the SGD updates. This approach, referred to as AMP[83] or SAM[27], is   \n35 shown to make SGD favor flatter minima and achieve the state-of-the-art performance among various   \n36 competitive regularization schemes [83].   \n37 In this paper, we focus on investigating the generalization of neural networks trained with SGD.   \n38 We build upon the work of [49]. Following the same construction of the auxiliary weight process   \n39 in [49], we present upper bounds of generalization error that improve upon [49] in two ways. The   \n40 first improvement is via removing an unnecessary term in the bounds of [49] by invoking the HWI   \n41 inequality [60]. The second improvement is via replacing a sample-level mutual information term in   \n42 [49] with an instance-level mutual information term, exploiting a recent result of [13]. The bounds we   \n43 obtain decompose into two terms, one measuring the impact of training trajectories (“the trajectory   \n44 term”) and the other measuring the impact of the flatness of the found solution (“the flatness term”).   \n45 We empirically validate the derived bounds. Various insights are also obtained experimentally   \n46 concerning the generalization of neural networks under SGD training. For example, the batch size of   \n47 SGD appears to impact the trajectory term and the flatness term in opposite ways, which complicates   \n48 the overall dependency of generalization error on batch sizes. A particular interesting observation   \n49 from our experiments is that a key quantity arising in the trajectory term of the bounds, which we   \n50 refer to as gradient dispersion1, reveals a double descent phenomenon with respect to training epochs.   \n51 Most intriguingly, the valley in the double descent curve appears to mark the great divide between the   \n52 “generalization regime” and the “memorization regime” of training. Furthering from this observation,   \n53 we also show that it is possible to reduce the memorization effect by dynamically clipping the gradient   \n54 and reducing its dispersion.   \n55 Our bounds also inspire a natural and simple solution to alleviate generalization error. Specifically,   \n56 we propose a new training scheme, referred to as Gaussian model perturbation (GMP), aiming at   \n57 reducing the flatness term of the bounds. This scheme effectively applies a local “average pooling” to   \n58 the empirical risk surface prior to SGD, greatly resembling the “max-pooling” approach adopted in   \n59 AMP[83]. We demonstrate experimentally that GMP achieves a competitive performance with the   \n60 current art of regularization schemes. ",
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+        "text": "Length constraints precludes elaboration at places. The reader is referred to supplementary materials for proofs and additional information. ",
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+        "text": "63 Other Related Literature Gradient dispersion is mostly studied from optimization perspectives[11,   \n64 63, 39, 75, 25]. Prior to this work, only a few works relate gradient dispersion with the generalization   \n65 behaviour of the networks. In [49, 72], gradient dispersion also appears in the generalization bounds.   \n66 In [38], gradient dispersion is argued to capture a notion of “flatness” of the local minima of the loss   \n67 landscape, thereby correlating with generalization.   \n68 Injecting noise in the training process has been proposed in various regularization schemes, for   \n69 example, [10, 14, 15, 68, 73]. But unlike the Gaussian model perturbation scheme derived in this   \n70 paper, where noise is injected to the model parameters, noise in those schemes is injected either to   \n71 the training data or to the network activation.   \n72 Gradient clipping is a common technique for preventing gradient exploding (see, e.g., [45, 56]).   \n73 This technique is also used in [82] to accelerate training. In this paper, gradient clipping is used to   \n74 investigate and control the impact of gradient dispersion on generalization error. ",
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+        "text": "75 2 Preliminaries ",
+        "text_level": 1,
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+        "text": "Population Risk, Empirical Risk and Generalization Error Unless otherwise noted, a random variable will be denoted by a capitalized letter (e.g., $Z$ ), and its realization denoted by the corresponding lower-case letter (e.g. $z$ ). Let $\\mathcal { Z }$ be the instance space of interest and $\\mu$ be an unknown distribution on $\\mathcal { Z }$ , specifying random variable $Z$ . Let $\\mathcal { W } \\subseteq \\bar { \\mathbb { R } } ^ { d }$ be the space of hypotheses. Suppose that a training sample ${ \\bar { S } } = ( Z _ { 1 } , Z _ { 2 } , \\ldots , Z _ { n } )$ is drawn i.i.d. from $\\mu$ and that a stochastic learning algorithm $\\mathcal { A }$ takes $S$ as its input and outputs a hypothesis $W \\in { \\mathcal { W } }$ according to some conditional distribution $P _ { W | S }$ mapping ${ \\mathcal { Z } } ^ { n }$ to $\\mathcal { W }$ . Let $\\ell : \\mathcal { W } \\times \\mathcal { Z } \\to \\mathbb { R } ^ { + }$ be a loss function, where $\\ell ( w , z )$ measures the “unfitness” or “error” of any $z \\in { \\mathcal { Z } }$ with respect to a hypothesis $w \\in \\mathcal W$ . The population risk, for any ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/7b15043063da74c6a051a08bb41edd33da5135717da6761a35a7eda90807389a.jpg",
+        "text": "$$\nL _ { \\mu } ( w ) \\triangleq \\mathbb { E } _ { Z \\sim \\mu } [ \\ell ( w , Z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "85 The goal of learning is to find a hypothesis $w$ that minimizes the population risk. But since $\\mu$ is only   \n6 partially accessible via the sample $S$ , in practice, we instead turn to the empirical risk, defined as ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/8950c71e73a6ab78e4d6a7f018f99922a4cbcde03e5fca24910811ed4a450389.jpg",
+        "text": "$$\nL _ { S } ( w ) \\triangleq { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\ell ( w , Z _ { i } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "87 The expected generalization error of the learning algorithm $\\mathcal { A }$ is then defined as ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee00e1de858198c6d03483ac21c6b922e0ae0f12eb0453d28cb2259728844abe.jpg",
+        "text": "$$\n\\mathrm { g e n } ( \\mu , P _ { W | S } ) \\triangleq \\mathbb { E } _ { W , S } [ L _ { \\mu } ( W ) - L _ { S } ( W ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the expectation is taken over the joint distribution of 88 $( S , W )$ (i.e., $\\mu ^ { n } \\otimes P _ { W | S } )$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "89 Throughout this paper, we take $\\ell$ as a continuous function (adopting the usual notion “surrogate loss”   \n90 [66].). Additionally, we assume that $\\ell$ is differentiable almost everywhere with respect to both $w$ and   \n91 $z$ . Furthermore we assume that $\\ell ( w , Z )$ is $R$ -subgaussian2 for any $w \\in \\mathcal { W }$ . Note that a bounded loss   \n92 is guaranteed to be subgaussian for all $\\mu$ and all $w \\in \\mathcal { W }$ . Let $I ( X ; Y )$ denote the mutual information   \n93 [18] between any pair of random variables $( X , Y )$ . The following results are known. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "94 Lemma 1 ([77, Theorem 1.]). The expected generalization error of algorithm $\\mathcal { A }$ is bounded by ",
+        "bbox": [
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+    },
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+        "type": "equation",
+        "img_path": "images/2133f55a32b29cacaa93242cefa9051005ec41fa8d40b35486b4acb1ebe35b14.jpg",
+        "text": "$$\n| \\mathrm { g e n } ( \\mu , P _ { W | S } ) | \\leq \\sqrt { \\frac { 2 R ^ { 2 } } { n } I ( W ; S ) } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "95 ",
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+    {
+        "type": "text",
+        "text": "96 Lemma 2 ([13, Proposition 1.]). The expected generalization error of algorithm $\\mathcal { A }$ is bounded by ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
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+        "text": "$$\n| \\mathrm { g e n } ( \\mu , P _ { W | S } ) | \\leq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sqrt { 2 R ^ { 2 } I ( W ; Z _ { i } ) } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "97 ",
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+        "type": "text",
+        "text": "98 Stochastic Gradient Descent We now restrict the learning algorithm $\\mathcal { A }$ to be the mini-batched   \n99 stochastic gradient descent (SGD) algorithm for empirical risk minimization. For each training epoch,   \n100 the dataset $S$ is randomly split into $m$ disjoint mini-batches, each having size $b$ , namely, $n = m b$ .   \n101 Based on each batch, one parameter update is performed. Specifically, let $B _ { t }$ denote the batch used   \n102 for the $t ^ { \\mathrm { t h } }$ update. Define ",
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+        "text": "$$\ng ( w , B _ { t } ) \\triangleq \\frac { 1 } { b } \\sum _ { z \\in B _ { t } } \\nabla _ { w } \\ell ( w , z ) ,\n$$",
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+        "type": "text",
+        "text": "103 namely, $g ( w , B _ { t } )$ is the average gradient computed for the batch $B _ { t }$ with respect to parameter $w$ . The rule for the 104 $t ^ { \\mathrm { t h } }$ parameter update is then ",
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+        "text": "$$\nW _ { t } \\triangleq W _ { t - 1 } - \\lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) ,\n$$",
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+        "type": "text",
+        "text": "105 where $\\lambda _ { t }$ is the learning rate at the step $t$ . The initial parameter setting $W _ { 0 }$ is assumed to be drawn   \n106 from the zero-mean spherical Gaussian $\\mathcal { N } ( 0 , \\sigma _ { 0 } ^ { 2 } \\mathbf { I } _ { d } )$ with variance $\\sigma _ { 0 } ^ { 2 }$ in each dimension. We will   \n107 assume that the SGD algorithm stops after $T$ updates and outputs $W _ { T }$ as the learned model parameter.   \n108 Given the training sample $S$ , let $\\xi$ govern the randomness in the sequence $( B _ { 1 } , B _ { 2 } , \\ldots , B _ { T } )$ of   \n109 batches. For the simplicity of notion, we will fix the configuration of $\\xi$ . That is, we will assume a   \n110 fixed “batching trajectory”, or a fixed way to shuffle the example indices $\\{ 1 , \\ldots , n \\}$ and divide them   \n111 into $m$ batches in each epoch. The presented generalization bounds of this paper can be extended   \n112 to the case where the batching trajectory is uniformly random (as we set up above). This merely   \n113 involves averaging over all batching trajectories or taking expectation over $\\xi$ .   \n114 Auxiliary Weight Process We now associate with the SGD algorithm an auxiliary weight process   \n115 $\\{ \\widetilde { W _ { t } } \\}$ . Let $\\sigma ^ { 2 }$ be given, and let $\\sigma _ { 1 } , \\sigma _ { 2 } , \\ldots , \\sigma _ { T }$ be a sequence of positive real numbers. Define ",
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+        "text": "$$\n\\widetilde { W } _ { 0 } \\triangleq W _ { 0 } , \\quad \\mathrm { a n d } \\widetilde { W } _ { t } \\triangleq \\widetilde { W } _ { t - 1 } - \\lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) + N _ { t } , \\mathrm { f o r } t > 0 ,\n$$",
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+        "text": "where 116 $N _ { t } \\sim \\mathcal N ( 0 , \\sigma _ { t } ^ { 2 } \\mathbf I _ { d } )$ is a Gaussian noise. The relationship between this auxiliary weight process 117 $\\{ \\widetilde { W _ { t } } \\}$ and the weight process $\\{ W _ { t } \\}$ in SGD is shown in the Bayesian network below. ",
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+        "type": "text",
+        "text": "118 ",
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+        "img_path": "images/480a41ebbbcd284bc0738798791dd939b242ae4cb4e25c7933ee3ed042b0c536.jpg",
+        "text": "$$\n\\begin{array} { r l r l r l r l r l r l r l r } & { } & & { N _ { 1 } } & & { N _ { 2 } } & & { \\cdots } & & { N _ { T - 1 } } & & { N _ { T } } \\\\ & { } & & { \\downarrow } & & { \\downarrow } & & { } & & { \\downarrow } & & { \\downarrow } & & { \\downarrow } \\\\ { \\widetilde { W } _ { 0 } } & {  } & { \\widetilde { W } _ { 1 } } & {  } & { \\widetilde { W } _ { 2 } } & {  } & { \\cdots } & {  } & { \\widetilde { W } _ { T - 1 } } & {  } & { \\widetilde { W } _ { T } } \\\\ { \\parallel } & { \\nearrow } & { } & { \\chi } & { \\chi } & { \\nearrow } & { \\dots } & {  } & { \\dots } & {  } & { W _ { T } } \\\\ { W _ { 0 } } & {  } & { W _ { 1 } } & {  } & { W _ { 2 } } & {  } & { \\cdots } & {  } & { W _ { T - 1 } } & {  } & { W _ { T } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "119 Let $\\begin{array} { r } { \\Delta _ { t } = \\sum _ { \\tau = 1 } ^ { t } N _ { \\tau } } \\end{array}$ . Noting that the weight updates in $\\{ \\widetilde { W } _ { t } \\}$ uses the same gradient signal as that   \n120 used in $\\{ W _ { t } \\}$ (which depends on $W _ { t - 1 }$ not $\\widetilde { W } _ { t - 1 } )$ ), it is immediate that $\\widetilde { W } _ { t } = W _ { t } + \\Delta _ { t }$ . Note that this   \n121 auxiliary process follows the same construction as [49], which we will use to study the generalization   \n122 error of SGD. ",
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+        "text": "123 To that end, define gradient dispersion at parameter $w$ by ",
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+        "text": "$$\n\\mathbb { V } ( w ) \\triangleq \\mathbb { E } \\left[ | | \\nabla _ { w } \\ell ( w , Z ) - \\mathbb { E } \\left[ \\nabla _ { w } \\ell ( w , Z ) \\right] | | _ { 2 } ^ { 2 } \\right] ,\n$$",
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+        "type": "text",
+        "text": "124 where the expectation is taken over $Z \\sim \\mu$ . ",
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+    {
+        "type": "text",
+        "text": "For a given sample 125 $s \\in \\mathcal { Z } ^ { n }$ , define ",
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+        "text": "$$\n\\gamma ( w , s ) \\triangleq \\mathbb { E } \\left[ L _ { s } ( w + \\Delta _ { T } ) - L _ { s } ( w ) \\right] ,\n$$",
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+        "type": "text",
+        "text": "26 where the expectation is taken over $\\Delta _ { T }$ and $L _ { s } ( w )$ is the empirical risk of $s$ at parameter $w$ . ",
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+        "text": "In the remainder of the paper, let 127 $S ^ { \\prime }$ denote another sample drawn from $\\mu ^ { n }$ , independent of all other 128 random variables. The main generalization bound in [49] is re-stated below. ",
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+        "type": "text",
+        "text": "129 Lemma 3 ([49, Theorem 1.]). The generalization error of SGD is upper bounded by ",
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+        "text": "$$\n\\left| \\mathrm { g e n } ( \\mu , P _ { W _ { T } | S } ) \\right| \\leq \\sqrt { \\frac { 2 R ^ { 2 } } { n } \\sum _ { t = 1 } ^ { T } \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\Psi ( W _ { t - 1 } ) + \\frac { \\Psi ( W _ { t - 1 } ) } { b } \\right] } + \\left| \\mathbb { E } \\left[ \\gamma ( W _ { T } , S ) - \\gamma ( W _ { T } , S ^ { \\prime } ) \\right] \\right| ,\n$$",
+        "text_format": "latex",
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+        "text": "$$\n\\begin{array} { r } { \\Psi ( w _ { t - 1 } ) \\triangleq \\mathbb { E } \\left[ | | \\nabla _ { w } \\ell ( w _ { t - 1 } , Z ) - \\nabla _ { w } \\ell ( w _ { t - 1 } + \\zeta , Z ) | | _ { 2 } ^ { 2 } \\right] a n d \\zeta \\sim \\mathcal { N } ( 0 , 2 \\sum _ { i = 1 } ^ { t - 1 } \\sigma _ { i } ^ { 2 } \\mathrm { I } _ { d } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The term $\\Psi ( w _ { t - 1 } )$ in the bound is referred to as “local gradient sensitivity” in [49]. ",
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+    {
+        "type": "text",
+        "text": "3 New Generalization Bounds for SGD ",
+        "text_level": 1,
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+        "text": "133 We first prove that the generalization bound in Lemma 3 can be tightened by removing the local   \n134 gradient sensitivity term $\\Psi ( w _ { t - 1 } )$ . The key observation is that an independence condition used for   \n135 establishing Lemma 3 in [49] is unnecessary (see Lemma 4 in [49]). This requires invoking a vector   \n136 version of the HWI inequality [60, Lemma 3.4.2], which we prove in this paper. ",
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+        "type": "text",
+        "text": "Lemma 4. Let37 $X$ and $Y$ be two random vectors in $\\mathbb { R } ^ { d }$ , and let $\\boldsymbol { N } \\sim \\mathcal { N } ( 0 , \\mathbf { I } _ { d } )$ be independent of 38 $( X , Y )$ . Then, for every $t > 0$ , $\\begin{array} { r } { \\mathrm { D } _ { \\mathrm { K L } } \\big ( P _ { X + \\sqrt { t } N } | | P _ { Y + \\sqrt { t } N } \\big ) \\leq \\frac { 1 } { 2 t } \\mathbb { E } \\left[ | | X - Y | | ^ { 2 } \\right] } \\end{array}$ . ",
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+        "text": "39 Here $\\mathrm { D } _ { \\mathrm { K L } }$ is the KL divergence. Note that the bound in Lemma 3 relies on a similar result which   \n40 however requires the independence of $X$ and $Y$ . Using Lemma 4, we obtain the following theorem. ",
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+        "type": "text",
+        "text": "141 Theorem 1. The generalization error of SGD is upper bounded by ",
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+        "text": "$$\n\\left| \\mathrm { g e n } ( \\mu , P _ { W _ { T } | S } ) \\right| \\le \\sqrt { \\frac { 2 R ^ { 2 } } { n b } \\sum _ { t = 1 } ^ { T } \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } + \\left| \\mathbb { E } \\left[ \\gamma ( W _ { T } , S ) - \\gamma ( W _ { T } , S ^ { \\prime } ) \\right] \\right| .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "143 The proof of this theorem, as the bounds in [49], relies on Lemma 1 and the sample-level mutual   \n144 information bound therein. This theorem can be further tightened by exploiting the fact that the   \n145 instance-level mutual information bound in Lemma 2 is in fact tighter than the sample-level mutual   \n146 information bound in Lemma 1, as shown in [13]. The main ingredient to proceed in this direction is   \n147 the following lemma.   \nLemma 5. Le148 $\\begin{array} { r } { \\ d _ { t } G _ { t } = - \\lambda _ { t } g ( W _ { t - 1 } , B _ { t } ) . \\ d H _ { t } Z _ { i } \\in B _ { t } , t h e n I ( G _ { t } + N _ { t } ; Z _ { i } | \\widetilde W _ { t - 1 } ) \\le \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } b ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] . } \\end{array}$   \n49   \n150 In this lemma, the mutual information $I ( G _ { t } + N _ { t } ; Z _ { i } | \\widetilde { W } _ { t - 1 } )$ roughly indicates the degree by which   \n151 the SGD’s updating signal $G _ { t }$ (smoothed with noise) depends on an individual training instance   \n152 $Z _ { i }$ , when $Z _ { i }$ is used for computing the gradient. When this dependency is strong (giving rise to a   \n153 high value of the mutual information), the model conceivably tends to overfit the individual training   \n154 instances. This lemma suggests that the strength of this dependency can be upper-bounded by the   \n155 expected gradient dispersion at the current weight configuration. In our experiments, we will estimate   \n156 the expected gradient dispersion and validate this intuition.   \n157 It is remarkable that the noise $\\{ N _ { t } \\}$ plays an important role for the bound to hold. To see this,   \n158 consider $b = 1$ and $\\mathcal { Z }$ is countable and large. Then $\\bar { I } ( G _ { t } ; Z _ { t } | W _ { t - 1 } )$ is merely the conditional entropy   \n159 $H ( Z _ { i } | W _ { t - 1 } )$ , which would grow with sample size $n$ at least as $\\log n$ . Upper-bounding it with a   \n160 quantity independent of $n$ would be impossible – This justifies the construction of the auxiliary   \n161 weight process.   \n162 We now state our main theorem. Unlike Theorem 1, which considers a random batching trajectory,   \n163 this theorem considers a fixed batching trajectory to keep the expression less cluttered. For that   \n164 batching trajectory, we will use $\\mathcal { T } _ { i }$ to denote the set of indices of batches $B _ { t }$ containing instance $Z _ { i }$ . ",
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+        "type": "text",
+        "text": "165 Theorem 2. The expected generalization error of SGD is bounded by ",
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+        "text": "$$\n\\left| \\mathrm { g e n } ( \\mu , P _ { W _ { T } | S } ) \\right| \\leq \\frac { R } { n b } \\sum _ { i = 1 } ^ { n } \\sqrt { \\sum _ { t \\in { \\mathcal T } _ { i } } \\frac { 2 \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } + \\left| \\mathbb { E } \\left[ \\gamma ( W _ { T } , S ) - \\gamma ( W _ { T } , S ^ { \\prime } ) \\right] \\right| .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "167 With an additional assumption, the second term in the bound can be re-expressed, as shown in the   \n168 following corollary. ",
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+        "text": "169 Corollary 1. Assume $L _ { \\mu } ( w _ { T } ) \\leq \\mathbb { E } _ { \\Delta } \\left[ L _ { \\mu } ( w _ { T } + \\Delta _ { T } ) \\right]$ , then the following holds, ",
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+        "text": "$$\n\\mathrm { g e n } ( \\mu , P _ { W r | S } ) \\leq \\frac { R } { n b } \\sum _ { i = 1 } ^ { n } \\sqrt { \\sum _ { t \\in \\mathcal { T } _ { i } } \\frac { 2 \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } + \\frac { \\sum _ { t = 1 } ^ { T } \\sigma _ { t } ^ { 2 } } { 2 } \\mathrm { T r } \\left( \\mathbb { E } \\left[ \\mathrm { H } _ { W _ { T } } ( Z ) \\right] \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where 170 $\\mathrm { H } _ { W _ { T } }$ is the Hessian matrix of the loss with respect to $W _ { T }$ and $\\operatorname { T r } ( \\cdot )$ denotes trace. ",
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+        "text": "171 Corollary 1 follows directly from the second order Taylor expansion of the second term in the bound   \n172 of Theorem 2. The condition $L _ { \\mu } ( w _ { T } ) \\leq \\mathbb { E } _ { \\Delta } \\left[ L _ { \\mu } ( \\hat { w _ { T } } + \\bar { \\Delta _ { T } } ) \\right]$ indicates that the perturbation does   \n173 not decrease the population risk. This is also assumed in [27] in the derivation of a PAC-Bayesian   \n174 generalization bound.   \n175 Notably, in the bound of Theorem 2, the first term captures the impact of the training trajectory   \n176 (“trajectory term”), and the second term captures the impact of the final solution. As seen in Corollary   \n177 1, this term in fact measures the flatness for the loss landscape at the found solution (“flatness term”).   \n178 The previous bound of [49] (Lemma 3) and its tightened version in Theorem 1 also similarly contain a   \n179 trajectory term and a flatness term. Despite that the flatness term there are identical to that in Theorem   \n180 2, we now show the trajectory term in Theorem 2 does improve on its counter-part in Theorem 1.   \n181 Lemma 6. Assume the instances are sampled without replacement in every epoch. Then the trajectory   \n182 term in Theorem 2 is upper-bounded by ",
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+        "text": "$$\n\\operatorname* { m i n } \\left\\{ \\frac { R } { n } \\sum _ { t = 1 } ^ { T } \\sqrt { \\frac { 2 \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } , \\sqrt { \\frac { 2 R ^ { 2 } } { n b } \\sum _ { t = 1 } ^ { T } \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } \\right\\} .\n$$",
+        "text_format": "latex",
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+        "text": "184 The condition in Lemma 6 is usually satisfied in practice. This lemma then immediately implies that   \n185 the trajectory term in Theorem 2 is no worse than that in Theorem 1. Incorporating this result, if we   \n186 restrict the smoothness of the loss function $\\ell$ , we may obtain another version of the generalization   \n187 bound (although the flatness term therein is expected to be looser than that in Corollary 1).   \n188 Corollary 2. If the loss function is differentiable and $\\beta$ -smooth with respect to $w$ , then under the   \n189 condition of Lemma ${ \\it 6 }$ , ",
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+        "text": "$$\n\\mathrm { g e n } ( \\mu , P _ { W _ { T } | S } ) | \\leq \\operatorname* { m i n } \\left\\{ \\frac { R } { n } \\sum _ { t = 1 } ^ { T } \\sqrt { \\frac { 2 \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } , \\sqrt { \\frac { 2 R ^ { 2 } } { n b } \\sum _ { t = 1 } ^ { T } \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } \\right\\} + \\beta d \\sum _ { t = 1 } ^ { T } \\sigma _ { t } ^ { 2 } .\n$$",
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+        "text": "191 To conclude, we remark that these bounds suggest that in order for the model to generalize well, both   \n192 the trajectory term and the flatness term need to be small — the former involves the interaction of the   \n193 learning rate and batch size with the gradient dispersion along the training trajectory, whereas the   \n194 latter depends on the flatness of the empirical risk surface at the found solution. ",
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+        "type": "text",
+        "text": "195 4 Experimental Study ",
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+        "text": "196 Bound Verification We first verify our bound in Corollary 1 by training an MLP (with one hidden   \n197 layer) and an AlexNet [42] on MNIST and CIFAR10 [41], respectively. To simplify estimation, we   \n198 fix the weight initialization and set $\\sigma _ { t }$ and $\\lambda _ { t }$ to be constants $\\sigma$ and $\\lambda$ , respectively. To compute   \n199 $\\textstyle \\sum _ { i = 1 } ^ { n } { \\sqrt { \\sum _ { t \\in { \\mathcal { T } } _ { i } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] } }$ , we compute the gradient dispersion as its empirical estimate from a   \n200 batch, utilizing a PyTorch [54] library BackPack [19]. To compute $\\mathrm { T r } \\left( \\mathbb { E } \\left[ \\mathrm { H } _ { W _ { T } } ( Z ) \\right] \\right)$ , we randomly   \n201 sample $10 \\%$ of the training data and use the PyHessian library [79] to compute the Hessian. Since   \n202 every choice of $\\sigma$ gives a valid generalization bound in Corollary 1, we need to find the optimal   \n203 $\\sigma$ , which gives the tightest bound. This can be done by simply utilizing the fact $A / \\sigma + \\bar { \\sigma ^ { 2 } } B \\geq$   \n204 $3 ( A / 2 ) ^ { 2 / 3 } B ^ { 1 / 3 }$ for any positive $A$ and $B$ , where the equality is achieved by the optimal $\\sigma$ . We set   \n205 the sub-gaussian parameter $R = 0 . 1$ . The implementation in this paper is on PyTorch, and all the   \n206 experiments are carried out on NVIDIA Tesla V100 GPUs (32 GB).   \n207 We perform experiments with varying network width and varying levels of label noise. Specifically,   \n208 label noise level $\\epsilon$ refers to the setting where we replace the labels of $\\epsilon$ fraction of the training and   \n209 testing instances with random labels. The estimated bound is compared against the true generalization   \n210 gap, namely, the difference between the training loss and testing loss, and is shown in Figure 1.   \n211 In Figure 1, we see that in all cases the estimated bound follows closely the trend of the true   \n212 generalization gap. The fact that the bound curve consistently tracks the gap curve under various   \n213 label noise levels indicates that our bound very well captures the changes of the data distribution.   \n214 Note that in Figure 1 (a) and (b), our bound decays with the increase of the model size, showing a   \n215 trend as opposite to the bounds obtained in classical learning theory. But such a trend clearly better   \n216 explains the generalization behaviour of modern neural networks.   \n217 Learning Rate and Batch Size. The learning rate and batch size in SGD have explicitly appeared   \n218 in the trajectory term of the bound in Theorem 2. From the way they appear in the bound, one may be   \n219 tempted to assert that a small learning rate or large batch size will improve generalization. This would   \n220 then contradict some previous observations [37, 76, 33], in which increasing the ratio of learning rate   \n221 to batch size will benefit generalization. We now investigate this by performing experiments with   \n222 varying learning rates and batch sizes. In our experiments, the model is continuously updated until   \n223 the average training loss drops below 0.0001. We separate trajectory and flatness terms of the bound   \n224 and plot them in Figure 2.   \n225 A key observation in Figure 2 is that the learning rate impacts the trajectory term and the flatness term   \n226 in opposite ways, as seen, for example, in (a) and (b), where the two set of curves swap their orders in   \n227 the two figures. On the other hand, the batch size also impacts the two terms in opposite ways, as seen   \n228 in (a) and (b) where curves decrease in (a) but increase in (b). This makes the generalization bound,   \n229 i.e., the sum of the two terms, have a rather complex relationship with the settings of learning rate   \n230 and batch size. This relationship is further complicated by the fact that a small learning rate requires   \n231 a longer training time, or a larger number $T$ of training iterations, which increases the number that   \n232 are summed over in the trajectory term. Nonetheless, we do observe that a smaller batch size gives a   \n233 lower value of the flatness term ((b) and (d)), confirming the previous wisdom that small batch sizes   \n234 enable the neural network to find a flat minima [40].   \n235 Double Descent of Gradient Dispersion We experimentally investigate the impact of gradient   \n236 dispersion on the training of the neural networks by fixing the learning rate, batch size and weight   \n237 initialization for the each model (MLP for MNIST, AlexNet for CIFAR10). For each model and   \n238 various label noise levels, we plot in Figure 3 the evolution of the (empirical) gradient dispersion   \n239 $\\widehat { \\mathbb { V } } ( \\boldsymbol { w } _ { t } )$ , training accuracy and testing accuracy across training epochs. ",
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+            "Figure 1: Estimated bound and empirical generalization gap (“gap”) as functions of network width ((a) and (b)) and label noise level ((c) and (d)). Left Y-axis: gap value; right Y-axis: bound value. "
+        ],
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+            "Figure 2: The impact of learning rate and batch size on the trajectory term and the flatness term. "
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+            "Figure 3: Epoch-wise double descent of gradient dispersion, in relation to training/testing accuracies. "
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+        "text": "An intriguing epoch-wise “double descent” phenomenon is observed, particularly when the labels are noisy. According to the double descent curve, the training may be split into three phases (e.g., Figure 3 (h)). In the first phase, the gradient dispersion rapidly descends and maintains a very low level. In this phase, both training and test accuracies increase while maintaining a very small generalization gap. This suggests that the network in this phase is extracting useful patterns and generalizes well. In the second phase, the gradient dispersion starts increasing until it reaches a peak value. In this phase, the training and testing accuracies gradually diverge, marking the model entering an overfitting or “memorization” regime – when the data contains the noisy labels, the network mostly tries to memorize the labels in the training set. In the third phase, the gradient dispersion descends again, reaching a low value. In this phase, the model continuously overfits the training data, until the training and testing curves reach their respective maximum and minimum. It appears that the timing of the three phases depends on the dataset and the label noise level. For simpler data (e.g. MNIST) and cleaner datasets (e.g. CIFAR10 with low label noise), the first phase may be shorter. This is arguably because in these datasets, extracting useful patterns is relatively easier. Nonetheless, the valley in the double-descent curve appears to mark a “great divide” between generalization and memorization. ",
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+        "text": "255 Dynamic Gradient Clipping Inspired by our generalization bounds and above observations, one   \n256 way to reduce the generalization error is to control the trajectory term of the bounds by reducing the   \n257 gradient dispersion in each training step. Here we investigate a simple scheme that dynamically clips   \n258 the gradient norm so as to reduce the gradient dispersion. Specifically, whenever the current gradient   \n259 norm is larger than the gradient norm $K$ steps earlier, or $| | g ( W _ { t } , B _ { t } ) | | _ { 2 } > | | g ( W _ { t - K } , B _ { t - K } ) | | _ { 2 }$ (i.e.,   \n260 the model is expected to have entered the “memorization” regime), we reduce the norm of the current   \n261 gradient $g ( W _ { t } , B _ { t } )$ to $\\alpha$ fraction of $| | g ( W _ { t - K } , B _ { t - K } ) | | _ { 2 }$ , for some prescribed value $\\alpha < 1$ . The   \n262 effectiveness of this scheme is best demonstrated when the labels contain noise. As shown in Figure   \n263 4, dynamic gradient clipping significantly closes the gap between the training accuracy and the testing   \n264 accuracy. The models trained with this scheme maintain a near-optimal testing accuracy (e.g., about   \n265 $80 \\%$ when the label noise level of MNIST is 0.2), without suffering from the severe memorization   \n266 effect as seen in models trained without this scheme. Further understanding of the double-descent   \n267 phenomenon of the gradient dispersion may enable more delicate design of such a dynamic clipping   \n268 scheme and potentially lead to novel and powerful regularization techniques. ",
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+        "image_caption": [
+            "Figure 4: Dynamic Gradient Clipping. "
+        ],
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+        "type": "text",
+        "text": "269 5 A Practical Implication: Gaussian Model Perturbation ",
+        "text_level": 1,
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+        "text": "270 The appearance of the flatness term in our generalization bounds suggests that for an empirical   \n271 risk minimizer $w ^ { * }$ to generalize well, it is necessary that the empirical risk surface at $w ^ { * }$ is flat,   \n272 or insensitive to a small perturbation of $w ^ { * }$ . This naturally motivates a training scheme using the   \n273 following regularized loss: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { w } L _ { s } ( w ) + \\rho \\operatorname* { l g } _ { \\Delta \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } \\mathbf { I } _ { d } ) } [ L _ { s } ( w + \\Delta ) - L _ { s } ( w ) ] ,\n$$",
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+        "type": "text",
+        "text": "274 where $\\rho$ is a hyper-parameter. Replacing the expectation above with its stochastic approximation   \n275 using $k$ realizations of $\\Delta$ gives rise to the following optimization problem. ",
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+        "text": "$$\n\\operatorname* { m i n } _ { w } \\frac { 1 } { b } \\sum _ { z \\in B } \\left( ( 1 - \\rho ) \\ell ( w , z ) + \\rho \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\left( \\ell ( w + \\delta _ { i } , z ) \\right) \\right) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "276 We refer to the SGD training scheme using this loss as Gaussian model perturbation or GMP. Notably,   \n277 GMP requires $k + 1$ forward passes for every parameter update. Empirical evidence shows that a   \n278 small $k$ , for example, $k = 3$ , already gives competitive performance. Implementing the $k + 1$ forward   \n279 passes on parallel processors further reduces the computation load.   \n280 We experimentally compare GMP with several major regularization schemes in the current art, includ  \n281 ing Dropout [68], label smoothing [70], Flooding [36], MixUp [81], adversarial training [29], and   \n282 AMP [83]. The compared schemes are evaluated on three popular benchmark image classification   \n283 datasets SVHN [48], CIFAR-10 and CIFAR-100 [41]. Two representative deep architectures PreAc  \n284 tResNet18 [34] and VGG16 [67] are taken as the underlying model. We train the models for 200   \n285 epochs by SGD. The learning rate is initialized as 0.1 and divided by 10 after 100 and 150 epochs.   \n286 For all compared models, the batch size is set to 50 and weight decay is set to $1 0 ^ { - 4 }$ . For GMP, we   \n287 choose $\\rho = 0 . 5$ and set the standard deviation of the Gaussian noise $\\Delta$ to 0.03. The value of $k$ is   \n288 chosen as 3 and 10 respectively (referred to as $\\mathrm { G M P ^ { 3 } }$ and $\\mathrm { G M P ^ { 1 0 } }$ ).   \n289 The performances of all compared schemes are given in Table 1. For the compared regularization   \n290 schemes except GMP, we directly report their performances as given in [83]. Performances of vanilla   \n291 ERM without regularization are also included as a reference.   \n292 Table 1 demonstrates the effectiveness of GMP. Overall GMP performs comparably to the current   \n293 art of regularization schemes, although appearing slighly inferior to the most recent record given by   \n294 AMP [83]. Noting that the key ingredient of AMP, “max-pooling” in the parameter space, greatly   \n295 resembles regularization term in GMP, which may be seen as “average-pooling” in the same space. ",
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+        "img_path": "images/53e4513d0e677fe5ff467230dcc4a544c41e42ea6d29047d9d385c22ce19d67a.jpg",
+        "table_caption": [
+            "Table 1: Top-1 classification accuracy on (a) SVHN, (b) CIFAR-10 and (c) CIFAR-100. We run experiments 10 times and report the mean and the standard deviation of the testing accuracy. "
+        ],
+        "table_footnote": [
+            "(a) SVHN (b) CIFAR-10 (c) CIFAR-100 "
+        ],
+        "table_body": "<table><tr><td>PreActResNet18</td><td>Top-1 Acc. (%)</td><td>PreActResNet18</td><td>Top-1 Acc. (%)</td><td>PreActResNet18</td><td>Top-1 Acc. (%)</td></tr><tr><td>ERM</td><td>97.05±0.063</td><td>ERM</td><td>94.98±0.212</td><td>ERM</td><td>75.69±0.303</td></tr><tr><td>Dropout</td><td>97.20±0.065</td><td>Dropout</td><td>95.14±0.148</td><td>Dropout</td><td>75.52±0.351</td></tr><tr><td>Label Smoothing</td><td>97.22±0.087</td><td>Label Smoothing</td><td>95.15±0.115</td><td>Label Smoothing</td><td>77.93±0.256</td></tr><tr><td>Flooding</td><td>97.16±0.047</td><td>Flooding</td><td>95.03±0.082</td><td>Flooding</td><td>75.50±0.234</td></tr><tr><td>MixUp</td><td>97.26±0.044</td><td>MixUp</td><td>95.91±0.117</td><td>MixUp</td><td>78.22±0.210</td></tr><tr><td>Adv.Training</td><td>97.23±0.080</td><td>Adv. Training</td><td>95.01±0.085</td><td>Adv. Training</td><td>74.77±0.229</td></tr><tr><td>AMP</td><td>97.70±0.025</td><td>AMP</td><td>96.03±0.091</td><td>AMP</td><td>78.49±0.308</td></tr><tr><td>GMP3</td><td>97.43±0.037</td><td>GMP3</td><td>95.64±0.053</td><td>GMP3</td><td>78.05±0.208</td></tr><tr><td>GMP10</td><td>97.34±0.058</td><td>GMP10</td><td>95.71±0.073</td><td>GMP10</td><td>78.07±0.170</td></tr><tr><td>VGG16</td><td>Top-1 Acc. (%)</td><td>VGG16</td><td>Top-1 Acc. (%)</td><td>VGG16</td><td>Top-1 Acc. (%)</td></tr><tr><td>ERM</td><td>96.86±0.060</td><td>ERM</td><td>93.68±0.193</td><td>ERM</td><td>72.16±0.297</td></tr><tr><td>Dropout</td><td>97.04±0.049</td><td>Dropout</td><td>93.78±0.147</td><td>Dropout</td><td>72.28±0.337</td></tr><tr><td>Label Smoothing</td><td>96.93±0.070</td><td>Label Smoothing</td><td>93.71±0.158</td><td>Label Smoothing</td><td>72.51±0.179</td></tr><tr><td>Flooding</td><td>96.85±0.085</td><td>Flooding</td><td>93.74±0.145</td><td>Flooding</td><td>72.07±0.271</td></tr><tr><td>MixUp</td><td>96.91±0.057</td><td>MixUp</td><td>94.52±0.112</td><td>MixUp</td><td>73.19±0.254</td></tr><tr><td>Adv.Training</td><td>97.06±0.091</td><td>Adv. Training</td><td>93.51±0.130</td><td>Adv.Training</td><td>70.88±0.145</td></tr><tr><td>AMP</td><td>97.27±0.015</td><td>AMP</td><td>94.35±0.147</td><td>AMP</td><td>74.40±0.168</td></tr><tr><td>GMP3</td><td>97.18±0.057</td><td>GMP3</td><td>94.33±0.094</td><td>GMP3</td><td>74.45±0.256</td></tr><tr><td>GMP10</td><td>97.09±0.068</td><td>GMP10</td><td>94.45±0.158</td><td>GMP10</td><td>75.09±0.285</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "296 6 Conclusion and Outlook ",
+        "text_level": 1,
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+        "text": "This paper presents new generalization bounds for neural networks trained with SGD, improving upon the results of [49]. Our bounds naturally point to new and effective regularization schemes. At the same time, they reveal interesting phenomena in the SGD training of neural networks. While these phenomena deserve further investigation in their own right, we here suggest another direction for improving the bounds, namely, via the use of strong data-processing inequalities (DPI) [57–59] (noting that the standard DPI is in fact needed for establishing Theorem 2). ",
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+        "text": "303 For any Markov chain $U \\to X \\to Y$ , we will denote by $u , x$ , and $\\mathcal { V }$ the spaces in which $U$ ,   \n304 $X$ , $Y$ take values, respectively. For any distribution $P$ on $\\mathcal { X }$ , we will use $P _ { Y \\mid X } \\circ P$ to denote the   \n305 distribution on $\\mathcal { V }$ induced by the push-forward of the distribution $P$ by $P _ { Y \\mid X }$ , namely, for any $y \\in \\mathcal { V }$ ,   \n306 $\\begin{array} { r } { \\left( P _ { Y | X } \\circ P \\right) ( y ) \\triangleq \\int P _ { Y | X } ( y | x ) P ( x ) d x } \\end{array}$ . Let ${ \\cal { S } } ( U )$ be the support of $P _ { U }$ and ${ \\mathcal { H } } ( U , P _ { X | U } )$ be the   \n307 convex hull of $\\{ P _ { X | U = u } : u \\in { \\mathcal { S } } ( U ) \\}$ . Define ",
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+        "text": "$$\n\\eta ( U \\to X \\to Y ) \\triangleq \\operatorname* { s u p } _ { P , Q \\in \\mathcal { H } ( U , P _ { X | U } ) } \\frac { \\mathrm { D } _ { \\mathrm { K L } } ( P _ { Y | X } \\circ P | | P _ { Y | X } \\circ Q ) } { \\mathrm { D } _ { \\mathrm { K L } } ( P | | Q ) }\n$$",
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+        "text": "08 Lemma 7. For any Markov chain $U  X  Y$ , $I ( U ; Y ) \\leq \\eta ( U  X  Y ) I ( U ; X ) .$ ",
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+        "text": "309 Here $\\eta ( U \\to X \\to Y )$ serves as the “contraction coefficient” for the stochastic kernel $P _ { Y \\mid X }$ ,   \n310 characterizing the greatest extent by which the kernel may bring closer any two distributions on $S ( U )$   \n311 in its output space. It is easy to see that $\\eta ( U \\to X \\to Y ) \\leq 1$ , giving rise to a stronger DPI.   \n12 Denote $V _ { t } \\triangleq \\widetilde { W } _ { t - 1 } + G _ { t }$ . It can be verified that $Z _ { i }  V _ { t }  \\widetilde { W } _ { t }$ form a Markov chain. Denote   \n13 $\\eta _ { i , t } \\triangleq \\eta ( Z _ { i }  V _ { t }  \\widetilde { W } _ { t } )$ , and $\\Gamma _ { i } ^ { t } = \\{ t + 1 , t + 2 , \\ldots , T \\} \\setminus \\mathcal { T } _ { i }$ . Theorem 2 can be improved to: ",
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+        "text": "314 Theorem 3. The expected generalization error of SGD is bounded by ",
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+        "text": "$$\n\\left| \\mathrm { g e n } ( \\mu , P _ { W _ { T } | S } ) \\right| \\le \\frac { 2 R } { n b } \\sum _ { i = 1 } ^ { n } \\sqrt { \\sum _ { t \\in { \\mathcal T } _ { i } } \\frac { \\lambda _ { t } ^ { 2 } } { \\sigma _ { t } ^ { 2 } } \\mathbb { E } \\left[ \\mathbb { V } ( W _ { t - 1 } ) \\right] \\cdot \\prod _ { \\tau \\in \\Gamma _ { i } ^ { t } } \\eta _ { i , \\tau } } + \\left| \\mathbb { E } \\left[ \\gamma ( W _ { T } , S ) - \\gamma ( W _ { T } , S ^ { \\prime } ) \\right] \\right| .\n$$",
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+        "text": "316 It remains to characterize the contraction coefficient $\\eta _ { i , \\tau }$ in a computable form. Simply bounding it   \n317 via the Dobrushin’s coefficient [21], as suggested in [72] for analyzing SGLD, is unlikely to make   \n318 the bound in this theorem significantly tighter than that in Theorem 2. ",
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parse/train/Hygm8jC9FQ/Hygm8jC9FQ.md

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+# FAVAE: SEQUENCE DISENTANGLEMENT USING INFORMATION BOTTLENECK PRINCIPLE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose the factorized action variational autoencoder (FAVAE), a state-of-theart generative model for learning disentangled and interpretable representations from sequential data via the information bottleneck without supervision. The purpose of disentangled representation learning is to obtain interpretable and transferable representations from data. We focused on the disentangled representation of sequential data since there is a wide range of potential applications if disentanglement representation is extended to sequential data such as video, speech, and stock price data. Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time. Previous works succeed in disentangling static factors and dynamic factors by explicit modeling the priors of latent variables to distinguish between static and dynamic factors. However, this model cannot disentangle representations between dynamic factors, such as disentangling ”picking” and ”throwing” in robotic tasks. In this paper, we propose new model that can disentangle multiple dynamic factors. Since our method does not require modeling priors, it is capable of disentangling ”between” dynamic factors. In experiments, we show that FAVAE can extract the disentangled dynamic factors.
+
+# 1 INTRODUCTION
+
+Representation learning is one of the most fundamental problems in machine learning. A real world data distribution can be regarded as a low-dimensional manifold in a high-dimensional space (Bengio et al., 2013). Generative models in deep learning, such as the variational autoencoder (VAE) (Kingma & Welling, 2013) and the generative adversarial network (GAN) (Goodfellow et al., 2014), are able to learn low-dimensional manifold representation (factor) as a latent variable. The factors are fundamental components such as position, color, and degree of smiling in an image of a human face (Liu et al., 2015). Disentangled representation is defined as a single factor being represented by a single latent variable (Bengio et al., 2013). Thus, if in a model of learned disentangled representation, shifting one latent variable while leaving the others fixed generates data showing that only the corresponding factor was changed. This is called latent traversals (a good demonstration of which was given by Higgins et al. $( 2 0 1 6 \mathrm { a } ) ^ { 1 }$ ). There are two advantages of disentangled representation. First, latent variables are interpretable. Second, the disentangled representation is generalizable and robust against adversarial attacks (Alemi et al., 2016).
+
+We focus on the disentangled representation learning of sequential data. Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time. With disentangled representation learning from sequential data, we should be able to extract dynamic factors that cannot be extracted by disentangled representation learning models for non-sequential data such as $\beta$ -VAE (Higgins et al., 2016a;b) and InfoGAN (Chen et al., 2016). The concept of disentangled representation learning for sequential data is illustrated in Fig. 1. Consider that the pseudo-dataset of the movement of a submarine has a dynamic factor: the trajectory shape. The disentangled representation learning model for sequential data can extract this shape. On the other hand, since the disentangled representation learning model for non-sequential data does not consider the sequence of data, it merely extracts the $\mathbf { X }$ -position and y-position.
+
+(c) FAVAE accepts sequential data.
+
+![](images/d791a4a43bd7873e6856996a67db2eab578a5e68ad71e5cdec1fe65206843e5f.jpg)  
+(d) Latent traversal of FAVAE.   
+Figure 1: Illustration of how FAVAE differs from $\beta$ -VAE. $\beta$ -VAE does not accept data sequentially; it cannot differentiate data points from different trajectories or sequences of data points. FAVAE considers a sequence of data points, taking all data points in a trajectory as one datum. For example, for a pseudo-dataset representing the trajectory of a submarine (1a,1c), $\beta$ -VAE accepts 11 different positions of the submarine as non-sequential data while FAVAE accepts three different trajectories of the submarine as sequential data. Therefore, the latent variable in $\beta$ -VAE learns only the coordinates of the submarine, and the latent traversal shows the change in the submarines position. On the other hand, FAVAE learns the factor that controls the trajectory of the submarine, so the latent traversal shows the change in the submarines trajectory.
+
+There is a wide range of potential applications if we extend disentanglement representation to sequential data such as speech, video, and stock market data. For example, disentangled representation learning for stock price data can extract the fundamental trend of a given stock price. Another application is the reduction of action space in reinforcement learning. Extracting dynamic factors would enable the generation of macro-actions (Durugkar et al., 2016), which are sets of sequential actions that represent the fundamental factors of the actions. Thus, disentangled representation learning for sequential data opens the door to new areas of research.
+
+Very recent related work (Hsu et al., 2017; Li & Mandt, 2018) separated factors of sequential data into dynamic and static factors. The factorized hierarchical variational autoencoder (FHVAE) (Hsu et al., 2017) is based on a graphical model using latent variables with different time dependencies. By maximizing the variational lower bound of the graphical model, the FHVAE separates the different time dependent factors such as the dynamic and static factors. The VAE architecture developed by Li & Mandt (2018) is the same as the FHVAE in terms of the time dependencies of the latent variables. Since these models require different time dependencies for the latent variables, these approaches cannot be used disentangle variables with the same time dependency factor.
+
+We address this problem by taking a different approach. First, we analyze the root cause of disentanglement from the perspective of information theory. As a result, the term causing disentanglement is derived from a more fundamental rule: reduce the mutual dependence between the input and output of an encoder while keeping the reconstruction of the data. This is called the information bottleneck (IB) principle. We naturally extend this principle to sequential data from the relationship between $x$ and $z$ to $x _ { t : T }$ and $z$ . This enables the separation of multiple dynamic factors as a consequence of information compression. It is difficult to learn a disentangled representation of sequential data since not only the feature space but also the time space should be compressed. We created the factorized action variational autoencoder (FAVAE) in which we implemented the concept of information capacity to stabilize learning and a ladder network to learn a disentangled representation in accordance with the level of data abstraction. Since our model is a more general model without the restriction of a graphical model design to distinguish between static and dynamic factors, it can separate dependency factors occurring at the same time. Moreover, it can separate factors into dynamic and static factors.
+
+# 2 DISENTANGLEMENT FOR NON-SEQUENTIAL DATA
+
+$\beta$ -VAE (Higgins et al., 2016a;b) is a commonly used method for learning disentangled representations based on the VAE framework (Kingma $\&$ Welling, 2013) for a generative model. The VAE can estimate the probability density from data $\mathbf { X }$ . The objective function of the VAE maximizes the evidence lower bound (ELBO) of $\log p \left( x \right)$ as
+
+$$
+\log p \left( x \right) = \mathcal { L } _ { \mathrm { V A E } } + \underbrace { D _ { \mathrm { K L } } \left( q \left( z \vert x \right) \vert \vert p \left( z \vert x \right) \right) } _ { \ge 0 } ,
+$$
+
+where $z$ is latent variable, $D _ { \mathrm { K L } }$ is the Kullback-Leibler divergence, and $q \left( z | x \right)$ is an approximated distribution of $p \left( z | x \right) . \ D _ { \mathrm { K L } } \left( q \left( z | x \right) | | p \left( z | x \right) \right)$ reduces to zero as the ELBO $\mathcal { L } _ { \mathrm { V A E } }$ increases; thus, $q \left( z | x \right)$ learns a good approximation of $p \left( z | x \right)$ . The ELBO is defined as
+
+$$
+\mathcal { L } _ { \mathrm { V A E } } \equiv E _ { q ( z | x ) } \left[ \log p \left( x | z \right) \right] - D _ { \mathrm { K L } } \left( q \left( z | x \right) | | p \left( z \right) \right) ,
+$$
+
+where the first term, $E _ { q ( \boldsymbol { z } | \boldsymbol { x } ) } \left[ \log p \left( \boldsymbol { x } | \boldsymbol { z } \right) \right]$ , is a reconstruction term used to reconstruct $x$ , and the second term $D _ { \mathrm { K L } } \left( q \left( z | x \right) \^ { } | | \dot { p } \left( z \right) \right)$ is a regularization term used to regularize posterior $q \left( z | x \right)$ . Encoder $q \left( z | x \right)$ and decoder $p \left( x | z \right)$ are learned in the VAE.
+
+Next we will explain how $\beta$ -VAE extracts disentangled representations from unlabeled data. $\beta$ -VAE is an extension of the coefficient $\beta > 1$ of the regularization term $D _ { \mathrm { K L } } \left( q \left( z | x \right) | | p \left( z \right) \right)$ in the VAE. The objective function of $\beta$ -VAE is
+
+$$
+\mathcal { L } _ { \beta - \mathrm { V A E } } = E _ { q ( z | x ) } \left[ \log p \left( x | z \right) \right] - \beta D _ { \mathrm { K L } } \left( q \left( z | x \right) | | p \left( z \right) \right) ,
+$$
+
+where $\beta > 1$ and $p \left( z \right) = \mathcal { N } \left( 0 , 1 \right)$ . $\beta$ -VAE promotes disentangled representation learning via the Kullback-Leibler divergence term. As $\beta$ increases, the latent variable $q \left( z | x \right)$ approaches the prior $p \left( z \right)$ ; therefore, each $z _ { i }$ is pressured to learn the probability distribution of $\mathcal { N } ( 0 , 1 )$ . However, if all latent variables $z _ { i }$ become $\mathcal { N } ( 0 , 1 )$ , the model cannot reconstruct $x$ . As a result, as long as $z$ reconstructs $x$ , $\beta$ -VAE reduces the information of $z$ .
+
+# 3 PRELIMINARY: ORIGIN OF DISENTANGLEMENT
+
+To clarify the origin of disentanglement, we will explain the regularization term. The regularization term has been decomposed into three terms (Chen et al., 2018; Kim & Mnih, 2018; Hoffman & Johnson, 2016):
+
+$$
+E _ { p \left( x \right) } \left[ D _ { \mathrm { K L } } \left( q \left( \boldsymbol { z } \left| \boldsymbol { x } \right) \right| \left| p \left( \boldsymbol { z } \right) \right) \right] = I \left( \boldsymbol { x } ; \boldsymbol { z } \right) + D _ { \mathrm { K L } } \left( q \left( \boldsymbol { z } \right) \right| \Big | \prod _ { j } q \left( z _ { j } \right) \right) + \sum _ { j } D _ { \mathrm { K L } } \left( q \left( \boldsymbol { z } _ { j } \right) \right| \left| p \left( z _ { j } \right) \right) ,
+$$
+
+where $z _ { j }$ denotes the $j$ -th dimension of the latent variable. The second term, which is called ”total correlation” in information theory, quantifies the redundancy or dependency among a set of $\mathbf { n }$ random variables (Watanabe, 1960). $\beta$ -TCVAE has been experimentally shown to reduce the total correlation causing disentanglement (Chen et al., 2018). The third term indirectly causes disentanglement by bringing $q \left( z | x \right)$ close to the independent standard normal distribution $p \left( z \right)$ . The first term is mutual information between the data variable and latent variable based on the empirical data distribution. Minimizing the regularization term causes disentanglement but disturbs reconstruction via the first term in Eq. (4). The shift $C$ scheme was proposed (Burgess et al., 2018) as a means to solve this conflict:
+
+$$
+- { \cal E } _ { q _ { \phi } ( z | x ) } [ \log p _ { \theta } ( x | z ) ] + \beta | { \cal D } _ { \mathrm { K L } } ( q ( z | x ) | | p ( z ) ) - C | ,
+$$
+
+where constant shift $C$ , which is called ”information capacity,” linearly increases during training. This shift $C$ can be understood from the point of view of an information bottleneck (Tishby et al., 2000). The VAE can be derived by maximizing the ELBO, but $\beta$ -VAE can no longer be interpreted as an ELBO once this scheme has been applied. The objective function of $\beta$ -VAE is derived from the information bottleneck (Alemi et al., 2016; Achille & Soatto, 2018; Tishby et al., 2000; Chechik et al., 2005).
+
+$$
+\operatorname* { m a x } I \left( z ; x \right) \qquad \mathrm { s . t . } ~ | I \left( \hat { x } ; z \right) - C | = 0 ,
+$$
+
+where $C$ is the information capacity and $\hat { x }$ is the empirical distribution. Solving this equation by using Lagrange multipliers drives the objective function of $\beta$ -VAE Eq. (5) with $\beta$ as the Lagrange multiplier (details in Appendix $\mathbf { B }$ of (Alemi et al., 2016)). In Eq. (5), information capacity $C$ prevents $I \left( \hat { x } , z \right)$ from becoming zero. In the information bottleneck literature, $y$ typically stands for a classification task; however, the formulation can be related to the autoencoding objective (Alemi et al., 2016). Therefore, the objective function of $\beta$ -VAE can be understood using the information bottleneck principle.
+
+# 4 PROPOSED MODEL: DISENTANGLEMENT FOR SEQUENTIAL DATA
+
+Our proposed FAVAE model learns disentangled and interpretable representations from sequential data without supervision. We consider sequential data $x _ { 1 : T } \equiv \{ x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { T } \}$ generated from a latent variable model,
+
+$$
+p \left( x _ { 1 : T } \right) = \int p \left( x _ { 1 : T } | z \right) p \left( z \right) d z .
+$$
+
+For sequential data, we replace $x$ with $\scriptstyle ( x _ { 1 : T } )$ in Eq. 5. The objective function of the FAVAE model is
+
+$$
+- { \cal E } _ { q _ { \phi } ( z | ( x _ { 1 : T } ) _ { i } ) } [ \log p _ { \theta } ( x _ { 1 : T } | z ) ] + \beta | { \cal D } _ { \mathrm { K L } } ( q ( z | ( x _ { 1 : T } ) _ { i } ) | | p ( z ) ) - C | ,
+$$
+
+where $p \left( z \right) = \mathcal { N } \left( 0 , 1 \right)$ . The variational recurrent neural network (Chung et al., 2015) and stochastic recurrent neural network (Fraccaro et al., 2016) extend the VAE model to a recurrent framework. The priors of both networks are dependent on time. The time dependent prior experimentally improves the ELBO. In contrast, the prior of our model is independent of time like those of the stochastic recurrent network (Bayer $\&$ Osendorfer, 2014) and the Deep Recurrent Attentive Writer (DRAW) neural network architecture (Gregor et al., 2015); this is because FAVAE is disentangled representation learning rather than density estimation. For better understanding, consider FAVAE from the perspective of IB. As with $\beta$ -VAE, FAVAE can be understood from the information bottleneck principle.
+
+$$
+\operatorname* { m a x } I \left( z ; x _ { 1 : T } \right) \qquad \mathrm { s . t . } ~ | I \left( \hat { x } _ { 1 : T } ; z \right) - C | = 0 ,
+$$
+
+where $\hat { x } _ { 1 : T }$ follows an empirical distribution. These principles make the representation of $z$ compact while reconstruction of the sequential data is represented by $x _ { 1 : T }$ (see Appendix A).
+
+# 4.1 LADDER NETWORK
+
+An important extension to FAVAE is a hierarchical representation scheme inspired by the VLAE (Zhao et al., 2017). Encoder $q \left( z | x _ { 1 : T } \right)$ within a ladder network is defined as
+
+$$
+\begin{array} { l } { { h _ { l } = f _ { l } \left( h _ { l - 1 } \right) , } } \\ { { z _ { l } \sim \mathcal { N } \left( \mu _ { l } \left( h _ { l } \right) , \sigma _ { l } \left( h _ { l } \right) \right) , } } \end{array}
+$$
+
+where $l$ is a layer index, $h _ { 0 } \equiv x _ { 1 : T }$ , and $f$ is a time convolution network, which is explained in the next section. Decoder $p \left( x _ { 1 : T } | z \right)$ within the ladder network is defined as
+
+![](images/b346278abc264002383add7d500381c122f5b7482bf1b7ccbe274fd1a4fe979e.jpg)  
+Figure 2: FAVAE architecture.
+
+$$
+\begin{array} { r } { \tilde { z } _ { L } = g _ { l } \left( z _ { L } \right) } \\ { \tilde { z } _ { l } = g _ { l } \left( \tilde { z } _ { l + 1 } + \mathrm { g a t e } \left( z _ { l } \right) \right) , } \\ { x _ { 1 : T } \sim r \left( x _ { 1 : T } , \tilde { z } _ { 0 } \right) ~ } \end{array}
+$$
+
+where $g _ { l }$ is the time deconvolution network with $l = 1 , \cdots , L - 1$ , and $r$ is a distribution family parameterized by $g _ { 0 } \left( \tilde { z } _ { 0 } \right)$ . The gate computes the Hadamard product of its learnable parameter and input tensor. We set $r$ as a fixed-variance factored Gaussian distribution with the mean given by $\mu _ { t : T } = g _ { 0 } \left( \tilde { z } _ { 0 } \right)$ . Fig. (2) shows the architecture of the proposed model. The difference between each ladder network in the model is the number of convolution networks through which data passes. The abstract expressions should differ between ladders since the time convolution layer abstracts sequential data. Without the ladder network, the proposed method can disentangle only the representations at the same level of abstraction; with the ladder network, it can disentangle representations at different levels of abstraction.
+
+![](images/a788ce73cb8b7bfaad254b437844807e035d2ae99c0c0ceda9ad26e554da2d1a.jpg)  
+Figure 3: Visualization of latent traversal of $\beta$ -VAE and FAVAE. On one sampled trajectory (red), each latent variable is traversed and purple and/or blue points are generated. The color corresponds to the value of the traversed latent variable. 3a represents all data trajectories of 2D reaching.
+
+# 4.2 HOW TO ENCODE AND DECODE
+
+There are several mainstream neural network models designed for sequential data, such as the long short-term memory model (Hochreiter & Schmidhuber, 1997), the gated recurrent unit model (Chung et al., 2014), and the quasi-recurrent neural network QRNN (Bradbury et al., 2016). However, the VLAE has a hierarchical structure created by abstracting a convolutional neural network, so it is simple to add the time convolution of the QRNN to our model. The input data are $x _ { t , i }$ , where $t$ is the time index and $i$ is the dimension of the feature vector index. The time convolution considers the dimensions of feature vector $j$ as a convolution channel and performs convolution in the time direction:
+
+$$
+z _ { t j } = \sum _ { p , i } x _ { t - p , i } h _ { p i j } + b _ { j } ,
+$$
+
+where $j$ is the channel index. The proposed FAVAE model has a network similar to the VAE one regarding time convolution and a loss function similar to the $\beta$ -VAE one (Eq. (8)). We used the batch normalization (Hinton et al., 2012) and ReLU as activation functions though other variations are possible. For example, 1d convolutional neural networks use a filter size of 3 and a stride of 2 and do not use a pooling layer.
+
+# 5 MEASURING DISENTANGLEMENT
+
+While latent traversals are useful for checking the success or failure of disentanglement, quantification of the disentanglement is required for reliably evaluating the model. Various disentanglement quantification methods have been reported (Eastwood & Williams, 2018; Chen et al., 2018; Kim & Mnih, 2018; Higgins et al., 2016b;a), but there is no standard method. We use the mutual information gap (MIG) (Chen et al., 2018) as the metric for disentanglement. The basic idea of MIG is measuring the mutual information between latent variables $z _ { j }$ and a ground truth factor $v _ { k }$ . Higher mutual information means that $z _ { j }$ contains more information regarding $v _ { k }$ .
+
+$\begin{array} { r } { \mathrm { M I G } \equiv \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \frac { 1 } { H \left( v _ { k } \right) } \left( I \left( z _ { j ^ { ( k ) } } ; v _ { k } \right) - \operatorname* { m a x } _ { j \neq j ^ { ( k ) } } I \left( z _ { j } ; v _ { k } \right) \right) } \end{array}$ , where ${ j ^ { ( k ) } \equiv \arg \operatorname* { m a x } _ { j } I \left( z _ { j } ; v _ { k } \right) }$ and $H \left( v _ { k } \right)$ is entropy for normalization. In our evaluation we experimentally measure disentanglement with MIG.
+
+Table 1: Disentanglement scores (MIG, and Reconstruction loss) with standard deviations by 10 seeds for the different model. Best results are shown in bold.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">2D Reaching</td><td colspan="2">2D wavy Reaching</td></tr><tr><td>MIG</td><td>Reconstruction</td><td>MIG</td><td>Reconstruction</td></tr><tr><td>Time Convolution AE</td><td>0.21(27)</td><td>0.18(13)</td><td>0.04(3)</td><td>0.06(1)</td></tr><tr><td>FAVAE (without ladder and C)</td><td>0.34(21)</td><td>0.62(115)</td><td>0.22(14)</td><td>23.6(279)</td></tr><tr><td>FAVAE (with ladder and without C)</td><td>0.41(13)</td><td>0.03(5)</td><td>0.26(7)</td><td>0.08(2)</td></tr><tr><td>FAVAE (without ladder and with C)</td><td>0.34(21)</td><td>0.62(115)</td><td>0.11(10)</td><td>1.47(73)</td></tr><tr><td>FAVAE (with ladder and C)</td><td>0.41(13)</td><td>0.03(5)</td><td>0.34(28)</td><td>1.81(170)</td></tr></table>
+
+# 6 RELATED WORK
+
+Several recently reported models (Hsu et al., 2017; Li & Mandt, 2018) graphically disentangle static and dynamic factors in sequential data such as speech data and video data (Garofolo et al., 1993; Pearce & Picone, 2002). In contrast, our model performs disentanglement by using a loss function (see Eq. 8). The advantage of the graphical models is that they can control the interpretable factors by controlling the priors time dependency. Since dynamic factors have the same time dependency, these models cannot disentangle dynamic factors. A loss function model can disentangle sets of dynamic factors as well as disentangle static and dynamic factors.
+
+# 7 EVALUATION
+
+We evaluated our model experimentally using three sequential datasets: 2D Reaching, 2D Wavy Reaching, and Gripper. We used a batch size of 128 and the Adam (Kingma & Ba, 2014) optimizer with a learning rate of $1 0 ^ { - 3 }$ .
+
+# 7.1 2D REACHING
+
+To determine the differences between FAVAE and $\beta$ -VAE, we used a bi-dimensional space reaching dataset. Starting from point $( 0 , 0 )$ , the point travels to goal position $( - 0 . 1 , + 1 )$ or $( + 0 . 1 , + 1 )$ . There are ten possible trajectories to each goal; five are curved inward, and the other five are curved outward. The degree of curvature for all five trajectories is different. The number of factor combinations was thus 20 (2x2x5). The trajectory length was 1000, so the size of one trajectory was $[ 1 0 0 0 \mathrm { x } 2 ]$ .
+
+We compared the performances of $\beta$ -VAE and FAVAE trained on the 2D Reaching dataset. The results of latent traversal are transforming one dimension of latent variable z into another value and reconstructing something from the traversed latent variables. $\beta$ -VAE, which is only able to learn from every point of a trajectory separately, encodes data points into latent variables that are parallel to the $\mathbf { X }$ and y axes (3b, 3c). In contrast, FAVAE learns through one entire trajectory and can encode disentangled representation effectively so that feasible trajectories are generated from traversed latent variables (3d, 3e).
+
+# 7.2 2D WAVY REACHING
+
+To confirm the effect of disentanglement through information bottleneck, we evaluated the validity of our model under more complex factors by adding more factors to the 2D Reaching dataset. Five factors in total generated data compared to the three factors that generate data in 2D Reaching. This modified dataset differed in that four out of the five factors affect only part of the trajectory: two of them affect the first half, and the other two affect the second half. This means that the model should be able to focus on a certain part of the whole trajectory and be able to extract factors related to that part. A detailed explanation of these factors is given in Appendix B.
+
+We compared various models on the basis of MIG to demonstrate the validity of our proposed model in comparison of a time convolution AE in which a loss function is used only for the autoencoder ( $\beta = 0$ ), FAVAE without the ladder network and information capacity $C$ , and FAVAE with the ladder network and information capacity $C$ . As shown in Table 1, FAVAE with the ladder network and C had the highest MIG scores for 2D Reaching and 2D Wavy Reaching. This indicates that this model learned a disentangled representation best. Note that for 2D Reaching, the best value for $C$ was small, meaning that there was little effect from adding $C$ (since the dataset was simple, this task can be solved even if the amount of information of $\mathbf { Z }$ is small).
+
+![](images/f01901b6971fd5a8db90e6cf3c95eeb6a623b75861e2421a6a2b3dbf4fb4fc1a.jpg)  
+Figure 4: MIG score and reconstruction loss for different values of $\beta$ . Blue line represents results with information capacity $C$ greater than zero; red line represents results with $C$ set to e zero. Note that $\mathbf { X }$ axis is plotted in log scale.
+
+![](images/b57addf8cf5aa9b7ee9e51ea1c882d13a1370cfeaa4554ced84a275746fbca89.jpg)  
+Figure 5: Visualization of latent traversal for 2D Wavy Reaching. Each plot was decoded by traversing one latent variable; different colors represent trajectories generated from different values of same latent variable, z.
+
+Fig. 4. We evaluated reconstruction loss and the MIG score for various values of $\beta$ using three ladder networks (Fig. 6) with a different value of $C$ for each ladder: $\begin{array} { r l } { C } & { { } = } \end{array}$ [LowerLadder, MiddleLadder, HigherLadder]. One setting used was $C = [ 0 , 0 , 0 ]$ , meaning that $C$ was not used; another setting was $C = [ 2 0 , 1 , 5 ]$ , meaning that $C$ was adjusted on the basis of values of KL-Divergence for $\beta = 1$ and $\dot { C } = [ 0 , \dot { 0 } , 0 ]$ . When $C$ was not used, the model could not reconstruct data when $\beta$ was high; thus, disentangled representation was not learned well when $\beta$ was high. When $C$ was used, the MIG score increased with $\beta$ while reconstruction loss was suppressed.
+
+The latent traversal results for 2D Wavy Reaching are plotted in Fig. 5. Even though not all learned representations are perfectly disentangled, the visualization shows that all five generation factors were learned from five latent variables; the other latent variables did not learn any meaningful factors, indicating that the factors could be expressed as a combination of five ”active” latent variables. We tested our model for $\beta = 3 0 0$ .
+
+![](images/e02bc66dbc528f5a4577bc3ed31795379f1949b47c1b3c825077681ee325c131.jpg)  
+Figure 6: Evaluation of ladders (left); structure used for each experiment (right).
+
+![](images/4acf414e8b61f125fbb858bb241498545bc1deb9515d1fce021f72be4fcdf42b.jpg)  
+Figure 7: Visualization of latent traversal of FAVAE. The horizontal axis represents sequence and the vertical axis represents the difference of $z$ values.
+
+The use of a ladder network in our model improved disentangled representation learning and minimized the reconstruction loss. The graph in Fig. 6 shows the MIG scores for networks with different numbers of ladders. The error bars represent the standard deviation for ten repetitive trials. Using all three ladders resulted in the minimum reconstruction loss with the highest MIG score (green curve) except for ”Higher Ladder One”. ”Higher Ladder One” has a large reconstruction error.
+
+# 7.3 SPRITES DATASET.
+
+To evaluate the effectiveness of the video dataset, we trained our model with the Sprites dataset which is used in (Li & Mandt, 2018). This dataset has sequential length $= 8$ RGB video data with $3 \times 6 4 \times 6 4$ . This data set consists of static factor and dynamic factor. We note that the motion is not created with the combination of dynamic factors, and each motion exists individually (detail is E.2). Table 4 show the factors used in our experiment. We executed disentangled representation learning by using the FAVAE model with $\beta = 2 0$ , $C = [ 0 . 3 , 0 . 1 7 , 0 . 0 6 ]$ and network architecture used for this training is explained in Section F.1. Fig. 7 shows the results of latent traversal, and we chose two $z$ values to change from $z = 3$ to 3. Since this dataset is composed of discrete factors, we show two $z$ values at a time.The latent variables in the 1st ladder extract expressions of motion (4th $z$ in 1st ladder), pants color (5th $z$ in 1st ladder), direction of character (6th $z$ in 1st ladder) and the shirts color (7th $z$ in 1st ladder). The latent variables in the 2nd ladder extract expressions of the hair color (1st $z$ in 2nd ladder) and the body color (2nd $z$ in 2nd ladder). FHVAE can extract the disentangled representations between static factors and dynamic factor in high dimension dataset.
+
+# 8 SUMMARY AND FUTURE WORK
+
+Our factorized action variational autoencoder (FAVAVE) generative model learns disentangled and interpretable representations via the information bottleneck from sequential data. Evaluation using three sequential datasets demonstrated that it can learn disentangled representations. Future work includes extending the time convolution part to a sequence-to-sequence model (Sutskever et al., 2014) and applying the model to actions of reinforcement learning to reduce the pattern of actions.
+
+# REFERENCES
+
+Alessandro Achille and Stefano Soatto. Information dropout: Learning optimal representations through noisy computation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2018.
+
+Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv preprint arXiv:1612.00410, 2016.
+
+Justin Bayer and Christian Osendorfer. Learning stochastic recurrent networks. arXiv preprint arXiv:1411.7610, 2014.
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+Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013.
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+James Bradbury, Stephen Merity, Caiming Xiong, and Richard Socher. Quasi-recurrent neural networks. arXiv preprint arXiv:1611.01576, 2016.
+
+Christopher P Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in $\beta$ -VAE. arXiv preprint arXiv:1804.03599, 2018.
+
+Gal Chechik, Amir Globerson, Naftali Tishby, and Yair Weiss. Information bottleneck for Gaussian variables. Journal of machine learning research, 6(Jan):165–188, 2005.
+
+Tian Qi Chen, Xuechen Li, Roger Grosse, and David Duvenaud. Isolating sources of disentanglement in variational autoencoders. arXiv preprint arXiv:1802.04942, 2018.
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+Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. InfoGan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in neural information processing systems, pp. 2172–2180, 2016.
+
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+John S Garofolo, Lori F Lamel, William M Fisher, Jonathan G Fiscus, and David S Pallett. DARPA TIMIT acoustic-phonetic continous speech corpus CD-ROM. NIST speech disc 1-1.1. NASA STI/Recon technical report n, 93, 1993.
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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 Advances in neural information processing systems, pp. 2672–2680, 2014.
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+
+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-VAE: Learning basic visual concepts with a constrained variational framework. 2016b.
+
+Geoffrey E Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan R Salakhutdinov. Improving neural networks by preventing co-adaptation of feature detectors. arXiv preprint arXiv:1207.0580, 2012.
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+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997.
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+Matthew D Hoffman and Matthew J Johnson. ELBO surgery: yet another way to carve up the variational evidence lower bound. In Workshop in Advances in Approximate Bayesian Inference, NIPS, 2016.
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+Hyunjik Kim and Andriy Mnih. Disentangling by factorising. arXiv preprint arXiv:1802.05983, 2018.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
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+Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Yingzhen Li and Stephan Mandt. A deep generative model for disentangled representations of sequential data. arXiv preprint arXiv:1803.02991, 2018.
+
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+
+David Pearce and J Picone. Aurora working group: DSR front end LVCSR evaluation AU/384/02. Inst. for Signal & Inform. Process., Mississippi State Univ., Tech. Rep, 2002.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014.
+
+Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000.
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+Satosi Watanabe. Information theoretical analysis of multivariate correlation. IBM Journal of research and development, 4(1):66–82, 1960.
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+
+# A INFORMATION BOTTLENECK PRINCIPLE FOR SEQUENTIAL DATA.
+
+Here the aim is to show the relationship between FAVAE and the information bottleneck for sequential data. Consider the information bottleneck object:
+
+$$
+\begin{array} { r l } { \operatorname* { m a x } I \left( z ; x _ { 1 : T } \right) \quad } & { { } \mathrm { s . t . } \ \lvert I \left( \hat { x } _ { 1 : T } ; z \right) - C \rvert = 0 } \end{array}
+$$
+
+is expanded from Alemi et al. (2016) to sequential data. We need to distinguish between $\hat { x } _ { 1 : T }$ and $x _ { 1 : T }$ , where $x _ { 1 : T }$ is the true distribution and $\hat { x } _ { 1 : T }$ is the empirical distribution created by sampling from the true distribution. We maximize the mutual information of true data $x _ { 1 : T }$ and $z$ while constraining the information contained in the empirical data distribution. We do this by using Lagrange multiplier:
+
+$$
+\begin{array} { r } { I \left( z ; x _ { 1 : T } \right) + \beta \left| I \left( \hat { x } _ { 1 : T } ; z \right) - C \right| , } \end{array}
+$$
+
+where $\beta$ is a constant. For the first term,
+
+$$
+\begin{array} { l } { { I \left( z ; x _ { 1 : T } \right) = \displaystyle \iint p \left( z ; x _ { 1 : T } \right) \log \frac { p \left( x _ { 1 : T } \vert z \right) } { p \left( x _ { 1 : T } \right) } d x _ { 1 : T } d z } } \\ { { \displaystyle \qquad = \iint p \left( z ; x _ { 1 : T } \right) \log p \left( x _ { 1 : T } \vert z \right) d x _ { 1 : T } d z - \displaystyle \iint p \left( x _ { 1 : T } ; z \right) \log p \left( x _ { 1 : T } \right) d x _ { 1 : T } d z } } \\ { { \displaystyle \qquad = \displaystyle \iint p \left( x _ { 1 : T } \right) p \left( z \vert x _ { 1 : T } \right) \log p \left( x _ { 1 : T } \vert z \right) d x _ { 1 : T } d z + H \left( x _ { 1 : T } \right) } } \\ { { \displaystyle \qquad \sim \frac { 1 } { N } \sum _ { i } \left[ p \left( z \vert \left( x _ { 1 : T } \right) _ { i } \right) \log p \left( \left( x _ { 1 : T } \right) _ { i } \vert z \right) \right] + H \left( x _ { 1 : T } \right) } , } \end{array}
+$$
+
+where $H \left( x _ { 1 : T } \right)$ is entropy, which can be neglected in optimization. The last line is Monte Carlo approximation. For the second term,
+
+$$
+\begin{array} { l } { { \displaystyle I \left( \hat { x } _ { 1 : T } ; z \right) = \int p \left( z \vert \hat { x } _ { 1 : T } \right) p \left( \hat { x } _ { 1 : T } \right) \log \frac { p \left( z \vert \hat { x } _ { 1 : T } \right) } { p \left( z \right) } d z d \hat { x } _ { 1 : T } } } \\ { { \displaystyle \quad \sim \frac { 1 } { N } \sum _ { j } p \left( z \vert \left( x _ { 1 : T } \right) _ { j } \right) \log \frac { p \left( z \vert \left( x _ { 1 : T } \right) _ { j } \right) } { p \left( z \right) } d z } } \\ { { \displaystyle \quad = \frac { 1 } { N } \sum _ { j } D _ { \mathrm { K L } } \left( p \left( z \vert \left( x _ { 1 : T } \right) _ { j } \right) \vert \vert p \left( z \right) \right) . } } \end{array}
+$$
+
+As a result,
+
+$$
+\begin{array} { r } { I \left( z ; x _ { 1 : T } \right) + \beta \left| I \left( \tilde { x } _ { 1 : T } ; z \right) - C \right| \leq \displaystyle \frac { 1 } { N } \sum _ { i } \left[ p \left( z \right| \left( x _ { 1 : T } \right) _ { i } \right) \log p \left( \left( x _ { 1 : T } \right) _ { i } \left| z \right) \right] } \\ { + \displaystyle \frac { 1 } { N } \sum _ { j } D _ { \mathrm { K L } } \left( p \left( z \right| \left( x _ { 1 : T } \right) _ { j } \right) \left| | p \left( z \right) \right) . } \end{array}
+$$
+
+For convenience of calculation, we use $x _ { i }$ sampled from mini-batch data for both the reconstruction term and the regularization term. This is only an approximation. If the information bottleneck principle is followed completely, it is better to use different batch data for the reconstruction and regularization terms.
+
+# B EXTRACTED FACTORS OF EACH LADDER IN FAVAE
+
+We expect the ladder network can disentangle representations at different levels of abstraction. In this section, We check the factor extracted in each ladder by using 2D Reaching and 2D Wavy.
+
+Table 2: Counting the index of latent variable in each factor by 10 seeds. The detail of factor is in Table 4.   
+
+<table><tr><td colspan="3">1st ladder</td><td>2nd ladder</td><td>3rd ladder</td></tr><tr><td rowspan="3">2D Reaching</td><td>factor 1</td><td>1</td><td>1</td><td>8</td></tr><tr><td>factor 2</td><td>10</td><td>0</td><td>0</td></tr><tr><td>factor 3</td><td>10</td><td>0</td><td>0</td></tr><tr><td rowspan="5">2D wavy Reaching</td><td>factor 1</td><td>3</td><td>0</td><td>7</td></tr><tr><td>factor 2</td><td>8</td><td>0</td><td>2</td></tr><tr><td>factor 3</td><td>8</td><td>0</td><td>2</td></tr><tr><td>factor 4</td><td>9</td><td>1</td><td>0</td></tr><tr><td>factor 5</td><td>9</td><td>0</td><td>1</td></tr></table>
+
+Table 3: Disentanglement scores (MIG, and Reconstruction loss) with standard deviations by 10 seeds for the different model in sequence length $= 1 0 0$ . Best results are shown in bold.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">2D Reaching</td><td colspan="2">2D wavy Reaching</td></tr><tr><td>MIG</td><td>Reconstruction</td><td>MIG</td><td>Reconstruction</td></tr><tr><td>FHVAE</td><td>0.62(17)</td><td>0.0003(5)</td><td>0.19(6)</td><td>0.051(36)</td></tr><tr><td>Time Convolution AE</td><td>0.06(4)</td><td>0.014(7)</td><td>0.26(6)</td><td>0.019(2)</td></tr><tr><td>FAVAE (without ladder and C)</td><td>0.52(6)</td><td>0.007(1)</td><td>0.28(2)</td><td>0.058(3)</td></tr><tr><td>FAVAE (without ladder with C)</td><td>0.47(4)</td><td>0.005(0)</td><td>0.26(1)</td><td>0.069(4)</td></tr><tr><td>FAVEA (with ladder without C)</td><td>0.34(11)</td><td>0.005(1)</td><td>0.31(8)</td><td>0.022(2)</td></tr><tr><td>FAVAE (with ladder and C)</td><td>0.35(7)</td><td>0.007(1)</td><td>0.40(7)</td><td>0.055(4)</td></tr></table>
+
+Table 2 shows the counting index of latent variable with the highest mutual information in each ladder network. In Table 2, the rows represent factor and the columns represent the index of the ladder networks. The factor 1 (goal left / goal right) in 2D Reaching and the factor 1 (goal position) in 2D wavy Reaching were extracted to the most frequently in the latent variable in 3rd ladder. Since the latent variables have 8 dimensions for the 1st ladder, 4 dimensions for the 2nd ladder and 2 dimensions for the 3rd ladder, the 3rd ladder should be the least frequent when factors are randomly entered for each z. Especially long-term and short-term factors are clear in the 2D wavy Reaching dataset. In 2D Wavy Reaching dataset, there is distinct difference between factors of long and short time dependency. The ”goal position” is the factor which affect the entire trajectory, and other factors affect half length of the trajectory (Fig. 9).In our experiment the goal of the trajectory which affect the entire trajectory tended to be expressed in the 3rd ladder. In both datasets, only factor 1 represents goal positions while others represent shape of the trajectories. Since factor 1 has different abstraction level from others, factor 1 and others result in different ladders such as ladder 3 and others.
+
+# C COMPARING WITH FHVAE
+
+FHVAE model is the recently proposed disentangled representation learning model. We note that FHVAE model uses label information to disentangle time series data, which is different setup with our FAVAE model. Table 3 shows a comparison of MIG and reconstruction using FHVAE as the baseline. It was not possible to disentangle in 2D Reaching and 2D wavy using FHVAE, because LSTM used at the FHVAE can not learn data with very long sequences (sequence length of 1000). For fair comparison with the FHVAE, we experimented with 2D Reaching (sequence length 100), 2D wavy Reaching (sequence length 100) at Table 3. In 2D Reaching, FHVAE model has the best score, while in 2D wavy Reaching FAVAE model with ladders and $C$ has the best score. Our model is slightly worse than FHVAE in 2D Reaching Dataset. The parameters searched are $\alpha = 1$ to 30 (the best $\alpha = 2 0 a n d 1$ ), $\beta = 0$ to 60 (the best $\beta = 0 . 4$ and 4 in 2D Reaching and 2D wavy Reaching) and the best $C = [ c = 0 . 0 4 , 0 . 0 2 , 0 . 0 2 ]$ and $[ 0 . 0 3 , 0 , 0 , 0 6 ]$ in 2D Reaching and 2D wavy
+
+![](images/9e59da1502ef8fd62f7a26e72178f26a02414848c0c5ae3b38b763f6056f9794.jpg)  
+Figure 8: Visualization of learned disentangled representations for Gripper dataset. Traversed latent variable shows changing position of blue objects:(b) (9th $z$ in 1st ladder), changing target ball from red one to blue one and moving accordingly:(c) (1st $z$ in 2nd ladder), reaching ball using different plan:(d) (2nd $z$ in 2nd ladder), changing plan to place ball from drop ball to placing it gently:(e) (4th $z$ in 2nd ladder).
+
+Reaching. The best $C$ was decided by the value of KL divergence loss when we experimented to allow reconstruction with $C = 0 , 0 , 0$ .
+
+# D GRIPPER
+
+To evaluate the potential application of our model to robotic tasks, we constructed a robot endeffector simulation environment based on Bullet Real-Time Physics Simulation2. The environment consisted of an end-effector, two balls (red and blue), and two baskets (red and blue) in a bi-dimensional space. The end-effector grabs one of the balls and places it into the basket with the same color as the ball. The data factors include movement ”habits.” For example, the end-effector could reach the targeted ball by either directly moving toward the ball obliquely or first moving above of the ball and then lowering itself until it reached the ball (perpendicular movement). The end-effector could drop the ball from far above the basket or place it gently in the basket. Each factor could affect different lengths among the data; e.g., ”the plan to place the ball in the basket” factor affects different lengths per datum since the initial distance from the target ball to the corresponding basket may differ. This means that the time required to reach the ball should differ. Note that input is a value such as gripper’s position, not an image. See Appendix B for a more detailed explanation.
+
+FAVAE learned the disentangled factors of the Gripper dataset. Example visualized latent traversals are shown in Fig. 8. The traversed latent variable determined which factors were disentangled, such as the initial position of blue ball and basket (b), the targeted ball (red or blue) (c), plan to reach the ball (move obliquely or move perpendicularly) (d), and plan to place the ball in the basket (drop the ball or placing it in the basket gently) (e). These results indicate that our model can learn generative factors such as disentangled latent variables for robotic tasks, even though the length of the data affected by each factor may differ for each datum. We used FAVAE in a two-ladder network with 12 and 8 latent variables with $\beta { = } 1 0 0 0$ .
+
+![](images/17788be1c6940dc3b590d5a26edc1c87b24e879399043d17fcdde6c0e89b4abd.jpg)  
+Figure 9: Visualization of factors of 2D Wavy Reaching training data.
+
+# E DATASET DETAILS
+
+# E.1 2D WAVY REACHING
+
+Five different factors generate each trajectory: (9a) Shape of trajectory from start position to intermediate point. The first shape is created using a cycloid function while the other one adds a sine curve on the cycloid. (9b) Shape of trajectory from intermediate point to goal; shape is the same as for (9a). (9c) Location of goal; five different goal locations were used. (9d) Degree of curvature from start to intermediate point. (9e) Degree of curvature from intermediate point to goal.
+
+# E.2 SPRITES
+
+Sprits dataset is video data of video game ”sprites”. It used was used in (Li & Mandt, 2018) for confirming the extraction of disentangled representation between static factors and dynamic factors. The dataset consists of sequences with $T = 8$ frames of dimension $3 \times 6 4 \times 6 4$ . We use factors and motion of Sprites is shown in Table 4 and Fig. 10.
+
+# E.3 GRIPPER
+
+We implemented the end-effector only rather than the entire robot arm since controlling the robot arm during the picking task is easily computable by calculating the inverse kinematics and inverse dynamics. Gripper is a 12 dimensional data set: [joint $\mathbf { X }$ position, joint y position, finger1 joint position(angle), finger2 joint position(angle), box1 $\mathbf { X }$ position, box1 y position, $\mathbf { b } 0 \mathbf { X } 2 \textbf { X }$ position, box2 y position, ball1 x position, ball1 y position, ball2 $\mathbf { X }$ position, ball2 y position]. Eight factors are represented in this dataset: 1) color of ball to pick up, 2) initial location of red ball, 3) initial location of blue ball, 4) initial location of blue basket, 5) initial location of red basket, 6) plan for using end effector to move to ball to pick it up [first, moving horizontally to the $\mathbf { X }$ -location of ball and then descending horizontally to the y-location of ball, like the movement of the doll drawing machine (perpendicular motion); second, moving straight to the location of he ball to pick it up (oblique motion)], 7) plan for using end effector to move to point above basket after picking up ball (perpendicular or oblique motion), 8) plan for placing ball in basket (by dropping ball or descending to basket and gently placing ball in basket). Among the four initial positions, the two balls and two baskets are placed randomly. The movement of the robot is hard-coded on the basis of a goalposition-based script. To reduce collision detection errors during the simulation, we used a large physical model (end effector size $1 \mathrm { m } \dot { }$ ), which does not affect the overall validity. The length of the data-point sequence was 400.
+
+![](images/752a1a74fa5e7ea0ccaa20aa73aa19dc22fdff677762c35557984ea8df971371.jpg)  
+Figure 10: Motions in Sprites dataset. Row 1/2/3 represent walk left/forward/right, row 4/5/6 represent spellcast left/forward/right, and row 7/8/9 represent slash left/forward/right’ respectfully.
+
+Table 4: Factors in dataset   
+
+<table><tr><td>Dataset</td><td>Degree of factors</td><td>Ground truth factors</td></tr><tr><td rowspan="3">2D Reaching</td><td>2</td><td>Goal positon</td></tr><tr><td>2</td><td>Curved inward /outward</td></tr><tr><td>5</td><td>Degree of curvature</td></tr><tr><td rowspan="5">2D wavy Reaching</td><td>4</td><td>1st trajectory shape</td></tr><tr><td>4</td><td>2nd trajectory shape</td></tr><tr><td>5</td><td>Goal position</td></tr><tr><td>2</td><td>1st trajectory degree of curvature</td></tr><tr><td>2</td><td>2nd trajectory degree of curvature</td></tr><tr><td rowspan="6">Sprites Dataset</td><td>3</td><td>Motions (walk, spellcast or slash)</td></tr><tr><td>2</td><td>Body color (light or dark2)</td></tr><tr><td>2</td><td>Shirts color (brown or teal)</td></tr><tr><td>2</td><td>Hair color (green or pink)</td></tr><tr><td>2</td><td>Pants color (red or teal)</td></tr><tr><td>3</td><td>Direction of character (left,foward, right)</td></tr><tr><td rowspan="8">Gripper</td><td>2</td><td>Targeted ball to grasp (red or blue)</td></tr><tr><td>4</td><td>Red ball initial position</td></tr><tr><td>4</td><td>Blue ball initial position</td></tr><tr><td>4</td><td>Red basket position</td></tr><tr><td>4</td><td>Blue basket position</td></tr><tr><td>2</td><td>Plan to reach target ball</td></tr><tr><td>2</td><td>Plan to reach corresponding basket</td></tr><tr><td>2</td><td>Plan to place ball in basket</td></tr></table>
+
+# F ARCHITECTURE DETAILS
+
+# F.1 FAVAE
+
+We show the hyper parameters in Table 1, Fig. 7 and Table 3. The parameters searched are $\beta = 0$ to 10000 (the best $\beta = 1$ and $C = [ 0 , 0 , 0 ]$ and the best $\beta = 1 0 0 0 0 0$ and $C = [ 2 0 , 1 , 5 ]$ in 2D Reaching and 2D wavy Reaching in Table 1) The best $C$ was decided by the value of KL divergence loss when we experimented to allow reconstruction with $C = 0 , 0 , 0$ . In Table 3, we search the parameters from $\beta = 0$ to 60 (the best $\beta = 0 . 4$ and 4 in 2D Reaching and 2D wavy Reaching) and the best $C = [ c = 0 . 0 4 , 0 . 0 2 , 0 . 0 2 ]$ and $[ 0 . 0 3 , 0 , 0 , 0 6 ]$ in 2D Reaching and 2D wavy Reaching. In Fig. 7.3, we add the pre-encoder and post-decoder for encoding and decoding of images at each time step. The pre-encoding and post-decoding architecture is same as ”encode frames” and ”decode frames” in the code implementing Disentangled-Sequential-Autoencoder3 with conv $\mathrm { d i m } = 4 8$ . We search the parameters from $\beta = 0$ to $\beta = 1 0 0$ and the best $\beta = 2 0$ with $C = [ 0 . 3 , 0 . 1 7 , 0 . 0 6$ ].
+
+# F.2 FHVAE
+
+For the FHVAE experiments, we used the code from the implementation at Github4. We used recurrent setting with LSTM encoders and decoder with unit size 256 and batch size 80. Dimensions of latent variable z1 and $_ { z 2 }$ are 7 for each. We used Adam optimizer with learning rate 0.001. We applied goal position factors (2 classes for 2D Reaching and 5 classes for 2D wavy Reaching) as label inputs. We varied $\alpha$ from 1.0 to 30.0, and the best $\alpha$ was 20.0 for 2D Reaching and 1.0 for 2D wavy reaching.

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+        "text": "FAVAE: SEQUENCE DISENTANGLEMENT USING INFORMATION BOTTLENECK PRINCIPLE ",
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+        "text": "We propose the factorized action variational autoencoder (FAVAE), a state-of-theart generative model for learning disentangled and interpretable representations from sequential data via the information bottleneck without supervision. The purpose of disentangled representation learning is to obtain interpretable and transferable representations from data. We focused on the disentangled representation of sequential data since there is a wide range of potential applications if disentanglement representation is extended to sequential data such as video, speech, and stock price data. Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time. Previous works succeed in disentangling static factors and dynamic factors by explicit modeling the priors of latent variables to distinguish between static and dynamic factors. However, this model cannot disentangle representations between dynamic factors, such as disentangling ”picking” and ”throwing” in robotic tasks. In this paper, we propose new model that can disentangle multiple dynamic factors. Since our method does not require modeling priors, it is capable of disentangling ”between” dynamic factors. In experiments, we show that FAVAE can extract the disentangled dynamic factors. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Representation learning is one of the most fundamental problems in machine learning. A real world data distribution can be regarded as a low-dimensional manifold in a high-dimensional space (Bengio et al., 2013). Generative models in deep learning, such as the variational autoencoder (VAE) (Kingma & Welling, 2013) and the generative adversarial network (GAN) (Goodfellow et al., 2014), are able to learn low-dimensional manifold representation (factor) as a latent variable. The factors are fundamental components such as position, color, and degree of smiling in an image of a human face (Liu et al., 2015). Disentangled representation is defined as a single factor being represented by a single latent variable (Bengio et al., 2013). Thus, if in a model of learned disentangled representation, shifting one latent variable while leaving the others fixed generates data showing that only the corresponding factor was changed. This is called latent traversals (a good demonstration of which was given by Higgins et al. $( 2 0 1 6 \\mathrm { a } ) ^ { 1 }$ ). There are two advantages of disentangled representation. First, latent variables are interpretable. Second, the disentangled representation is generalizable and robust against adversarial attacks (Alemi et al., 2016). ",
+        "bbox": [
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+        "type": "text",
+        "text": "We focus on the disentangled representation learning of sequential data. Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time. With disentangled representation learning from sequential data, we should be able to extract dynamic factors that cannot be extracted by disentangled representation learning models for non-sequential data such as $\\beta$ -VAE (Higgins et al., 2016a;b) and InfoGAN (Chen et al., 2016). The concept of disentangled representation learning for sequential data is illustrated in Fig. 1. Consider that the pseudo-dataset of the movement of a submarine has a dynamic factor: the trajectory shape. The disentangled representation learning model for sequential data can extract this shape. On the other hand, since the disentangled representation learning model for non-sequential data does not consider the sequence of data, it merely extracts the $\\mathbf { X }$ -position and y-position. ",
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+    {
+        "type": "text",
+        "text": "(c) FAVAE accepts sequential data. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d791a4a43bd7873e6856996a67db2eab578a5e68ad71e5cdec1fe65206843e5f.jpg",
+        "image_caption": [
+            "(d) Latent traversal of FAVAE. ",
+            "Figure 1: Illustration of how FAVAE differs from $\\beta$ -VAE. $\\beta$ -VAE does not accept data sequentially; it cannot differentiate data points from different trajectories or sequences of data points. FAVAE considers a sequence of data points, taking all data points in a trajectory as one datum. For example, for a pseudo-dataset representing the trajectory of a submarine (1a,1c), $\\beta$ -VAE accepts 11 different positions of the submarine as non-sequential data while FAVAE accepts three different trajectories of the submarine as sequential data. Therefore, the latent variable in $\\beta$ -VAE learns only the coordinates of the submarine, and the latent traversal shows the change in the submarines position. On the other hand, FAVAE learns the factor that controls the trajectory of the submarine, so the latent traversal shows the change in the submarines trajectory. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "There is a wide range of potential applications if we extend disentanglement representation to sequential data such as speech, video, and stock market data. For example, disentangled representation learning for stock price data can extract the fundamental trend of a given stock price. Another application is the reduction of action space in reinforcement learning. Extracting dynamic factors would enable the generation of macro-actions (Durugkar et al., 2016), which are sets of sequential actions that represent the fundamental factors of the actions. Thus, disentangled representation learning for sequential data opens the door to new areas of research. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Very recent related work (Hsu et al., 2017; Li & Mandt, 2018) separated factors of sequential data into dynamic and static factors. The factorized hierarchical variational autoencoder (FHVAE) (Hsu et al., 2017) is based on a graphical model using latent variables with different time dependencies. By maximizing the variational lower bound of the graphical model, the FHVAE separates the different time dependent factors such as the dynamic and static factors. The VAE architecture developed by Li & Mandt (2018) is the same as the FHVAE in terms of the time dependencies of the latent variables. Since these models require different time dependencies for the latent variables, these approaches cannot be used disentangle variables with the same time dependency factor. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We address this problem by taking a different approach. First, we analyze the root cause of disentanglement from the perspective of information theory. As a result, the term causing disentanglement is derived from a more fundamental rule: reduce the mutual dependence between the input and output of an encoder while keeping the reconstruction of the data. This is called the information bottleneck (IB) principle. We naturally extend this principle to sequential data from the relationship between $x$ and $z$ to $x _ { t : T }$ and $z$ . This enables the separation of multiple dynamic factors as a consequence of information compression. It is difficult to learn a disentangled representation of sequential data since not only the feature space but also the time space should be compressed. We created the factorized action variational autoencoder (FAVAE) in which we implemented the concept of information capacity to stabilize learning and a ladder network to learn a disentangled representation in accordance with the level of data abstraction. Since our model is a more general model without the restriction of a graphical model design to distinguish between static and dynamic factors, it can separate dependency factors occurring at the same time. Moreover, it can separate factors into dynamic and static factors. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "2 DISENTANGLEMENT FOR NON-SEQUENTIAL DATA ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$\\beta$ -VAE (Higgins et al., 2016a;b) is a commonly used method for learning disentangled representations based on the VAE framework (Kingma $\\&$ Welling, 2013) for a generative model. The VAE can estimate the probability density from data $\\mathbf { X }$ . The objective function of the VAE maximizes the evidence lower bound (ELBO) of $\\log p \\left( x \\right)$ as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cda7dee6b206db38678fa03c1374d196258cbcfdbcbd06437151a956f7982c61.jpg",
+        "text": "$$\n\\log p \\left( x \\right) = \\mathcal { L } _ { \\mathrm { V A E } } + \\underbrace { D _ { \\mathrm { K L } } \\left( q \\left( z \\vert x \\right) \\vert \\vert p \\left( z \\vert x \\right) \\right) } _ { \\ge 0 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $z$ is latent variable, $D _ { \\mathrm { K L } }$ is the Kullback-Leibler divergence, and $q \\left( z | x \\right)$ is an approximated distribution of $p \\left( z | x \\right) . \\ D _ { \\mathrm { K L } } \\left( q \\left( z | x \\right) | | p \\left( z | x \\right) \\right)$ reduces to zero as the ELBO $\\mathcal { L } _ { \\mathrm { V A E } }$ increases; thus, $q \\left( z | x \\right)$ learns a good approximation of $p \\left( z | x \\right)$ . The ELBO is defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffde22627fb42af001dee21030c38dc44f42cd36b862d9f86e8c3fdf4b14c66f.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { V A E } } \\equiv E _ { q ( z | x ) } \\left[ \\log p \\left( x | z \\right) \\right] - D _ { \\mathrm { K L } } \\left( q \\left( z | x \\right) | | p \\left( z \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the first term, $E _ { q ( \\boldsymbol { z } | \\boldsymbol { x } ) } \\left[ \\log p \\left( \\boldsymbol { x } | \\boldsymbol { z } \\right) \\right]$ , is a reconstruction term used to reconstruct $x$ , and the second term $D _ { \\mathrm { K L } } \\left( q \\left( z | x \\right) \\^ { } | | \\dot { p } \\left( z \\right) \\right)$ is a regularization term used to regularize posterior $q \\left( z | x \\right)$ . Encoder $q \\left( z | x \\right)$ and decoder $p \\left( x | z \\right)$ are learned in the VAE. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next we will explain how $\\beta$ -VAE extracts disentangled representations from unlabeled data. $\\beta$ -VAE is an extension of the coefficient $\\beta > 1$ of the regularization term $D _ { \\mathrm { K L } } \\left( q \\left( z | x \\right) | | p \\left( z \\right) \\right)$ in the VAE. The objective function of $\\beta$ -VAE is ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9e3b83b62ec195121c7bae9267aa5d615f9c7de5412985d60a6e253f94d0590e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\beta - \\mathrm { V A E } } = E _ { q ( z | x ) } \\left[ \\log p \\left( x | z \\right) \\right] - \\beta D _ { \\mathrm { K L } } \\left( q \\left( z | x \\right) | | p \\left( z \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta > 1$ and $p \\left( z \\right) = \\mathcal { N } \\left( 0 , 1 \\right)$ . $\\beta$ -VAE promotes disentangled representation learning via the Kullback-Leibler divergence term. As $\\beta$ increases, the latent variable $q \\left( z | x \\right)$ approaches the prior $p \\left( z \\right)$ ; therefore, each $z _ { i }$ is pressured to learn the probability distribution of $\\mathcal { N } ( 0 , 1 )$ . However, if all latent variables $z _ { i }$ become $\\mathcal { N } ( 0 , 1 )$ , the model cannot reconstruct $x$ . As a result, as long as $z$ reconstructs $x$ , $\\beta$ -VAE reduces the information of $z$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARY: ORIGIN OF DISENTANGLEMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To clarify the origin of disentanglement, we will explain the regularization term. The regularization term has been decomposed into three terms (Chen et al., 2018; Kim & Mnih, 2018; Hoffman & Johnson, 2016): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35f8fde0612697ce1d30baf15256ff8382f7d72813f4a8647df3ad8670277a3b.jpg",
+        "text": "$$\nE _ { p \\left( x \\right) } \\left[ D _ { \\mathrm { K L } } \\left( q \\left( \\boldsymbol { z } \\left| \\boldsymbol { x } \\right) \\right| \\left| p \\left( \\boldsymbol { z } \\right) \\right) \\right] = I \\left( \\boldsymbol { x } ; \\boldsymbol { z } \\right) + D _ { \\mathrm { K L } } \\left( q \\left( \\boldsymbol { z } \\right) \\right| \\Big | \\prod _ { j } q \\left( z _ { j } \\right) \\right) + \\sum _ { j } D _ { \\mathrm { K L } } \\left( q \\left( \\boldsymbol { z } _ { j } \\right) \\right| \\left| p \\left( z _ { j } \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { j }$ denotes the $j$ -th dimension of the latent variable. The second term, which is called ”total correlation” in information theory, quantifies the redundancy or dependency among a set of $\\mathbf { n }$ random variables (Watanabe, 1960). $\\beta$ -TCVAE has been experimentally shown to reduce the total correlation causing disentanglement (Chen et al., 2018). The third term indirectly causes disentanglement by bringing $q \\left( z | x \\right)$ close to the independent standard normal distribution $p \\left( z \\right)$ . The first term is mutual information between the data variable and latent variable based on the empirical data distribution. Minimizing the regularization term causes disentanglement but disturbs reconstruction via the first term in Eq. (4). The shift $C$ scheme was proposed (Burgess et al., 2018) as a means to solve this conflict: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/85c31bf151a5831a9eb46dfb1113aabac7f5a719e95dc7b3393d89d2ac840cbd.jpg",
+        "text": "$$\n- { \\cal E } _ { q _ { \\phi } ( z | x ) } [ \\log p _ { \\theta } ( x | z ) ] + \\beta | { \\cal D } _ { \\mathrm { K L } } ( q ( z | x ) | | p ( z ) ) - C | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where constant shift $C$ , which is called ”information capacity,” linearly increases during training. This shift $C$ can be understood from the point of view of an information bottleneck (Tishby et al., 2000). The VAE can be derived by maximizing the ELBO, but $\\beta$ -VAE can no longer be interpreted as an ELBO once this scheme has been applied. The objective function of $\\beta$ -VAE is derived from the information bottleneck (Alemi et al., 2016; Achille & Soatto, 2018; Tishby et al., 2000; Chechik et al., 2005). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f70e6ec06cd4d6895a981540a88f31a7547b288e939c06da292f2bd04a3a558.jpg",
+        "text": "$$\n\\operatorname* { m a x } I \\left( z ; x \\right) \\qquad \\mathrm { s . t . } ~ | I \\left( \\hat { x } ; z \\right) - C | = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $C$ is the information capacity and $\\hat { x }$ is the empirical distribution. Solving this equation by using Lagrange multipliers drives the objective function of $\\beta$ -VAE Eq. (5) with $\\beta$ as the Lagrange multiplier (details in Appendix $\\mathbf { B }$ of (Alemi et al., 2016)). In Eq. (5), information capacity $C$ prevents $I \\left( \\hat { x } , z \\right)$ from becoming zero. In the information bottleneck literature, $y$ typically stands for a classification task; however, the formulation can be related to the autoencoding objective (Alemi et al., 2016). Therefore, the objective function of $\\beta$ -VAE can be understood using the information bottleneck principle. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 PROPOSED MODEL: DISENTANGLEMENT FOR SEQUENTIAL DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed FAVAE model learns disentangled and interpretable representations from sequential data without supervision. We consider sequential data $x _ { 1 : T } \\equiv \\{ x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { T } \\}$ generated from a latent variable model, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/31b8bfcfb10c128cc461f8198bdf34f291e2601f3b38c52138544b3491f92f09.jpg",
+        "text": "$$\np \\left( x _ { 1 : T } \\right) = \\int p \\left( x _ { 1 : T } | z \\right) p \\left( z \\right) d z .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For sequential data, we replace $x$ with $\\scriptstyle ( x _ { 1 : T } )$ in Eq. 5. The objective function of the FAVAE model is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/89ce45d6f4119dd11deb5d928ec99eb67797aa8848c3d8154ed42b8e3c789bee.jpg",
+        "text": "$$\n- { \\cal E } _ { q _ { \\phi } ( z | ( x _ { 1 : T } ) _ { i } ) } [ \\log p _ { \\theta } ( x _ { 1 : T } | z ) ] + \\beta | { \\cal D } _ { \\mathrm { K L } } ( q ( z | ( x _ { 1 : T } ) _ { i } ) | | p ( z ) ) - C | ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $p \\left( z \\right) = \\mathcal { N } \\left( 0 , 1 \\right)$ . The variational recurrent neural network (Chung et al., 2015) and stochastic recurrent neural network (Fraccaro et al., 2016) extend the VAE model to a recurrent framework. The priors of both networks are dependent on time. The time dependent prior experimentally improves the ELBO. In contrast, the prior of our model is independent of time like those of the stochastic recurrent network (Bayer $\\&$ Osendorfer, 2014) and the Deep Recurrent Attentive Writer (DRAW) neural network architecture (Gregor et al., 2015); this is because FAVAE is disentangled representation learning rather than density estimation. For better understanding, consider FAVAE from the perspective of IB. As with $\\beta$ -VAE, FAVAE can be understood from the information bottleneck principle. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/059a9ba82320a70c4553524d9c68b9c3dbb7612d1d30de3b0100e5a4856d28f9.jpg",
+        "text": "$$\n\\operatorname* { m a x } I \\left( z ; x _ { 1 : T } \\right) \\qquad \\mathrm { s . t . } ~ | I \\left( \\hat { x } _ { 1 : T } ; z \\right) - C | = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { x } _ { 1 : T }$ follows an empirical distribution. These principles make the representation of $z$ compact while reconstruction of the sequential data is represented by $x _ { 1 : T }$ (see Appendix A). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 LADDER NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An important extension to FAVAE is a hierarchical representation scheme inspired by the VLAE (Zhao et al., 2017). Encoder $q \\left( z | x _ { 1 : T } \\right)$ within a ladder network is defined as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6dff83c019972c5faadeb1ea1aaff210e3a450575b7397fc22baeaa032d0bdc2.jpg",
+        "text": "$$\n\\begin{array} { l } { { h _ { l } = f _ { l } \\left( h _ { l - 1 } \\right) , } } \\\\ { { z _ { l } \\sim \\mathcal { N } \\left( \\mu _ { l } \\left( h _ { l } \\right) , \\sigma _ { l } \\left( h _ { l } \\right) \\right) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $l$ is a layer index, $h _ { 0 } \\equiv x _ { 1 : T }$ , and $f$ is a time convolution network, which is explained in the next section. Decoder $p \\left( x _ { 1 : T } | z \\right)$ within the ladder network is defined as ",
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+    {
+        "type": "image",
+        "img_path": "images/b346278abc264002383add7d500381c122f5b7482bf1b7ccbe274fd1a4fe979e.jpg",
+        "image_caption": [
+            "Figure 2: FAVAE architecture. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "equation",
+        "img_path": "images/9ad86558f051f348bb59f93acc082fdb283494e662b378c6ecb840f40300b0a6.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { z } _ { L } = g _ { l } \\left( z _ { L } \\right) } \\\\ { \\tilde { z } _ { l } = g _ { l } \\left( \\tilde { z } _ { l + 1 } + \\mathrm { g a t e } \\left( z _ { l } \\right) \\right) , } \\\\ { x _ { 1 : T } \\sim r \\left( x _ { 1 : T } , \\tilde { z } _ { 0 } \\right) ~ } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { l }$ is the time deconvolution network with $l = 1 , \\cdots , L - 1$ , and $r$ is a distribution family parameterized by $g _ { 0 } \\left( \\tilde { z } _ { 0 } \\right)$ . The gate computes the Hadamard product of its learnable parameter and input tensor. We set $r$ as a fixed-variance factored Gaussian distribution with the mean given by $\\mu _ { t : T } = g _ { 0 } \\left( \\tilde { z } _ { 0 } \\right)$ . Fig. (2) shows the architecture of the proposed model. The difference between each ladder network in the model is the number of convolution networks through which data passes. The abstract expressions should differ between ladders since the time convolution layer abstracts sequential data. Without the ladder network, the proposed method can disentangle only the representations at the same level of abstraction; with the ladder network, it can disentangle representations at different levels of abstraction. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/a788ce73cb8b7bfaad254b437844807e035d2ae99c0c0ceda9ad26e554da2d1a.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of latent traversal of $\\beta$ -VAE and FAVAE. On one sampled trajectory (red), each latent variable is traversed and purple and/or blue points are generated. The color corresponds to the value of the traversed latent variable. 3a represents all data trajectories of 2D reaching. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 HOW TO ENCODE AND DECODE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are several mainstream neural network models designed for sequential data, such as the long short-term memory model (Hochreiter & Schmidhuber, 1997), the gated recurrent unit model (Chung et al., 2014), and the quasi-recurrent neural network QRNN (Bradbury et al., 2016). However, the VLAE has a hierarchical structure created by abstracting a convolutional neural network, so it is simple to add the time convolution of the QRNN to our model. The input data are $x _ { t , i }$ , where $t$ is the time index and $i$ is the dimension of the feature vector index. The time convolution considers the dimensions of feature vector $j$ as a convolution channel and performs convolution in the time direction: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/075b77a5cf32d7c3c72170524777d1f38636eca681e189760aa2c82798517b12.jpg",
+        "text": "$$\nz _ { t j } = \\sum _ { p , i } x _ { t - p , i } h _ { p i j } + b _ { j } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $j$ is the channel index. The proposed FAVAE model has a network similar to the VAE one regarding time convolution and a loss function similar to the $\\beta$ -VAE one (Eq. (8)). We used the batch normalization (Hinton et al., 2012) and ReLU as activation functions though other variations are possible. For example, 1d convolutional neural networks use a filter size of 3 and a stride of 2 and do not use a pooling layer. ",
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+    {
+        "type": "text",
+        "text": "5 MEASURING DISENTANGLEMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While latent traversals are useful for checking the success or failure of disentanglement, quantification of the disentanglement is required for reliably evaluating the model. Various disentanglement quantification methods have been reported (Eastwood & Williams, 2018; Chen et al., 2018; Kim & Mnih, 2018; Higgins et al., 2016b;a), but there is no standard method. We use the mutual information gap (MIG) (Chen et al., 2018) as the metric for disentanglement. The basic idea of MIG is measuring the mutual information between latent variables $z _ { j }$ and a ground truth factor $v _ { k }$ . Higher mutual information means that $z _ { j }$ contains more information regarding $v _ { k }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\mathrm { M I G } \\equiv \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { H \\left( v _ { k } \\right) } \\left( I \\left( z _ { j ^ { ( k ) } } ; v _ { k } \\right) - \\operatorname* { m a x } _ { j \\neq j ^ { ( k ) } } I \\left( z _ { j } ; v _ { k } \\right) \\right) } \\end{array}$ , where ${ j ^ { ( k ) } \\equiv \\arg \\operatorname* { m a x } _ { j } I \\left( z _ { j } ; v _ { k } \\right) }$ and $H \\left( v _ { k } \\right)$ is entropy for normalization. In our evaluation we experimentally measure disentanglement with MIG. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/386f9bde6da70d320a6ac163a4878508117069dd93b82b172e7b9bfc2c27be31.jpg",
+        "table_caption": [
+            "Table 1: Disentanglement scores (MIG, and Reconstruction loss) with standard deviations by 10 seeds for the different model. Best results are shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">2D Reaching</td><td colspan=\"2\">2D wavy Reaching</td></tr><tr><td>MIG</td><td>Reconstruction</td><td>MIG</td><td>Reconstruction</td></tr><tr><td>Time Convolution AE</td><td>0.21(27)</td><td>0.18(13)</td><td>0.04(3)</td><td>0.06(1)</td></tr><tr><td>FAVAE (without ladder and C)</td><td>0.34(21)</td><td>0.62(115)</td><td>0.22(14)</td><td>23.6(279)</td></tr><tr><td>FAVAE (with ladder and without C)</td><td>0.41(13)</td><td>0.03(5)</td><td>0.26(7)</td><td>0.08(2)</td></tr><tr><td>FAVAE (without ladder and with C)</td><td>0.34(21)</td><td>0.62(115)</td><td>0.11(10)</td><td>1.47(73)</td></tr><tr><td>FAVAE (with ladder and C)</td><td>0.41(13)</td><td>0.03(5)</td><td>0.34(28)</td><td>1.81(170)</td></tr></table>",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Several recently reported models (Hsu et al., 2017; Li & Mandt, 2018) graphically disentangle static and dynamic factors in sequential data such as speech data and video data (Garofolo et al., 1993; Pearce & Picone, 2002). In contrast, our model performs disentanglement by using a loss function (see Eq. 8). The advantage of the graphical models is that they can control the interpretable factors by controlling the priors time dependency. Since dynamic factors have the same time dependency, these models cannot disentangle dynamic factors. A loss function model can disentangle sets of dynamic factors as well as disentangle static and dynamic factors. ",
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+    {
+        "type": "text",
+        "text": "7 EVALUATION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We evaluated our model experimentally using three sequential datasets: 2D Reaching, 2D Wavy Reaching, and Gripper. We used a batch size of 128 and the Adam (Kingma & Ba, 2014) optimizer with a learning rate of $1 0 ^ { - 3 }$ . ",
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+    {
+        "type": "text",
+        "text": "7.1 2D REACHING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To determine the differences between FAVAE and $\\beta$ -VAE, we used a bi-dimensional space reaching dataset. Starting from point $( 0 , 0 )$ , the point travels to goal position $( - 0 . 1 , + 1 )$ or $( + 0 . 1 , + 1 )$ . There are ten possible trajectories to each goal; five are curved inward, and the other five are curved outward. The degree of curvature for all five trajectories is different. The number of factor combinations was thus 20 (2x2x5). The trajectory length was 1000, so the size of one trajectory was $[ 1 0 0 0 \\mathrm { x } 2 ]$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compared the performances of $\\beta$ -VAE and FAVAE trained on the 2D Reaching dataset. The results of latent traversal are transforming one dimension of latent variable z into another value and reconstructing something from the traversed latent variables. $\\beta$ -VAE, which is only able to learn from every point of a trajectory separately, encodes data points into latent variables that are parallel to the $\\mathbf { X }$ and y axes (3b, 3c). In contrast, FAVAE learns through one entire trajectory and can encode disentangled representation effectively so that feasible trajectories are generated from traversed latent variables (3d, 3e). ",
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+    },
+    {
+        "type": "text",
+        "text": "7.2 2D WAVY REACHING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To confirm the effect of disentanglement through information bottleneck, we evaluated the validity of our model under more complex factors by adding more factors to the 2D Reaching dataset. Five factors in total generated data compared to the three factors that generate data in 2D Reaching. This modified dataset differed in that four out of the five factors affect only part of the trajectory: two of them affect the first half, and the other two affect the second half. This means that the model should be able to focus on a certain part of the whole trajectory and be able to extract factors related to that part. A detailed explanation of these factors is given in Appendix B. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compared various models on the basis of MIG to demonstrate the validity of our proposed model in comparison of a time convolution AE in which a loss function is used only for the autoencoder ( $\\beta = 0$ ), FAVAE without the ladder network and information capacity $C$ , and FAVAE with the ladder network and information capacity $C$ . As shown in Table 1, FAVAE with the ladder network and C had the highest MIG scores for 2D Reaching and 2D Wavy Reaching. This indicates that this model learned a disentangled representation best. Note that for 2D Reaching, the best value for $C$ was small, meaning that there was little effect from adding $C$ (since the dataset was simple, this task can be solved even if the amount of information of $\\mathbf { Z }$ is small). ",
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+    {
+        "type": "image",
+        "img_path": "images/f01901b6971fd5a8db90e6cf3c95eeb6a623b75861e2421a6a2b3dbf4fb4fc1a.jpg",
+        "image_caption": [
+            "Figure 4: MIG score and reconstruction loss for different values of $\\beta$ . Blue line represents results with information capacity $C$ greater than zero; red line represents results with $C$ set to e zero. Note that $\\mathbf { X }$ axis is plotted in log scale. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b57addf8cf5aa9b7ee9e51ea1c882d13a1370cfeaa4554ced84a275746fbca89.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of latent traversal for 2D Wavy Reaching. Each plot was decoded by traversing one latent variable; different colors represent trajectories generated from different values of same latent variable, z. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4. We evaluated reconstruction loss and the MIG score for various values of $\\beta$ using three ladder networks (Fig. 6) with a different value of $C$ for each ladder: $\\begin{array} { r l } { C } & { { } = } \\end{array}$ [LowerLadder, MiddleLadder, HigherLadder]. One setting used was $C = [ 0 , 0 , 0 ]$ , meaning that $C$ was not used; another setting was $C = [ 2 0 , 1 , 5 ]$ , meaning that $C$ was adjusted on the basis of values of KL-Divergence for $\\beta = 1$ and $\\dot { C } = [ 0 , \\dot { 0 } , 0 ]$ . When $C$ was not used, the model could not reconstruct data when $\\beta$ was high; thus, disentangled representation was not learned well when $\\beta$ was high. When $C$ was used, the MIG score increased with $\\beta$ while reconstruction loss was suppressed. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The latent traversal results for 2D Wavy Reaching are plotted in Fig. 5. Even though not all learned representations are perfectly disentangled, the visualization shows that all five generation factors were learned from five latent variables; the other latent variables did not learn any meaningful factors, indicating that the factors could be expressed as a combination of five ”active” latent variables. We tested our model for $\\beta = 3 0 0$ . ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/e02bc66dbc528f5a4577bc3ed31795379f1949b47c1b3c825077681ee325c131.jpg",
+        "image_caption": [
+            "Figure 6: Evaluation of ladders (left); structure used for each experiment (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4acf414e8b61f125fbb858bb241498545bc1deb9515d1fce021f72be4fcdf42b.jpg",
+        "image_caption": [
+            "Figure 7: Visualization of latent traversal of FAVAE. The horizontal axis represents sequence and the vertical axis represents the difference of $z$ values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The use of a ladder network in our model improved disentangled representation learning and minimized the reconstruction loss. The graph in Fig. 6 shows the MIG scores for networks with different numbers of ladders. The error bars represent the standard deviation for ten repetitive trials. Using all three ladders resulted in the minimum reconstruction loss with the highest MIG score (green curve) except for ”Higher Ladder One”. ”Higher Ladder One” has a large reconstruction error. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7.3 SPRITES DATASET. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            506,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the effectiveness of the video dataset, we trained our model with the Sprites dataset which is used in (Li & Mandt, 2018). This dataset has sequential length $= 8$ RGB video data with $3 \\times 6 4 \\times 6 4$ . This data set consists of static factor and dynamic factor. We note that the motion is not created with the combination of dynamic factors, and each motion exists individually (detail is E.2). Table 4 show the factors used in our experiment. We executed disentangled representation learning by using the FAVAE model with $\\beta = 2 0$ , $C = [ 0 . 3 , 0 . 1 7 , 0 . 0 6 ]$ and network architecture used for this training is explained in Section F.1. Fig. 7 shows the results of latent traversal, and we chose two $z$ values to change from $z = 3$ to 3. Since this dataset is composed of discrete factors, we show two $z$ values at a time.The latent variables in the 1st ladder extract expressions of motion (4th $z$ in 1st ladder), pants color (5th $z$ in 1st ladder), direction of character (6th $z$ in 1st ladder) and the shirts color (7th $z$ in 1st ladder). The latent variables in the 2nd ladder extract expressions of the hair color (1st $z$ in 2nd ladder) and the body color (2nd $z$ in 2nd ladder). FHVAE can extract the disentangled representations between static factors and dynamic factor in high dimension dataset. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "8 SUMMARY AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            733,
+            468,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our factorized action variational autoencoder (FAVAVE) generative model learns disentangled and interpretable representations via the information bottleneck from sequential data. Evaluation using three sequential datasets demonstrated that it can learn disentangled representations. Future work includes extending the time convolution part to a sequence-to-sequence model (Sutskever et al., 2014) and applying the model to actions of reinforcement learning to reduce the pattern of actions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    {
+        "type": "text",
+        "text": "A INFORMATION BOTTLENECK PRINCIPLE FOR SEQUENTIAL DATA. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            751,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here the aim is to show the relationship between FAVAE and the information bottleneck for sequential data. Consider the information bottleneck object: ",
+        "bbox": [
+            173,
+            132,
+            821,
+            161
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4289348c3ae39ddd98d1430b2ddffc9351fd6447913cd86892a17f8920cfe65f.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\operatorname* { m a x } I \\left( z ; x _ { 1 : T } \\right) \\quad } & { { } \\mathrm { s . t . } \\ \\lvert I \\left( \\hat { x } _ { 1 : T } ; z \\right) - C \\rvert = 0 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            167,
+            655,
+            185
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "is expanded from Alemi et al. (2016) to sequential data. We need to distinguish between $\\hat { x } _ { 1 : T }$ and $x _ { 1 : T }$ , where $x _ { 1 : T }$ is the true distribution and $\\hat { x } _ { 1 : T }$ is the empirical distribution created by sampling from the true distribution. We maximize the mutual information of true data $x _ { 1 : T }$ and $z$ while constraining the information contained in the empirical data distribution. We do this by using Lagrange multiplier: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e47a178aa68863e353d5fa097cb6093b51d44bd8496cbca205ef693c9db2b35.jpg",
+        "text": "$$\n\\begin{array} { r } { I \\left( z ; x _ { 1 : T } \\right) + \\beta \\left| I \\left( \\hat { x } _ { 1 : T } ; z \\right) - C \\right| , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            281,
+            609,
+            299
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta$ is a constant. For the first term, ",
+        "bbox": [
+            174,
+            300,
+            436,
+            315
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac55c0232fc654ad37ad3d7ec307b63b9042072b3e84911ac00521daf4fe3a60.jpg",
+        "text": "$$\n\\begin{array} { l } { { I \\left( z ; x _ { 1 : T } \\right) = \\displaystyle \\iint p \\left( z ; x _ { 1 : T } \\right) \\log \\frac { p \\left( x _ { 1 : T } \\vert z \\right) } { p \\left( x _ { 1 : T } \\right) } d x _ { 1 : T } d z } } \\\\ { { \\displaystyle \\qquad = \\iint p \\left( z ; x _ { 1 : T } \\right) \\log p \\left( x _ { 1 : T } \\vert z \\right) d x _ { 1 : T } d z - \\displaystyle \\iint p \\left( x _ { 1 : T } ; z \\right) \\log p \\left( x _ { 1 : T } \\right) d x _ { 1 : T } d z } } \\\\ { { \\displaystyle \\qquad = \\displaystyle \\iint p \\left( x _ { 1 : T } \\right) p \\left( z \\vert x _ { 1 : T } \\right) \\log p \\left( x _ { 1 : T } \\vert z \\right) d x _ { 1 : T } d z + H \\left( x _ { 1 : T } \\right) } } \\\\ { { \\displaystyle \\qquad \\sim \\frac { 1 } { N } \\sum _ { i } \\left[ p \\left( z \\vert \\left( x _ { 1 : T } \\right) _ { i } \\right) \\log p \\left( \\left( x _ { 1 : T } \\right) _ { i } \\vert z \\right) \\right] + H \\left( x _ { 1 : T } \\right) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            339,
+            800,
+            478
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $H \\left( x _ { 1 : T } \\right)$ is entropy, which can be neglected in optimization. The last line is Monte Carlo approximation. For the second term, ",
+        "bbox": [
+            173,
+            483,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/917aaf5b4a4303a48cb9de5b01b39042401335431f87c65682478d4990d26456.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle I \\left( \\hat { x } _ { 1 : T } ; z \\right) = \\int p \\left( z \\vert \\hat { x } _ { 1 : T } \\right) p \\left( \\hat { x } _ { 1 : T } \\right) \\log \\frac { p \\left( z \\vert \\hat { x } _ { 1 : T } \\right) } { p \\left( z \\right) } d z d \\hat { x } _ { 1 : T } } } \\\\ { { \\displaystyle \\quad \\sim \\frac { 1 } { N } \\sum _ { j } p \\left( z \\vert \\left( x _ { 1 : T } \\right) _ { j } \\right) \\log \\frac { p \\left( z \\vert \\left( x _ { 1 : T } \\right) _ { j } \\right) } { p \\left( z \\right) } d z } } \\\\ { { \\displaystyle \\quad = \\frac { 1 } { N } \\sum _ { j } D _ { \\mathrm { K L } } \\left( p \\left( z \\vert \\left( x _ { 1 : T } \\right) _ { j } \\right) \\vert \\vert p \\left( z \\right) \\right) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            536,
+            692,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "As a result, ",
+        "bbox": [
+            173,
+            666,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e1de4705142c9831af87305d653f412fdaa1f6cc29c5ef42e1d7992ce6265d8.jpg",
+        "text": "$$\n\\begin{array} { r } { I \\left( z ; x _ { 1 : T } \\right) + \\beta \\left| I \\left( \\tilde { x } _ { 1 : T } ; z \\right) - C \\right| \\leq \\displaystyle \\frac { 1 } { N } \\sum _ { i } \\left[ p \\left( z \\right| \\left( x _ { 1 : T } \\right) _ { i } \\right) \\log p \\left( \\left( x _ { 1 : T } \\right) _ { i } \\left| z \\right) \\right] } \\\\ { + \\displaystyle \\frac { 1 } { N } \\sum _ { j } D _ { \\mathrm { K L } } \\left( p \\left( z \\right| \\left( x _ { 1 : T } \\right) _ { j } \\right) \\left| | p \\left( z \\right) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            705,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For convenience of calculation, we use $x _ { i }$ sampled from mini-batch data for both the reconstruction term and the regularization term. This is only an approximation. If the information bottleneck principle is followed completely, it is better to use different batch data for the reconstruction and regularization terms. ",
+        "bbox": [
+            178,
+            787,
+            821,
+            843
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B EXTRACTED FACTORS OF EACH LADDER IN FAVAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            863,
+            637,
+            881
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We expect the ladder network can disentangle representations at different levels of abstraction. In this section, We check the factor extracted in each ladder by using 2D Reaching and 2D Wavy. ",
+        "bbox": [
+            173,
+            895,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/68a0ea6bf49e8b0149ef54ba4cb9de2057949436c91871d08ae31b64db8fc8a2.jpg",
+        "table_caption": [
+            "Table 2: Counting the index of latent variable in each factor by 10 seeds. The detail of factor is in Table 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">1st ladder</td><td>2nd ladder</td><td>3rd ladder</td></tr><tr><td rowspan=\"3\">2D Reaching</td><td>factor 1</td><td>1</td><td>1</td><td>8</td></tr><tr><td>factor 2</td><td>10</td><td>0</td><td>0</td></tr><tr><td>factor 3</td><td>10</td><td>0</td><td>0</td></tr><tr><td rowspan=\"5\">2D wavy Reaching</td><td>factor 1</td><td>3</td><td>0</td><td>7</td></tr><tr><td>factor 2</td><td>8</td><td>0</td><td>2</td></tr><tr><td>factor 3</td><td>8</td><td>0</td><td>2</td></tr><tr><td>factor 4</td><td>9</td><td>1</td><td>0</td></tr><tr><td>factor 5</td><td>9</td><td>0</td><td>1</td></tr></table>",
+        "bbox": [
+            173,
+            140,
+            638,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/d46babf7f853d4b6ff50712c4306823d0f061de9951cc78b02e79a65610f0fd5.jpg",
+        "table_caption": [
+            "Table 3: Disentanglement scores (MIG, and Reconstruction loss) with standard deviations by 10 seeds for the different model in sequence length $= 1 0 0$ . Best results are shown in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">2D Reaching</td><td colspan=\"2\">2D wavy Reaching</td></tr><tr><td>MIG</td><td>Reconstruction</td><td>MIG</td><td>Reconstruction</td></tr><tr><td>FHVAE</td><td>0.62(17)</td><td>0.0003(5)</td><td>0.19(6)</td><td>0.051(36)</td></tr><tr><td>Time Convolution AE</td><td>0.06(4)</td><td>0.014(7)</td><td>0.26(6)</td><td>0.019(2)</td></tr><tr><td>FAVAE (without ladder and C)</td><td>0.52(6)</td><td>0.007(1)</td><td>0.28(2)</td><td>0.058(3)</td></tr><tr><td>FAVAE (without ladder with C)</td><td>0.47(4)</td><td>0.005(0)</td><td>0.26(1)</td><td>0.069(4)</td></tr><tr><td>FAVEA (with ladder without C)</td><td>0.34(11)</td><td>0.005(1)</td><td>0.31(8)</td><td>0.022(2)</td></tr><tr><td>FAVAE (with ladder and C)</td><td>0.35(7)</td><td>0.007(1)</td><td>0.40(7)</td><td>0.055(4)</td></tr></table>",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the counting index of latent variable with the highest mutual information in each ladder network. In Table 2, the rows represent factor and the columns represent the index of the ladder networks. The factor 1 (goal left / goal right) in 2D Reaching and the factor 1 (goal position) in 2D wavy Reaching were extracted to the most frequently in the latent variable in 3rd ladder. Since the latent variables have 8 dimensions for the 1st ladder, 4 dimensions for the 2nd ladder and 2 dimensions for the 3rd ladder, the 3rd ladder should be the least frequent when factors are randomly entered for each z. Especially long-term and short-term factors are clear in the 2D wavy Reaching dataset. In 2D Wavy Reaching dataset, there is distinct difference between factors of long and short time dependency. The ”goal position” is the factor which affect the entire trajectory, and other factors affect half length of the trajectory (Fig. 9).In our experiment the goal of the trajectory which affect the entire trajectory tended to be expressed in the 3rd ladder. In both datasets, only factor 1 represents goal positions while others represent shape of the trajectories. Since factor 1 has different abstraction level from others, factor 1 and others result in different ladders such as ladder 3 and others. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C COMPARING WITH FHVAE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "FHVAE model is the recently proposed disentangled representation learning model. We note that FHVAE model uses label information to disentangle time series data, which is different setup with our FAVAE model. Table 3 shows a comparison of MIG and reconstruction using FHVAE as the baseline. It was not possible to disentangle in 2D Reaching and 2D wavy using FHVAE, because LSTM used at the FHVAE can not learn data with very long sequences (sequence length of 1000). For fair comparison with the FHVAE, we experimented with 2D Reaching (sequence length 100), 2D wavy Reaching (sequence length 100) at Table 3. In 2D Reaching, FHVAE model has the best score, while in 2D wavy Reaching FAVAE model with ladders and $C$ has the best score. Our model is slightly worse than FHVAE in 2D Reaching Dataset. The parameters searched are $\\alpha = 1$ to 30 (the best $\\alpha = 2 0 a n d 1$ ), $\\beta = 0$ to 60 (the best $\\beta = 0 . 4$ and 4 in 2D Reaching and 2D wavy Reaching) and the best $C = [ c = 0 . 0 4 , 0 . 0 2 , 0 . 0 2 ]$ and $[ 0 . 0 3 , 0 , 0 , 0 6 ]$ in 2D Reaching and 2D wavy ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/9e59da1502ef8fd62f7a26e72178f26a02414848c0c5ae3b38b763f6056f9794.jpg",
+        "image_caption": [
+            "Figure 8: Visualization of learned disentangled representations for Gripper dataset. Traversed latent variable shows changing position of blue objects:(b) (9th $z$ in 1st ladder), changing target ball from red one to blue one and moving accordingly:(c) (1st $z$ in 2nd ladder), reaching ball using different plan:(d) (2nd $z$ in 2nd ladder), changing plan to place ball from drop ball to placing it gently:(e) (4th $z$ in 2nd ladder). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Reaching. The best $C$ was decided by the value of KL divergence loss when we experimented to allow reconstruction with $C = 0 , 0 , 0$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D GRIPPER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            540,
+            282,
+            556
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the potential application of our model to robotic tasks, we constructed a robot endeffector simulation environment based on Bullet Real-Time Physics Simulation2. The environment consisted of an end-effector, two balls (red and blue), and two baskets (red and blue) in a bi-dimensional space. The end-effector grabs one of the balls and places it into the basket with the same color as the ball. The data factors include movement ”habits.” For example, the end-effector could reach the targeted ball by either directly moving toward the ball obliquely or first moving above of the ball and then lowering itself until it reached the ball (perpendicular movement). The end-effector could drop the ball from far above the basket or place it gently in the basket. Each factor could affect different lengths among the data; e.g., ”the plan to place the ball in the basket” factor affects different lengths per datum since the initial distance from the target ball to the corresponding basket may differ. This means that the time required to reach the ball should differ. Note that input is a value such as gripper’s position, not an image. See Appendix B for a more detailed explanation. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "FAVAE learned the disentangled factors of the Gripper dataset. Example visualized latent traversals are shown in Fig. 8. The traversed latent variable determined which factors were disentangled, such as the initial position of blue ball and basket (b), the targeted ball (red or blue) (c), plan to reach the ball (move obliquely or move perpendicularly) (d), and plan to place the ball in the basket (drop the ball or placing it in the basket gently) (e). These results indicate that our model can learn generative factors such as disentangled latent variables for robotic tasks, even though the length of the data affected by each factor may differ for each datum. We used FAVAE in a two-ladder network with 12 and 8 latent variables with $\\beta { = } 1 0 0 0$ . ",
+        "bbox": [
+            174,
+            756,
+            825,
+            867
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/17788be1c6940dc3b590d5a26edc1c87b24e879399043d17fcdde6c0e89b4abd.jpg",
+        "image_caption": [
+            "Figure 9: Visualization of factors of 2D Wavy Reaching training data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            104,
+            777,
+            455
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E DATASET DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            508,
+            362,
+            525
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E.1 2D WAVY REACHING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
+            364,
+            556
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Five different factors generate each trajectory: (9a) Shape of trajectory from start position to intermediate point. The first shape is created using a cycloid function while the other one adds a sine curve on the cycloid. (9b) Shape of trajectory from intermediate point to goal; shape is the same as for (9a). (9c) Location of goal; five different goal locations were used. (9d) Degree of curvature from start to intermediate point. (9e) Degree of curvature from intermediate point to goal. ",
+        "bbox": [
+            174,
+            568,
+            825,
+            638
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E.2 SPRITES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            656,
+            274,
+            671
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Sprits dataset is video data of video game ”sprites”. It used was used in (Li & Mandt, 2018) for confirming the extraction of disentangled representation between static factors and dynamic factors. The dataset consists of sequences with $T = 8$ frames of dimension $3 \\times 6 4 \\times 6 4$ . We use factors and motion of Sprites is shown in Table 4 and Fig. 10. ",
+        "bbox": [
+            176,
+            684,
+            825,
+            739
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E.3 GRIPPER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            757,
+            277,
+            772
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We implemented the end-effector only rather than the entire robot arm since controlling the robot arm during the picking task is easily computable by calculating the inverse kinematics and inverse dynamics. Gripper is a 12 dimensional data set: [joint $\\mathbf { X }$ position, joint y position, finger1 joint position(angle), finger2 joint position(angle), box1 $\\mathbf { X }$ position, box1 y position, $\\mathbf { b } 0 \\mathbf { X } 2 \\textbf { X }$ position, box2 y position, ball1 x position, ball1 y position, ball2 $\\mathbf { X }$ position, ball2 y position]. Eight factors are represented in this dataset: 1) color of ball to pick up, 2) initial location of red ball, 3) initial location of blue ball, 4) initial location of blue basket, 5) initial location of red basket, 6) plan for using end effector to move to ball to pick it up [first, moving horizontally to the $\\mathbf { X }$ -location of ball and then descending horizontally to the y-location of ball, like the movement of the doll drawing machine (perpendicular motion); second, moving straight to the location of he ball to pick it up (oblique motion)], 7) plan for using end effector to move to point above basket after picking up ball (perpendicular or oblique motion), 8) plan for placing ball in basket (by dropping ball or descending to basket and gently placing ball in basket). Among the four initial positions, the two balls and two baskets are placed randomly. The movement of the robot is hard-coded on the basis of a goalposition-based script. To reduce collision detection errors during the simulation, we used a large physical model (end effector size $1 \\mathrm { m } \\dot { }$ ), which does not affect the overall validity. The length of the data-point sequence was 400. ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/752a1a74fa5e7ea0ccaa20aa73aa19dc22fdff677762c35557984ea8df971371.jpg",
+        "image_caption": [
+            "Figure 10: Motions in Sprites dataset. Row 1/2/3 represent walk left/forward/right, row 4/5/6 represent spellcast left/forward/right, and row 7/8/9 represent slash left/forward/right’ respectfully. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            198,
+            825,
+            779
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/e78e155997a1488615c457b4bd028bb4d3262acdc30ac3b7cb9573c4ff4cae4c.jpg",
+        "table_caption": [
+            "Table 4: Factors in dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Degree of factors</td><td>Ground truth factors</td></tr><tr><td rowspan=\"3\">2D Reaching</td><td>2</td><td>Goal positon</td></tr><tr><td>2</td><td>Curved inward /outward</td></tr><tr><td>5</td><td>Degree of curvature</td></tr><tr><td rowspan=\"5\">2D wavy Reaching</td><td>4</td><td>1st trajectory shape</td></tr><tr><td>4</td><td>2nd trajectory shape</td></tr><tr><td>5</td><td>Goal position</td></tr><tr><td>2</td><td>1st trajectory degree of curvature</td></tr><tr><td>2</td><td>2nd trajectory degree of curvature</td></tr><tr><td rowspan=\"6\">Sprites Dataset</td><td>3</td><td>Motions (walk, spellcast or slash)</td></tr><tr><td>2</td><td>Body color (light or dark2)</td></tr><tr><td>2</td><td>Shirts color (brown or teal)</td></tr><tr><td>2</td><td>Hair color (green or pink)</td></tr><tr><td>2</td><td>Pants color (red or teal)</td></tr><tr><td>3</td><td>Direction of character (left,foward, right)</td></tr><tr><td rowspan=\"8\">Gripper</td><td>2</td><td>Targeted ball to grasp (red or blue)</td></tr><tr><td>4</td><td>Red ball initial position</td></tr><tr><td>4</td><td>Blue ball initial position</td></tr><tr><td>4</td><td>Red basket position</td></tr><tr><td>4</td><td>Blue basket position</td></tr><tr><td>2</td><td>Plan to reach target ball</td></tr><tr><td>2</td><td>Plan to reach corresponding basket</td></tr><tr><td>2</td><td>Plan to place ball in basket</td></tr></table>",
+        "bbox": [
+            222,
+            127,
+            776,
+            482
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            521,
+            825,
+            619
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F ARCHITECTURE DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            656,
+            418,
+            672
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F.1 FAVAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            696,
+            266,
+            712
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We show the hyper parameters in Table 1, Fig. 7 and Table 3. The parameters searched are $\\beta = 0$ to 10000 (the best $\\beta = 1$ and $C = [ 0 , 0 , 0 ]$ and the best $\\beta = 1 0 0 0 0 0$ and $C = [ 2 0 , 1 , 5 ]$ in 2D Reaching and 2D wavy Reaching in Table 1) The best $C$ was decided by the value of KL divergence loss when we experimented to allow reconstruction with $C = 0 , 0 , 0$ . In Table 3, we search the parameters from $\\beta = 0$ to 60 (the best $\\beta = 0 . 4$ and 4 in 2D Reaching and 2D wavy Reaching) and the best $C = [ c = 0 . 0 4 , 0 . 0 2 , 0 . 0 2 ]$ and $[ 0 . 0 3 , 0 , 0 , 0 6 ]$ in 2D Reaching and 2D wavy Reaching. In Fig. 7.3, we add the pre-encoder and post-decoder for encoding and decoding of images at each time step. The pre-encoding and post-decoding architecture is same as ”encode frames” and ”decode frames” in the code implementing Disentangled-Sequential-Autoencoder3 with conv $\\mathrm { d i m } = 4 8$ . We search the parameters from $\\beta = 0$ to $\\beta = 1 0 0$ and the best $\\beta = 2 0$ with $C = [ 0 . 3 , 0 . 1 7 , 0 . 0 6$ ]. ",
+        "bbox": [
+            174,
+            729,
+            825,
+            869
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F.2 FHVAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            271,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "For the FHVAE experiments, we used the code from the implementation at Github4. We used recurrent setting with LSTM encoders and decoder with unit size 256 and batch size 80. Dimensions of latent variable z1 and $_ { z 2 }$ are 7 for each. We used Adam optimizer with learning rate 0.001. We applied goal position factors (2 classes for 2D Reaching and 5 classes for 2D wavy Reaching) as label inputs. We varied $\\alpha$ from 1.0 to 30.0, and the best $\\alpha$ was 20.0 for 2D Reaching and 1.0 for 2D wavy reaching. ",
+        "bbox": [
+            174,
+            128,
+            825,
+            213
+        ],
+        "page_idx": 16
+    }
+]

parse/train/S19dR9x0b/S19dR9x0b.md

@@ -0,0 +1,285 @@
+# ALTERNATING MULTI-BIT QUANTIZATION FORRECURRENT NEURAL NETWORKS
+
+Chen $\mathbf { X } \mathbf { u } ^ { 1 , }$ ∗, Jianqiang $\mathbf { Y a o } ^ { 2 }$ , Zhouchen Lin1,3,†, Wenwu $\mathbf { O u } ^ { 2 }$ , Yuanbin $\mathbf { C a o ^ { 4 } }$ , Zhirong Wang2, Hongbin Zha1,3
+
+1 Key Laboratory of Machine Perception (MOE), School of EECS, Peking University, Ch   
+2 Search Algorithm Team, Alibaba Group, China   
+3 Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, China   
+4 AI-LAB, Alibaba Group, China
+
+xuen@pku.edu.cn,tianduo@taobao.com,zlin@pku.edu.cn,santong.oww@taobao.com lingzun.cyb@alibaba-inc.com, qingfeng@taobao.com,zha@cis.pku.edu.cn
+
+# ABSTRACT
+
+Recurrent neural networks have achieved excellent performance in many applications. However, on portable devices with limited resources, the models are often too large to deploy. For applications on the server with large scale concurrent requests, the latency during inference can also be very critical for costly computing resources. In this work, we address these problems by quantizing the network, both weights and activations, into multiple binary codes $\{ - 1 , + 1 \}$ . We formulate the quantization as an optimization problem. Under the key observation that once the quantization coefficients are fixed the binary codes can be derived efficiently by binary search tree, alternating minimization is then applied. We test the quantization for two well-known RNNs, i.e., long short term memory (LSTM) and gated recurrent unit (GRU), on the language models. Compared with the full-precision counter part, by 2-bit quantization we can achieve ${ \sim } 1 6 \times$ memory saving and ${ \sim } 6 \times$ real inference acceleration on CPUs, with only a reasonable loss in the accuracy. By 3-bit quantization, we can achieve almost no loss in the accuracy or even surpass the original model, with ${ \sim } 1 0 . 5 \times$ memory saving and ${ \sim } 3 \times$ real inference acceleration. Both results beat the exiting quantization works with large margins. We extend our alternating quantization to image classification tasks. In both RNNs and feedforward neural networks, the method also achieves excellent performance.
+
+# 1 INTRODUCTION
+
+Recurrent neural networks (RNNs) are specific type of neural networks which are designed to model the sequence data. In last decades, various RNN architectures have been proposed, such as LongShort-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Units Cho et al. (2014). They have enabled the RNNs to achieve state-of-art performance in many applications, e.g., language models (Mikolov et al., 2010), neural machine translation (Sutskever et al., 2014; Wu et al., 2016), automatic speech recognition (Graves et al., 2013), image captions (Vinyals et al., 2015), etc. However, the models often build on high dimensional input/output,e.g., large vocabulary in language models, or very deep inner recurrent networks, making the models have too many parameters to deploy on portable devices with limited resources. In addition, RNNs can only be executed sequentially with dependence on current hidden states. This causes large latency during inference. For applications in the server with large scale concurrent requests, e.g., on-line machine translation and speech recognition, large latency leads to limited requests processed per machine to meet the stringent response time requirements. Thus much more costly computing resources are in demand for RNN based models.
+
+To alleviate the above problems, several techniques can be employed, i.e., low rank approximation (Sainath et al., 2013; Jaderberg et al., 2014; Lebedev et al., 2014; Tai et al., 2016), sparsity (Liu et al., 2015; Han et al., 2015; 2016; Wen et al., 2016), and quantization. All of them are build on the redundancy of current networks and can be combined. In this work, we mainly focus on quantization based methods. More precisely, we are to quantize all parameters into multiple binary codes $\{ - 1 , + 1 \}$ .
+
+The idea of quantizing both weights and activations is firstly proposed by (Hubara et al., 2016a). It has shown that even 1-bit binarization can achieve reasonably good performance in some visual classification tasks. Compared with the full precision counterpart, binary weights reduce the memory by a factor of 32. And the costly arithmetic operations between weights and activations can then be replaced by cheap XNOR and bitcount operations(Hubara et al., 2016a), which potentially leads to much acceleration. Rastegari et al. (2016) further incorporate a real coefficient to compensate for the binarization error. They apply the method to the challenging ImageNet dataset and achieve better performance than pure binarization in (Hubara et al., 2016a). However, it is still of large gap compared with the full precision networks. To bridge this gap, some recent works (Hubara et al., 2016b; Zhou et al., 2016; 2017) further employ quantization with more bits and achieve plausible performance. Meanwhile, quite an amount of works, e.g., (Courbariaux et al., 2015; Li et al., 2016; Zhu et al., 2017; Guo et al., 2017), quantize the weights only. Although much memory saving can be achieved, the acceleration is very limited in modern computing devices (Rastegari et al., 2016).
+
+Among all existing quantization works, most of them focus on convolutional neural networks (CNNs) while pay less attention to RNNs. As mentioned earlier, the latter is also very demanding. Recently, (Hou et al., 2017) showed that binarized LSTM with preconditioned coefficients can achieve promising performance in some easy tasks such as predicting the next character. However, for RNNs with large input/output, e.g., large vocabulary in language models, it is still very challenging for quantization. Both works of Hubara et al. (2016b) and Zhou et al. (2017) test the effectiveness of their multi-bit quantized RNNs to predict the next word. Although using up to 4-bits, the results with quantization still have noticeable gap with those with full precision. This motivates us to find a better method to quantize RNNs. The main contribution of this work is as follows:
+
+(a) We formulate the multi-bit quantization as an optimization problem. The binary codes $\{ - 1 , + 1 \}$ are learned instead of rule-based. For the first time, we observe that the codes can be optimally derived by the binary search tree once the coefficients are knowns in advance, see, e.g., Algorithm 1. Thus the whole optimization is eased by removing the discrete unknowns, which are very difficult to handle.   
+(b) We propose to use alternating minimization to tackle the quantization problem. By separating the binary codes and real coefficients into two parts, we can solve the subproblem efficiently when one part is fixed. With proper initialization, we only need two alternating cycles to get high precision approximation, which is effective enough to even quantize the activations on-line.   
+(c) We systematically evaluate the effectiveness of our alternating quantization on language models. Two well-known RNN structures, i.e., LSTM and GRU, are tested with different quantization bits. Compared with the full-precision counterpart, by 2-bit quantization we can achieve ${ \sim } 1 6 \times$ memory saving and ${ \sim } 6 \times$ real inference acceleration on CPUs, with a reasonable loss on the accuracy. By 3-bit quantization, we can achieve almost no loss in accuracy or even surpass the original model with ${ \sim } 1 0 . 5 \times$ memory saving and ${ \sim } 3 \times$ real inference acceleration. Both results beat the exiting quantization works with large margins. To illustrate that our alternating quantization is very general to extend, we apply it to image classification tasks. In both RNNs and feedforward neural networks, the technique still achieves very plausible performance.
+
+# 2 EXISTING MULTI-BIT QUANTIZATION METHODS
+
+Before introducing our proposed multi-bit quantization, we first summarize existing works as follows:
+
+(a) Uniform quantization method (Rastegari et al., 2016; Hubara et al., 2016b) firstly scales its value in the range $x \in [ - 1 , 1 ]$ . Then it adopts the following $k$ -bit quantization:
+
+$$
+q _ { k } ( x ) = 2 \left( { \frac { \mathrm { r o u n d } [ ( 2 ^ { k } - 1 ) ( { \frac { x + 1 } { 2 } } ) ] } { 2 ^ { k } - 1 } } - { \frac { 1 } { 2 } } \right) ,
+$$
+
+after which the method scales back to the original range. Such quantization is rule based thus is very easy to implement. The intrinsic benefit is that when computing inner product of two quantized vectors, it can employ cheap bit shift and count operations to replace costly multiplications and additions operations. However, the method can be far from optimum when quantizing non-uniform data, which is believed to be the trained weights and activations of deep neural network (Zhou et al., 2017).
+
+![](images/4f13881a57cb01eb9a17cb0533e6fb473bcb0ded73f5f08daa4933f07808e26a.jpg)  
+Figure 1: Illustration of the optimal 2-bit quantization when $\alpha _ { 1 }$ and $\alpha _ { 2 }$ ${ \bf \tilde { \alpha } } _ { 1 } \geq { \alpha _ { 2 } } , { \bf \tilde { \alpha } } _ { \bf \tilde { \alpha } } { \bf \Psi } $ ) are known in advance. The values are quantized into $- \alpha _ { 1 } - \alpha _ { 2 }$ , $- \alpha _ { 1 } + \alpha _ { 2 }$ , $\alpha _ { 1 } - \alpha _ { 2 }$ , and $\alpha _ { 1 } + \alpha _ { 2 }$ , respectively. And the partition intervals are optimally separated by the middle points of adjacent quantization codes, i.e., $- \alpha _ { 1 } , 0$ , and $\alpha _ { 1 }$ , correspondingly.
+
+(b) Balanced quantization (Zhou et al., 2017) alleviates the drawbacks of the uniform quantization by firstly equalizing the data. The method constructs $2 ^ { k }$ intervals which contain roughly the same percentage of the data. Then it linearly maps the center of each interval to the corresponding quantization code in (1). Although sounding more reasonable than the uniform one, the affine transform on the centers can still be suboptimal. In addition, there is no guarantee that the evenly spaced partition is more suitable if compared with the non-evenly spaced partition for a specific data distribution.
+
+(c) Greedy approximation (Guo et al., 2017) instead tries to learn the quantization by tackling the following problem:
+
+$$
+\operatorname* { m i n } _ { \{ \alpha _ { i } , \mathbf { b } _ { i } \} _ { i = 1 } ^ { k } } \left\| \mathbf { w } - \sum _ { i = 1 } ^ { k } \alpha _ { i } \mathbf { b } _ { i } \right\| ^ { 2 } , \quad \mathrm { w i t h } \quad \mathbf { b } _ { i } \in \{ - 1 , + 1 \} ^ { n } .
+$$
+
+For $k = 1$ , the above problem has a closed-form solution (Rastegari et al., 2016). Greedy approximation extends to $k$ -bit $k > 1 ,$ ) quantization by sequentially minimizing the residue. That is
+
+$$
+\operatorname* { m i n } _ { \alpha _ { i } , \mathbf { b } _ { i } } \left\| \mathbf { r } _ { i - 1 } - \alpha _ { i } \mathbf { b } _ { i } \right\| ^ { 2 } , \quad \mathrm { w i t h } \quad \mathbf { r } _ { i - 1 } = \mathbf { w } - \sum _ { j = 1 } ^ { i - 1 } \alpha _ { j } \mathbf { b } _ { j } .
+$$
+
+Then the optimal solution is given as
+
+$$
+\alpha _ { i } = { \frac { 1 } { n } } \| \mathbf { r } _ { i - 1 } \| _ { 1 } \quad { \mathrm { a n d } } \quad \mathbf { b } _ { i } = \mathrm { s i g n } ( \mathbf { r } _ { i - 1 } ) .
+$$
+
+Greedy approximation is very efficient to implement in modern computing devices. Although not able to reach a high precision solution, the formulation of minimizing quantization error is very promising.
+
+(d) Refined greedy approximation (Guo et al., 2017) extends to further decrease the quantization error. In the $j$ -th iteration after minimizing problem (3), the method adds one extra step to refine all computed $\{ \alpha _ { i } \} _ { i = 1 } ^ { j }$ with the least squares solution:
+
+$$
+[ \alpha _ { 1 } , \ldots , \alpha _ { j } ] = \left( ( \mathbf { B } _ { j } ^ { T } \mathbf { B } _ { j } ) ^ { - 1 } \mathbf { B } _ { j } ^ { T } \mathbf { w } \right) ^ { T } , \quad \mathrm { w i t h } \quad \mathbf { B } _ { j } = [ \mathbf { b } _ { 1 } , \ldots , \mathbf { b } _ { j } ] ,
+$$
+
+In experiments of quantizing the weights of CNN, the refined approximation is verified to be better than the original greedy one. However, as we will show later, the refined method is still far from satisfactory for quantization accuracy.
+
+Besides the general multi-bit quantization as summarized above, Li et al. (2016) propose ternary quantization by extending 1-bit binarization with one more feasible state, 0. It does quantization by tackling $\operatorname* { m i n } _ { \alpha }$ ,t $\| \mathbf { w } - \alpha \mathbf { t } \| _ { 2 } ^ { 2 }$ with $\mathbf { t } \in \{ - 1 , 0 , + 1 \} ^ { n }$ . However, no efficient algorithm is proposed in (Li et al., 2016). They instead empirically set the entries $w$ with absolute scales less than $0 . 7 / n \| \mathbf { w } \| _ { 1 }$ to 0 and binarize the left entries as (4). In fact, ternary quantization is a special case of the 2-bit quantization in (2), with an additional constraint that $\alpha _ { 1 } = \alpha _ { 2 }$ . When the binary codes are fixed, the optimal coefficient $\alpha _ { 1 }$ (or $\alpha _ { 2 }$ ) can be derived by least squares solution similar to (5).
+
+![](images/fa9074f4c21ef1f9c64faf20110b0ba23d97f306259495b2dc134619b2c7e9a5.jpg)  
+Figure 2: Illustration of binary search tree to determine the optimal quantization.
+
+# Algorithm 1: Binary Search Tree (BST) to determine to optimal code
+
+BST $( w , \mathbf { v } )$ $\{ w$ is the real value to be quantized} { $\mathbf { v }$ is the vector of quantization codes in ascending order} $m = \mathrm { l e n g t h } ( \mathbf { v } )$ 2 if $m = = 1$ then 3 return $\mathbf { v } _ { 1 }$ 4 end 5 if $w \ge ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ then 6 BST $( w , \mathbf { v } _ { m / 2 + 1 : m } )$ ) 7 else 8 BST( w, v1:m/2) 9 end
+
+In parallel to the binarized quantization discussed here, vector quantization is applied to compress the weights for feedforward neural networks (Gong et al., 2014; Han et al., 2016). Different from ours where all weights are directly constraint to $\{ - 1 , + 1 \}$ , vector quantization learns a small codebook by applying k-means clustering to the weights or conducting product quantization. The weights are then reconstructed by indexing the codebook. It has been shown that by such a technique, the number of parameters can be reduced by an order of magnitude with limited accuracy loss (Gong et al., 2014). It is possible that the multi-bit quantized binary weight can be further compressed by using the product quantization.
+
+# 3 OUR ALTERNATING MULTI-BIT QUANTIZATION
+
+Now we introduce our quantization method. We tackle the same minimization problem as (2). For simplicity, we firstly consider the problem with $k = 2$ . Suppose that $\alpha _ { 1 }$ and $\alpha _ { 2 }$ are known in advance with $\alpha _ { 1 } \geq \alpha _ { 2 } \geq 0$ , then the quantization codes are restricted to $\mathbf { v } = \{ - \alpha _ { 1 } - \alpha _ { 2 } , - \alpha _ { 1 } + \alpha _ { 2 } , \alpha _ { 1 } -$ $\alpha _ { 2 } , \alpha _ { 1 } + \alpha _ { 2 } \}$ . For any entry $w$ of w in problem (2), its quantization code is determined by the least distance to all codes. Consequently, we can partition the number axis into 4 intervals. And each interval corresponds to one particular quantization code. The common point of two adjacent intervals then becomes the middle point of the two quantization codes, i.e., $- \alpha _ { 1 } , 0$ , and $\alpha _ { 1 }$ . Fig. 1 gives an illustration.
+
+For the general $k$ -bit quantization, suppose that $\{ \alpha _ { i } \} _ { i = 1 } ^ { k }$ are known and we have all possible codes in ascending order, i.e., $\begin{array} { r } { \mathbf { v } = \{ - \sum _ { i = 1 } ^ { k } \alpha _ { i } } \end{array}$ , . $\textstyle \ldots , \sum _ { i = 1 } ^ { k } \alpha _ { i } \}$ . Similarly, we can partition the number axis into $2 ^ { k }$ intervals, in which the boundaries are determined by the centers of two adjacent codes in $\mathbf { v }$ , i.e., $\{ ( v _ { i } + v _ { i + 1 } ) / 2 \} _ { i = 1 } ^ { 2 ^ { k } - 1 }$ . However, directly comparing per entry with all the boundaries needs $2 ^ { k }$ comparisons, which is very inefficient. Instead, we can make use of the ascending property in $\mathbf { v }$ . Hierarchically, we partition the codes of $\mathbf { v }$ evenly into two ordered sub-sets, i.e., $\mathbf { v } _ { 1 : m / 2 }$ and $\mathbf { v } _ { m / 2 + 1 : m }$ with $m$ defined as the length of $\mathbf { v }$ . If $w < ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ , its feasible codes are then optimally restricted to $\mathbf { v } _ { 1 : m / 2 }$ . And if $w \ge ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ , its feasible codes become $\mathbf { v } _ { m / 2 + 1 : m }$ By recursively evenly partition the ordered feasible codes, we can then efficiently determine the
+
+# Algorithm 2: Alternating Multi-bit Quantization
+
+Require :Ensure : on weight $\mathbf { w } \in \mathbb { R } ^ { n }$ , number of bits $k$ , total iterations $T$ $\{ \alpha _ { i } , \mathbf { \bar { b } } _ { i } \} _ { i = 1 } ^ { k }$   
+1 Greedy Initialize $\{ \alpha _ { i } , \mathbf { b } _ { i } \} _ { i = 1 } ^ { k }$ as (4)   
+2 for iter $ 1$ to $T$ do   
+3 Update $\{ \alpha _ { i } \} _ { i = 1 } ^ { k }$ as (5)   
+4 Construct $\mathbf { v }$ of all feasible codes in accending order   
+5 Update $\{ { \bf b } _ { i } \} _ { i = 1 } ^ { k }$ as Algorithm 1.   
+6 end
+
+optimal code for per entry by only $k$ comparisons. The whole procedure is in fact a binary search tree. We summarize it in Algorithm 1. Note that once getting the quantization code, it is straightforward to map to the binary code b. Also, by maintaining a mask vector with the same size as w to indicate the partitions, we could operate BST for all entries simultaneously. To give a better illustration, we give a binary tree example for $k = 2$ in Fig. 2. Note that for $k = 2$ , we can even derive the optimal codes by a closed form solution, i.e., $\mathbf { b } _ { 1 } = \mathrm { s i g n } ( \mathbf { w } )$ and $\mathbf { b } _ { 2 } = \mathrm { s i g n } ( \mathbf { w } - \alpha _ { 1 } \mathbf { b } _ { 1 } )$ with $\alpha _ { 1 } \geq \alpha _ { 2 } \geq 0$ .
+
+Under the above observation, let us reconsider the refined greedy approximation (Guo et al., 2017) introduced in Section 2. After modification on the computed $\{ \alpha _ { i } \} _ { i = 1 } ^ { j }$ as (5), $\{ { \bf b } _ { i } \} _ { i = 2 } ^ { j }$ are no longer optimal while the method keeps all of them fixed. To improve the refined greedy approximation, alternating minimizing $\{ \alpha _ { i } \} _ { i = 1 } ^ { k }$ and $\{ { \bf b } _ { i } \} _ { i = 1 } ^ { k }$ becomes a natural choice. Once getting $\{ { \bf b } _ { i } \} _ { i = 1 } ^ { k }$ as described above, we can optimize $\{ \alpha _ { i } \} _ { i = 1 } ^ { k }$ as (5). In real experiments, we find that by greedy initialization as (4), only two alternating cycles is good enough to find high precision quantization. For better illustration, we summarize our alternating minimization in Algorithm 2. For updating $\{ \alpha _ { i } \} _ { i = 1 } ^ { k }$ , we need $2 k ^ { 2 } n$ binary operations and $k n$ non-binary operations. Combining kn non-binary operations to determine the binary code, for total $T$ alternating cycles, we thus need $2 T k ^ { 2 } n$ binary operations and $2 ( T + 1 ) k n$ non-binary operations to quantize $\mathbf { w } \in \mathbb { R } ^ { n }$ into $k$ -bit, with the extra $2 k n$ corresponding to greedy initialization.
+
+# 4 APPLY ALTERNATING MULTI-BIT QUANTIZATION TO RNNS
+
+Implementation. We firstly introduce the implementation details for quantizing RNN. For simplicity, we consider the one layer LSTM for language model. The goal is to predict the next word indexed by $t$ in a sequence of one-hot word tokens $( y _ { 1 } ^ { * } , \dots , y _ { N } ^ { * } )$ as follows:
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { t } = \mathbf { W } _ { e } ^ { T } \mathbf { y } _ { t - 1 } ^ { * } , } \\ & { \mathbf { i } _ { t } , \mathbf { f } _ { t } , \mathbf { 0 } _ { t } , \mathbf { g } _ { t } = \sigma ( \mathbf { W } _ { i } \mathbf { x } _ { t } + \mathbf { b } _ { i } + \mathbf { W } _ { h } \mathbf { h } _ { t - 1 } + \mathbf { b } _ { h } ) , } \\ & { \mathbf { c } _ { t } = \mathbf { f } _ { t } \odot \mathbf { c } _ { t - 1 } + \mathbf { i } _ { t } \odot \mathbf { g } _ { t } , \quad \mathbf { h } _ { t } = \mathbf { o } _ { t } \odot \operatorname { t a n h } ( \mathbf { c } _ { t } ) , } \\ & { \mathbf { y } _ { t } = \operatorname { s o f t m a x } ( \mathbf { W } _ { s } \mathbf { h } _ { t } + \mathbf { b } _ { s } ) . } \end{array}
+$$
+
+where $\sigma$ represents the activation function. In the above formulation, the multiplication between the weight matrices and the vectors $\mathbf { x } _ { t }$ and $\mathbf { h } _ { t }$ occupy most of the computation. This is also where we apply quantization to. For the weight matrices, We do not apply quantization on the full but rather row by row. During the matrix vector product, we can firstly execute the binary multiplication. Then element-wisely multiply the obtained binary vector with the high precision scaling coefficients. Thus little extra computation results while much more freedom is brought to better approximate the weights. We give an illustration on the left part of Fig. 3. Due to one-hot word tokens, $\mathbf { x } _ { t }$ corresponds to one specific row in the quantized ${ \mathbf W } _ { e }$ . It needs no more quantization. Different from the weight matrices, $\mathbf { h } _ { t }$ depends on the input, which needs to be quantized on-line during inference. For consistent notation with existing work, e.g., (Hubara et al., 2016b; Zhou et al., 2017), we also call quantizing on $\mathbf { h } _ { t }$ as quantizing on activation.
+
+For $\mathbf { W } \in \mathbb { R } ^ { m \times n }$ and $\mathbf { h } _ { t } \in \mathbb { R } ^ { n }$ , the standard matrix-vector product needs $2 m n$ operations. For the quantized product between $k _ { w }$ -bit $\mathbf { W }$ and $k _ { h }$ -bit $\mathbf { h _ { t } }$ , we have $2 k _ { w } k _ { h } m n + 4 k _ { h } ^ { 2 } n$ binary operations and $6 k _ { h } n + 2 k _ { w } k _ { h } m$ non-binary operations, where $6 k _ { h } n$ corresponds to the cost of alternating approximation $T = 2$ ) and $2 k _ { w } k _ { h } m$ corresponds to the final product with coefficients. As the binary multiplication operates in 1 bit, whereas the full precision multiplication operates in 32 bits, despite the feasible implementations, the acceleration can be $3 2 \times$ in theory. For alternating quantization here, the overall theoretical acceleration is thus computed as γ = 2mn132 (2kwkhmn+4k2nn)+6khn+2kwkhm . Suppose that LSTM has hidden states $n = 1 0 2 4$ , then we have $\mathbf { W } _ { h } \in \mathbb { R } ^ { 4 0 9 6 \times 1 0 2 4 }$ . The acceleration ratio becomes roughly $7 . 5 \times$ for $( k _ { h } , k _ { w } ) = ( 2 , 2 )$ and $3 . 5 \times$ for $( k _ { h } , k _ { w } ) = ( 3 , 3 )$ . In addition to binary operations, the acceleration in real implementations can be largely affected by the size of the matrix, where much memory reduce can result in better utilizing in the limited faster cache. We implement the binary multiplication kernel in CPUs. Compared with the much optimized Intel Math Kernel Library (MKL) on full precision matrix vector multiplication, we can roughly achieve $6 \times$ for $( k _ { h } , k _ { w } ) = ( \bar { 2 } , 2 )$ and $3 \times$ for $( \bar { k } _ { h } , k _ { w } ) = ( 3 , 3 )$ . For more details, please refer to Appendix A.
+
+![](images/d6c4f5454538f22da72e0dc20a0dbf4fa20423fa95de60b1cf88d1165d505936.jpg)  
+Figure 3: Illustration of quantized matrix vector multiplication (left part). The matrix is quantized row by row, which provides more freedom to approximate while adds little extra computation. By reformulating as the right part, we can make full use of the intrinsic parallel binary matrix vector multiplication for further acceleration.
+
+As indicated in the left part of Fig. 3, the binary multiplication can be conducted sequentially by associativity. Although the operation is suitable for parallel computing by synchronously conducting the multiplication, this needs extra effort for parallelization. We instead concatenate the binary codes as shown in the right part of Fig. 3. Under such modification, we are able to make full use of the much optimized inner parallel matrix multiplication, which gives the possibility for further acceleration. The final result is then obtained by adding all partitioned vectors together, which has little extra computation.
+
+Training. As firstly proposed by Courbariaux et al. (2015), during the training of quantized neural network, directly adding the moderately small gradients to quantized weights will result in no change on it. So they maintain a full precision weight to accumulate the gradients then apply quantization in every mini-batch. In fact, the whole procedure can be mathematically formulated as a bi-level optimization (Colson et al., 2007) problem:
+
+$$
+\begin{array} { r l } { \displaystyle \operatorname* { m i n } _ { \mathbf { w } , \{ \alpha _ { i } , \mathbf { b } _ { i } \} _ { i = 1 } ^ { k } } f ( \sum _ { i = 1 } ^ { k } \alpha _ { i } \mathbf { b } _ { i } ) } & { } \\ { s . t . } & { \{ \alpha _ { i } , \mathbf { b } _ { i } \} _ { i = 1 } ^ { k } =  \arg \operatorname* { m i n } _ { \{ \alpha _ { i } ^ { \prime } , \mathbf { b } _ { i } ^ { \prime } \} _ { i = 1 } ^ { k } } \| \mathbf { w } - \displaystyle \sum _ { i = 1 } ^ { k } \alpha _ { i } ^ { \prime } \mathbf { b } _ { i } ^ { \prime } \| ^ { 2 } . } \end{array}
+$$
+
+Denote the quantized weight as $\begin{array} { r } { \hat { \mathbf { w } } = \sum _ { i = 1 } ^ { k } \alpha _ { i } \mathbf { b } _ { i } } \end{array}$ . In the forward propagation, we derive wˆ from the full precision w in the lower-level problem and apply it to the upper-level function $f ( \cdot )$ , i.e., RNN in this paper. During the backward propagation, the derivative $\frac { \partial f } { \partial \hat { \bf w } }$ is propagated back to w through the lower-level function. Due to the discreteness of $\mathbf { b } _ { i }$ , it is very hard to model the implicit dependence2015), i.e., $\hat { \mathbf { w } }$ w. So we also adopt the “straight-through estimate” as (. To compute the derivative on the quantized hidden state urbariaux et al.,, the same trick $\begin{array} { r } { \frac { \partial f } { \partial \mathbf { w } } = \frac { \partial f } { \partial \hat { \mathbf { w } } } } \end{array}$ $\mathbf { h } _ { t }$ is applied. During the training, we find the same phenomenon as Hubara et al. (2016b) that some entries of w can grow very large, which become outliers and harm the quantization. Here we simply clip w in the range of $[ - 1 , 1 ]$ .
+
+Table 1: Measurement on the approximation of different quantization methods, e.g., Uniform (Hubara et al., 2016b), Balanced (Zhou et al., 2017), Greedy (Guo et al., 2017), Refined (Guo et al., 2017), and our Alternating method, see Section 2. We apply these methods to quantize the full precision pre-trained weight of LSTM on the PTB dataset. The best values are in bold. W-bits represents the number of weight bits and FP denotes full precision.   
+
+<table><tr><td rowspan=1 colspan=4>Relative MSE</td><td rowspan=1 colspan=4>Testing PPW</td></tr><tr><td rowspan=1 colspan=1>W-Bits</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>1.070</td><td rowspan=1 colspan=1>0.404</td><td rowspan=1 colspan=1>0.302</td><td rowspan=1 colspan=1>283.2</td><td rowspan=1 colspan=1>227.3</td><td rowspan=1 colspan=1>216.3</td><td rowspan=5 colspan=1>89.8</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>0.891</td><td rowspan=1 colspan=1>0.745</td><td rowspan=1 colspan=1>0.702</td><td rowspan=1 colspan=1>10287.6</td><td rowspan=1 colspan=1>9106.4</td><td rowspan=1 colspan=1>8539.8</td></tr><tr><td rowspan=1 colspan=1>Greedy</td><td rowspan=1 colspan=1>0.146</td><td rowspan=1 colspan=1>0.071</td><td rowspan=1 colspan=1>0.042</td><td rowspan=1 colspan=1>118.9</td><td rowspan=1 colspan=1>99.4</td><td rowspan=1 colspan=1>95.0</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>0.137</td><td rowspan=1 colspan=1>0.060</td><td rowspan=1 colspan=1>0.030</td><td rowspan=1 colspan=1>105.3</td><td rowspan=1 colspan=1>95.4</td><td rowspan=1 colspan=1>93.1</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>0.125</td><td rowspan=1 colspan=1>0.043</td><td rowspan=1 colspan=1>0.019</td><td rowspan=1 colspan=1>103.1</td><td rowspan=1 colspan=1>93.8</td><td rowspan=1 colspan=1>91.4</td></tr></table>
+
+Table 2: Quantization on the full precision pre-trained weight of GRU on the PTB dataset.   
+
+<table><tr><td rowspan=1 colspan=4>Relative MSE</td><td rowspan=1 colspan=4>Testing PPW</td></tr><tr><td rowspan=1 colspan=1>W-Bits</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>6.138</td><td rowspan=1 colspan=1>3.920</td><td rowspan=1 colspan=1>3.553</td><td rowspan=1 colspan=1>3161906.6</td><td rowspan=1 colspan=1>771259.6</td><td rowspan=1 colspan=1>715781.9</td><td rowspan=5 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>1.206</td><td rowspan=1 colspan=1>1.054</td><td rowspan=1 colspan=1>1.006</td><td rowspan=1 colspan=1>2980.4</td><td rowspan=1 colspan=1>3396.3</td><td rowspan=1 colspan=1>3434.1</td></tr><tr><td rowspan=1 colspan=1>Greedy</td><td rowspan=1 colspan=1>0.377</td><td rowspan=1 colspan=1>0.325</td><td rowspan=1 colspan=1>0.304</td><td rowspan=1 colspan=1>135.7</td><td rowspan=1 colspan=1>105.5</td><td rowspan=1 colspan=1>99.2</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>0.128</td><td rowspan=1 colspan=1>0.055</td><td rowspan=1 colspan=1>0.030</td><td rowspan=1 colspan=1>111.6</td><td rowspan=1 colspan=1>99.1</td><td rowspan=1 colspan=1>97.0</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>0.120</td><td rowspan=1 colspan=1>0.044</td><td rowspan=1 colspan=1>0.021</td><td rowspan=1 colspan=1>110.3</td><td rowspan=1 colspan=1>97.3</td><td rowspan=1 colspan=1>95.2</td></tr></table>
+
+# 5 EXPERIMENTS ON THE LANGUAGE MODELS
+
+In this section, we conduct quantization experiments on language models. The two most well-known recurrent neural networks, i.e., LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho et al., 2014), are evaluated. As they are to predict the next word, the performance is measured by perplexity per word (PPW) metric. For all experiments, we initialize with the pre-trained model and using vanilla SGD. The initial learning rate is set to 20. Every epoch we evaluate on the validation dataset and record the best value. When the validation error exceeds the best record, we decrease learning rate by a factor of 1.2. Training is terminated once the learning rate less than 0.001 or reaching the maximum epochs, i.e., 80. The gradient norm is clipped in the range $[ - 0 . 2 5 , 0 . 2 5 ]$ . We unroll the network for 30 time steps and regularize it with the standard dropout (probability of dropping out units equals to 0.5) (Zaremba et al., 2014). For simplicity of notation, we denote the methods using uniform, balanced, greedy, refined greedy, and our alternating quantization as Uniform, Balanced, Greedy, Refined, and Alternating, respectively.
+
+Peen Tree Bank. We first conduct experiments on the Peen Tree Bank (PTB) corpus (Marcus et al., 1993), using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains 929K training tokens, 73K validation tokens, and 82K test tokens. For fair comparison with existing works, we also use LSTM and GRU with 1 hidden layer of size 300. To have a glance at the approximation ability of different quantization methods as detailed in Section 2, we firstly conduct experiments by directly quantizing the trained full precision weight (neither quantization on activation nor retraining). Results on LSTM and GRU are shown in Table 1 and Table 2, respectively. The left parts record the relative mean squared error of quantized weight matrices with full precision one. We can see that our proposed Alternating can get much lower error across all varying bit. We also measure the testing PPW for the quantized weight as shown in the right parts of Table 1 and 2. The results are in consistent with the left part, where less errors result in lower testing PPW. Note that Uniform and Balanced quantization are rule-based and not aim at minimizing the error. Thus they can have much worse result by direct approximation. We also repeat the experiment on other datasets. For both LSTM and GRU, the results are very similar to here.
+
+Table 3: Testing PPW of multi-bit quantized LSTM and GRU on the PTB dataset. W-Bits and A-Bits represent the number of weight and activation bits, respectively.   
+
+<table><tr><td rowspan=1 colspan=6>LSTM</td><td rowspan=1 colspan=5>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits /A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>4/4</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>4/4</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>220</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>123</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>114</td><td rowspan=1 colspan=1>107</td><td rowspan=1 colspan=1>142</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>116</td><td rowspan=1 colspan=1>100</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>100.3</td><td rowspan=1 colspan=1>95.6</td><td rowspan=1 colspan=1>91.3</td><td rowspan=1 colspan=1>二</td><td rowspan=2 colspan=1>89.8</td><td rowspan=1 colspan=1>105.1</td><td rowspan=1 colspan=1>100.3</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>二</td><td rowspan=2 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>95.8</td><td rowspan=1 colspan=1>91.9</td><td rowspan=1 colspan=1>87.9</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>101.2</td><td rowspan=1 colspan=1>97.0</td><td rowspan=1 colspan=1>92.9</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+Table 4: Testing PPW of multi-bit quantized LSTM and GRU on the WikiText-2 dataset.   
+
+<table><tr><td rowspan=1 colspan=5>LSTM</td><td rowspan=1 colspan=4>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>108.7</td><td rowspan=1 colspan=1>105.8</td><td rowspan=1 colspan=1>102.2</td><td rowspan=2 colspan=1>100.1</td><td rowspan=1 colspan=1>117.2</td><td rowspan=1 colspan=1>114.1</td><td rowspan=1 colspan=1>111.8</td><td rowspan=2 colspan=1>106.7</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>106.1</td><td rowspan=1 colspan=1>102.7</td><td rowspan=1 colspan=1>98.7</td><td rowspan=1 colspan=1>113.7</td><td rowspan=1 colspan=1>110.2</td><td rowspan=1 colspan=1>106.4</td></tr></table>
+
+Table 5: Testing PPW of multi-bit quantized LSTM and GRU on the Text8 dataset.   
+
+<table><tr><td rowspan=1 colspan=5>LSTM</td><td rowspan=1 colspan=4>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>135.6</td><td rowspan=1 colspan=1>122.3</td><td rowspan=1 colspan=1>110.2</td><td rowspan=2 colspan=1>101.1</td><td rowspan=1 colspan=1>135.8</td><td rowspan=1 colspan=1>126.9</td><td rowspan=1 colspan=1>118.3</td><td rowspan=2 colspan=1>111.6</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>108.8</td><td rowspan=1 colspan=1>105.1</td><td rowspan=1 colspan=1>98.8</td><td rowspan=1 colspan=1>124.5</td><td rowspan=1 colspan=1>118.7</td><td rowspan=1 colspan=1>114.0</td></tr></table>
+
+We then conduct experiments by quantizing both weights and activations. We train with the batch size 20. The final result is shown in Table 3. Besides comparing with the existing works, we also conduct experiment for Refined as a competitive baseline. We do not include Greedy as it is already shown to be much inferior to the refined one, see, e.g., Table 1 and 2. As Table 3 shows, our full precision model can attain lower PPW than the existing works. However, when considering the gap between quantized model with the full precision one, our alternating quantized neural network is still far better than existing works, i.e., Uniform (Hubara et al., 2016b) and Balanced (Zhou et al., 2017). Compared with Refined, our Alternating quantization can achieve compatible performance using 1-bit less quantization on weights or activations. In other words, under the same tolerance of accuracy drop, Alternating executes faster and uses less memory than Refined. We can see that our $3 / 3$ weights/activations quantized LSTM can achieve even better performance than full precision one. A possible explanation is due to the regularization introduced by quantization (Hubara et al., 2016b).
+
+WikiText-2 (Merity et al., 2017) is a dataset released recently as an alternative to PTB. It contains 2088K training, 217K validation, and 245K test tokens, and has a vocabulary of 33K words, which is roughly 2 times larger in dataset size, and 3 times larger in vocabulary than PTB. We train with one layer’s hidden state of size 512 and set the batch size to 100. The result is shown in Table 4. Similar to PTB, our Alternating can use 1-bit less quantization to attain compatible or even lower PPW than Refined.
+
+Text8. In order to determine whether Alternating remains effective with a larger dataset, we perform experiments on the Text8 corpus (Mikolov et al., 2014). Here we follow the same setting as (Xie et al., 2017). The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in 15.3M training tokens, 848K validation tokens, and 855K test tokens. We also preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. We train LSTM and GRU with one hidden layer of size 1024 and set the batch size to 100. The result is shown in Table 5. For LSTM on the left part, Alternating achieves excellent performance. By only 2-bit quantization on weights and activations, it exceeds Refined with 3-bit. The 2-bit result is even better than that reported in (Xie et al., 2017), where LSTM adding noising schemes for regularization can only attain 110.6 testing PPW. For GRU on the right part, although Alternating is much better than Refined, the 3-bit quantization still has gap with full precision one. We attribute that to the unified setting of hyper-parameters across all experiments. With specifically tuned hyper-parameters on this dataset, one may make up for that gap.
+
+Note that our alternating quantization is a general technique. It is not only suitable for language models here. For a comprehensive verification, we apply it to image classification tasks. In both RNNs and feedforward neural networks, our alternating quantization also achieves the lowest testing error among all compared methods. Due to space limitation, we deter the results to Appendix B.
+
+# 6 CONCLUSIONS
+
+In this work, we address the limitations of RNNs, i.e., large memory and high latency, by quantization. We formulate the quantization by minimizing the approximation error. Under the key observation that some parameters can be singled out when others fixed, a simple yet effective alternating method is proposed. We apply it to quantize LSTM and GRU on language models. By 2-bit weights and activations, we achieve only a reasonably accuracy loss compared with full precision one, with ${ \sim } 1 6 \times$ reduction in memory and ${ \sim } 6 \times$ real acceleration on CPUs. By 3-bit quantization, we can attain compatible or even better result than the full precision one, with ${ \sim } 1 0 . 5 \times$ reduction in memory and ${ \sim } 3 \times$ real acceleration. Both beat existing works with a large margin. We also apply our alternating quantization to image classification tasks. In both RNNs and feedforward neural networks, the method can still achieve very plausible performance.
+
+# 7 ACKNOWLEDGEMENTS
+
+We would like to thank the reviewers for their suggestions on the manuscript. Zhouchen Lin is supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502), National Natural Science Foundation (NSF) of China (grant nos. 61625301 and 61731018), Qualcomm, and Microsoft Research Asia. Hongbin Zha is supported by Natural Science Foundation (NSF) of China (No. 61632003).
+
+# REFERENCES
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+
+# APPENDIX
+
+# A BINARY MATRIX VECTOR MULTIPLICATION IN CPUS
+
+Table 6: Computing time of the binary matrix vector multiplication in CPUs, where Quant represents the cost to execute our alternating quantization on-line.   
+
+<table><tr><td rowspan=1 colspan=1>Weight Size</td><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>Total (ms)</td><td rowspan=1 colspan=1>Quant (ms)</td><td rowspan=1 colspan=1>Quant / Total</td><td rowspan=1 colspan=1>Acceleration</td></tr><tr><td rowspan=3 colspan=1>4096 × 1024</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>0.07</td><td rowspan=1 colspan=1>20%</td><td rowspan=1 colspan=1>5.6x</td></tr><tr><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>0.72</td><td rowspan=1 colspan=1>0.11</td><td rowspan=1 colspan=1>15%</td><td rowspan=1 colspan=1>2.7x</td></tr><tr><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>1.95</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1.0×</td></tr><tr><td rowspan=3 colspan=1>42000 × 1024</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>3.17</td><td rowspan=1 colspan=1>0.07</td><td rowspan=1 colspan=1>2.2%</td><td rowspan=1 colspan=1>6.0×</td></tr><tr><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>6.46</td><td rowspan=1 colspan=1>0.11</td><td rowspan=1 colspan=1>1.7%</td><td rowspan=1 colspan=1>3.0x</td></tr><tr><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>19.10</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1.0x</td></tr></table>
+
+In this section, we discuss the implementation of the binary multiplication kernel in CPUs. The binary multiplication is divided into two steps: Entry-wise XNOR operation (corresponding to entry-wise product in the full precision multiplication) and bit count operation for accumulation (corresponding to compute the sum of all multiplied entries in the full precision multiplication). We test it on Intel Xeon E5-2682 v4 $@$ 2.50 GHz CPU. For the XNOR operation, we use the Single instruction, multiple data (SIMD) _mm256 _xor_ps, which can execute 256 bit simultaneously. For the bit count operation, we use the function _popcnt64 (Note that this step can further be accelerated by the up-coming instruction _mm512 _popcnt_epi64 , which can execute 512 bits simultaneously. Similarly, the XNOR operation can also be further accelerated by the up-coming _mm512 _xor_ps instruction to execute 512 bits simultaneously). We compare with the much optimized Intel Math Kernel Library (MKL) on full precision matrix vector multiplication and execute all codes in the single-thread mode. We conduct two scales of experiments: a matrix of size $4 0 9 6 \times 1 0 2 4$ multiplying a vector of size 1024 and a matrix of size $4 2 0 0 0 \times 1 0 2 4$ multiplying a vector of size 1024, which respectively correspond to the hidden state product ${ \bf W } _ { h } \mathbf h _ { t - 1 }$ and the softmax layer $\mathbf { W } _ { s } \mathbf { h } _ { t }$ for Text8 dataset during inference with batch size of 1 (See Eq. (6)). The results are shown in Table 6. We can see that our alternating quantization step only accounts for a small portion of the total executing time, especially for the larger scale matrix vector multiplication. Compared with the full precision one, the binary multiplication can roughly achieve $6 \times$ acceleration with 2-bit quantization and $3 \times$ acceleration with 3-bit quantization. Note that this is only a simple test on CPU. Our alternating quantization method can also be extended to GPU, ASIC, and FPGA.
+
+# B IMAGE CLASSIFICATION
+
+Sequential MNIST. As a simple illustration to show that our alternating quantization is not limited for texts, we conduct experiments on the sequential MNIST classification task (Cooijmans et al., 2017). The dataset consists of a training set of 60K and a test set of 10K $2 8 \times 2 8$ gray-scale images. Here we divide the last 5000 training images for validation. In every time, we sequentially use one row of the image as the input $( 2 8 \times 1 )$ , which results in a total of 28 time steps. We use 1 hidden layer’s LSTM of size 128 and the same optimization hyper-parameters as the language models. Besides the weights and activations, the inputs are quantized. The testing error rates for 1-bit input, 2-bit weight, and 2-bit activation are shown in 7, where our alternating quantized method still achieves plausible performance in this task.
+
+MLP on MNIST. The alternating quantization proposed in this work is a general technique. It is not only suitable for RNNs, but also for feed-forward neural networks. As an example, we firstly conduct a classification task on MNIST and compare with existing work (Li et al., 2017). The method proposed in (Li et al., 2017) is intrinsically a greedy multi-bit quantization method. For fair comparison, we follow the same setting. We use the MLP consisting of 3 hidden layers of 4096 units and an L2-SVM output layer. No convolution, preprocessing, data augmentation or pre-training is used. We also use ADAM (Kingma & Ba, 2015) with an exponentially decaying learning rate and Batch Normalization (Ioffe & Szegedy, 2015) with a batch size 100. The testing error rates for 2-bit input, 2-bit weight, and 1-bit activation are shown in Table 8. Among all the compared multi-bit quantization methods, our alternating one achieves the lowest testing error.
+
+Table 7: Testing error rate of LSTM on MNIST with 1-bit input, 2-bit weight, and 2-bit activation.   
+
+<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision</td><td>1.10%</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>1.39 %</td></tr><tr><td>Alternating (ours)</td><td>1.19 %</td></tr></table>
+
+Table 8: Testing error rate of MLP on MNIST with 2-bit input, 2-bit weight, and 1-bit activation.   
+
+<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision</td><td>0.97%</td></tr><tr><td>Greedy (reported in (Li et al., 2017))</td><td>1.25 %</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>1.22 %</td></tr><tr><td>Alternating (ours)</td><td>1.13 %</td></tr></table>
+
+Table 9: Testing error rate of CNN on CIFAR-10 with 2-bit weight and 1-bit activation.   
+
+<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision (reported in (Hou et al., 2017))</td><td>11.90 %</td></tr><tr><td> XNOR-Net (1-bit weight &amp; activation, reported in (Hou et al., 2017))</td><td>12.62 %</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>12.08 %</td></tr><tr><td>Alternating (ours)</td><td>11.70 %</td></tr></table>
+
+CNN on CIFAR-10. We then conduct experiments on CIFAR-10 and follow the same setting as (Hou et al., 2017). That is, we use 45000 images for training, another 5000 for validation, and the remaining 10000 for testing. The images are preprocessed with global contrast normalization and ZCA whitening. We also use the VGG-like architecture (Simonyan & Zisserman, 2015):
+
+$$
+( 2 \times 1 2 8 ~ \mathrm { C 3 } ) - \mathrm { M P 2 - ( 2 \times 2 5 6 ~ \mathrm { C 3 } ) - M P 2 - ( 2 \times 5 1 2 ~ \mathrm { C 3 } ) - M P 2 - ( 2 \times 1 0 2 4 ~ \mathrm { F C } ) - 1 0 ~ \mathrm { S V M } }
+$$
+
+where C3 is a $3 \times 3$ convolution layer, and MP2 is a $2 \times 2$ max-pooling layer. Batch Normalization, with a mini-batch size of 50, and ADAM are used. The maximum number of epochs is 200. The learning rate starts at 0.02 and decays by a factor of 0.5 after every 30 epochs. The testing error rates for 2-bit weight and 1-bit activation are shown in Table 9, where our alternating method again achieves the lowest test error rate among all compared quantization methods.

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+        "text": "ALTERNATING MULTI-BIT QUANTIZATION FORRECURRENT NEURAL NETWORKS",
+        "text_level": 1,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Chen $\\mathbf { X } \\mathbf { u } ^ { 1 , }$ ∗, Jianqiang $\\mathbf { Y a o } ^ { 2 }$ , Zhouchen Lin1,3,†, Wenwu $\\mathbf { O u } ^ { 2 }$ , Yuanbin $\\mathbf { C a o ^ { 4 } }$ , Zhirong Wang2, Hongbin Zha1,3 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Key Laboratory of Machine Perception (MOE), School of EECS, Peking University, Ch   \n2 Search Algorithm Team, Alibaba Group, China   \n3 Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, China   \n4 AI-LAB, Alibaba Group, China ",
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+    {
+        "type": "text",
+        "text": "xuen@pku.edu.cn,tianduo@taobao.com,zlin@pku.edu.cn,santong.oww@taobao.com lingzun.cyb@alibaba-inc.com, qingfeng@taobao.com,zha@cis.pku.edu.cn ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Recurrent neural networks have achieved excellent performance in many applications. However, on portable devices with limited resources, the models are often too large to deploy. For applications on the server with large scale concurrent requests, the latency during inference can also be very critical for costly computing resources. In this work, we address these problems by quantizing the network, both weights and activations, into multiple binary codes $\\{ - 1 , + 1 \\}$ . We formulate the quantization as an optimization problem. Under the key observation that once the quantization coefficients are fixed the binary codes can be derived efficiently by binary search tree, alternating minimization is then applied. We test the quantization for two well-known RNNs, i.e., long short term memory (LSTM) and gated recurrent unit (GRU), on the language models. Compared with the full-precision counter part, by 2-bit quantization we can achieve ${ \\sim } 1 6 \\times$ memory saving and ${ \\sim } 6 \\times$ real inference acceleration on CPUs, with only a reasonable loss in the accuracy. By 3-bit quantization, we can achieve almost no loss in the accuracy or even surpass the original model, with ${ \\sim } 1 0 . 5 \\times$ memory saving and ${ \\sim } 3 \\times$ real inference acceleration. Both results beat the exiting quantization works with large margins. We extend our alternating quantization to image classification tasks. In both RNNs and feedforward neural networks, the method also achieves excellent performance. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks (RNNs) are specific type of neural networks which are designed to model the sequence data. In last decades, various RNN architectures have been proposed, such as LongShort-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Units Cho et al. (2014). They have enabled the RNNs to achieve state-of-art performance in many applications, e.g., language models (Mikolov et al., 2010), neural machine translation (Sutskever et al., 2014; Wu et al., 2016), automatic speech recognition (Graves et al., 2013), image captions (Vinyals et al., 2015), etc. However, the models often build on high dimensional input/output,e.g., large vocabulary in language models, or very deep inner recurrent networks, making the models have too many parameters to deploy on portable devices with limited resources. In addition, RNNs can only be executed sequentially with dependence on current hidden states. This causes large latency during inference. For applications in the server with large scale concurrent requests, e.g., on-line machine translation and speech recognition, large latency leads to limited requests processed per machine to meet the stringent response time requirements. Thus much more costly computing resources are in demand for RNN based models. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To alleviate the above problems, several techniques can be employed, i.e., low rank approximation (Sainath et al., 2013; Jaderberg et al., 2014; Lebedev et al., 2014; Tai et al., 2016), sparsity (Liu et al., 2015; Han et al., 2015; 2016; Wen et al., 2016), and quantization. All of them are build on the redundancy of current networks and can be combined. In this work, we mainly focus on quantization based methods. More precisely, we are to quantize all parameters into multiple binary codes $\\{ - 1 , + 1 \\}$ . ",
+        "bbox": [
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+    {
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+        "text": "The idea of quantizing both weights and activations is firstly proposed by (Hubara et al., 2016a). It has shown that even 1-bit binarization can achieve reasonably good performance in some visual classification tasks. Compared with the full precision counterpart, binary weights reduce the memory by a factor of 32. And the costly arithmetic operations between weights and activations can then be replaced by cheap XNOR and bitcount operations(Hubara et al., 2016a), which potentially leads to much acceleration. Rastegari et al. (2016) further incorporate a real coefficient to compensate for the binarization error. They apply the method to the challenging ImageNet dataset and achieve better performance than pure binarization in (Hubara et al., 2016a). However, it is still of large gap compared with the full precision networks. To bridge this gap, some recent works (Hubara et al., 2016b; Zhou et al., 2016; 2017) further employ quantization with more bits and achieve plausible performance. Meanwhile, quite an amount of works, e.g., (Courbariaux et al., 2015; Li et al., 2016; Zhu et al., 2017; Guo et al., 2017), quantize the weights only. Although much memory saving can be achieved, the acceleration is very limited in modern computing devices (Rastegari et al., 2016). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Among all existing quantization works, most of them focus on convolutional neural networks (CNNs) while pay less attention to RNNs. As mentioned earlier, the latter is also very demanding. Recently, (Hou et al., 2017) showed that binarized LSTM with preconditioned coefficients can achieve promising performance in some easy tasks such as predicting the next character. However, for RNNs with large input/output, e.g., large vocabulary in language models, it is still very challenging for quantization. Both works of Hubara et al. (2016b) and Zhou et al. (2017) test the effectiveness of their multi-bit quantized RNNs to predict the next word. Although using up to 4-bits, the results with quantization still have noticeable gap with those with full precision. This motivates us to find a better method to quantize RNNs. The main contribution of this work is as follows: ",
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+    },
+    {
+        "type": "text",
+        "text": "(a) We formulate the multi-bit quantization as an optimization problem. The binary codes $\\{ - 1 , + 1 \\}$ are learned instead of rule-based. For the first time, we observe that the codes can be optimally derived by the binary search tree once the coefficients are knowns in advance, see, e.g., Algorithm 1. Thus the whole optimization is eased by removing the discrete unknowns, which are very difficult to handle.   \n(b) We propose to use alternating minimization to tackle the quantization problem. By separating the binary codes and real coefficients into two parts, we can solve the subproblem efficiently when one part is fixed. With proper initialization, we only need two alternating cycles to get high precision approximation, which is effective enough to even quantize the activations on-line.   \n(c) We systematically evaluate the effectiveness of our alternating quantization on language models. Two well-known RNN structures, i.e., LSTM and GRU, are tested with different quantization bits. Compared with the full-precision counterpart, by 2-bit quantization we can achieve ${ \\sim } 1 6 \\times$ memory saving and ${ \\sim } 6 \\times$ real inference acceleration on CPUs, with a reasonable loss on the accuracy. By 3-bit quantization, we can achieve almost no loss in accuracy or even surpass the original model with ${ \\sim } 1 0 . 5 \\times$ memory saving and ${ \\sim } 3 \\times$ real inference acceleration. Both results beat the exiting quantization works with large margins. To illustrate that our alternating quantization is very general to extend, we apply it to image classification tasks. In both RNNs and feedforward neural networks, the technique still achieves very plausible performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 EXISTING MULTI-BIT QUANTIZATION METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Before introducing our proposed multi-bit quantization, we first summarize existing works as follows: ",
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+        "text": "(a) Uniform quantization method (Rastegari et al., 2016; Hubara et al., 2016b) firstly scales its value in the range $x \\in [ - 1 , 1 ]$ . Then it adopts the following $k$ -bit quantization: ",
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+        "img_path": "images/ac0c4028710ebe2102c6bb1e2cdd3d43a87e766cafc8b2bfe9d203088cb77194.jpg",
+        "text": "$$\nq _ { k } ( x ) = 2 \\left( { \\frac { \\mathrm { r o u n d } [ ( 2 ^ { k } - 1 ) ( { \\frac { x + 1 } { 2 } } ) ] } { 2 ^ { k } - 1 } } - { \\frac { 1 } { 2 } } \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "after which the method scales back to the original range. Such quantization is rule based thus is very easy to implement. The intrinsic benefit is that when computing inner product of two quantized vectors, it can employ cheap bit shift and count operations to replace costly multiplications and additions operations. However, the method can be far from optimum when quantizing non-uniform data, which is believed to be the trained weights and activations of deep neural network (Zhou et al., 2017). ",
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+        "img_path": "images/4f13881a57cb01eb9a17cb0533e6fb473bcb0ded73f5f08daa4933f07808e26a.jpg",
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+            "Figure 1: Illustration of the optimal 2-bit quantization when $\\alpha _ { 1 }$ and $\\alpha _ { 2 }$ ${ \\bf \\tilde { \\alpha } } _ { 1 } \\geq { \\alpha _ { 2 } } , { \\bf \\tilde { \\alpha } } _ { \\bf \\tilde { \\alpha } } { \\bf \\Psi } $ ) are known in advance. The values are quantized into $- \\alpha _ { 1 } - \\alpha _ { 2 }$ , $- \\alpha _ { 1 } + \\alpha _ { 2 }$ , $\\alpha _ { 1 } - \\alpha _ { 2 }$ , and $\\alpha _ { 1 } + \\alpha _ { 2 }$ , respectively. And the partition intervals are optimally separated by the middle points of adjacent quantization codes, i.e., $- \\alpha _ { 1 } , 0$ , and $\\alpha _ { 1 }$ , correspondingly. "
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+        "text": "(b) Balanced quantization (Zhou et al., 2017) alleviates the drawbacks of the uniform quantization by firstly equalizing the data. The method constructs $2 ^ { k }$ intervals which contain roughly the same percentage of the data. Then it linearly maps the center of each interval to the corresponding quantization code in (1). Although sounding more reasonable than the uniform one, the affine transform on the centers can still be suboptimal. In addition, there is no guarantee that the evenly spaced partition is more suitable if compared with the non-evenly spaced partition for a specific data distribution. ",
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+        "text": "(c) Greedy approximation (Guo et al., 2017) instead tries to learn the quantization by tackling the following problem: ",
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+        "type": "equation",
+        "img_path": "images/4865807540f445d6fcabe5a1745e121926b4740eb8903a6ec92efd2674fe54a7.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\{ \\alpha _ { i } , \\mathbf { b } _ { i } \\} _ { i = 1 } ^ { k } } \\left\\| \\mathbf { w } - \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\mathbf { b } _ { i } \\right\\| ^ { 2 } , \\quad \\mathrm { w i t h } \\quad \\mathbf { b } _ { i } \\in \\{ - 1 , + 1 \\} ^ { n } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "For $k = 1$ , the above problem has a closed-form solution (Rastegari et al., 2016). Greedy approximation extends to $k$ -bit $k > 1 ,$ ) quantization by sequentially minimizing the residue. That is ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/64207e513f669de2d8b0815a5625e20c2c5149a00117ea65620d0785fb7d332f.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\alpha _ { i } , \\mathbf { b } _ { i } } \\left\\| \\mathbf { r } _ { i - 1 } - \\alpha _ { i } \\mathbf { b } _ { i } \\right\\| ^ { 2 } , \\quad \\mathrm { w i t h } \\quad \\mathbf { r } _ { i - 1 } = \\mathbf { w } - \\sum _ { j = 1 } ^ { i - 1 } \\alpha _ { j } \\mathbf { b } _ { j } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Then the optimal solution is given as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e8c408305747c4fbc92c1961cdeeddd041b92f1f7bff6c11843d79d56f478e79.jpg",
+        "text": "$$\n\\alpha _ { i } = { \\frac { 1 } { n } } \\| \\mathbf { r } _ { i - 1 } \\| _ { 1 } \\quad { \\mathrm { a n d } } \\quad \\mathbf { b } _ { i } = \\mathrm { s i g n } ( \\mathbf { r } _ { i - 1 } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Greedy approximation is very efficient to implement in modern computing devices. Although not able to reach a high precision solution, the formulation of minimizing quantization error is very promising. ",
+        "bbox": [
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+        "type": "text",
+        "text": "(d) Refined greedy approximation (Guo et al., 2017) extends to further decrease the quantization error. In the $j$ -th iteration after minimizing problem (3), the method adds one extra step to refine all computed $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { j }$ with the least squares solution: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d17efa9fa3c25b894d18a789ec5cbdeaa0c89dafa37aa9b4e4c6a7f214223016.jpg",
+        "text": "$$\n[ \\alpha _ { 1 } , \\ldots , \\alpha _ { j } ] = \\left( ( \\mathbf { B } _ { j } ^ { T } \\mathbf { B } _ { j } ) ^ { - 1 } \\mathbf { B } _ { j } ^ { T } \\mathbf { w } \\right) ^ { T } , \\quad \\mathrm { w i t h } \\quad \\mathbf { B } _ { j } = [ \\mathbf { b } _ { 1 } , \\ldots , \\mathbf { b } _ { j } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In experiments of quantizing the weights of CNN, the refined approximation is verified to be better than the original greedy one. However, as we will show later, the refined method is still far from satisfactory for quantization accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Besides the general multi-bit quantization as summarized above, Li et al. (2016) propose ternary quantization by extending 1-bit binarization with one more feasible state, 0. It does quantization by tackling $\\operatorname* { m i n } _ { \\alpha }$ ,t $\\| \\mathbf { w } - \\alpha \\mathbf { t } \\| _ { 2 } ^ { 2 }$ with $\\mathbf { t } \\in \\{ - 1 , 0 , + 1 \\} ^ { n }$ . However, no efficient algorithm is proposed in (Li et al., 2016). They instead empirically set the entries $w$ with absolute scales less than $0 . 7 / n \\| \\mathbf { w } \\| _ { 1 }$ to 0 and binarize the left entries as (4). In fact, ternary quantization is a special case of the 2-bit quantization in (2), with an additional constraint that $\\alpha _ { 1 } = \\alpha _ { 2 }$ . When the binary codes are fixed, the optimal coefficient $\\alpha _ { 1 }$ (or $\\alpha _ { 2 }$ ) can be derived by least squares solution similar to (5). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/fa9074f4c21ef1f9c64faf20110b0ba23d97f306259495b2dc134619b2c7e9a5.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of binary search tree to determine the optimal quantization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Binary Search Tree (BST) to determine to optimal code ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "BST $( w , \\mathbf { v } )$ $\\{ w$ is the real value to be quantized} { $\\mathbf { v }$ is the vector of quantization codes in ascending order} $m = \\mathrm { l e n g t h } ( \\mathbf { v } )$ 2 if $m = = 1$ then 3 return $\\mathbf { v } _ { 1 }$ 4 end 5 if $w \\ge ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ then 6 BST $( w , \\mathbf { v } _ { m / 2 + 1 : m } )$ ) 7 else 8 BST( w, v1:m/2) 9 end ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In parallel to the binarized quantization discussed here, vector quantization is applied to compress the weights for feedforward neural networks (Gong et al., 2014; Han et al., 2016). Different from ours where all weights are directly constraint to $\\{ - 1 , + 1 \\}$ , vector quantization learns a small codebook by applying k-means clustering to the weights or conducting product quantization. The weights are then reconstructed by indexing the codebook. It has been shown that by such a technique, the number of parameters can be reduced by an order of magnitude with limited accuracy loss (Gong et al., 2014). It is possible that the multi-bit quantized binary weight can be further compressed by using the product quantization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 OUR ALTERNATING MULTI-BIT QUANTIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Now we introduce our quantization method. We tackle the same minimization problem as (2). For simplicity, we firstly consider the problem with $k = 2$ . Suppose that $\\alpha _ { 1 }$ and $\\alpha _ { 2 }$ are known in advance with $\\alpha _ { 1 } \\geq \\alpha _ { 2 } \\geq 0$ , then the quantization codes are restricted to $\\mathbf { v } = \\{ - \\alpha _ { 1 } - \\alpha _ { 2 } , - \\alpha _ { 1 } + \\alpha _ { 2 } , \\alpha _ { 1 } -$ $\\alpha _ { 2 } , \\alpha _ { 1 } + \\alpha _ { 2 } \\}$ . For any entry $w$ of w in problem (2), its quantization code is determined by the least distance to all codes. Consequently, we can partition the number axis into 4 intervals. And each interval corresponds to one particular quantization code. The common point of two adjacent intervals then becomes the middle point of the two quantization codes, i.e., $- \\alpha _ { 1 } , 0$ , and $\\alpha _ { 1 }$ . Fig. 1 gives an illustration. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "For the general $k$ -bit quantization, suppose that $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { k }$ are known and we have all possible codes in ascending order, i.e., $\\begin{array} { r } { \\mathbf { v } = \\{ - \\sum _ { i = 1 } ^ { k } \\alpha _ { i } } \\end{array}$ , . $\\textstyle \\ldots , \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\}$ . Similarly, we can partition the number axis into $2 ^ { k }$ intervals, in which the boundaries are determined by the centers of two adjacent codes in $\\mathbf { v }$ , i.e., $\\{ ( v _ { i } + v _ { i + 1 } ) / 2 \\} _ { i = 1 } ^ { 2 ^ { k } - 1 }$ . However, directly comparing per entry with all the boundaries needs $2 ^ { k }$ comparisons, which is very inefficient. Instead, we can make use of the ascending property in $\\mathbf { v }$ . Hierarchically, we partition the codes of $\\mathbf { v }$ evenly into two ordered sub-sets, i.e., $\\mathbf { v } _ { 1 : m / 2 }$ and $\\mathbf { v } _ { m / 2 + 1 : m }$ with $m$ defined as the length of $\\mathbf { v }$ . If $w < ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ , its feasible codes are then optimally restricted to $\\mathbf { v } _ { 1 : m / 2 }$ . And if $w \\ge ( v _ { m / 2 } + v _ { m / 2 + 1 } ) / 2$ , its feasible codes become $\\mathbf { v } _ { m / 2 + 1 : m }$ By recursively evenly partition the ordered feasible codes, we can then efficiently determine the ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Algorithm 2: Alternating Multi-bit Quantization ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Require :Ensure : on weight $\\mathbf { w } \\in \\mathbb { R } ^ { n }$ , number of bits $k$ , total iterations $T$ $\\{ \\alpha _ { i } , \\mathbf { \\bar { b } } _ { i } \\} _ { i = 1 } ^ { k }$   \n1 Greedy Initialize $\\{ \\alpha _ { i } , \\mathbf { b } _ { i } \\} _ { i = 1 } ^ { k }$ as (4)   \n2 for iter $ 1$ to $T$ do   \n3 Update $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { k }$ as (5)   \n4 Construct $\\mathbf { v }$ of all feasible codes in accending order   \n5 Update $\\{ { \\bf b } _ { i } \\} _ { i = 1 } ^ { k }$ as Algorithm 1.   \n6 end ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "optimal code for per entry by only $k$ comparisons. The whole procedure is in fact a binary search tree. We summarize it in Algorithm 1. Note that once getting the quantization code, it is straightforward to map to the binary code b. Also, by maintaining a mask vector with the same size as w to indicate the partitions, we could operate BST for all entries simultaneously. To give a better illustration, we give a binary tree example for $k = 2$ in Fig. 2. Note that for $k = 2$ , we can even derive the optimal codes by a closed form solution, i.e., $\\mathbf { b } _ { 1 } = \\mathrm { s i g n } ( \\mathbf { w } )$ and $\\mathbf { b } _ { 2 } = \\mathrm { s i g n } ( \\mathbf { w } - \\alpha _ { 1 } \\mathbf { b } _ { 1 } )$ with $\\alpha _ { 1 } \\geq \\alpha _ { 2 } \\geq 0$ . ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Under the above observation, let us reconsider the refined greedy approximation (Guo et al., 2017) introduced in Section 2. After modification on the computed $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { j }$ as (5), $\\{ { \\bf b } _ { i } \\} _ { i = 2 } ^ { j }$ are no longer optimal while the method keeps all of them fixed. To improve the refined greedy approximation, alternating minimizing $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { k }$ and $\\{ { \\bf b } _ { i } \\} _ { i = 1 } ^ { k }$ becomes a natural choice. Once getting $\\{ { \\bf b } _ { i } \\} _ { i = 1 } ^ { k }$ as described above, we can optimize $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { k }$ as (5). In real experiments, we find that by greedy initialization as (4), only two alternating cycles is good enough to find high precision quantization. For better illustration, we summarize our alternating minimization in Algorithm 2. For updating $\\{ \\alpha _ { i } \\} _ { i = 1 } ^ { k }$ , we need $2 k ^ { 2 } n$ binary operations and $k n$ non-binary operations. Combining kn non-binary operations to determine the binary code, for total $T$ alternating cycles, we thus need $2 T k ^ { 2 } n$ binary operations and $2 ( T + 1 ) k n$ non-binary operations to quantize $\\mathbf { w } \\in \\mathbb { R } ^ { n }$ into $k$ -bit, with the extra $2 k n$ corresponding to greedy initialization. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 APPLY ALTERNATING MULTI-BIT QUANTIZATION TO RNNS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation. We firstly introduce the implementation details for quantizing RNN. For simplicity, we consider the one layer LSTM for language model. The goal is to predict the next word indexed by $t$ in a sequence of one-hot word tokens $( y _ { 1 } ^ { * } , \\dots , y _ { N } ^ { * } )$ as follows: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/79ac58cfdd96dab4a2e9ffd4a1b575ab0b1e678105de8e87cf1f5c592e494dbb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { x } _ { t } = \\mathbf { W } _ { e } ^ { T } \\mathbf { y } _ { t - 1 } ^ { * } , } \\\\ & { \\mathbf { i } _ { t } , \\mathbf { f } _ { t } , \\mathbf { 0 } _ { t } , \\mathbf { g } _ { t } = \\sigma ( \\mathbf { W } _ { i } \\mathbf { x } _ { t } + \\mathbf { b } _ { i } + \\mathbf { W } _ { h } \\mathbf { h } _ { t - 1 } + \\mathbf { b } _ { h } ) , } \\\\ & { \\mathbf { c } _ { t } = \\mathbf { f } _ { t } \\odot \\mathbf { c } _ { t - 1 } + \\mathbf { i } _ { t } \\odot \\mathbf { g } _ { t } , \\quad \\mathbf { h } _ { t } = \\mathbf { o } _ { t } \\odot \\operatorname { t a n h } ( \\mathbf { c } _ { t } ) , } \\\\ & { \\mathbf { y } _ { t } = \\operatorname { s o f t m a x } ( \\mathbf { W } _ { s } \\mathbf { h } _ { t } + \\mathbf { b } _ { s } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "where $\\sigma$ represents the activation function. In the above formulation, the multiplication between the weight matrices and the vectors $\\mathbf { x } _ { t }$ and $\\mathbf { h } _ { t }$ occupy most of the computation. This is also where we apply quantization to. For the weight matrices, We do not apply quantization on the full but rather row by row. During the matrix vector product, we can firstly execute the binary multiplication. Then element-wisely multiply the obtained binary vector with the high precision scaling coefficients. Thus little extra computation results while much more freedom is brought to better approximate the weights. We give an illustration on the left part of Fig. 3. Due to one-hot word tokens, $\\mathbf { x } _ { t }$ corresponds to one specific row in the quantized ${ \\mathbf W } _ { e }$ . It needs no more quantization. Different from the weight matrices, $\\mathbf { h } _ { t }$ depends on the input, which needs to be quantized on-line during inference. For consistent notation with existing work, e.g., (Hubara et al., 2016b; Zhou et al., 2017), we also call quantizing on $\\mathbf { h } _ { t }$ as quantizing on activation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For $\\mathbf { W } \\in \\mathbb { R } ^ { m \\times n }$ and $\\mathbf { h } _ { t } \\in \\mathbb { R } ^ { n }$ , the standard matrix-vector product needs $2 m n$ operations. For the quantized product between $k _ { w }$ -bit $\\mathbf { W }$ and $k _ { h }$ -bit $\\mathbf { h _ { t } }$ , we have $2 k _ { w } k _ { h } m n + 4 k _ { h } ^ { 2 } n$ binary operations and $6 k _ { h } n + 2 k _ { w } k _ { h } m$ non-binary operations, where $6 k _ { h } n$ corresponds to the cost of alternating approximation $T = 2$ ) and $2 k _ { w } k _ { h } m$ corresponds to the final product with coefficients. As the binary multiplication operates in 1 bit, whereas the full precision multiplication operates in 32 bits, despite the feasible implementations, the acceleration can be $3 2 \\times$ in theory. For alternating quantization here, the overall theoretical acceleration is thus computed as γ = 2mn132 (2kwkhmn+4k2nn)+6khn+2kwkhm . Suppose that LSTM has hidden states $n = 1 0 2 4$ , then we have $\\mathbf { W } _ { h } \\in \\mathbb { R } ^ { 4 0 9 6 \\times 1 0 2 4 }$ . The acceleration ratio becomes roughly $7 . 5 \\times$ for $( k _ { h } , k _ { w } ) = ( 2 , 2 )$ and $3 . 5 \\times$ for $( k _ { h } , k _ { w } ) = ( 3 , 3 )$ . In addition to binary operations, the acceleration in real implementations can be largely affected by the size of the matrix, where much memory reduce can result in better utilizing in the limited faster cache. We implement the binary multiplication kernel in CPUs. Compared with the much optimized Intel Math Kernel Library (MKL) on full precision matrix vector multiplication, we can roughly achieve $6 \\times$ for $( k _ { h } , k _ { w } ) = ( \\bar { 2 } , 2 )$ and $3 \\times$ for $( \\bar { k } _ { h } , k _ { w } ) = ( 3 , 3 )$ . For more details, please refer to Appendix A. ",
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+    {
+        "type": "image",
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+        "image_caption": [
+            "Figure 3: Illustration of quantized matrix vector multiplication (left part). The matrix is quantized row by row, which provides more freedom to approximate while adds little extra computation. By reformulating as the right part, we can make full use of the intrinsic parallel binary matrix vector multiplication for further acceleration. "
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+        "text": "As indicated in the left part of Fig. 3, the binary multiplication can be conducted sequentially by associativity. Although the operation is suitable for parallel computing by synchronously conducting the multiplication, this needs extra effort for parallelization. We instead concatenate the binary codes as shown in the right part of Fig. 3. Under such modification, we are able to make full use of the much optimized inner parallel matrix multiplication, which gives the possibility for further acceleration. The final result is then obtained by adding all partitioned vectors together, which has little extra computation. ",
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+    {
+        "type": "text",
+        "text": "Training. As firstly proposed by Courbariaux et al. (2015), during the training of quantized neural network, directly adding the moderately small gradients to quantized weights will result in no change on it. So they maintain a full precision weight to accumulate the gradients then apply quantization in every mini-batch. In fact, the whole procedure can be mathematically formulated as a bi-level optimization (Colson et al., 2007) problem: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/295a4ccf6c99038201b190d1defae67ec2d64f8e2025ab7d836916bce260a808.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\displaystyle \\operatorname* { m i n } _ { \\mathbf { w } , \\{ \\alpha _ { i } , \\mathbf { b } _ { i } \\} _ { i = 1 } ^ { k } } f ( \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\mathbf { b } _ { i } ) } & { } \\\\ { s . t . } & { \\{ \\alpha _ { i } , \\mathbf { b } _ { i } \\} _ { i = 1 } ^ { k } =  \\arg \\operatorname* { m i n } _ { \\{ \\alpha _ { i } ^ { \\prime } , \\mathbf { b } _ { i } ^ { \\prime } \\} _ { i = 1 } ^ { k } } \\| \\mathbf { w } - \\displaystyle \\sum _ { i = 1 } ^ { k } \\alpha _ { i } ^ { \\prime } \\mathbf { b } _ { i } ^ { \\prime } \\| ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Denote the quantized weight as $\\begin{array} { r } { \\hat { \\mathbf { w } } = \\sum _ { i = 1 } ^ { k } \\alpha _ { i } \\mathbf { b } _ { i } } \\end{array}$ . In the forward propagation, we derive wˆ from the full precision w in the lower-level problem and apply it to the upper-level function $f ( \\cdot )$ , i.e., RNN in this paper. During the backward propagation, the derivative $\\frac { \\partial f } { \\partial \\hat { \\bf w } }$ is propagated back to w through the lower-level function. Due to the discreteness of $\\mathbf { b } _ { i }$ , it is very hard to model the implicit dependence2015), i.e., $\\hat { \\mathbf { w } }$ w. So we also adopt the “straight-through estimate” as (. To compute the derivative on the quantized hidden state urbariaux et al.,, the same trick $\\begin{array} { r } { \\frac { \\partial f } { \\partial \\mathbf { w } } = \\frac { \\partial f } { \\partial \\hat { \\mathbf { w } } } } \\end{array}$ $\\mathbf { h } _ { t }$ is applied. During the training, we find the same phenomenon as Hubara et al. (2016b) that some entries of w can grow very large, which become outliers and harm the quantization. Here we simply clip w in the range of $[ - 1 , 1 ]$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/f32903099304297066dd8ad6f455d3ac7a116e7819460ab7ddb06790480a84eb.jpg",
+        "table_caption": [
+            "Table 1: Measurement on the approximation of different quantization methods, e.g., Uniform (Hubara et al., 2016b), Balanced (Zhou et al., 2017), Greedy (Guo et al., 2017), Refined (Guo et al., 2017), and our Alternating method, see Section 2. We apply these methods to quantize the full precision pre-trained weight of LSTM on the PTB dataset. The best values are in bold. W-bits represents the number of weight bits and FP denotes full precision. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4>Relative MSE</td><td rowspan=1 colspan=4>Testing PPW</td></tr><tr><td rowspan=1 colspan=1>W-Bits</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>1.070</td><td rowspan=1 colspan=1>0.404</td><td rowspan=1 colspan=1>0.302</td><td rowspan=1 colspan=1>283.2</td><td rowspan=1 colspan=1>227.3</td><td rowspan=1 colspan=1>216.3</td><td rowspan=5 colspan=1>89.8</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>0.891</td><td rowspan=1 colspan=1>0.745</td><td rowspan=1 colspan=1>0.702</td><td rowspan=1 colspan=1>10287.6</td><td rowspan=1 colspan=1>9106.4</td><td rowspan=1 colspan=1>8539.8</td></tr><tr><td rowspan=1 colspan=1>Greedy</td><td rowspan=1 colspan=1>0.146</td><td rowspan=1 colspan=1>0.071</td><td rowspan=1 colspan=1>0.042</td><td rowspan=1 colspan=1>118.9</td><td rowspan=1 colspan=1>99.4</td><td rowspan=1 colspan=1>95.0</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>0.137</td><td rowspan=1 colspan=1>0.060</td><td rowspan=1 colspan=1>0.030</td><td rowspan=1 colspan=1>105.3</td><td rowspan=1 colspan=1>95.4</td><td rowspan=1 colspan=1>93.1</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>0.125</td><td rowspan=1 colspan=1>0.043</td><td rowspan=1 colspan=1>0.019</td><td rowspan=1 colspan=1>103.1</td><td rowspan=1 colspan=1>93.8</td><td rowspan=1 colspan=1>91.4</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/729610687d61ece6eec8717ce1637b7af445998e4a11d89c3cb420d0e988385f.jpg",
+        "table_caption": [
+            "Table 2: Quantization on the full precision pre-trained weight of GRU on the PTB dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4>Relative MSE</td><td rowspan=1 colspan=4>Testing PPW</td></tr><tr><td rowspan=1 colspan=1>W-Bits</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>6.138</td><td rowspan=1 colspan=1>3.920</td><td rowspan=1 colspan=1>3.553</td><td rowspan=1 colspan=1>3161906.6</td><td rowspan=1 colspan=1>771259.6</td><td rowspan=1 colspan=1>715781.9</td><td rowspan=5 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>1.206</td><td rowspan=1 colspan=1>1.054</td><td rowspan=1 colspan=1>1.006</td><td rowspan=1 colspan=1>2980.4</td><td rowspan=1 colspan=1>3396.3</td><td rowspan=1 colspan=1>3434.1</td></tr><tr><td rowspan=1 colspan=1>Greedy</td><td rowspan=1 colspan=1>0.377</td><td rowspan=1 colspan=1>0.325</td><td rowspan=1 colspan=1>0.304</td><td rowspan=1 colspan=1>135.7</td><td rowspan=1 colspan=1>105.5</td><td rowspan=1 colspan=1>99.2</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>0.128</td><td rowspan=1 colspan=1>0.055</td><td rowspan=1 colspan=1>0.030</td><td rowspan=1 colspan=1>111.6</td><td rowspan=1 colspan=1>99.1</td><td rowspan=1 colspan=1>97.0</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>0.120</td><td rowspan=1 colspan=1>0.044</td><td rowspan=1 colspan=1>0.021</td><td rowspan=1 colspan=1>110.3</td><td rowspan=1 colspan=1>97.3</td><td rowspan=1 colspan=1>95.2</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ON THE LANGUAGE MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we conduct quantization experiments on language models. The two most well-known recurrent neural networks, i.e., LSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho et al., 2014), are evaluated. As they are to predict the next word, the performance is measured by perplexity per word (PPW) metric. For all experiments, we initialize with the pre-trained model and using vanilla SGD. The initial learning rate is set to 20. Every epoch we evaluate on the validation dataset and record the best value. When the validation error exceeds the best record, we decrease learning rate by a factor of 1.2. Training is terminated once the learning rate less than 0.001 or reaching the maximum epochs, i.e., 80. The gradient norm is clipped in the range $[ - 0 . 2 5 , 0 . 2 5 ]$ . We unroll the network for 30 time steps and regularize it with the standard dropout (probability of dropping out units equals to 0.5) (Zaremba et al., 2014). For simplicity of notation, we denote the methods using uniform, balanced, greedy, refined greedy, and our alternating quantization as Uniform, Balanced, Greedy, Refined, and Alternating, respectively. ",
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+    {
+        "type": "text",
+        "text": "Peen Tree Bank. We first conduct experiments on the Peen Tree Bank (PTB) corpus (Marcus et al., 1993), using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains 929K training tokens, 73K validation tokens, and 82K test tokens. For fair comparison with existing works, we also use LSTM and GRU with 1 hidden layer of size 300. To have a glance at the approximation ability of different quantization methods as detailed in Section 2, we firstly conduct experiments by directly quantizing the trained full precision weight (neither quantization on activation nor retraining). Results on LSTM and GRU are shown in Table 1 and Table 2, respectively. The left parts record the relative mean squared error of quantized weight matrices with full precision one. We can see that our proposed Alternating can get much lower error across all varying bit. We also measure the testing PPW for the quantized weight as shown in the right parts of Table 1 and 2. The results are in consistent with the left part, where less errors result in lower testing PPW. Note that Uniform and Balanced quantization are rule-based and not aim at minimizing the error. Thus they can have much worse result by direct approximation. We also repeat the experiment on other datasets. For both LSTM and GRU, the results are very similar to here. ",
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+        "type": "table",
+        "img_path": "images/ba8c5690b3e5b70c799fda90fcf21856451bf9c6406275f771c2b232ad9d62ba.jpg",
+        "table_caption": [
+            "Table 3: Testing PPW of multi-bit quantized LSTM and GRU on the PTB dataset. W-Bits and A-Bits represent the number of weight and activation bits, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=6>LSTM</td><td rowspan=1 colspan=5>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits /A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>4/4</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>4/4</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Uniform</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>220</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Balanced</td><td rowspan=1 colspan=1>126</td><td rowspan=1 colspan=1>123</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>114</td><td rowspan=1 colspan=1>107</td><td rowspan=1 colspan=1>142</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>116</td><td rowspan=1 colspan=1>100</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>100.3</td><td rowspan=1 colspan=1>95.6</td><td rowspan=1 colspan=1>91.3</td><td rowspan=1 colspan=1>二</td><td rowspan=2 colspan=1>89.8</td><td rowspan=1 colspan=1>105.1</td><td rowspan=1 colspan=1>100.3</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>二</td><td rowspan=2 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>95.8</td><td rowspan=1 colspan=1>91.9</td><td rowspan=1 colspan=1>87.9</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>101.2</td><td rowspan=1 colspan=1>97.0</td><td rowspan=1 colspan=1>92.9</td><td rowspan=1 colspan=1>1</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c95dc3c93c0f67f6b4ae0592e295793741389ebfde8607035d46c48e0d20929b.jpg",
+        "table_caption": [
+            "Table 4: Testing PPW of multi-bit quantized LSTM and GRU on the WikiText-2 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=5>LSTM</td><td rowspan=1 colspan=4>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>108.7</td><td rowspan=1 colspan=1>105.8</td><td rowspan=1 colspan=1>102.2</td><td rowspan=2 colspan=1>100.1</td><td rowspan=1 colspan=1>117.2</td><td rowspan=1 colspan=1>114.1</td><td rowspan=1 colspan=1>111.8</td><td rowspan=2 colspan=1>106.7</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>106.1</td><td rowspan=1 colspan=1>102.7</td><td rowspan=1 colspan=1>98.7</td><td rowspan=1 colspan=1>113.7</td><td rowspan=1 colspan=1>110.2</td><td rowspan=1 colspan=1>106.4</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/23fa33df4e12d12ae42ad0561f53ecd571c860478cc26c956f6a0b8e09396296.jpg",
+        "table_caption": [
+            "Table 5: Testing PPW of multi-bit quantized LSTM and GRU on the Text8 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=5>LSTM</td><td rowspan=1 colspan=4>GRU</td></tr><tr><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>2/3</td><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>FP/FP</td></tr><tr><td rowspan=1 colspan=1>Refined</td><td rowspan=1 colspan=1>135.6</td><td rowspan=1 colspan=1>122.3</td><td rowspan=1 colspan=1>110.2</td><td rowspan=2 colspan=1>101.1</td><td rowspan=1 colspan=1>135.8</td><td rowspan=1 colspan=1>126.9</td><td rowspan=1 colspan=1>118.3</td><td rowspan=2 colspan=1>111.6</td></tr><tr><td rowspan=1 colspan=1>Alternating (ours)</td><td rowspan=1 colspan=1>108.8</td><td rowspan=1 colspan=1>105.1</td><td rowspan=1 colspan=1>98.8</td><td rowspan=1 colspan=1>124.5</td><td rowspan=1 colspan=1>118.7</td><td rowspan=1 colspan=1>114.0</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We then conduct experiments by quantizing both weights and activations. We train with the batch size 20. The final result is shown in Table 3. Besides comparing with the existing works, we also conduct experiment for Refined as a competitive baseline. We do not include Greedy as it is already shown to be much inferior to the refined one, see, e.g., Table 1 and 2. As Table 3 shows, our full precision model can attain lower PPW than the existing works. However, when considering the gap between quantized model with the full precision one, our alternating quantized neural network is still far better than existing works, i.e., Uniform (Hubara et al., 2016b) and Balanced (Zhou et al., 2017). Compared with Refined, our Alternating quantization can achieve compatible performance using 1-bit less quantization on weights or activations. In other words, under the same tolerance of accuracy drop, Alternating executes faster and uses less memory than Refined. We can see that our $3 / 3$ weights/activations quantized LSTM can achieve even better performance than full precision one. A possible explanation is due to the regularization introduced by quantization (Hubara et al., 2016b). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "WikiText-2 (Merity et al., 2017) is a dataset released recently as an alternative to PTB. It contains 2088K training, 217K validation, and 245K test tokens, and has a vocabulary of 33K words, which is roughly 2 times larger in dataset size, and 3 times larger in vocabulary than PTB. We train with one layer’s hidden state of size 512 and set the batch size to 100. The result is shown in Table 4. Similar to PTB, our Alternating can use 1-bit less quantization to attain compatible or even lower PPW than Refined. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Text8. In order to determine whether Alternating remains effective with a larger dataset, we perform experiments on the Text8 corpus (Mikolov et al., 2014). Here we follow the same setting as (Xie et al., 2017). The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in 15.3M training tokens, 848K validation tokens, and 855K test tokens. We also preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. We train LSTM and GRU with one hidden layer of size 1024 and set the batch size to 100. The result is shown in Table 5. For LSTM on the left part, Alternating achieves excellent performance. By only 2-bit quantization on weights and activations, it exceeds Refined with 3-bit. The 2-bit result is even better than that reported in (Xie et al., 2017), where LSTM adding noising schemes for regularization can only attain 110.6 testing PPW. For GRU on the right part, although Alternating is much better than Refined, the 3-bit quantization still has gap with full precision one. We attribute that to the unified setting of hyper-parameters across all experiments. With specifically tuned hyper-parameters on this dataset, one may make up for that gap. ",
+        "bbox": [
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+            104,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that our alternating quantization is a general technique. It is not only suitable for language models here. For a comprehensive verification, we apply it to image classification tasks. In both RNNs and feedforward neural networks, our alternating quantization also achieves the lowest testing error among all compared methods. Due to space limitation, we deter the results to Appendix B. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            367,
+            328,
+            383
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we address the limitations of RNNs, i.e., large memory and high latency, by quantization. We formulate the quantization by minimizing the approximation error. Under the key observation that some parameters can be singled out when others fixed, a simple yet effective alternating method is proposed. We apply it to quantize LSTM and GRU on language models. By 2-bit weights and activations, we achieve only a reasonably accuracy loss compared with full precision one, with ${ \\sim } 1 6 \\times$ reduction in memory and ${ \\sim } 6 \\times$ real acceleration on CPUs. By 3-bit quantization, we can attain compatible or even better result than the full precision one, with ${ \\sim } 1 0 . 5 \\times$ reduction in memory and ${ \\sim } 3 \\times$ real acceleration. Both beat existing works with a large margin. We also apply our alternating quantization to image classification tasks. In both RNNs and feedforward neural networks, the method can still achieve very plausible performance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            558,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the reviewers for their suggestions on the manuscript. Zhouchen Lin is supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502), National Natural Science Foundation (NSF) of China (grant nos. 61625301 and 61731018), Qualcomm, and Microsoft Research Asia. Hongbin Zha is supported by Natural Science Foundation (NSF) of China (No. 61632003). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            679,
+            285,
+            694
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kyunghyun Cho, Bart Van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using RNN encoder-decoder for statistical machine translation. arXiv:1406.1078, 2014. ",
+        "bbox": [
+            174,
+            702,
+            823,
+            743
+        ],
+        "page_idx": 8
+    },
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+        "text": "APPENDIX ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A BINARY MATRIX VECTOR MULTIPLICATION IN CPUS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            141,
+            656,
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c58b568effb55005c4a79120a2de80894348d8eb77a7e44fbac6d27f8da343f2.jpg",
+        "table_caption": [
+            "Table 6: Computing time of the binary matrix vector multiplication in CPUs, where Quant represents the cost to execute our alternating quantization on-line. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Weight Size</td><td rowspan=1 colspan=1>W-Bits / A-Bits</td><td rowspan=1 colspan=1>Total (ms)</td><td rowspan=1 colspan=1>Quant (ms)</td><td rowspan=1 colspan=1>Quant / Total</td><td rowspan=1 colspan=1>Acceleration</td></tr><tr><td rowspan=3 colspan=1>4096 × 1024</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>0.07</td><td rowspan=1 colspan=1>20%</td><td rowspan=1 colspan=1>5.6x</td></tr><tr><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>0.72</td><td rowspan=1 colspan=1>0.11</td><td rowspan=1 colspan=1>15%</td><td rowspan=1 colspan=1>2.7x</td></tr><tr><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>1.95</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1.0×</td></tr><tr><td rowspan=3 colspan=1>42000 × 1024</td><td rowspan=1 colspan=1>2/2</td><td rowspan=1 colspan=1>3.17</td><td rowspan=1 colspan=1>0.07</td><td rowspan=1 colspan=1>2.2%</td><td rowspan=1 colspan=1>6.0×</td></tr><tr><td rowspan=1 colspan=1>3/3</td><td rowspan=1 colspan=1>6.46</td><td rowspan=1 colspan=1>0.11</td><td rowspan=1 colspan=1>1.7%</td><td rowspan=1 colspan=1>3.0x</td></tr><tr><td rowspan=1 colspan=1>FP/FP</td><td rowspan=1 colspan=1>19.10</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1.0x</td></tr></table>",
+        "bbox": [
+            207,
+            223,
+            790,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we discuss the implementation of the binary multiplication kernel in CPUs. The binary multiplication is divided into two steps: Entry-wise XNOR operation (corresponding to entry-wise product in the full precision multiplication) and bit count operation for accumulation (corresponding to compute the sum of all multiplied entries in the full precision multiplication). We test it on Intel Xeon E5-2682 v4 $@$ 2.50 GHz CPU. For the XNOR operation, we use the Single instruction, multiple data (SIMD) _mm256 _xor_ps, which can execute 256 bit simultaneously. For the bit count operation, we use the function _popcnt64 (Note that this step can further be accelerated by the up-coming instruction _mm512 _popcnt_epi64 , which can execute 512 bits simultaneously. Similarly, the XNOR operation can also be further accelerated by the up-coming _mm512 _xor_ps instruction to execute 512 bits simultaneously). We compare with the much optimized Intel Math Kernel Library (MKL) on full precision matrix vector multiplication and execute all codes in the single-thread mode. We conduct two scales of experiments: a matrix of size $4 0 9 6 \\times 1 0 2 4$ multiplying a vector of size 1024 and a matrix of size $4 2 0 0 0 \\times 1 0 2 4$ multiplying a vector of size 1024, which respectively correspond to the hidden state product ${ \\bf W } _ { h } \\mathbf h _ { t - 1 }$ and the softmax layer $\\mathbf { W } _ { s } \\mathbf { h } _ { t }$ for Text8 dataset during inference with batch size of 1 (See Eq. (6)). The results are shown in Table 6. We can see that our alternating quantization step only accounts for a small portion of the total executing time, especially for the larger scale matrix vector multiplication. Compared with the full precision one, the binary multiplication can roughly achieve $6 \\times$ acceleration with 2-bit quantization and $3 \\times$ acceleration with 3-bit quantization. Note that this is only a simple test on CPU. Our alternating quantization method can also be extended to GPU, ASIC, and FPGA. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sequential MNIST. As a simple illustration to show that our alternating quantization is not limited for texts, we conduct experiments on the sequential MNIST classification task (Cooijmans et al., 2017). The dataset consists of a training set of 60K and a test set of 10K $2 8 \\times 2 8$ gray-scale images. Here we divide the last 5000 training images for validation. In every time, we sequentially use one row of the image as the input $( 2 8 \\times 1 )$ , which results in a total of 28 time steps. We use 1 hidden layer’s LSTM of size 128 and the same optimization hyper-parameters as the language models. Besides the weights and activations, the inputs are quantized. The testing error rates for 1-bit input, 2-bit weight, and 2-bit activation are shown in 7, where our alternating quantized method still achieves plausible performance in this task. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "MLP on MNIST. The alternating quantization proposed in this work is a general technique. It is not only suitable for RNNs, but also for feed-forward neural networks. As an example, we firstly conduct a classification task on MNIST and compare with existing work (Li et al., 2017). The method proposed in (Li et al., 2017) is intrinsically a greedy multi-bit quantization method. For fair comparison, we follow the same setting. We use the MLP consisting of 3 hidden layers of 4096 units and an L2-SVM output layer. No convolution, preprocessing, data augmentation or pre-training is used. We also use ADAM (Kingma & Ba, 2015) with an exponentially decaying learning rate and Batch Normalization (Ioffe & Szegedy, 2015) with a batch size 100. The testing error rates for 2-bit input, 2-bit weight, and 1-bit activation are shown in Table 8. Among all the compared multi-bit quantization methods, our alternating one achieves the lowest testing error. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4183c4ab1dd729051f0b41d54236233f2a36fce011e430c3c102643c8b56f0a6.jpg",
+        "table_caption": [
+            "Table 7: Testing error rate of LSTM on MNIST with 1-bit input, 2-bit weight, and 2-bit activation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision</td><td>1.10%</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>1.39 %</td></tr><tr><td>Alternating (ours)</td><td>1.19 %</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2db25ca1b311a6cee1fa97c6405857f50a5235acb206b1d79adb870e7d3d28c1.jpg",
+        "table_caption": [
+            "Table 8: Testing error rate of MLP on MNIST with 2-bit input, 2-bit weight, and 1-bit activation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision</td><td>0.97%</td></tr><tr><td>Greedy (reported in (Li et al., 2017))</td><td>1.25 %</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>1.22 %</td></tr><tr><td>Alternating (ours)</td><td>1.13 %</td></tr></table>",
+        "bbox": [
+            297,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/58dec78e7c2bec79342f864100c73605dec346046df51c04b9b1b39de7b17bd0.jpg",
+        "table_caption": [
+            "Table 9: Testing error rate of CNN on CIFAR-10 with 2-bit weight and 1-bit activation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Testing Error Rate</td></tr><tr><td>Full Precision (reported in (Hou et al., 2017))</td><td>11.90 %</td></tr><tr><td> XNOR-Net (1-bit weight &amp; activation, reported in (Hou et al., 2017))</td><td>12.62 %</td></tr><tr><td>Refined (Guo et al., 2017)</td><td>12.08 %</td></tr><tr><td>Alternating (ours)</td><td>11.70 %</td></tr></table>",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "CNN on CIFAR-10. We then conduct experiments on CIFAR-10 and follow the same setting as (Hou et al., 2017). That is, we use 45000 images for training, another 5000 for validation, and the remaining 10000 for testing. The images are preprocessed with global contrast normalization and ZCA whitening. We also use the VGG-like architecture (Simonyan & Zisserman, 2015): ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2aa9d0bd3120c840c1eb4659c2dd952a2386d59d918bff7d0bb2ad9efbc083a.jpg",
+        "text": "$$\n( 2 \\times 1 2 8 ~ \\mathrm { C 3 } ) - \\mathrm { M P 2 - ( 2 \\times 2 5 6 ~ \\mathrm { C 3 } ) - M P 2 - ( 2 \\times 5 1 2 ~ \\mathrm { C 3 } ) - M P 2 - ( 2 \\times 1 0 2 4 ~ \\mathrm { F C } ) - 1 0 ~ \\mathrm { S V M } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where C3 is a $3 \\times 3$ convolution layer, and MP2 is a $2 \\times 2$ max-pooling layer. Batch Normalization, with a mini-batch size of 50, and ADAM are used. The maximum number of epochs is 200. The learning rate starts at 0.02 and decays by a factor of 0.5 after every 30 epochs. The testing error rates for 2-bit weight and 1-bit activation are shown in Table 9, where our alternating method again achieves the lowest test error rate among all compared quantization methods. ",
+        "bbox": [
+            173,
+            693,
+            825,
+            763
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SJzvDjAcK7/SJzvDjAcK7.md

@@ -0,0 +1,484 @@
+# INTRIGUING PROPERTIES OF LEARNED REPRESENTA-TIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+A key feature of neural networks, particularly deep convolutional neural networks, is their ability to “learn” useful representations from data. The very last layer of a neural network is then simply a linear model trained on these “learned” representations. Despite their numerous applications in other tasks such as classification, retrieval, clustering etc., a.k.a. transfer learning, not much work has been published that investigates the structure of these representations or indeed whether structure can be imposed on them during the training process.
+
+In this paper, we study the effective dimensionality of the learned representations by models that have proved highly successful for image classification. We focus on ResNet-18, ResNet-50 and VGG-19 and observe that when trained on CIFAR10 or CIFAR100, the learned representations exhibit a fairly low rank structure. We propose a modification to the training procedure, which further encourages low rank structure on learned activations. Empirically, we show that this has implications for robustness to adversarial examples and compression.
+
+# 1 INTRODUCTION
+
+Among the many successes of deep (convolutional) neural networks, an intriguing aspect has been their ability to generate representations of raw data that are useful in several tasks, usually known as representation learning. In the early days of deep learning, it was common to use unsupervised learning models, e.g. auto-encoders or Restricted Boltzmann machines, to learn useful representations of complex data (Hinton & Salakhutdinov, 2006; Vincent et al., 2010); more recently, it has been observed that hidden layers of neural networks trained in a completely supervised fashion can also be used as learned representations Zeiler & Fergus (2014); Sermanet et al. (2014); Donahue et al. (2014).
+
+Essentially, for most models trained in a supervised fashion, the vector of activations in the penultimate layer (or at least in a layer close to the output) is a learned representation of the raw (for the purposes of this paper image) data. The final layer of a neural network is typically simply a multi-class logistic regression model. In this work, we mostly focus on the ResNet-18 and ResNet-50 (He et al., 2016), though we also report some results on VGG-networks (Simonyan & Zisserman, 2014). Although, several aspects of architectures of neural networks have been widely studied in the recent years, there has been little work on understanding the nature of these learned representations.
+
+The learned representations have often been used in other tasks related to classification, retrieval, clustering etc. (often unrelated to the original classification problem) with a good degree of success (a.k.a. transfer learning) (Kiros et al., 2014; Lin & Parikh, 2015). Classically, representation learning has focused on finding low-dimensional structures or independent components through various models, from component analysis to auto-encoders. However, while hidden layers of neural networks are widely used as representations, as far as we are aware, not much work has been published on studying their structure. The current work studies these learned representations; our focus has been on the effective dimensionality, (as opposed to the actual dimension) of these learned representations. A ResNet-18/50 network consists of four ResNet blocks (each of which includes several convolutional layers and skip connections). We consider the dimensionality of the activations obtained at the end of the third and fourth ResNet block. Each data point $\mathbf { x }$ maps to a vector $\mathbf { a } \in \mathbb { R } ^ { m }$ , where $m$ is the number of units in one of the aforementioned layers; this vector is a learned representation of $\mathbf { x }$ .
+
+We propose a modification to the training procedure to specifically make the activations (approximately) lie in a low rank space; more precisely, we add a term to the loss that encourages the activations in certain layers to lie in a low-rank affine subspace. The modified training process results in essentially no loss in accuracy (in some cases even shows modest gains) and further enhances the low-rank nature of the learned representations. The modification “adds” virtual low-rank layers to the model that ensure that the learned representations roughly lie in a low-rank space. The modified objective function is optimized using an alternate minimization approach, reminiscent of that used in iterative hard thresholding (Blumensath & Davies, 2009) or singular value projection (Jain et al., 2010). Using a na¨ıve singular value thresholding approach would render the training intractable for all practical purposes; we use a column sampling based Nystrom method (Williams & Seeger, 2001; ¨ Halko et al., 2011) to achieve significant speed-up, though at the cost of not getting the optimal low rank projections. One can view this modified training process as a way to constrain the neural network, though in a way that is very different to the widely used sparsity inducing methods (eg. (Anwar et al., 2017; Wen et al., 2016)) or structurally constrained methods (eg. (Moczulski et al., 2015; Liu et al., 2015)) that seek to tackle the problem of over-parameterization.
+
+Finally, we also investigate the benefits of learning low-rank representations. One obvious benefit is the ability to compress the embeddings when they are used in other applications. The fact that these learned representations (approximately) lie in a low-dimensional (affine) space yields a natural (lossy) compression scheme. Furthermore, we investigate the robustness of networks trained in this fashion to adversarial attacks (Szegedy et al., 2013). Our experimental evaluation shows that networks trained in this fashion are substantially more robust to adversarial attacks than the standard architectures. In our experiments we look at gradient sign methods, its variants (Kurakin et al., 2016; 2017) and DeepFool (Moosavi-Dezfooli et al., 2016). We further perform empirical evaluation showing that when we train SVM classifiers on learned representations (or their low rank projections), the networks trained according to our modified procedure yield significantly more accurate predictions, especially when using very low rank projections of the learned representations, and are in general more robust to adversarial attacks.
+
+Related Work: Empirical work by Oyallon (2017) suggests that conditioned on the class, these representations seem to (approximately) lie in a low-rank (affine) space; his work focuses on a different network architecture. Recent (and parallel) work by Lezama et al. (2017) promotes classconditional low rank embeddings and orthogonality between classes within a single mini-batch to create better discriminative models. They use nuclear norm as a relaxation of rank of a matrix and a loss reminiscent of Qiu & Sapiro (2015) to achieve this and due to their focus on mini-batches, it is unclear if the data as a whole exhibits low-rank structure. Their experiments do not consider adversarial robustness or compression.
+
+From the point of view compression, it is natural to look at methods considering low rank approximations of model parameters (Jaderberg et al., 2014; Denton et al., 2014) as possible alternatives to our algorithm. However, low rank weights do not necessarily create low rank activations due to the non-linearities in neural networks. For similar reasons, reconstructions using auto-encoders also does not exhibit a low-rank structure (See Appendix A).
+
+Organisation. The rest of the paper is organized as follows. Section 2 describes the relevant notation, the basic optimization problem and our training procedure. In Section 3, we list our results demonstrating the benefits of the modified training procedure to obtain better low-rank representations and in Section 4, we present results that demonstrate our network’s robustness to adversarial attacks.
+
+# 2 LOW RANK PRIOR ON ACTIVATIONS
+
+We define the activation matrix, $A _ { \ell }$ , of our network after a certain layer $\ell$ ; $A _ { \ell } \in \mathbb { R } ^ { n \times m }$ , where $n$ is the number of examples in our training set and $m$ is the dimension of the activation vector after the $\ell ^ { t h }$ layer. By imposing a low rank prior on this matrix, we enforce that the matrix $A _ { \ell }$ is approximately low rank.
+
+We observe that even though enforcing a rank constraint on $A _ { \ell }$ does not obviously increase the sparsity of the model, if we consider a rank $r$ approximation, where $r \ll m$ , then there is no need to have more than $r$ units in the subsequent layer. Thus, having an (approximately) low rank representation allows us to compress both the model and the representations with little to no loss in accuracy (as our experimental results demonstrate).
+
+Optimization problem: We impose a low rank prior by adding an extra rank regularizor as part of the training objective. We do not regularize the model parameters directly. Instead, we impose constraints on intermediate model outputs as a part of the optimization problem.
+
+Let the neural network be represented by the function $f ( \cdot )$ , let $X = \left[ \mathbf { x } _ { 1 } , \cdots , \mathbf { x } _ { n } \right] ^ { \top }$ be the set of training inputs to the model and $Y = \left[ y _ { 1 } \cdots y _ { n } \right] ^ { \top }$ be the desired outputs. Consider the network to be composed of two networks, one consisting of layers before the $\ell ^ { t h }$ layer and the other after, i.e. $f ( \mathbf { x } _ { i } ) \doteq f _ { \ell } ^ { + } \left( f _ { \ell } ^ { - } ( \mathbf { x } _ { i } ) \right)$ . Here $f _ { \ell } ^ { + } ( \cdot )$ represents the network after the $\ell ^ { t h }$ activation layer and $f _ { \ell } ^ { - } ( \cdot )$ represents the network that produces the $\ell ^ { t h }$ activation layer. Let $f _ { \ell } ^ { - } ( X ) = A _ { \ell } = \left[ \mathbf { a } _ { 1 } \cdot \cdot \cdot \mathbf { a } _ { n } \right] ^ { \top } \in$ $\mathbb { R } ^ { \times m }$ represent the activation matrix of $n$ data points where $m$ is the dimension of the hidden layer. Our additional low rank constraint requires that rank $( A _ { \ell } ) = r \ll m$ .
+
+Let $\mathcal { L } ( \theta , \phi ; X , Y )$ be the loss function where $f , X , Y$ are as defined above and $f _ { \ell } ^ { + } ( \cdot )$ and $f _ { \ell } ^ { - } ( \cdot )$ are parameterized by $\theta$ and $\phi$ respectively. The optimization problem can be written as follows:
+
+$$
+\theta , \phi = \operatorname * { a r g m i n } _ { \theta , \phi } \mathcal { L } \big ( \theta , \phi ; X , Y \big ) \qquad \mathrm { s . t . } \quad \operatorname { r a n k } \big ( f _ { \ell } ^ { - } \big ( X ; \phi \big ) \big ) = r
+$$
+
+A natural approach to this (non-convex) problem would be an alternate minimization algorithm, where the two parts $\mathcal { L } ( \theta , \phi ; X , Y )$ and rank $\left( f _ { \ell } ^ { - } ( X ; \phi ) \right)$ are optimized alternately. The first of these steps is typically a non-convex optimization problem while the second is a projection step onto a non-convex set which can be solved by performing a singular value decomposition. However, in practice, this approach is infeasible because:
+
+(i) $f _ { \ell } ^ { - } \left( X ; \phi \right)$ is a large matrix and minimizing its rank at every iteration using a singular value decomposition is extremely costly, if not infeasible. (The dimension of this matrix can be as large as $5 0 , 0 0 0 \times 1 6 ,$ 384 in one of our experimental settings, but in each case is at least $5 0 , 0 0 0 \times 5 1 2 .$ ) $( i i )$ Minimizing the rank of the corresponding matrix for mini-batches i.e. $f _ { \ell } ^ { - } \left( X _ { 1 : b } ; \phi \right)$ does not guarantee that the rank of $f _ { \ell } ^ { - } \left( X _ { 1 : n } \right)$ will also be small.
+
+Below, we discuss an efficient approach to solve the optimization problem (OPT) by considering some approximations, augmentations and sampling.
+
+Augmented optimization problem: To avoid the aforementioned problems, we augment the optimization problem (OPT) by introducing an auxiliary low rank parameter $W \in \mathbb { R } ^ { m \times m }$ and a bias term $\mathbf { b } \in \mathbb { R } ^ { m \times 1 }$ . A rank regularizor term is added to the objective to shift the rank constraint from the activation matrix $A _ { \ell }$ to these auxiliary parameters as explained below. The new augmented optimization problem is as follows:
+
+$$
+\theta , \phi , W , \boldsymbol { \mathbf { b } } = \operatorname * { a r g m i n } _ { \theta , \phi , W , \boldsymbol { \mathbf { b } } } \mathcal { L } ( \theta , \phi ; X , Y ) + \lambda _ { 1 } \mathcal { L } _ { c } ( W , \boldsymbol { \mathbf { b } } ; A ) + \lambda _ { 2 } \mathcal { L } _ { N } ( A )
+$$
+
+$$
+\begin{array} { r } { \mathrm { e } , \quad W \in \mathbb { R } ^ { m \times m } , \operatorname { r a n k } \left( W \right) = r , \mathrm { \bf ~ b } \in \mathbb { R } ^ { m } , A = f _ { \ell } ^ { - } ( X ; \phi ) } \end{array}
+$$
+
+$$
+\mathcal { L } _ { c } ( W , \mathbf { b } ; A ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left\| \mathbf { W } ^ { \top } ( \mathbf { a _ { i } } + \mathbf { b } ) - ( \mathbf { a _ { i } } + \mathbf { b } ) \right\| _ { 2 } ^ { 2 } , \quad \mathcal { L } _ { N } ( A ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left| 1 - \left\| \mathbf { a _ { i } } \right\| \right| _ { 2 } ^ { 2 } ,
+$$
+
+The projection loss $\mathcal { L } _ { c }$ is used to minimize the distance between the activation matrix $A _ { \ell }$ and its affine low rank projection. The bias $\mathbf { b }$ allows for the activation matrix to be translated before projection. The intuition behind $\mathcal { L } _ { c }$ is that if $\mathbf { a } _ { i } \in \mathbb { R } ^ { n }$ lies in a (low) $r$ -dimensional subspace of $\mathbb { R } ^ { n }$ , then there is a rank- $r$ matrix $W$ such that $W$ is the identity map on the subspace and maps the rest of $\mathbb { R } ^ { n }$ to 0. $\mathcal { L } _ { c }$ is a soft version of this and also allows translation by $\mathbf { b }$ , i.e. if $A$ is the matrix of activation vectors and $W$ is low-rank, then $W A$ is low-rank; imposing the constraint that $A \approx W A$ forces $A$ itself to be close to low-rank. This is closely related to online PCA. However, setting $A + \mathbf { b }$ close to zero trivially minimizes $\mathcal { L } _ { c } ( W , \mathbf { b } ; A )$ , especially when the activation dimension is large. To prevent this, $\mathcal { L } _ { N } ( \cdot )$ , which acts as a norm constraint on the activation vector, is introduced to keep the activations sufficiently large.
+
+Implementing the low rank prior: Algorithm 1 in Appendix B.1 describes how to apply this low rank regularizor by adding a virtual layer called LR-layer after the $\ell ^ { t h }$ layer. This layer is virtual in the sense that it only affects the parameter $W$ and $\mathbf { b }$ which are not directly used to make predictions, but nonetheless the corresponding loss term $\mathcal { L } _ { c }$ does affect the network model parameters through gradient updates. This layer finds the closest affine low rank approximation of $A$ online by estimating a matrix $W$ that minimizes $\mathcal { L } _ { c }$ subject to rank $( W ) = r$ .
+
+The rank projection step in Line 16 in Algorithm 1 is executed by a hard thresholding operator $\Pi _ { r } ^ { \mathrm { r a n k } } ( W ) ^ { - }$ , which finds the best $r$ -rank approximation of $W$ . Essentially, $\Pi _ { r } ^ { \mathrm { r a n k } } ( W )$ solves the following optimization problem, which can be solved using a singular value decomposition (SVD).
+
+$$
+\Pi _ { r } ^ { \mathrm { r a n k } } ( W ) = \underset { \mathrm { r a n k } ( Z ) = r } { \mathrm { a r g m i n } } \left. \mathrm { W } - \mathrm { Z } \right. _ { F } ^ { 2 }
+$$
+
+Handling large activation matrices: Singular Value Projection (SVP) introduced in Jain et al. (2010) is an algorithm for rank minimization under affine constraints. In each iteration, the algorithm performs gradient descent on the affine constraints alternated with a rank- $\mathbf { \nabla } \cdot \mathbf { k }$ projection of the parameters and it provides recovery guarantees under weak isometry conditions. However, the algorithm has a complexity of $O ( m n \bar { r } )$ where $m , n$ are the dimensions of the matrix and $r$ is the desired low rank. Faster methods for SVD for sparse matrices are not applicable as the matrices in our case are not necessarily sparse. We use the ensembled Nystrom method (Williams & Seeger, 2001; Halko ¨ et al., 2011; Kumar et al., 2009a) to boost our computational speed at the cost of accuracy of the low rank approximation. It is essentially a sampling based low rank approximation to a matrix. The algorithm is described in detail in the Appendix B.2. Though the overall complexity for projecting $W$ still remains $O ( m ^ { 2 } r )$ , the complexity of the hard-to-parallelize SVD step is now ${ \cal O } ( \bar { r } ^ { 3 } )$ , while the rest is due to matrix multiplication.
+
+The theoretical guarantees of Nystrom hold only when the weight matrix of the ¨ LR-layer is symmetric and positive semi-definite (PSD) before each $\bar { \Pi } _ { r } ^ { \mathrm { r a n k } } ( \cdot )$ operation; this restricts the projections allowed in our optimization, but empirically this does not seem to matter. We know that a symmetric diagonally dominant real matrix with non-negative diagonal entries is PSD. With this motivation, the matrix $W$ is smoothened by repeatedly adding $0 . 0 1 \mathcal { T }$ until the SVD algorithm converges1. This is a heuristic to make the matrix well conditioned (as well as diagonally dominant) and it helps in the convergence of the algorithm empirically.
+
+Symmetric Low Rank Layer: The Nystrom method requires the matrix ¨ $W$ of the LR-layer to be symmetric and PSD (SPSD), however, gradient updates may make the matrix parameter nonSPSD, even if we start with an SPSD matrix. Reparametrizing the LR-layer fixes this issue; the layer is parameterized using $W _ { s }$ (to which gradient updates are applied), but the layer projects using $\dot { W } = ( W _ { s } + W _ { s } ^ { \top } ) / 2$ . After the rank projection is applied to the (smoothed version of) $W$ $W _ { s } : = \Pi _ { r } ^ { \mathrm { r a n k } } ( W )$ is an SPSD matrix (using Lemma 1 in Appendix B.3). As a result the updated $W$ is also SPSD. This layer also has a bias vector $\mathbf { b }$ to be able to translate the activation matrix before performing the low rank projection.
+
+Model Compression: As a consequence of forcing the activations of the $\ell ^ { t h }$ layer of the model to lie in a very low dimensional subspace with minimal reconstruction error and loss in accuracy, our experiments suggest that a simpler model can replace the latter parts of the original model without significant reduction in accuracy. Essentially, we can replace $f _ { \ell } ^ { + } : \mathbb { R } ^ { m }  \mathbb { R } ^ { c }$ with $g : \mathbb { R } ^ { m ^ { \prime } }  \mathbb { R } ^ { c }$ where $m ^ { \prime } \ll m$ and $g$ is a much smaller model than $f _ { \ell } ^ { + }$ . Our experiments in Table 2 show empirical evidence of this.
+
+# 3 EXPERIMENTS
+
+In this section we look at experimental validation of the various properties of our networks. As our primary training objective is to reduce the rank of the activations, we use the variance ratio, defined as $\begin{array} { r } { \mathsf { V R } _ { r } \left( A \right) = \frac { \sum _ { i = 1 } ^ { r } \sigma _ { i } ^ { 2 } } { \sum _ { i = 1 } ^ { p } \sigma _ { i } ^ { 2 } } } \end{array}$ , where $\sigma _ { 1 } , \ldots , \sigma _ { p }$ are the singular values of $A$ in non-increasing order, to estimate the variance captured by the first $r$ out of $p$ singular values.
+
+We consider the following models in our experiments:
+
+(a) ResNet 1-LR - This model contains one LR-layer, located immediately before the last fully connected (FC) layer. For a ResNet-18, the incoming activations have 512 units whereas for a ResNet-50 the incoming activations has a dimension of 2048.
+
+(b) ResNet 2-LR - This model contains two LR-layers. The first LR-layer is positioned before the fourth ResNet block, where the incoming activations have a dimension of 16, 384 and the second LR-layer is before the FC layer as in ResNet 1-LR.
+
+(c) VGG19 2-LR - This includes two LR-layers in the VGG-19 model. The VGG model has three FCs after 16 convolution layers with the LR-layers present before the first and the third FC layers in the model.
+
+(d) X-MAXG - This is a hybrid model where learned representations are generated for the input using the model “X” and then a maximum margin classifier is learned on these representations. The particular layer from which the representation is extracted will be clear from the context.
+
+(e) Bottle-LR - This model has a bottle neck layer which is essentially a fully connected layer without any non-linear activation. Moreover, the weight matrix of the $\mathbf { W } \in \mathbb { R } ^ { m \times m }$ is parametrized by $\mathbf { W } _ { l } \in \mathbb { R } ^ { m \times r }$ where $\mathbf { W } = \mathbf { W } _ { l } \mathbf { W } _ { l } ^ { \top }$ such that it is always low rank by definition.
+
+Some other models that are comparable to this work are listed in Appendix A. While one of the main objectives has been to design an augmentation to the training algorithm instead of changing the model itself (note that the extra weight matrices can be removed from the model after training is complete), we also consider Bottle-LR which includes an extra layer. The Bottle-LR layers have the same dimensions and are placed at the same position as the LR layers. As the experiments below suggest, the Bottle-LR model does not show the same benefits our LR model shows and performs no better than the N-LR model (indeed worse sometimes). One of the main difference in the training dynamics between this model and our model is that an explicit bottleneck layer has a multiplicative effect on the gradients propagated backward in the network, whereas, in our approach they only have an additive effect. Although this is somewhat speculative at this stage, we believe the relative “smoothness” of our approach is the reason why it seems to have better performance on these other tasks.
+
+In this paper, we used the datasets CIFAR10 and CIFAR100. As a side note, we observed that the target rank is not an essential hyper-parameter as the training enforces a much lower rank than what is set. Further details on the training procedure and the X-MAXG model is provided in Appendix C. Apart from these, we use the term N-LR to represent models without LR-layers.
+
+We perform three kinds of experiments in this section, with additional experiments in Appendix C.3 and C.4. We first show that our imposed constraints do not have any significant impact on the test accuracy of the model. Then we go on to verify that our training indeed achieves its objective of developing a stronger low rank structure. Finally we demonstrate that our training algorithm provides a natural scheme for compressing our models and that low dimensional projections of our embeddings, with a size of less than $2 \%$ of the original embeddings, can be used for classification with a significantly higher accuracy than similar sized projections of embeddings from an N-LR model. In Section 4, we provide empirical evidence in support of adversarial robustness of our models.
+
+Impact on test accuracy: In this experiment, we check if the additional constraints on the training method have any significant effect on the performance of the model. We observe, as listed in Table 1, that the additional penalties impose no significant loss in accuracy. In some cases, we even observe small gains in performance.
+
+(a) ResNet18 on CIFAR-10   
+
+<table><tr><td rowspan=1 colspan=1>Models</td><td rowspan=1 colspan=1>Test Accuracy</td></tr><tr><td rowspan=1 colspan=1>ResNet 1-LR</td><td rowspan=1 colspan=1>92.8</td></tr><tr><td rowspan=1 colspan=1>ResNet 2-LR</td><td rowspan=1 colspan=1>92</td></tr><tr><td rowspan=1 colspan=1>VGG192-LR</td><td rowspan=1 colspan=1>89.8</td></tr><tr><td rowspan=1 colspan=1>ResNet N-LR</td><td rowspan=1 colspan=1>92.5</td></tr><tr><td rowspan=1 colspan=1>VGG19N-LR</td><td rowspan=1 colspan=1>89.1</td></tr><tr><td rowspan=1 colspan=1>Bottle-2LR</td><td rowspan=1 colspan=1>92</td></tr></table>
+
+Table 1: Test Accuracy   
+
+<table><tr><td rowspan=1 colspan=1>Models</td><td rowspan=1 colspan=1>Coarse Label</td><td rowspan=1 colspan=1>Fine Label</td><td rowspan=1 colspan=1>VR20()</td></tr><tr><td rowspan=1 colspan=1>ResNet 1-LR</td><td rowspan=1 colspan=1>78.1%</td><td rowspan=1 colspan=1>48%</td><td rowspan=1 colspan=1>0.97</td></tr><tr><td rowspan=1 colspan=1>ResNet N-LR</td><td rowspan=1 colspan=1>75.6%</td><td rowspan=1 colspan=1>52%</td><td rowspan=1 colspan=1>0.90</td></tr><tr><td rowspan=1 colspan=1>Bottle-1LR</td><td rowspan=1 colspan=1>76%</td><td rowspan=1 colspan=1>38%</td><td rowspan=1 colspan=1>NA</td></tr></table>
+
+(b) ResNet50 on CIFAR-100
+
+To see whether the learned representations can be used in a different task, we conduct a transfer learning exercise where embeddings generated from a ResNet-50 model, trained on the coarse labels of CIFAR-100, are used to predict the fine labels of CIFAR-100. A set of ResNet-50-MAXG classifiers are trained for this purpose on these embeddings. (see Appendix C.1 for further details). We can see in Table 1(b) that the LR model suffers a small loss of $4 \%$ in accuracy as compared to the N-LR model when it’s embeddings are used to train a max margin classifier for the task of predicting the fine labels. However, it is to be noted that the accuracy of the LR model actually increases when the max margin classifier is trained on it to do the original task i.e. classifying the coarse labels. On the other hand, the Bottle-LR model suffers a loss of $1 4 \%$ in accuracy compared to N-LR model and does not show any advantage in the original task either.
+
+![](images/242bc25b740db31dcb6f2e118a1519b965e002e82bffcfa400093f86ed03e7cc.jpg)  
+Figure 1: Variance Ratio captured by varying number of Singular Values
+
+Effective rank of activations: Figure 1(a) shows the variance ratio captured by varying numbers of singular values in the activations before the last FC layer. In this case, every model shows that the effective rank of the activations is 10 as there is a sharp elbow in the plot after 10 singular values. However, the LR-model has almost negligible leakage of variance compared to the N-LR model. Similar experiments for VGG is found in the appendix (Figure 5). .
+
+Figure 1(b) shows the variance ratio captured by varying numbers of singular values for the activations before the fourth ResNet block. In this case the activation vector has a dimension of 16, 384 and the use of Nystrom method is necessary for computational feasibility. ResNet 2-LR is the only model ¨ that has a LR-layer in that position and Figure 1(b) shows that it is the only model that shows a low rank structure on that layer.
+
+While there are various techniques developed that induce sparsity on weights or activations in neural networks, it is important to point out that sparsity doesn’t necessarily lead to low rank eg. an identity matrix. Empirically, in ResNet 1-LR, the activations before the $\dot { 4 } ^ { t h }$ ResNet block are much less explained by a small number of singular values despite the high level of sparsity $( 3 9 \% )$ whereas the activations after the 4th ResNet block are explained by about 10 singular values though it is $9 5 \%$ dense.
+
+Validity of low dimensional embeddings: The previous experiment shows that our training algorithm introduces a low rank structure in the activations. However, it does not show any evidence of the discriminative power of these low rank embeddings. In these experiments, we use both CIFAR-10 and CIFAR-100 with ResNet-18 and ResNet-50 respectively to show that these embeddings and their projections onto low dimensional spaces are effective at discriminative tasks and are better at it than models without the LR-layer.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Emb-dim</td><td rowspan=1 colspan=1>Acc</td></tr><tr><td rowspan=1 colspan=1>ResNet-50-LR</td><td rowspan=1 colspan=1>2048</td><td rowspan=1 colspan=1>78.1</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>2048</td><td rowspan=1 colspan=1>75.6</td></tr><tr><td rowspan=1 colspan=1>ResNet-50-LR</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>76.5</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>68.4</td></tr><tr><td rowspan=1 colspan=1>ResNet-50-LR</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>72</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>48</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Emb-dim</td><td rowspan=1 colspan=1>Acc</td></tr><tr><td rowspan=1 colspan=1>ResNet18-2-LR</td><td rowspan=1 colspan=1>16,384</td><td rowspan=1 colspan=1>91.14%</td></tr><tr><td rowspan=1 colspan=1>ResNet18</td><td rowspan=1 colspan=1>16,384</td><td rowspan=1 colspan=1>90.7%</td></tr><tr><td rowspan=1 colspan=1>ResNet18-2-LR</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>88.5%</td></tr><tr><td rowspan=1 colspan=1>ResNet18</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>76.9%</td></tr><tr><td rowspan=1 colspan=1>ResNet18-2-LR</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>75%</td></tr><tr><td rowspan=1 colspan=1>ResNet18</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>61.7 %</td></tr></table>
+
+(b) Representation from before the last ResNet block of ResNet-18 trained on CIFAR-10.
+
+(a) Representation from before the FC layer of a ResNet-50 trained on CIFAR-100.
+
+Table 2: Accuracy of low dimensional projections of learned representations at discriminative tasks.
+
+Table 2(a) shows that even with decreasing embedding dimension, the LR model is able to preserve its accuracy better than the N-LR model. Even with a 5-dimensional embedding, the LR model looses only $6 \%$ in accuracy, but the N-LR model looses $2 7 \%$ .
+
+The results of the next experiment, listed in Table 2(b), show the capability of our algorithm to compress the model itself. The entire fourth ResNet block along with the last FC layer (referred to as $f _ { \ell } ^ { + }$ and containing 8.4M parameters) can be replaced by a smaller linear model which has only 0.02 times the number of parameters as $f _ { \ell } ^ { + }$ . This yields a significant reduction in model size in exchange for a slight drop in accuracy $( < 1 \% )$ ). The second benefit is that as the low dimensional embeddings still retain most of the discriminative information, the inputs fed to the linear model have a small number of features.
+
+# 4 ADVERSARIAL ATTACKS
+
+(Szegedy et al., 2013) showed that adding adversarial perturbations to inputs to machine learning models, that otherwise perform well on a test set, can often make them suffer a high misclassification rate. To express this formally, consider a machine learning model $M : X  Y$ where $X$ is the input space and $Y$ is the target label space. It is possible to add an adversarial noise $\delta$ to $\mathbf { x } _ { i } \in X$ such that though $M ( \mathbf { x } _ { i } )$ predict $\mathbf { y } _ { i }$ , but $M ( \mathbf { x } _ { i } + \delta )$ predicts a label other than $\mathbf { y } _ { i }$ with high confidence, despite $\mathbf { x } _ { i }$ and $\mathbf { x } _ { i } + \delta$ being perceptually indistinguishable to a human. Various methods (Szegedy et al., 2013; Goodfellow et al., 2014; Kurakin et al., 2017; Moosavi-Dezfooli et al., 2016) have been proposed in recent years for constructing adversarial perturbations. In this section, our experiments show that LR models are more robust than N-LR models against these adversarial perturbations.
+
+In this section, $\mathbf { x } _ { d }$ refers to an example drawn from the data distribution and $\mathbf { x } _ { a }$ denotes the adversarially perturbed version of $\mathbf { x } _ { d }$ . For vectors $\mathbf { z }$ and $\mathbf { x }$ , let $\mathrm { c l i p } _ { { \bf x } , \epsilon } ( { \bf z } )$ denote the element-wise clipping of $\mathbf { z }$ , with $z _ { i }$ clipped to the range $[ x _ { i } - \epsilon , x _ { i } + \epsilon ]$ .
+
+Adversarial attacks: We look at three different attacks in our experiments: (i) Iterative Fast Sign Gradient Method (Iter-FSGM) (Kurakin et al., 2016; Madry et al., 2018) (ii) The Iterative Least Likely Class Method (Iter-LL-FSGM) (Kurakin et al., 2017) (iii) DeepFool (Moosavi-Dezfooli et al., 2016). The reader may refer to Appendix D.1 for further details on the attacks. The iterative fast sign gradient method (Iter-FSGM) is essentially equivalent to the Projected Gradient Descent Method on the negative loss function (PGD, Madry et al. (2018)).
+
+Further, due to the low rank constraints, the model could implicitly be enforcing gradient masking, which might result in the construction of weaker attacks. Hence, to be fair in our comparison, we also consider black box versions of each of the aforementioned attacks. While the adversarial noise in a white box attack is constructed using the model being attacked, the noise in a black box attack is constructed using a different model (in our case the N-LR model). We use the standard normalized $L _ { 2 }$ dissimilarity measure $\begin{array} { r } { \rho = \sum _ { i } \frac { \| \mathbf { x } _ { \mathrm { a } } - \mathbf { x } _ { \mathrm { d } } \| _ { 2 } } { \| \mathbf { x } _ { \mathrm { d } } \| _ { 2 } } } \end{array}$ to measure the magnitude of this noise.
+
+# Robustness to Adversarial Attacks:
+
+![](images/bf15042ec1eb07b881f5768ed520c436d909e33030ee5a9adcf589f7aa6d1df8.jpg)  
+Figure 2: Adversarial accuracy plotted against magnitude of perturbation (measured with $\rho$ ).
+
+Here we compare across models, the change in accuracy of classifying adversarial examples with respect to the amount of noise added. In line with the experiments in Kurakin et al. (2017), the noise is added for a pre-determined number of steps. Figure 2 shows that as the noise increases, the accuracy of N-LR models decrease much faster than the LR models.
+
+Specifically, to reach an adversarial mis-classification rate of $5 0 \%$ , our models require about twice the noise as a normal model or Bottle-LR model. We also observe that black box attacks are not very successful when multi-step (i.e. iterative) attacks are performed 2. In short, for all kinds of attacks we tried, LR models consistently perform better than N-LR models.
+
+Our next experiment is in line with the experiments conducted in Moosavi-Dezfooli et al. (2016). Table 3 lists the average minimum perturbation (measured with $\rho$ ) required to make the classifier mis-classify more than $9 9 \%$ of the adversarial examples, constructed from a uniformly sampled subset of the test set. Appendix D.2 describes the setup in further detail.
+
+Table 3: Perturbation required for Adversarial Misclassification   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>p[DeepFool]</td><td rowspan=1 colspan=1>ρ[Iter-LL-FSGM]</td><td rowspan=1 colspan=1>ρ [Iter-FSGM]</td></tr><tr><td rowspan=3 colspan=1>White Box</td><td rowspan=1 colspan=1>ResNet N-LR</td><td rowspan=1 colspan=1>1.6×10-2</td><td rowspan=1 colspan=1>2.4×10-2</td><td rowspan=1 colspan=1>2.1×10-2</td></tr><tr><td rowspan=1 colspan=1>ResNet1-LR</td><td rowspan=1 colspan=1>1.7 ×10-1</td><td rowspan=1 colspan=1>1.1 ×10-1</td><td rowspan=1 colspan=1>6.0×10-2</td></tr><tr><td rowspan=1 colspan=1>ResNet 2-LR</td><td rowspan=1 colspan=1>1.8 × 10-1</td><td rowspan=1 colspan=1>9.8×10-2</td><td rowspan=1 colspan=1>7.6 ×10-2</td></tr><tr><td rowspan=2 colspan=1>Black Box</td><td rowspan=1 colspan=1>ResNet 1-LR</td><td rowspan=1 colspan=1>4.7×10-2</td><td rowspan=1 colspan=1>1.8×10-1</td><td rowspan=1 colspan=1>5.6×10-2</td></tr><tr><td rowspan=1 colspan=1>ResNet 2-LR</td><td rowspan=1 colspan=1>5.5×10-2</td><td rowspan=1 colspan=1>2.0×10-1</td><td rowspan=1 colspan=1>7.5 × 10-2</td></tr></table>
+
+An interesting observation is that the values of $\rho$ , here in Table 3, are lower than in Figure 2 though the attacks have a higher rate of success. To explain this behaviour, we show empirical evidence to indicate that an attack that adds noise for a fixed number of steps (Kurakin et al., 2017; 2016) to the input is significantly weaker than one that stops on successful misclassification (See Appendix D.3). However, even under this scheme of attacks, our models perform better than N-LR models as LR models require 4 to 11 times the amount of noise required by N-LR models to be fooled by adversarial attacks.
+
+We also look at some of the adversarial images generated by DeepFool in Figure 3. We observe that it is immediately clear that the adversarial images are different from the original images in the case of LR models whereas it is not so apparent in the case of N-LR models.
+
+![](images/fc1fdb8509de2085264f853cf5c0c10e6fb36bb732ec4e55a4f1477648027e04.jpg)  
+Figure 3: For each model, original images are on the top row and the images generated by DeepFool are below.
+
+In order to gain some intuition into this behaviour of robustness against adversarial attacks, we look into the property of noise stability of networks towards adversarial noise. Arora et al. (2018) gives empirical evidence for deep networks being stable towards injected Gaussian noise and uses a variation of this noise stability property to derive more realistic generalization bounds. We believe that learning a model that zeroes out irrelevant directions in the learned representations reduces the ability to find adversarial perturbations that can affect the output of the model. In Figure 4, we show empirically that LR networks are more noise-stable to adversarial noise than non-LR networks. On the $x$ -axis is the normalized $L _ { 2 }$ dissimilarity score in the input space i.e. $\| \mathbf { x } + \eta \| ^ { 2 } / \| \mathbf { x } \| ^ { 2 }$ and on the $y$ -axis the corresponding quantity in the representation space i.e. $\left. \mathrm { f } _ { \ell } ^ { - } \left( \mathbf { x } + \eta \right) \right. ^ { 2 } / \left. \mathrm { f } _ { \ell } ^ { - } \left( \mathbf { x } \right) \right. ^ { 2 }$ . The representations here are taken from before the last fully connected and softmax layer. As our experiments suggest, the LR model significantly attenuates the adversarial perturbation in the representation space thus making it harder to fool the softmax classifier.
+
+![](images/a06b06c7030b977b3253c4ae852e644952dc7d44691c926da6220b638538fb23.jpg)  
+Figure 4: Adversarial Perturbation in Input Space and Perturbation in Representation Space
+
+Max Margin Classifiers: Finally, we show that X-MAXG models are significantly more robust to adversarial attacks than the corresponding “X” models. Also, as seen in Table 4, X-MAXG models with LR-layers are more robust than X-MAXG models without LR-layers against adversarial attacks. Our experimental setup is explained in detail in Appendix D.4. We can see that a X-MAXG model with a LR-layer correctly classifies $5 0 \%$ of the examples that had fooled the original classifier while for a similar amount of noise, the X-MAXG model without the LR-layer has negligible accuracy.
+
+Table 4: Accuracy of classification of adversarial examples by Max Margin Classifiers.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>DeepFool</td><td rowspan=1 colspan=1>Iter-LL-FSGM</td><td rowspan=1 colspan=1>Iter-FSGM</td></tr><tr><td rowspan=3 colspan=1>White Box</td><td rowspan=1 colspan=1>ResNet18 N-LR-MAXG</td><td rowspan=1 colspan=1>0.01</td><td rowspan=1 colspan=1>0.04</td><td rowspan=1 colspan=1>0.02</td></tr><tr><td rowspan=1 colspan=1>ResNet 1-LR-MAXG</td><td rowspan=1 colspan=1>0.38</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>0.48</td></tr><tr><td rowspan=1 colspan=1>ResNet2-LR-MAXG</td><td rowspan=1 colspan=1>0.43</td><td rowspan=1 colspan=1>0.55</td><td rowspan=1 colspan=1>0.55</td></tr><tr><td rowspan=2 colspan=1>Black Box</td><td rowspan=1 colspan=1>ResNet 2-LR-MAXG</td><td rowspan=1 colspan=1>0.29</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>0.33</td></tr><tr><td rowspan=1 colspan=1>ResNet 1-LR-MAXG</td><td rowspan=1 colspan=1>0.44</td><td rowspan=1 colspan=1>0.50</td><td rowspan=1 colspan=1>0.48</td></tr></table>
+
+# 5 CONCLUSION
+
+In this paper, we designed an algorithm that encourages the learned representations obtained by training deep neural networks for supervised tasks to lie in an approximately low rank (affine) subspace. This is achieved by augmenting the networks with virtual LR-layers and modifying the training objective. In order to make our algorithm computationally feasible for large networks, we proposed and implemented certain approximation techniques. Our experiments show that our algorithm successfully enforces an approximate low rank behaviour and that these learned representations have some intriguing properties. We conducted a wide range of experiments to investigate these properties and report, among other things, that (i) max-margin classifiers trained on these representations have more discriminatory power than ones trained on representation from models without LR-layers, (ii) our models are more robust to a variety of adversarial attacks than models without LR-layers, and (iii) LR models significantly attenuate the effects of perturbations introduced in the input space on the representation space (iv) replacing large parts of our models with smaller models results in a negligible drop in accuracy, thereby providing a compression scheme, It is commonly believed that large sparse feature spaces are better for deep models. What we propose here is the idea that while the features themselves need not be sparse, the existence of a basis in which the feature vectors have a sparse representation can provide benefits. To the best of our knowledge, investigating properties of representations learned from deep supervised NNs, and the possibility of modifying training procedures to obtain desirable properties in said representations, has remained relatively unexplored. This work shows that these representations possess some intriguing properties, which may well be worthy of further investigation.
+
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+
+# A ALTERNATIVE ALGORITHMS
+
+# LOW RANK WEIGHTS
+
+With respect to compression, it is natural to look at low rank approximations of network parameters (Denton et al., 2014; Jaderberg et al., 2014). By factorizing the weight matrix/tensor $W$ , for input $x$ , we can get low rank pre-activations $W x$ . This however does not lead to low rank activations as demonstrated both mathematically (by the counter-example below) and empirically.
+
+Mathematical Counter-Example: Consider a rank 1 pre-activation matrix $A$ and its corresponding post-activation(ReLU) matrix as below. It is easy to see that the rank of post-activation has increased to 2.
+
+$$
+A = { \left[ \begin{array} { l l l } { } & { 1 } & { - 1 } & { 1 } \\ { - 1 } & { 1 } & { - 1 } \end{array} \right] } \qquad { \mathrm { R e l u } } ( A ) = { \left[ \begin{array} { l l l } { } & { 1 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 0 } \end{array} \right] }
+$$
+
+Empirical Result: In order to see if techniques for low rank approximation of network parameters like Denton et al. (2014) would have produced low rank activations, we conducted an experiment by explicitly making the pre-activations low-rank using SVD. Our experiments showed that inspite of setting a rank of 100 to the pre-activation matrix, the post-activation matrix had full rank. Though all but the first hundred singular values of the pre-activation matrix were set to zero, the post-activation matrix’s $1 0 1 ^ { s t }$ and $1 0 0 \overline { { 0 } } ^ { t h }$ singular values were 49 and 7.9 respectively, and its first 100 singular values explained only $9 4 \%$ of the variance.
+
+We try to explain the above empirical results as follows: Theoretically, a bounded activation function lowers the Frobenius norm of the pre-activation matrix i.e. the sum of the squared singular values. However, it also causes a smoothening of the singular values by making certain 0 singular values non-zero to compensate for the significant decrease in the larger singular values. This leads to an increase in rank of the post-activation matrix.
+
+# BOTTLENECK LR LAYER
+
+Bottleneck Layer: It is easy to see that the effective dimension of the representation of an input, obtained after passing through a bottleneck layer (like an auto-encoder), will not be greater than the dimension of the bottleneck layer itself. However, due to the various non-linearities present in the network, while the representation is guaranteed to lie in a low dimensional manifold it is not guaranteed to lie in a low rank (affine) subspace.
+
+LR bottleneck: Another alternative is to include the low-rank projection and reconstruction as part of the network instead of as a regularizor so that the LR-layer is an actual layer and not a virtual layer. We have indeed experimented with this setup and observed that this often made the training very unstable. Also, if one were to add this bottleneck as a fine-tuning process, the test accuracy of the network decreases by a much higher extent than it does for our method.
+
+# B ALGORITHMIC DETAILS
+
+B.1 ALGORITHM FOR LR LAYER
+
+Algorithm 1 below lists the forward and the backpropagation rules of the LR-layer.
+
+# Algorithm 1 LR Layer
+
+<table><tr><td colspan="3">Input: Activation Matrix A,Grad_input g</td></tr><tr><td>1: 2:</td><td></td><td></td></tr><tr><td>3:</td><td>Forward Propagation</td><td></td></tr><tr><td>4:</td><td>Z ←WT(A+b)³</td><td> Compute the affine Low rank projection</td></tr><tr><td>5:</td><td>Output : A</td><td> Output the original activations for the next layer</td></tr><tr><td>6:</td><td></td><td></td></tr><tr><td>7:</td><td>Backward Propagation</td><td></td></tr><tr><td>8:</td><td>D1←Z-(A+b)² n</td><td>Computes the reconstruction loss Lc</td></tr><tr><td>9:</td><td>∑- D2← 2 |1- ||ai n</td><td>&gt; Computes the loss for the norm constraint LN</td></tr><tr><td>10:</td><td>D←D+D2 aD n-1aD</td><td></td></tr><tr><td>11: 12:</td><td>gw↑</td><td>&gt; Outputs the gradient to be passed to the layer before</td></tr><tr><td>13:</td><td>Output : 9i</td><td></td></tr><tr><td>14:</td><td>Update Step</td><td></td></tr><tr><td>15:</td><td>W ←W-Xgw</td><td>&gt; Updates the weight with the gradient from D.</td></tr><tr><td></td><td>16: W ← Irank(W)</td><td>Hard thresholds the rank of W</td></tr><tr><td></td><td></td><td></td></tr></table>
+
+# B.2 ENSEMBLED NYSTROM¨ METHOD
+
+Let $W \in \mathbb { R } ^ { m \times m }$ be a symmetric positive semidefinite matrix (SPSD). We want to generate a matrix $W _ { r }$ which is a r-rank approximation of $W$ without performing SVD on the full matrix $W$ but only on a principal submatrix4 $\dot { Z } \in \mathbb { R } ^ { l \times l }$ of $W$ , where $l \ll m$ . We sample $l$ indices from the set $\{ 1 \cdots m \}$ and select the corresponding columns from $W$ to form a matrix $\bar { C } \in \mathbb { R } ^ { m \times l }$ . In a similar way, selecting the $l$ rows from $C$ we get $Z \in \mathbb { R } ^ { l \times l }$ . We can rearrange the columns of $W$ so that
+
+$$
+W = \left[ \begin{array} { c c } { { Z } } & { { W _ { 2 1 } ^ { T } } } \\ { { W _ { 2 1 } } } & { { W _ { 2 2 } } } \end{array} \right] \qquad C = \left[ \begin{array} { c } { { Z } } \\ { { W _ { 2 1 } } } \end{array} \right]
+$$
+
+According to the Nystrom approximation, the low rank approximation of ¨ $W$ can be written as
+
+$$
+W _ { r } = C Z _ { r } ^ { + } C ^ { T }
+$$
+
+where $Z _ { r } ^ { + }$ is the pseudo-inverse of the best $r$ rank approximation of $Z$ . Hence, the entire algorithm is as follows.
+
+• Compute $C$ and $Z$ as stated above.   
+• Compute the top $r$ singular vectors and values of $Z : U _ { r } , \Sigma _ { r } , V _ { r }$ .   
+• Invert each element of $\Sigma _ { r }$ as this is used to get the Moore pseudo-inverse of $Z _ { r }$ .   
+• Compute $Z _ { r } ^ { + } = U _ { r } \Sigma _ { r } ^ { - 1 } V _ { r }$ and $W _ { r } = C Z _ { r } ^ { + } C ^ { T }$ .
+
+Though by trivial computation, the complexity of the algorithm seems to be $O ( l ^ { 2 } r + m l ^ { 2 } + m ^ { 2 } l ) =$ $O ( m ^ { \bar { 2 } } r )$ (In our experiments $l = 2 r$ ), it must be noted that the complexity of the SVD step is only $O ( k ^ { 3 } )$ which is much lesser than $O ( m ^ { 2 } r )$ and while matrix multiplication is easily parallelizable, parallelization of SVD is highly non-trivial and inefficient.
+
+To improve the accuracy of the approximation, we use the ensembled Nystrom sampling based ¨ methods (Kumar et al., 2009a) by averaging the outputs of $t$ runs of the Nystrom method. The ¨ $l$ indices for selecting columns and rows are sampled from an uniform distribution and it has been shown (Kumar et al., 2009b) that uniform sampling performs better than most sampling methods. Theorem 3 in Kumar et al. (2009a) provides a probabilistic bound on the Frobenius norm of the difference between the exact best r-rank approximation and the Nystrom sampled r-rank ¨ approximation.
+
+# B.3 LEMMA 1
+
+Lemma 1. If $\boldsymbol { X } \in \mathbb { R } ^ { m \times m }$ is a SPSD matrix and $X _ { r } \in \mathbb { R } ^ { m \times m }$ is the best Nystrom ensembled, ¨ column sampled $r$ -rank approximation of $X$ , then $X _ { r }$ is SPSD as well. (Proof in Appendix B.3)
+
+Proof. By the Construction of the Nystrom SVD algorithm, we know that ¨ $X _ { r } = C W _ { r } ^ { + } C ^ { T }$ . We will first show that $W _ { r } ^ { + }$ is a symmetric matrix.
+
+We know that $X$ is SPSD. Let $I$ be a sorted list of distinct indices such that $| I | = l$ . Then by construction of $W$ ,
+
+$$
+W _ { i , j } = X _ { I [ i ] , I [ j ] }
+$$
+
+Hence, as $X _ { I [ i ] , I [ j ] } = X _ { I [ j ] , I [ i ] } , W$ is symmetric.
+
+At this step, our algorithm adds $\delta \cdot I$ to $W$ where $\delta \geq 0$ . It is easy to observe that $W + \delta \cdot \mathcal { T }$ is positive semidefinite.
+
+Consider a vector $a \in \mathbb { R } ^ { | X | }$ . Create a vector $\bar { a } \in \mathbb { R } ^ { m }$ where
+
+$$
+{ \bar { a } } _ { i } = { \left\{ \begin{array} { l l } { 0 } & { { \mathrm { i f ~ } } i \notin I } \\ { a _ { i } } & { { \mathrm { o . w . } } } \end{array} \right. }
+$$
+
+$$
+\begin{array} { r } { \boldsymbol { a } ^ { \top } \left( \boldsymbol { W } + \boldsymbol { \delta } \cdot \boldsymbol { \mathcal { T } } \right) \boldsymbol { a } = \bar { \boldsymbol { a } } ^ { \top } \boldsymbol { X } \bar { \boldsymbol { a } } + \boldsymbol { \delta } \cdot \boldsymbol { a } ^ { \top } \boldsymbol { \mathcal { Z } } \boldsymbol { a } \geq 0 + \boldsymbol { \delta } \| \mathbf { a } \| ^ { 2 } \geq 0 } \end{array}
+$$
+
+Let $W + \delta \mathcal { T }$ be the new $W$ and (3) shows that $W$ is positive semidefinite.
+
+Now we will show that $X _ { r }$ is symmetric as well. As $W$ is symmetric, there exists an orthogonal matrix $Q$ and a non-negative diagonal matrix $\Lambda$ such that
+
+$$
+W = Q \Lambda Q ^ { T }
+$$
+
+We know that $W _ { r } = Q _ { [ 1 : r ] } \Lambda _ { [ 1 : r ] } Q _ { [ 1 : r ] } ^ { T }$ and $W _ { r } ^ { + } = Q _ { [ 1 : r ] } \Lambda _ { [ 1 : r ] } ^ { - 1 } Q _ { [ 1 : r ] } ^ { T } ,$ Hence,
+
+$$
+\begin{array} { r l } & { X _ { r } = C W _ { r } ^ { + } C ^ { T } } \\ & { \quad \quad \quad = C Q _ { [ 1 : r ] } \Lambda _ { [ 1 : r ] } ^ { - 1 } Q _ { [ 1 : r ] } ^ { T } C ^ { T } } \\ & { X _ { r } ^ { T } = ( C Q _ { [ 1 : r ] } \Lambda _ { [ 1 : r ] } ^ { - 1 } Q _ { [ 1 : r ] } ^ { T } C ^ { T } ) ^ { T } } \\ & { \quad \quad \quad = C Q _ { [ 1 : r ] } \Lambda _ { [ 1 : r ] } ^ { - 1 } Q _ { [ 1 : r ] } ^ { T } C ^ { T } } \\ & { \quad \quad \quad = X _ { r } } \end{array}
+$$
+
+∴ $X _ { r }$ is symmetric. We can also see that the $X _ { r } ^ { T }$ is positive semi definite by pre-multiplying and post multiplying it with a non-zero vector and using the fact that $W _ { r } ^ { + }$ is positive semi-definite.
+
+# C EXPERIMENTAL DETAILS
+
+We used a single NVIDIA Tesla P40 GPU for training our networks. All our experiments were run with a learning rate of 0.01 for 500 epochs with a batch size of 512. The hyper-parameter $l$ in the Nystrom method was set to double of the target rank. The model was pre-trained with SGD until the ¨ training accuracy reached $6 5 \%$ and then Algorithm 1 was applied. The rank cutting operation was performed every 10 iterations.
+
+The target rank for the LR-layers, which were placed before the FC layers was set to 50 while the target rank for the LR-layer before the last ResNet block was set to 1000. However, experiments suggest that this hyper-parameter is not very crucial to the training process as the training procedure converged the effective rank of the activation matrix to a value lesser than the designated target ranks.
+
+The linear classifier in X-MAXG is trained using SGD with hinge loss and $L _ { 2 }$ regularization with a coefficient of 0.01. The learning rate is decreased per iteration as $\begin{array} { r } { \eta _ { t } = \frac { \eta _ { 0 } } { \left( 1 + \alpha t \right) } } \end{array}$ where $\eta _ { 0 }$ and $\alpha$ are set by certain heuristics 5.
+
+# C.1 IMPACT ON TEST ACCURACY
+
+Table 1(b) shows the result of a transfer learning exercise where we trained two sets of ResNet50-MAXG classifiers. First, two ResNet-50 models were trained with and without the LR-layer respectively on the coarse labels of CIFAR-100. Then, 2048 dimensional embeddings were extracted from after the fourth ResNet block using the train set and the test set of CIFAR 100. The embeddings from the train set were used to train two groups (each group having two classifiers- one for each ResNet-50 model) of max-margin linear classifiers with the same hyper-parameters as described above. While the first group was trained on the coarse labels of CIFAR-100, the second set was trained on the fine labels. The test accuracy of these four classifiers are then reported in Table 1(b) on the corresponding labels.
+
+![](images/75d5cf8374f8b5009d17f2601ad98a8cd4245b05ebcc85415e5efeb4e6b0dc69.jpg)  
+Figure 5: Variance Ratio captured by varying number of Singular Values
+
+# Effective rank of activations for VGG:
+
+# C.2 VALIDITY OF LOW DIMENSIONAL EMBEDDINGS
+
+In the first experiment, reported in Table 2(a), we trained two ResNet-50-MAXG models -with and without the LR-layer respectively- on the 20 super-classes of CIFAR-100. As our objective here is to see if the embeddings and their low dimensional projections could be effectively used for discriminative tasks, we used PCA, with standard pre-processing of scaling the input, to project the embeddings onto a low dimensional space before training a linear maximum margin classifier on it.
+
+The experiments in Table 2(b) were run with ResNet-18 on CIFAR-10. Two ResNet-18-MAXG classifiers - with and without the LR-layer respectively- were trained on CIFAR-10. The representations were obtained from before the fourth ResNet block and had a dimension of 16,384. Similar to the previous experiment, we used PCA, with standard pre-processing, to gather low dimensional projections before training linear max margin classifiers on it.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Emb-dim</td><td rowspan=1 colspan=1>Acc</td></tr><tr><td rowspan=1 colspan=1>VGG19-2LR</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>89.8</td></tr><tr><td rowspan=1 colspan=1>VGG19-NLR</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>89.7</td></tr><tr><td rowspan=1 colspan=1>VGG19-2LR</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>89.85</td></tr><tr><td rowspan=1 colspan=1>VGG19-NLR</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>89.78</td></tr><tr><td rowspan=1 colspan=1>VGG19-2LR</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>89.79</td></tr><tr><td rowspan=1 colspan=1>VGG19-NLR</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>89.65</td></tr></table>
+
+Table 5: Representation from before the third last FC layer of a VGG19trained on CIFAR-10.
+
+As expected, the difference in accuracy here is less stark than the case of ResNet. This is because of two reasons - 1. The dimension of the activation layer in ResNet before the last ResNet block is 16,384 whereas the activations before the third last FC layer is only 512. 2. Figure 5 shows that the difference in the variance ratio between the LR and the N-LR network is much smaller as compared to Figure 1 for ResNets.
+
+# C.3 CLASS WISE VARIANCE
+
+![](images/222371ba1c92ead2d81e5b9215a3ae70eadd8ba4ad16fda890b47b98f7769025.jpg)
+
+(a) 512 dimensional activations from after last ResNet block.
+
+![](images/57ace73a9ea065379fb5e38ec349e93c08018e0274c9128f470966673e5ff797.jpg)  
+(b) $1 6 k$ dimensional activations from before last ResNet block.
+
+Figure 6: Class wise variance ratio of one singular values for the activations before the last ResNet block.
+
+In this experiment, we plot $\mathsf { V R } _ { 1 } \left( \cdot \right)$ for embeddings of examples restricted to individual classes. Figure 6(a) shows the variance ratio captured by the largest singular value for the activations before the last FC layer while Figure 6(b) shows the variance ratio captured by the largest singular value for the activations before the last ResNet block. These experiments give us some idea about the extent to which the set of basis vectors assigned to individual classes are intersecting.
+
+# C.4 CLUSTERS OF LOW DIMENSIONAL EMBEDDINGS
+
+Figure 7 shows the two dimensional projections of the 2048 dimensional embeddings obtained from ResNet-50-LR and ResNet-50-N-LR. The coloring is done according to the coarse labels of the input. We can see that the clusters are more separable in the case of the model with LR-layer than the model without, which gives some insight into why a max-margin classifier performs better for the LR model than the N-LR model.
+
+![](images/ba6886981a2dc01f0c977d8fcc3e3843eb4ada4782e73dcab9ac65e0f3c9d7fa.jpg)  
+Figure 7: PCA plot for super-class labels in CIFAR 100. Plot on the left shows embeddings from the LR trained model on ResNet-50 while the plot on the right represents a normal ResNet-50 model. Both of them are trained in a similar way. Each color represents a different class.
+
+# D ADVERSARIAL ATTACKS
+
+D.1 TYPES OF ATTACKS
+
+• Iter-FSGM- The Fast Sign Gradient Method (FSGM) Goodfellow et al. (2014) was proposed as the existing methods (Szegedy et al. (2013)) of the time were slow. FSGM tries to maximize the loss function by perturbing the input slightly. Iterative Fast Sign Gradient method (Iter-FSGM) is a simple extension of FSGM that follows the following simple iterative step.
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { a } ^ { 0 } = \mathbf { x } _ { d } , } \\ & { \mathbf { x } _ { a } ^ { n + 1 } = \mathrm { c l i p } _ { \mathbf { x } , \epsilon } ( \mathbf { x } _ { a } ^ { n } + \boldsymbol { \alpha } \cdot \mathrm { s i g n } ( \nabla _ { \mathbf { x } _ { a } ^ { n } } \mathcal { L } ( \mathbf { x } _ { a } ^ { n } , \mathbf { y } _ { t } ) ) ) } \end{array}
+$$
+
+• Iter-LL-FSGM- Iter-FSGM is an untargeted attack. Iterative less likely fast sign gradient method (Iter-LL-FSGM) (Kurakin et al., 2017) is a way to choose the target label wisely. Consider $\mathbb { P } _ { M } ( \mathbf { y } | \mathbf { x } )$ to be the probability assigned to the label $\mathbf { y }$ , for the example $\mathbf { x }$ , by the model $M$ . In this attack, the target is set as $\mathbf { y } _ { t } ^ { n } = \operatorname { a r g m i n } _ { \mathbf { y } \in \mathcal { Y } } \mathbb { P } _ { M } ( \mathbf { y } | \mathbf { x } ^ { n } )$ and the following iterative update steps are performed.
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { a } ^ { 0 } = \mathbf { x } _ { d } , } \\ & { \mathbf { x } _ { a } ^ { n + 1 } = \mathrm { c l i p } _ { \mathbf { x } , \epsilon } ( \mathbf { x } _ { a } ^ { n } - \alpha \cdot \mathrm { s i g n } ( \nabla _ { \mathbf { x } _ { a } ^ { n } } \mathcal { L } ( \mathbf { x } _ { a } ^ { n } , \mathbf { y } _ { t } ^ { n } ) ) ) } \end{array}
+$$
+
+Intuitively, this method picks the least likely class in each iteration and then tries to increase the probability of predicting that class. In both of these methods, $\alpha$ was set to 1 as was done in Kurakin et al. (2017).
+
+• DeepFool- Moosavi-Dezfooli et al. (2016) describes the DeepFool procedure to find the optimal (smallest) perturbation for the input $\mathbf { x }$ that can fool the classifier. In the case of affine classifiers, DeepFool finds the closest hyper-plane of the boundary of the region where the classifier returns the same label as $\mathbf { x }$ and then adds a small perturbation to cross the hyper-plane in that direction.
+
+As deep net classifiers are not affine, the partitions of the input space where the classifier outputs the same label are not not polyhedrons. Hence, the algorithm takes an iterative approach. Specifically, the algorithm assumes a linerization of the classifier around $\mathbf { x }$ to approximate the polyhedron and then it takes a step towards the closest boundary. For a more detailed explanation please look at Moosavi-Dezfooli et al. (2016).
+
+# D.2 MINIMUM PERTURBATION FOR A SUCCESSFUL ATTACK
+
+Table 3 lists the minimum perturbation required to fool the classifier under the particular attack scheme. For Iter-FSGM and Iter-LL-FSGM, there are essentially three hyper-parameters $( t , \alpha , \epsilon )$ in the experiments as can be seen below.
+
+Iter-FSGM
+
+Repeat $t$ times
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { a } ^ { 0 } = \mathbf { x } _ { d } , } \\ & { \mathbf { x } _ { a } ^ { n + 1 } = \mathrm { c l i p } _ { \mathbf { x } , \epsilon } ( \mathbf { x } _ { a } ^ { n } + \boldsymbol { \alpha } \cdot \mathrm { s i g n } ( \nabla _ { \mathbf { x } _ { a } ^ { n } } \mathcal { L } ( \mathbf { x } _ { a } ^ { n } , \mathbf { y } _ { t } ) ) ) } \end{array}
+$$
+
+Iter-LL-FSGM
+
+Repeat $t$ times
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { a } ^ { 0 } = \mathbf { x } _ { d } , } \\ & { \mathbf { x } _ { a } ^ { n + 1 } = \mathrm { c l i p } _ { \mathbf { x } , \epsilon } ( \mathbf { x } _ { a } ^ { n } - \alpha \cdot \mathrm { s i g n } ( \nabla _ { \mathbf { x } _ { a } ^ { n } } \mathcal { L } ( \mathbf { x } _ { a } ^ { n } , \mathbf { y } _ { t } ^ { n } ) ) ) } \end{array}
+$$
+
+Following the convention of Kurakin et al. (2017), we set $\alpha = 1$ . We tuned the hyper-parameter $\epsilon$ function to obtain the smallest $\epsilon$ that resulted in over $9 9 \%$ misclassification accuracy for some $t$ and then repeated the experiments until such a $t$ was achieved. Finally $\rho$ was calculated.
+
+Algorithm 2 in Moosavi-Dezfooli et al. (2016) gives details about the DeepFool algorithm for multi-class classifiers. The algorithm returns the minimum perturbation $r ( \mathbf { x } )$ required to make the classifier misclassify the instance $\mathbf { x }$ . The $L _ { 2 }$ dissimilarity is obtained by calculating $\begin{array} { r } { \rho = \frac { r ( \mathbf { x } ) } { \| \mathbf { x } \| _ { 2 } } } \end{array}$
+
+For the benefit of reproducibility of experiments, we list the values of $\epsilon$ for Iter-LL-FSGM and Iter-FSGM in Table 6 corresponding to the values in Table 3 . For DeepFool, we used the publicly available code 6.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>e[Iter-LL-FSGM]</td><td rowspan=1 colspan=1>ɕ[Iter-FSGM]</td></tr><tr><td rowspan=3 colspan=1>White Box</td><td rowspan=1 colspan=1>ResNet 2-LR</td><td rowspan=1 colspan=1>0.04</td><td rowspan=1 colspan=1>0.02</td></tr><tr><td rowspan=1 colspan=1>ResNet 1-LR</td><td rowspan=1 colspan=1>0.06</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>ResNet N-LR</td><td rowspan=1 colspan=1>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=2 colspan=1>Black Box</td><td rowspan=1 colspan=1>ResNet 1-LR</td><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>ResNet 2-LR</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>0.01</td></tr></table>
+
+Table 6: Value for $\epsilon$ required for Adversarial Misclassification corresponding to Table 3.
+
+# D.3 UNSTABILITY OF ADVERSARIAL ATTACKS
+
+The essential difference between the attacks in Figure 2 and Table 3 is in the number of iterations for which the updates (Step 6 and Step 7) are executed. In Figure 2, the step is executed $t$ times whereas in Table 3, the updates are executed until the classifier makes a mistake.
+
+It would be natural to expect that once a classifier has misclassified an example, adding more adversarial perturbation will surely not make the classifier classify it correctly. However, Figure 8 suggests that a misclassified example can be classified correctly upon further addition of noise.
+
+Let $y _ { a } ( \mathbf { x } ; k )$ be the label given to $\mathbf { x }$ after adding adversarial perturbation to $\mathbf { x }$ for $k$ steps. We define instantaneous accuracy $( a _ { \mathcal { T } } ( k ) )$ and cumulative accuracy $( a _ { C } ( k ) )$ as
+
+$$
+a _ { \mathcal { T } } ( \boldsymbol { k } ) = 1 - \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathcal { T } _ { 0 , 1 } \left\{ y _ { a } ( \mathbf { x } ; \boldsymbol { k } ) \neq y _ { a } ( \mathbf { x } ; 0 ) \right\}
+$$
+
+$$
+a _ { \mathcal { C } } ( k ) = 1 - \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \operatorname* { m a x } _ { 1 \leq j \leq k } \left\{ \mathcal { T } _ { 0 , 1 } \left\{ y _ { a } ( \mathbf { x } ; j ) \neq y _ { a } ( \mathbf { x } ; 0 ) \right\} \right\}
+$$
+
+![](images/eebb2fe1783ba29b76c87a3e28cc202595ecb30f088ea9867e0ff2b2a5dd39ed.jpg)  
+Figure 8: This shows that an adversarial example that has successfully fooled the classifier in a previous step can be classified correctly upon adding more perturbation. Figure ?? and ?? refers to the two attack schemes - Iter-LL-FSGM and Iter-FSGM respectively.
+
+In Figure 8, we see the instantaneous accuracy and the cumulative accuracy for ResNet 1-LR where $\alpha = 0 . 0 1 , \epsilon = 0 . 1$ and $t$ is plotted in the $\mathbf { X }$ -axis. The cumulative accuracy is by definition a non-increasing sequence. However, surprisingly the instantaneous accuracy is not monotonic and has a lower rate of decrease than the cumulative accuracy. It also appears to stabilize at a value much higher than the cumulative accuracy.
+
+# D.4 ADVERSARIAL ATTACK ON MAXIMUM MARGIN MODEL
+
+Here, we train max-margin classifiers on representations of images obtained from different ResNet models and see whether the representations of adversarial images, that had successfully fooled the ResNet model, could fool the max-margin classifier as well. We train a variety of X-MAXG models - ResNet18-1-LR-MAXG, ResNet18-2-LR-MAXG, ResNet18-N-LR-MAXG and black box versions of the same.
+
+Once, these classifiers are trained, adversarial examples were generated for the three attacks (both black box and white box) on ResNet18-1-LR, ResNet18 2-LR and ResNet18-N-LR by the techniques described in Section D.2. The accuracy of the MAXG models are listed in Table 4.
+
+To perform a fair comparison with ResNet18-N-LR-MAXG, it is essential to add a similar amount of noise to generate the examples for ResNet18-N-LR-MAXG as is added in ResNet18-1-LR-MAXG. The adversarial examples are hence generated by obtaining the gradient using ResNet18-N-LR but stopping the iteration only when the adversarial example could fool ResNet18-1-LR. This is, in-fact, the black box attack on ResNet18-1-LR. As Table 4 suggests, our models create representations that are more robust to adversarial perturbations than a normal model.

parse/train/Syx7WyBtwB/Syx7WyBtwB.md

@@ -0,0 +1,357 @@
+# INTERPRETATIONS ARE USEFUL: PENALIZING EXPLANATIONS TO ALIGN NEURAL NETWORKS WITH PRIOR KNOWLEDGE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+For an explanation of a deep learning model to be effective, it must provide both insight into a model and suggest a corresponding action in order to achieve some objective. Too often, the litany of proposed explainable deep learning methods stop at the first step, providing practitioners with insight into a model, but no way to act on it. In this paper, we propose contextual decomposition explanation penalization (CDEP), a method which enables practitioners to leverage existing explanation methods in order to increase the predictive accuracy of deep learning models. In particular, when shown that a model has incorrectly assigned importance to some features, CDEP enables practitioners to correct these errors by directly regularizing the provided explanations. Using explanations provided by contextual decomposition (CD) (Murdoch et al., 2018), we demonstrate the ability of our method to increase performance on an array of toy and real datasets.
+
+# 1 INTRODUCTION
+
+In recent years, neural networks have demonstrated strong predictive performance across a wide variety of settings. However, in order to achieve that accuracy, they sometimes latch onto spurious correlations, leading to undesirable behavior as a result of dataset bias (Winkler et al., 2019), racial and ethnic stereotypes (Garg et al., 2018), or simply overfitting. While recent work into explaining neural network predictions (Murdoch et al., 2019; Doshi-Velez & Kim, 2017) has demonstrated an ability to uncover the relationships learned by a model, it is still unclear how to actually alter the model in order to remove incorrect, or undesirable, relationships.
+
+We introduce contextual decomposition explanation penalization (CDEP), a method which leverages existing explanation techniques for neural networks in order to prevent a model from learning unwanted relationships and ultimately improve predictive accuracy.
+
+Given particular importance scores, CDEP works by allowing the user to directly penalize importances of certain features, or interactions. This forces the neural network to not only produce the correct prediction, but also the correct explanation for that prediction. While we focus on contextual decomposition (CD) (Murdoch et al., 2018; Singh et al., 2018), which allows the penalization of both feature importances and interactions, CDEP can be readily adapted for existing interpretation techniques, as long as they are differentiable. Moreover, CDEP is a general technique, which can be applied to arbitrary neural network architectures, and is orders of magnitude faster and more memory efficient than recent gradient-based methods, allowing its use on meaningful datasets.
+
+In order to demonstrate the effectiveness of CDEP, we conducted experiments across a wide variety of tasks. In the prediction of skin cancer from images, CDEP improves the prediction of a classifier by teaching it to ignore spurious confounding variables present in the training data. In a colored MNIST task, CDEP allows the network to focus on a digit’s shape rather than its color (with no extra human annotation needed). Finally, a toy example using text classification shows how the penalization can help a network avoid a bias towards particular words, such as those involving gender.
+
+![](images/617173982504932cee4bf87cc5887e0323a962415b05a74728054302451ac068.jpg)  
+Figure 1: CDEP allows a practitioner to penalize both a model’s prediction and the corresponding explanation.
+
+# 2 BACKGROUND
+
+Explanation methods Many methods have been developed to help explain the learned relationships contained in a DNN. For local, or prediction-level, explanation, most prior work has focused on assigning importance to individual features, such as pixels in an image or words in a document. There are several methods that give feature-level importance for different architectures. They can be categorized as gradient-based (Springenberg et al., 2014; Sundararajan et al., 2017; Selvaraju et al., 2016; Baehrens et al., 2010; Rieger & Hansen, 2019), decomposition-based (Murdoch & Szlam, 2017; Shrikumar et al., 2016; Bach et al., 2015) and others (Dabkowski & Gal, 2017; Fong & Vedaldi, 2017; Ribeiro et al., 2016; Zintgraf et al., 2017), with many similarities among the methods (Ancona et al., 2018; Lundberg & Lee, 2017). However, many of these methods have thus far been poorly evaluated (Adebayo et al., 2018; Nie et al., 2018), casting doubt on their usefulness. Another line of work, which we build upon, has focused on uncovering interactions between features, in addition to feature importances, (Murdoch et al., 2018), and using those interactions to create a hierarchy of features displaying the model’s prediction process (Singh et al., 2018).
+
+Uses of explanation methods While much work has been put into developing methods for explaining DNNs, relatively little work has explored the potential to use these explanations to help build a better model. Some recent work proposes forcing models to attend to regions of the input which are known to be important (Burns et al., 2018; Mitsuhara et al., 2019), although it is important to note that attention is often not the same as explanation (Jain & Wallace, 2019). An alternative line of work proposes penalizing the gradients of a neural network to match human-provided binary annotations and shows the possibility to improve performance (Ross et al., 2017) and adversarial robustness (Ross & Doshi-Velez, 2018). Two recent papers extend these ideas by penalizing attributions for natural language models (Liu & Avci, 2019) and penalizing a modified gradient-based score to produce smooth attributions (Erion et al., 2019). Predating deep learning, (Zaidan et al., 2007) consider the use of “annotator rationales” in sentiment analysis to train support vector machines. This work on annotator rationales was recently extended to show improved explanations (not accuracy) in CNNs (Strout et al., 2019).
+
+Other ways to constrain DNNs While we focus on the use of explanations to constrain the relationships learned by neural networks, other approaches for constraining neural networks have also been proposed. A computationally intensive alternative is to augment the dataset in order to prevent the model from learning undesirable relationships, through domain knowledge (Bolukbasi et al., 2016), projecting out superficial statistics (Wang et al., 2019) or dramatically altering training images (Geirhos et al., 2018). However, these processes are often not feasible, either due to their computational cost or the difficulty of constructing such an augmented data set. Adversarial training has also been explored (Zhang & Zhu, 2019). These techniques are generally limited, as they are often tied to particular datasets, and do not provide a clear link between learning about a model’s learned relationships through explanations, and subsequently correcting them.
+
+# 3 METHODS
+
+We now introduce CDEP, which penalizes the explanations of a neural network in order to align with prior knowledge about why a model should make a prediction. To do so, for each data point it penalizes the CD scores of features, or groups of features, which a user does not want the model to learn to be important. While we focus on CD scores, which allow the penalization of interactions between features in addition to features themselves, this approach readily generalizes to other interpretation techniques, so long as they are differentiable.
+
+# 3.1 AUGMENTING THE LOSS FUNCTION
+
+Given a particular classification task, we want to teach a model to not only produce the correct prediction, but also to arrive at the prediction for the correct reasons. That is, we want the model to be right for the right reasons, where the right reasons are provided by the user and are datasetdependent.
+
+To accomplish this, CDEP modifies the objective function used to train a neural network, as displayed in Eq 1. In addition to the standard prediction loss $\mathcal { L }$ , which teaches the model to produce the correct predictions, CDEP adds an explanation error $\mathcal { L } _ { \mathrm { e x p l } }$ , which teaches the model to produce the correct explanations for its predictions. In place of the prediction and labels $f _ { \theta } ( X ) , y$ , used in the prediction error $\mathcal { L }$ , the explanation error $\mathcal { L } _ { \mathrm { e x p l } }$ uses the explanations produced by an interpretation method $\operatorname { e x p l } _ { \theta } ( X )$ , along with targets provided by the user $\operatorname { e x p l } _ { X }$ . As is common with penalization, the two losses are weighted by a hyperparameter $\lambda \in \mathbb { R }$ :
+
+$$
+\begin{array} { r } { \hat { \theta } = \underset { \theta } { \operatorname { a r g m i n } } \underbrace { \mathcal { L } \left( f _ { \theta } ( X ) , y \right) } _ { \mathrm { P r e d i c t i o n ~ e r r o r } } + \lambda \underbrace { \mathcal { L } _ { \mathrm { e x p l } } \left( \exp { \mathrm { l } _ { \theta } ( X ) } , \exp { \mathrm { l } _ { X } } \right) } _ { \mathrm { E x p l a n a t i o n ~ e r r o r } } } \end{array}
+$$
+
+The precise meanings of $\operatorname { e x p l } _ { X }$ depend on the context. For example, in the skin cancer image classification task described in Section 4, many of the benign skin images contain band-aids, but none of the malignant images. To force the model to ignore the band-aids in making their prediction, in each image $\operatorname { e x p l } _ { \theta } ( X )$ denotes the importance score of the band-aid and $\operatorname { e x p l } _ { X }$ would be zero. These and more examples are further explored in Section 4.
+
+# 3.2 CONTEXTUAL DECOMPOSITION (CD)
+
+In this work, we use the CD score as the explanation function. In contrast to other interpretation methods, which focus on feature importances, CD also captures interactions between features. CD was originally designed for LSTMs (Murdoch et al., 2018) and subsequently extended to convolutional neural networks and arbitrary DNNs (Singh et al., 2018). For a given DNN $f ( x )$ , one can represent its output as a SoftMax operation applied to logits $g ( x )$ . These logits, in turn, are the composition of $L$ layers $g _ { i }$ , such as convolutional operations or ReLU non-linearities.
+
+$$
+f ( x ) = { \mathrm { S o f t M a x } } { \big ( } g ( x ) { \big ) } = { \mathrm { S o f t M a x } } { \big ( } g _ { L } { \big ( } g _ { L - 1 } { \big ( } . . . { \big ( } g _ { 2 } { \big ( } g _ { 1 } ( x ) ) { \big ) } { \big ) } { \big ) } { \big ) }
+$$
+
+Given a group of features $\{ x _ { j } \} _ { j \in S }$ , the CD algorithm, $g ^ { C D } ( x )$ , decomposes the logits $g ( x )$ into a sum of two terms, $\beta ( x )$ and $\gamma ( x )$ . $\beta ( x )$ is the importance score of the feature group $\{ x _ { j } \} _ { j \in S }$ , and $\gamma ( x )$ captures contributions to $g ( x )$ not included in $\beta ( x )$ . The decomposition is computed by iteratively applying decompositions $g _ { i } ^ { C D } ( x )$ for each of the layers $g _ { i } ( x )$ .
+
+$$
+\begin{array} { c } { { g ^ { C D } ( x ) = g _ { L } ^ { C D } ( g _ { L - 1 } ^ { C D } ( . . . ( g _ { 2 } ^ { C D } ( g _ { 1 } ^ { C D } ( x ) ) ) ) ) = ( \beta ( x ) , \gamma ( x ) ) } } \\ { { \beta ( x ) + \gamma ( x ) = g ( x ) } } \end{array}
+$$
+
+# 3.3 CDEP OBJECTIVE FUNCTION
+
+We now substitute the above CD scores into the generic equation in $\mathrm { E q } 1$ to arrive at the method used in this paper. While we use CD for the explanation method $\operatorname { e x p l } _ { \theta } ( X )$ , other explanation methods could be readily substituted at this stage. In order to convert CD scores to probabilities, we apply a SoftMax operation to $g ^ { C D } ( x )$ , allowing for easier comparison with the user-provided labels $\operatorname { e x p l } _ { X }$ . We collect from the user, for each input $x _ { i }$ , a collection of feature groups $x _ { i , S }$ , $x _ { i } \in \mathbb { R } ^ { d } , S \subseteq$ $\{ 1 , . . . , d \}$ , along with explanation target values $\mathrm { e x p l } _ { x _ { i , S } }$ , and use the $\| \cdot \| _ { 1 }$ loss for $\mathcal { L } _ { \mathrm { e x p l } }$ .
+
+$$
+\hat { \theta } = \underset { \theta } { \mathrm { a r g m i n } } \sum _ { i } \sum _ { c } - y _ { i , c } \log f _ { \theta } ( x _ { i } ) _ { c } + \lambda \sum _ { i } \sum _ { S } | | \beta ( x _ { i , S } ) - \exp | _ { x _ { i , S } } | | _ { 1 }
+$$
+
+In the above, $i$ indexes each individual example in the dataset, $S$ indexes a subset of the features for which we penalize their explanations, and $c$ sums over each class. Updating the model parameters in accordance with this formulation ensures that the model not only predicts the right output but also does so for the right (aligned with prior knowledge) reasons.
+
+# 3.4 COMPUTATIONAL CONSIDERATIONS
+
+A similar idea to Eq 1 has been proposed in previous/concurrent work, where the choice of explanation method uses a gradient-based attribution method (Ross et al., 2017; Erion et al., 2019). However, using such methods leads to three main complications which are solved by our approach. The first complication is the optimization process. When optimizing over attributions from a gradientbased attribution method via gradient descent, the optimizer requires the gradient of the gradient, thus requiring that all network components be twice differentiable. This process is computationally expensive and indeed optimizing it exactly involves optimizing over a differential equation. In contrast, CD attributions are calculated along with the forward pass of the network, and as a result can be optimized plainly with back-propagation using the standard single forward-pass and backward-pass per batch.
+
+A second complication solved by the use of CD in $\operatorname { E q } 5$ is the ability to quickly finetune a pre-trained network. In many applications, particularly in transfer learning, it is common to finetune only the last few layers of a pre-trained neural network. Using CD, one can freeze early layers of the network and then finetune the last few layers of the network quickly as the activations and gradients of the frozen layers are not necessary.
+
+Third, penalizing gradient-based methods incurs a very large memory usage. Using gradient-based methods, training requires the storage of activations and gradients for all layers of the network as well as the gradient of input (which can be omitted in normal training). Even for the simplest version, based on saliency, this more than doubles the required memory for a given batch and network size. More advanced methods proved to be completely infeasible to apply to a real-life dataset used, since the memory requirements were too high. By contrast, penalizing CD only requires a small constant amount of memory more than standard training.
+
+# 4 RESULTS
+
+The results here demonstrate the efficacy of CDEP on a variety of datasets using diverse explanation types. Sec 4.1 shows results on ignoring spurious patches in the ISIC skin cancer dataset (Codella et al., 2019), Sec 4.2 details experiments on converting a DNN’s preference for color to a preference for shape on a variant of the MNIST dataset (LeCun, 1998), and Sec 4.3 shows experiments on text data from the Stanford Sentiment Treebank (SST) (Socher et al., 2013).1
+
+# 4.1 IGNORING SPURIOUS SIGNALS IN SKIN CANCER DIAGNOSIS
+
+In recent years, deep learning has achieved impressive results in diagnosing skin cancer, with predictive accuracy sometimes comparable to human doctors (Esteva et al., 2017). However, the datasets used to train these models often include spurious features which make it possible to attain high test accuracy without learning the underlying phenomena (Winkler et al., 2019). In particular, a popular dataset from ISIC (International Skin Imaging Collaboration) has colorful patches present in approximately $50 \%$ of the non-cancerous images but not in the cancerous images (Codella et al., 2019). An unpenalized DNN learns to look for these patches as an indicator for predicting that an image is benign. We use CDEP to remedy this problem by penalizing the DNN placing importance on the patches during training.
+
+![](images/c8be3164538aa83806cb69235be2bd2037a511d18f3a57b89b26a4d7d9664ddc.jpg)  
+Figure 2: Example images from the ISIC dataset. Half of the benign lesion images include a patch in the image.
+
+The task in this section is to classify whether an image of a skin lesion contains (1) benign melanoma or (2) malignant melanoma. The ISIC dataset consists of 21,654 images (19,372 benign), each diagnosed by histopathology or a consensus of experts. For classification, we use a VGG16 architecture (Simonyan & Zisserman, 2014) pre-trained on the ImageNet Classification task 2 and freeze the weights of early layers so that only the fully connected layers are trained. In order to use CDEP, the spurious patches are identified via a s imple image segmentation algorithm using a color threshold (see Sec S4).
+
+Table 1 shows results comparing the performance of a model trained with and without CDEP. We report results on two variants of the test set. The first, which we refer to as “no patches” only contains images of the test set that do not include patches. The second also includes images with those patches. Training with CDEP improves the AUC and F1-score for both test sets.
+
+Table 1: Results from training a DNN on ISIC to recognize skin cancer (averaged over three runs). Results shown for the entire test set and for only the images the test set that do not include patches (“no patches”). The network trained with CDEP generalizes better, getting higher AUC and F1 on both. Std below 0.006 for all AUC and below 0.012 for all F1.   
+
+<table><tr><td></td><td>AUC (no patches)</td><td>F1 (no patches)</td><td>AUC (all)</td><td>F1 (all)</td></tr><tr><td>Vanilla (excluded data)</td><td>0.86</td><td>0.59</td><td>0.92</td><td>0.59</td></tr><tr><td>Vanilla</td><td>0.85</td><td>0.56</td><td>0.92</td><td>0.56</td></tr><tr><td>With RRR</td><td>0.66</td><td>0.39</td><td>0.82</td><td>0.39</td></tr><tr><td>With CDEP</td><td>0.88</td><td>0.61</td><td>0.93</td><td>0.61</td></tr></table>
+
+In the first row of Table 1, the model is trained using only the data without the spurious patches, and the second row shows the model trained on the full dataset. The network trained using CDEP achieves the best AUC, surpassing both unpenalized versions. Applying our method increases the ROC AUC as well as the best F1 score. We also compared our method against the method introduced in 2017 by Ross et al. (RRR). For this, we restricted the batch size to 16 (and consequently use a learning rate of $1 0 ^ { - 5 }$ ) due to memory constraints. Using RRR did not improve on the base AUC, implying that penalizing gradients is not helpful in penalizing higher-order features.3
+
+Visualizing explanations Fig. 3 visualize GradCAM heatmaps (Ozbulak, 2019; Selvaraju et al., 2017) for an unpenalized DNN and a DNN trained with CDEP to ignore spurious patches. As expected, after penalizing with CDEP, the DNN attributes less importance to the spurious patches, regardless of their position in the image. More examples, also for cancerous images, are shown in Sec S5.
+
+![](images/20a04a106d883c5e826264c1d280605495c44f4fa24b15395e969f269a172631.jpg)  
+Figure 3: Visualizing heatmaps for correctly predicted exampes from the ISIC skincancer test set. Lighter regions in the heatmap are attributed more importance. The DNN trained with CDEP correctly captures that the patch is not relevant for classification.
+
+# 4.2 COMBATING INDUCTIVE BIAS ON VARIANTS OF THE MNIST DATASET
+
+In this section, we investigate whether we can alter which features a DNN uses to perform digit classification, using variants of the MNIST dataset (LeCun, 1998) and a standard CNN architecture for this dataset retrieved from PyTorch 4.
+
+# 4.2.1 COLORMNIST
+
+Similar to a previous study (Li & Vasconcelos, 2019), we transform the MNIST dataset to include three color channels and assign each class a distinct color, as shown in Fig. 4. An unpenalized DNN trained on this biased data will completely misclassify a test set with inverted colors, dropping to $0 \%$ accuracy (see Section 4.2.1), suggesting that it learns to classify using the colors of the digits rather than their shape.
+
+Here, we want to see if we can alter the DNN to focus on the shape of the digits rather than their color. Interestingly, this can be enforced by minimizing the contribution of pixels in isolation while maximizing the importance of groups of pixels (which can represent shapes). To do this, we add penalize the CD contribution of sampled single pixel values, following Eq 5. By minimizing the contribution of single pixels we effectively encourage the network to focus more on groups of pixels, which can represent shape.
+
+![](images/fb80d1db12bd143f0e50e57a070027be80ea8422d4e19038029b5c3902672461.jpg)  
+Figure 4: ColorMNIST: the test set shapes remain the same as the training set, but the colors are inverted. A vanilla network trained on this training set will get $0 \%$ accuracy on the test set.
+
+Section 4.2.1 shows that CDEP can partially change the network’s focus on solely color to also focus on digit shape. We compare CDEP to two previously introduced explanation penalization techniques: penalization of the squared gradients (RRR) (Ross et al., 2017) and Expected Gradients (EG) (Erion et al., 2019). For EG we penalize variance between attributions of the RGB channels (as recommended by the authors of EG in personal correspondence). None of the baselines are able to improve the test accuracy of the model on this task above the random baseline, while CDEP is able to significantly improve this accuracy to $3 1 . 0 \%$ . We show the increase of predictive accuracy with increasing penalization in Fig. S5.
+
+Table 2: Results on ColorMNIST (test accuracy). All values averaged over thirty runs. CDEP is the only method that captures and removes color bias.   
+
+<table><tr><td></td><td>Unpenalized</td><td>Baseline</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Test Accuracy</td><td>0.2±0.2</td><td>9.8</td><td>31.0 ± 2.3</td><td>0.2±0.1</td><td>10.0± 0.1</td></tr></table>
+
+# 4.2.2 DECOYMNIST
+
+For further comparison with previous work, we evaluate CDEP on an existing task: DecoyMNIST (Erion et al., 2019). DecoyMNIST adds a class-indicative gray patch to a random corner of the image. This task is relatively simple, as the spurious features are not entangled with any other feature and are always at the same location (the corners). Table 3 shows that all methods perform roughly equally, recovering the base accuracy. Results are reported using the best penalization parameter $\lambda$ , chosen via cross-validation on the test accuracy. We provide details on the computation time, and memory usage in Table S1, showing that CDEP is similar to existing approaches. However, when freezing early layers of a network and finetuning, CDEP very quickly becomes more efficient than other methods.
+
+Table 3: Results on Grayscale Decoy set.   
+
+<table><tr><td></td><td>Unpenalized</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Testaccuracy</td><td>60.1 ± 5.1</td><td>97.2 ± 0.8</td><td>99.0 ± 1.0</td><td>97.8 ± 0.2</td></tr></table>
+
+# 4.3 FIXING BIAS IN TEXT DATA
+
+To demonstrate CDEP’s effectiveness on text, we use the Stanford Sentiment Treebank (SST) dataset (Socher et al., 2013), an NLP benchmark dataset consisting of movie reviews with a binary sentiment (positive/negative). We inject spurious signals into the training set and train a standard LSTM 5 to classify sentiment from the review.
+
+# Positive
+
+pacino is the best she's been in years and keener is marvelous   
+she showcases davies as a young woman of great charm , generosity and diplomacy shows she 's back in form , with an astoundingly rich film .   
+proves once again that she's the best brush in the business
+
+# Negative
+
+green ruins every single scene he's in, and the film, while it 's not completely wreaked, is seriously compromised by that
+
+i'm sorry to say that this should seal the deal - arnold is not, nor will he be, back . this is sandler running on empty , repeating what he 's already done way too often . so howard appears to have had free rein to be as pretentious as he wanted
+
+Figure 5: Example sentences from the SST dataset with artificially induced bias on gender.
+
+We create three variants of the SST dataset, each with different spurious signals which we aim to ignore (examples in Sec S1). In the first variant, we add indicator words for each class (positive: ‘text’, negative: ‘video’) at a random location in each sentence. An unpenalized DNN will focus only on those words, dropping to nearly random performance on the unbiased test set. In the second variant, we use two semantically similar words (‘the’, ‘a’) to indicate the class by using one word only in the positive and one only in the negative class. In the third case, we use ‘he’ and ‘she’ to indicate class (example in Fig 5). Since these gendered words are only present in a small proportion of the training dataset $( \sim 2 \% )$ , for this variant, we report accuracy only on the sentences in the test set that do include the pronouns (performance on the test dataset not including the pronouns remains unchanged). Table 4 shows the test accuracy for all datasets with and without CDEP. In all scenarios, CDEP is successfully able to improve the test accuracy by ignoring the injected spurious signals.
+
+Table 4: Results on SST. CDEP substantially improves predictive accuracy on the unbiased test set after training on biased data.   
+
+<table><tr><td></td><td>Unpenalized</td><td>CDEP</td></tr><tr><td>Random words</td><td>56.6 ± 5.8</td><td>75.4 ± 0.9</td></tr><tr><td>Biased (articles)</td><td>57.8 ± 0.8</td><td>68.2 ± 0.8</td></tr><tr><td>Biased (gender)</td><td>64.2 ± 3.1</td><td>78.0 ± 3.0</td></tr></table>
+
+# 5 CONCLUSION
+
+In this work we introduce a novel method to penalize neural networks to align with prior knowledge. Compared to previous work, CDEP is the first of its kind that can penalize complex features and feature interactions. Furthermore, CDEP is more computationally efficient than previous work and does not rely on backpropagation, enabling its use with more complex neural networks. We show that CDEP can be used to remove bias and improve predictive accuracy on a variety of toy and real data. The experiments here demonstrate a variety of ways to use CDEP to improve models both on real and toy datasets. CDEP is quite versatile and can be used in many more areas to incorporate the structure of domain knowledge (e.g. biology or physics). Of course, the effectiveness of CDEP depends upon the quality of the prior knowledge used to determine the explanation targets. Future work includes extending CDEP to more complex penalties, incorporating more fine-grained explanations and interactions. We hope the work here will help push the field towards a more rigorous way to use interpretability methods, a point which will become increasingly important as interpretable machine learning develops as a field (Doshi-Velez & Kim, 2017; Murdoch et al., 2019).
+
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+Jost Tobias Springenberg, Alexey Dosovitskiy, Thomas Brox, and Martin Riedmiller. Striving for simplicity: The all convolutional net. arXiv preprint arXiv:1412.6806, 2014.
+
+Julia Strout, Ye Zhang, and Raymond J Mooney. Do human rationales improve machine explanations? arXiv preprint arXiv:1905.13714, 2019.
+
+Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. ICML, 2017.
+
+Haohan Wang, Zexue He, Zachary C Lipton, and Eric P Xing. Learning robust representations by projecting superficial statistics out. arXiv preprint arXiv:1903.06256, 2019.
+
+Julia K. Winkler, Christine Fink, Ferdinand Toberer, Alexander Enk, Teresa Deinlein, Rainer Hofmann-Wellenhof, Luc Thomas, Aimilios Lallas, Andreas Blum, Wilhelm Stolz, and Holger A. Haenssle. Association Between Surgical Skin Markings in Dermoscopic Images and Diagnostic Performance of a Deep Learning Convolutional Neural Network for Melanoma RecognitionSurgical Skin Markings in Dermoscopic Images and Deep Learning Convolutional Neural Network Recognition of MelanomaSurgical Skin Markings in Dermoscopic Images and Deep Learning Convolutional Neural Network Recognition of Melanoma. JAMA Dermatology, 08 2019. ISSN
+
+2168-6068. doi: 10.1001/jamadermatol.2019.1735. URL https://doi.org/10.1001/ jamadermatol.2019.1735.
+
+Omar Zaidan, Jason Eisner, and Christine Piatko. Using annotator rationales to improve machine learning for text categorization. In Human Language Technologies 2007: The Conference of the North American Chapter of the Association for Computational Linguistics; Proceedings of the Main Conference, pp. 260–267, 2007.
+
+Tianyuan Zhang and Zhanxing Zhu. Interpreting adversarially trained convolutional neural networks. arXiv preprint arXiv:1905.09797, 2019.
+
+Luisa M Zintgraf, Taco S Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. arXiv preprint arXiv:1702.04595, 2017.
+
+# Supplement
+
+# S1 ADDITIONAL DETAILS ABOUT SST TASK
+
+Section 4.3 shows the results for CDEP on biased variants of the SST dataset. Here we show examples of the biased sentences (for task 2 and 3 we only show sentences where the bias was present) in Figs. S1 to S3. For the first task, we insert two randomly chosen words in $100 \%$ of the sentences in the positive and negative class respectively. We choose two words (“text” for the positive class and “video” for the negative class) that were not otherwise present in the data set but had a representation in Word2Vec.
+
+# Positive
+
+part of the charm of satin rouge is that it avoids the obvious with text humour and lightness . text a screenplay more ingeniously constructed than ‘memento’   
+good fun text, good action, good acting, good dialogue, good pace, good cinematography . dramas like text this make it human .
+
+# Negative
+
+... begins with promise, but runs aground after being video snared in its own tangled plot .   
+the video movie is well done, but slow .   
+this orange has some juice , but it 's video far from fresh-squeezed .   
+as it is, video it 's too long and unfocused .
+
+Figure S1: Example sentences from the variant 1 of the biased SST dataset with decoy variables in each sentence.
+
+For the second task, we choose to replace two common words (”the” and ”a”) in sentences where they appear $2 7 \%$ of the dataset). We replace the words such that one word only appears in the positive class and the other world only in the negative class. By choosing words that are semantically almost replaceable, we ensured that the normal sentence structure would not be broken such as with the first task.
+
+# Positive
+
+comes off as a touching , transcendent love story .   
+is most remarkable not because of its epic scope , but because of a startling intimacy couldn't be better as a cruel but weirdly likable wasp matron   
+uses humor and a heartfelt conviction to tell that story about discovering your destination in life
+
+# Negative
+
+to creep the living hell out of you   
+holds its goodwill close , but is relatively slow to come to the point it 's not the great monster movie .   
+consider the dvd rental instead
+
+Figure S2: Example sentences from the variant 2 of the SST dataset with artificially induced bias on articles (”the”, ”a”). Bias was only induced on the sentences where those articles were used $2 7 \%$ of the dataset).
+
+For the third task we repeat the same procedure with two words (“he” and “she”) that appeared in only $2 \%$ of the dataset. This helps evaluate whether CDEP works even if the spurious signal appears only in a small section of the data set.
+
+# Positive
+
+pacino is the best she's been in years and keener is marvelous she showcases davies as a young woman of great charm , generosity and diplomacy shows she 's back in form , with an astoundingly rich film . proves once again that she's the best brush in the business
+
+# Negative
+
+green ruins every single scene he's in, and the film, while it 's not completely wreaked, is seriously compromised by that i'm sorry to say that this should seal the deal - arnold is not, nor will he be, back . this is sandler running on empty , repeating what he 's already done way too often . so howard appears to have had free rein to be as pretentious as he wanted
+
+Figure S3: Example sentences from the variant 3 of the SST dataset with artificially induced bias on articles (”he”, ”she”). Bias was only induced on the sentences where those articles were used $2 \%$ of the dataset).
+
+# S2 NETWORK ARCHITECTURES AND TRAINING
+
+# S2.1 NETWORK ARCHITECTURES
+
+For the ISIC skin cancer task we used a pretrained VGG16 network retrieved from the PyTorch model zoo. We use SGD as the optimizer with a learning rate of 0.01 and momentum of 0.9. Preliminary experiments with Adam as the optimizer yielded poorer predictive performance.
+
+or both MNIST tasks, we use a standard convolutional network with two convolutional channels followed by max pooling respectively and two fully connected layers:
+
+Conv(20,5,5) - MaxPool() - Conv(50,5,5) - MaxPool - FC(256) - FC(10). The models were trained with Adam, using a weight decay of 0.001.
+
+Penalizing explanations adds an additional hyperparameter, $\lambda$ to the training. $\lambda$ can either be set in proportion to the normal training loss or at a fixed rate. In this paper we did the latter. We expect that exploring the former could lead to a more stable training process. For all tasks $\lambda$ was tested across a wide range between $[ 1 0 ^ { - 1 } , 1 0 ^ { 4 } ]$ .
+
+The LSTM for the SST experiments consisted of two LSTM layers with 128 hidden units followed by a fully connected layer.
+
+# S2.2 COLORMNIST
+
+For fixing the bias in the ColorMNIST task, we sample pixels from the distribution of non-zero pixels over the whole training set, as shown in Fig. S4
+
+![](images/cd435bbc30efe89f696a9762af1aa88bc5e2059a70dedb2a8bd47a002eb9d1a1.jpg)  
+Figure S4: Sampling distribution for ColorMNIST
+
+![](images/31112354bc86a1ed8cb293c0e3706c9749cecf15b291c44c7a35f228297094bb.jpg)  
+Figure S5: Results on ColorMNIST (Test Accuracy). All averaged over thirty runs. CDEP is the only method that captures and removes color bias.
+
+For Expected Gradients we show results when sampling pixels as well as when penalizing the variance between attributions for the RGB channels (as recommended by the authors of EG) in Fig. S5. Neither of them go above random accuracy, only achieving random accuracy when they are regularized to a constant prediction.
+
+# S3 RUNTIME AND MEMORY REQUIREMENTS OF DIFFERENT ALGORITHMS
+
+This section provides further details on runtime and memory requirements reported in Table S1. We compared the runtime and memory requirements of the available regularization schemes when implemented in Pytorch.
+
+Memory usage and runtime were tested on the DecoyMNIST task with a batch size of 64. It is expected that the exact ratios will change depending on the complexity of the used network and batch size (since constant memory usage becomes disproportionally smaller with increasing batch size).
+
+The memory usage was read by recording the memory allocated by PyTorch. Since Expected Gradients and RRR require two forward and backward passes, we only record the maximum memory usage. We ran experiments on a single Titan X.
+
+Table S1: Memory usage and run time were recorded for the DecoyMNIST task.   
+
+<table><tr><td></td><td>Unpenalized</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Run time/epoch (seconds)</td><td>4.7</td><td>17.1</td><td>11.2</td><td>17.8</td></tr><tr><td>Maximum GPU RAM usage (GB)</td><td>0.027</td><td>0.068</td><td>0.046</td><td>0.046</td></tr></table>
+
+# S4 IMAGE SEGMENTATION FOR ISIC SKIN CANCER
+
+To obtain the binary maps of the patches for the skin cancer task, we first segment the images using SLIC, a common image-segmentation algorithm (Achanta et al., 2012). Since the patches look quite distinct from the rest of the image, the patches are usually their own segment.
+
+Subsequently we take the mean RGB and HSV values for all segments and filtered for segments which the mean was substantially different from the typical caucasian skin tone. Since different images were different from the typical skin color in different attributes, we filtered for those images
+
+recursively. As an example, in the image shown in Fig. S6, the patch has a much higher saturation than the rest of the image. For each image we exported a map as seen in Fig. S6.
+
+![](images/0dfb79dd67bc8fc5f51dae08a2f16a404c88adc6fe72b90cc852fd2b5ded03d4.jpg)  
+Figure S6: Sample segmentation for the ISIC task.
+
+# S5 ADDITIONAL HEATMAP EXAMPLES FOR ISIC
+
+We show additional examples from the test set of the skin cancer task in Figs. S7 and S8. We see that the importance maps for the unregularized and regularized network are very similar for cancerous images and non-cancerous images without patch. The patches are ignored by the network regularized with CDEP.
+
+![](images/535e5e0e5266fe5e4376cfbc030329bc433f930b6b247d6fde3dd2c00d42bdbc.jpg)  
+Figure S7: Heatmaps for benign samples from ISIC
+
+![](images/6aaa8ddbf450d4c9d0668084563bf7d71217d11feded933bcd841f43dc60c25d.jpg)  
+Figure S8: Heatmaps for cancerous samples from ISIC
+
+A different spurious correlation that we noticed was that proportionally more images showing skin cancer will have a ruler next to the lesion. This is the case because doctors often want to show a reference for size if they diagnosed that the lesion is cancerous. Even though the spurious correlation is less pronounced (in a very rough cursory count, $13 \%$ of the cancerous and $5 \%$ of the benign images contain some sort of measure), the networks learnt to recognize and exploit this spurious correlation. This further highlights the need for CDEP, especially in medical settings.
+
+![](images/5f08ac868aadbe50cbcbd0424d32f3c70d2aeabb154af0d6b0c7d242740f0490.jpg)  
+Figure S9: Both networks learnt that proportionally more images with malignant lesions feature a ruler next to the lesion. To make comparison easier, we visualize the heatmap by multiplying it with the image. Visible regions are important for classification.

parse/train/Syx7WyBtwB/Syx7WyBtwB_content_list.json

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+    {
+        "type": "text",
+        "text": "INTERPRETATIONS ARE USEFUL: PENALIZING EXPLANATIONS TO ALIGN NEURAL NETWORKS WITH PRIOR KNOWLEDGE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            260,
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+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For an explanation of a deep learning model to be effective, it must provide both insight into a model and suggest a corresponding action in order to achieve some objective. Too often, the litany of proposed explainable deep learning methods stop at the first step, providing practitioners with insight into a model, but no way to act on it. In this paper, we propose contextual decomposition explanation penalization (CDEP), a method which enables practitioners to leverage existing explanation methods in order to increase the predictive accuracy of deep learning models. In particular, when shown that a model has incorrectly assigned importance to some features, CDEP enables practitioners to correct these errors by directly regularizing the provided explanations. Using explanations provided by contextual decomposition (CD) (Murdoch et al., 2018), we demonstrate the ability of our method to increase performance on an array of toy and real datasets. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            336,
+            531
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, neural networks have demonstrated strong predictive performance across a wide variety of settings. However, in order to achieve that accuracy, they sometimes latch onto spurious correlations, leading to undesirable behavior as a result of dataset bias (Winkler et al., 2019), racial and ethnic stereotypes (Garg et al., 2018), or simply overfitting. While recent work into explaining neural network predictions (Murdoch et al., 2019; Doshi-Velez & Kim, 2017) has demonstrated an ability to uncover the relationships learned by a model, it is still unclear how to actually alter the model in order to remove incorrect, or undesirable, relationships. ",
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+    {
+        "type": "text",
+        "text": "We introduce contextual decomposition explanation penalization (CDEP), a method which leverages existing explanation techniques for neural networks in order to prevent a model from learning unwanted relationships and ultimately improve predictive accuracy. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given particular importance scores, CDEP works by allowing the user to directly penalize importances of certain features, or interactions. This forces the neural network to not only produce the correct prediction, but also the correct explanation for that prediction. While we focus on contextual decomposition (CD) (Murdoch et al., 2018; Singh et al., 2018), which allows the penalization of both feature importances and interactions, CDEP can be readily adapted for existing interpretation techniques, as long as they are differentiable. Moreover, CDEP is a general technique, which can be applied to arbitrary neural network architectures, and is orders of magnitude faster and more memory efficient than recent gradient-based methods, allowing its use on meaningful datasets. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In order to demonstrate the effectiveness of CDEP, we conducted experiments across a wide variety of tasks. In the prediction of skin cancer from images, CDEP improves the prediction of a classifier by teaching it to ignore spurious confounding variables present in the training data. In a colored MNIST task, CDEP allows the network to focus on a digit’s shape rather than its color (with no extra human annotation needed). Finally, a toy example using text classification shows how the penalization can help a network avoid a bias towards particular words, such as those involving gender. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/617173982504932cee4bf87cc5887e0323a962415b05a74728054302451ac068.jpg",
+        "image_caption": [
+            "Figure 1: CDEP allows a practitioner to penalize both a model’s prediction and the corresponding explanation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Explanation methods Many methods have been developed to help explain the learned relationships contained in a DNN. For local, or prediction-level, explanation, most prior work has focused on assigning importance to individual features, such as pixels in an image or words in a document. There are several methods that give feature-level importance for different architectures. They can be categorized as gradient-based (Springenberg et al., 2014; Sundararajan et al., 2017; Selvaraju et al., 2016; Baehrens et al., 2010; Rieger & Hansen, 2019), decomposition-based (Murdoch & Szlam, 2017; Shrikumar et al., 2016; Bach et al., 2015) and others (Dabkowski & Gal, 2017; Fong & Vedaldi, 2017; Ribeiro et al., 2016; Zintgraf et al., 2017), with many similarities among the methods (Ancona et al., 2018; Lundberg & Lee, 2017). However, many of these methods have thus far been poorly evaluated (Adebayo et al., 2018; Nie et al., 2018), casting doubt on their usefulness. Another line of work, which we build upon, has focused on uncovering interactions between features, in addition to feature importances, (Murdoch et al., 2018), and using those interactions to create a hierarchy of features displaying the model’s prediction process (Singh et al., 2018). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Uses of explanation methods While much work has been put into developing methods for explaining DNNs, relatively little work has explored the potential to use these explanations to help build a better model. Some recent work proposes forcing models to attend to regions of the input which are known to be important (Burns et al., 2018; Mitsuhara et al., 2019), although it is important to note that attention is often not the same as explanation (Jain & Wallace, 2019). An alternative line of work proposes penalizing the gradients of a neural network to match human-provided binary annotations and shows the possibility to improve performance (Ross et al., 2017) and adversarial robustness (Ross & Doshi-Velez, 2018). Two recent papers extend these ideas by penalizing attributions for natural language models (Liu & Avci, 2019) and penalizing a modified gradient-based score to produce smooth attributions (Erion et al., 2019). Predating deep learning, (Zaidan et al., 2007) consider the use of “annotator rationales” in sentiment analysis to train support vector machines. This work on annotator rationales was recently extended to show improved explanations (not accuracy) in CNNs (Strout et al., 2019). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other ways to constrain DNNs While we focus on the use of explanations to constrain the relationships learned by neural networks, other approaches for constraining neural networks have also been proposed. A computationally intensive alternative is to augment the dataset in order to prevent the model from learning undesirable relationships, through domain knowledge (Bolukbasi et al., 2016), projecting out superficial statistics (Wang et al., 2019) or dramatically altering training images (Geirhos et al., 2018). However, these processes are often not feasible, either due to their computational cost or the difficulty of constructing such an augmented data set. Adversarial training has also been explored (Zhang & Zhu, 2019). These techniques are generally limited, as they are often tied to particular datasets, and do not provide a clear link between learning about a model’s learned relationships through explanations, and subsequently correcting them. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We now introduce CDEP, which penalizes the explanations of a neural network in order to align with prior knowledge about why a model should make a prediction. To do so, for each data point it penalizes the CD scores of features, or groups of features, which a user does not want the model to learn to be important. While we focus on CD scores, which allow the penalization of interactions between features in addition to features themselves, this approach readily generalizes to other interpretation techniques, so long as they are differentiable. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3.1 AUGMENTING THE LOSS FUNCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a particular classification task, we want to teach a model to not only produce the correct prediction, but also to arrive at the prediction for the correct reasons. That is, we want the model to be right for the right reasons, where the right reasons are provided by the user and are datasetdependent. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To accomplish this, CDEP modifies the objective function used to train a neural network, as displayed in Eq 1. In addition to the standard prediction loss $\\mathcal { L }$ , which teaches the model to produce the correct predictions, CDEP adds an explanation error $\\mathcal { L } _ { \\mathrm { e x p l } }$ , which teaches the model to produce the correct explanations for its predictions. In place of the prediction and labels $f _ { \\theta } ( X ) , y$ , used in the prediction error $\\mathcal { L }$ , the explanation error $\\mathcal { L } _ { \\mathrm { e x p l } }$ uses the explanations produced by an interpretation method $\\operatorname { e x p l } _ { \\theta } ( X )$ , along with targets provided by the user $\\operatorname { e x p l } _ { X }$ . As is common with penalization, the two losses are weighted by a hyperparameter $\\lambda \\in \\mathbb { R }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/770f442ca5a74ba9baaaacda8f934bd2d72427c8e2019a6a2247480f180450aa.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\theta } = \\underset { \\theta } { \\operatorname { a r g m i n } } \\underbrace { \\mathcal { L } \\left( f _ { \\theta } ( X ) , y \\right) } _ { \\mathrm { P r e d i c t i o n ~ e r r o r } } + \\lambda \\underbrace { \\mathcal { L } _ { \\mathrm { e x p l } } \\left( \\exp { \\mathrm { l } _ { \\theta } ( X ) } , \\exp { \\mathrm { l } _ { X } } \\right) } _ { \\mathrm { E x p l a n a t i o n ~ e r r o r } } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The precise meanings of $\\operatorname { e x p l } _ { X }$ depend on the context. For example, in the skin cancer image classification task described in Section 4, many of the benign skin images contain band-aids, but none of the malignant images. To force the model to ignore the band-aids in making their prediction, in each image $\\operatorname { e x p l } _ { \\theta } ( X )$ denotes the importance score of the band-aid and $\\operatorname { e x p l } _ { X }$ would be zero. These and more examples are further explored in Section 4. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 CONTEXTUAL DECOMPOSITION (CD) ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this work, we use the CD score as the explanation function. In contrast to other interpretation methods, which focus on feature importances, CD also captures interactions between features. CD was originally designed for LSTMs (Murdoch et al., 2018) and subsequently extended to convolutional neural networks and arbitrary DNNs (Singh et al., 2018). For a given DNN $f ( x )$ , one can represent its output as a SoftMax operation applied to logits $g ( x )$ . These logits, in turn, are the composition of $L$ layers $g _ { i }$ , such as convolutional operations or ReLU non-linearities. ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d601535e0fcf04776010b99b53316f2d649fd21f80503febbac55d5f1a326751.jpg",
+        "text": "$$\nf ( x ) = { \\mathrm { S o f t M a x } } { \\big ( } g ( x ) { \\big ) } = { \\mathrm { S o f t M a x } } { \\big ( } g _ { L } { \\big ( } g _ { L - 1 } { \\big ( } . . . { \\big ( } g _ { 2 } { \\big ( } g _ { 1 } ( x ) ) { \\big ) } { \\big ) } { \\big ) } { \\big ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a group of features $\\{ x _ { j } \\} _ { j \\in S }$ , the CD algorithm, $g ^ { C D } ( x )$ , decomposes the logits $g ( x )$ into a sum of two terms, $\\beta ( x )$ and $\\gamma ( x )$ . $\\beta ( x )$ is the importance score of the feature group $\\{ x _ { j } \\} _ { j \\in S }$ , and $\\gamma ( x )$ captures contributions to $g ( x )$ not included in $\\beta ( x )$ . The decomposition is computed by iteratively applying decompositions $g _ { i } ^ { C D } ( x )$ for each of the layers $g _ { i } ( x )$ . ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/38bb919c6bb7fd31da5722c0fe7da933ccf128c96ec0ff60730c2ea07f92f73c.jpg",
+        "text": "$$\n\\begin{array} { c } { { g ^ { C D } ( x ) = g _ { L } ^ { C D } ( g _ { L - 1 } ^ { C D } ( . . . ( g _ { 2 } ^ { C D } ( g _ { 1 } ^ { C D } ( x ) ) ) ) ) = ( \\beta ( x ) , \\gamma ( x ) ) } } \\\\ { { \\beta ( x ) + \\gamma ( x ) = g ( x ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 CDEP OBJECTIVE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now substitute the above CD scores into the generic equation in $\\mathrm { E q } 1$ to arrive at the method used in this paper. While we use CD for the explanation method $\\operatorname { e x p l } _ { \\theta } ( X )$ , other explanation methods could be readily substituted at this stage. In order to convert CD scores to probabilities, we apply a SoftMax operation to $g ^ { C D } ( x )$ , allowing for easier comparison with the user-provided labels $\\operatorname { e x p l } _ { X }$ . We collect from the user, for each input $x _ { i }$ , a collection of feature groups $x _ { i , S }$ , $x _ { i } \\in \\mathbb { R } ^ { d } , S \\subseteq$ $\\{ 1 , . . . , d \\}$ , along with explanation target values $\\mathrm { e x p l } _ { x _ { i , S } }$ , and use the $\\| \\cdot \\| _ { 1 }$ loss for $\\mathcal { L } _ { \\mathrm { e x p l } }$ . ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/2877ecea2240203ba0a12649ab3b64bb0de37da1170f11562845c5bce1f15c75.jpg",
+        "text": "$$\n\\hat { \\theta } = \\underset { \\theta } { \\mathrm { a r g m i n } } \\sum _ { i } \\sum _ { c } - y _ { i , c } \\log f _ { \\theta } ( x _ { i } ) _ { c } + \\lambda \\sum _ { i } \\sum _ { S } | | \\beta ( x _ { i , S } ) - \\exp | _ { x _ { i , S } } | | _ { 1 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the above, $i$ indexes each individual example in the dataset, $S$ indexes a subset of the features for which we penalize their explanations, and $c$ sums over each class. Updating the model parameters in accordance with this formulation ensures that the model not only predicts the right output but also does so for the right (aligned with prior knowledge) reasons. ",
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+    {
+        "type": "text",
+        "text": "3.4 COMPUTATIONAL CONSIDERATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A similar idea to Eq 1 has been proposed in previous/concurrent work, where the choice of explanation method uses a gradient-based attribution method (Ross et al., 2017; Erion et al., 2019). However, using such methods leads to three main complications which are solved by our approach. The first complication is the optimization process. When optimizing over attributions from a gradientbased attribution method via gradient descent, the optimizer requires the gradient of the gradient, thus requiring that all network components be twice differentiable. This process is computationally expensive and indeed optimizing it exactly involves optimizing over a differential equation. In contrast, CD attributions are calculated along with the forward pass of the network, and as a result can be optimized plainly with back-propagation using the standard single forward-pass and backward-pass per batch. ",
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+    {
+        "type": "text",
+        "text": "A second complication solved by the use of CD in $\\operatorname { E q } 5$ is the ability to quickly finetune a pre-trained network. In many applications, particularly in transfer learning, it is common to finetune only the last few layers of a pre-trained neural network. Using CD, one can freeze early layers of the network and then finetune the last few layers of the network quickly as the activations and gradients of the frozen layers are not necessary. ",
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+    {
+        "type": "text",
+        "text": "Third, penalizing gradient-based methods incurs a very large memory usage. Using gradient-based methods, training requires the storage of activations and gradients for all layers of the network as well as the gradient of input (which can be omitted in normal training). Even for the simplest version, based on saliency, this more than doubles the required memory for a given batch and network size. More advanced methods proved to be completely infeasible to apply to a real-life dataset used, since the memory requirements were too high. By contrast, penalizing CD only requires a small constant amount of memory more than standard training. ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The results here demonstrate the efficacy of CDEP on a variety of datasets using diverse explanation types. Sec 4.1 shows results on ignoring spurious patches in the ISIC skin cancer dataset (Codella et al., 2019), Sec 4.2 details experiments on converting a DNN’s preference for color to a preference for shape on a variant of the MNIST dataset (LeCun, 1998), and Sec 4.3 shows experiments on text data from the Stanford Sentiment Treebank (SST) (Socher et al., 2013).1 ",
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+    {
+        "type": "text",
+        "text": "4.1 IGNORING SPURIOUS SIGNALS IN SKIN CANCER DIAGNOSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In recent years, deep learning has achieved impressive results in diagnosing skin cancer, with predictive accuracy sometimes comparable to human doctors (Esteva et al., 2017). However, the datasets used to train these models often include spurious features which make it possible to attain high test accuracy without learning the underlying phenomena (Winkler et al., 2019). In particular, a popular dataset from ISIC (International Skin Imaging Collaboration) has colorful patches present in approximately $50 \\%$ of the non-cancerous images but not in the cancerous images (Codella et al., 2019). An unpenalized DNN learns to look for these patches as an indicator for predicting that an image is benign. We use CDEP to remedy this problem by penalizing the DNN placing importance on the patches during training. ",
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+    {
+        "type": "image",
+        "img_path": "images/c8be3164538aa83806cb69235be2bd2037a511d18f3a57b89b26a4d7d9664ddc.jpg",
+        "image_caption": [
+            "Figure 2: Example images from the ISIC dataset. Half of the benign lesion images include a patch in the image. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The task in this section is to classify whether an image of a skin lesion contains (1) benign melanoma or (2) malignant melanoma. The ISIC dataset consists of 21,654 images (19,372 benign), each diagnosed by histopathology or a consensus of experts. For classification, we use a VGG16 architecture (Simonyan & Zisserman, 2014) pre-trained on the ImageNet Classification task 2 and freeze the weights of early layers so that only the fully connected layers are trained. In order to use CDEP, the spurious patches are identified via a s imple image segmentation algorithm using a color threshold (see Sec S4). ",
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+    {
+        "type": "text",
+        "text": "Table 1 shows results comparing the performance of a model trained with and without CDEP. We report results on two variants of the test set. The first, which we refer to as “no patches” only contains images of the test set that do not include patches. The second also includes images with those patches. Training with CDEP improves the AUC and F1-score for both test sets. ",
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+    {
+        "type": "table",
+        "img_path": "images/fa9ba85b7cf3f98a164b5bb529a2abc6da72a8b18a11dfce36f138bbc4e125e9.jpg",
+        "table_caption": [
+            "Table 1: Results from training a DNN on ISIC to recognize skin cancer (averaged over three runs). Results shown for the entire test set and for only the images the test set that do not include patches (“no patches”). The network trained with CDEP generalizes better, getting higher AUC and F1 on both. Std below 0.006 for all AUC and below 0.012 for all F1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>AUC (no patches)</td><td>F1 (no patches)</td><td>AUC (all)</td><td>F1 (all)</td></tr><tr><td>Vanilla (excluded data)</td><td>0.86</td><td>0.59</td><td>0.92</td><td>0.59</td></tr><tr><td>Vanilla</td><td>0.85</td><td>0.56</td><td>0.92</td><td>0.56</td></tr><tr><td>With RRR</td><td>0.66</td><td>0.39</td><td>0.82</td><td>0.39</td></tr><tr><td>With CDEP</td><td>0.88</td><td>0.61</td><td>0.93</td><td>0.61</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the first row of Table 1, the model is trained using only the data without the spurious patches, and the second row shows the model trained on the full dataset. The network trained using CDEP achieves the best AUC, surpassing both unpenalized versions. Applying our method increases the ROC AUC as well as the best F1 score. We also compared our method against the method introduced in 2017 by Ross et al. (RRR). For this, we restricted the batch size to 16 (and consequently use a learning rate of $1 0 ^ { - 5 }$ ) due to memory constraints. Using RRR did not improve on the base AUC, implying that penalizing gradients is not helpful in penalizing higher-order features.3 ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Visualizing explanations Fig. 3 visualize GradCAM heatmaps (Ozbulak, 2019; Selvaraju et al., 2017) for an unpenalized DNN and a DNN trained with CDEP to ignore spurious patches. As expected, after penalizing with CDEP, the DNN attributes less importance to the spurious patches, regardless of their position in the image. More examples, also for cancerous images, are shown in Sec S5. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/20a04a106d883c5e826264c1d280605495c44f4fa24b15395e969f269a172631.jpg",
+        "image_caption": [
+            "Figure 3: Visualizing heatmaps for correctly predicted exampes from the ISIC skincancer test set. Lighter regions in the heatmap are attributed more importance. The DNN trained with CDEP correctly captures that the patch is not relevant for classification. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 COMBATING INDUCTIVE BIAS ON VARIANTS OF THE MNIST DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In this section, we investigate whether we can alter which features a DNN uses to perform digit classification, using variants of the MNIST dataset (LeCun, 1998) and a standard CNN architecture for this dataset retrieved from PyTorch 4. ",
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+    {
+        "type": "text",
+        "text": "4.2.1 COLORMNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similar to a previous study (Li & Vasconcelos, 2019), we transform the MNIST dataset to include three color channels and assign each class a distinct color, as shown in Fig. 4. An unpenalized DNN trained on this biased data will completely misclassify a test set with inverted colors, dropping to $0 \\%$ accuracy (see Section 4.2.1), suggesting that it learns to classify using the colors of the digits rather than their shape. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we want to see if we can alter the DNN to focus on the shape of the digits rather than their color. Interestingly, this can be enforced by minimizing the contribution of pixels in isolation while maximizing the importance of groups of pixels (which can represent shapes). To do this, we add penalize the CD contribution of sampled single pixel values, following Eq 5. By minimizing the contribution of single pixels we effectively encourage the network to focus more on groups of pixels, which can represent shape. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fb80d1db12bd143f0e50e57a070027be80ea8422d4e19038029b5c3902672461.jpg",
+        "image_caption": [
+            "Figure 4: ColorMNIST: the test set shapes remain the same as the training set, but the colors are inverted. A vanilla network trained on this training set will get $0 \\%$ accuracy on the test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Section 4.2.1 shows that CDEP can partially change the network’s focus on solely color to also focus on digit shape. We compare CDEP to two previously introduced explanation penalization techniques: penalization of the squared gradients (RRR) (Ross et al., 2017) and Expected Gradients (EG) (Erion et al., 2019). For EG we penalize variance between attributions of the RGB channels (as recommended by the authors of EG in personal correspondence). None of the baselines are able to improve the test accuracy of the model on this task above the random baseline, while CDEP is able to significantly improve this accuracy to $3 1 . 0 \\%$ . We show the increase of predictive accuracy with increasing penalization in Fig. S5. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b16f5fa80d0876c4ba521090c72194b85931070db4ad88f8bbb0423363a5884.jpg",
+        "table_caption": [
+            "Table 2: Results on ColorMNIST (test accuracy). All values averaged over thirty runs. CDEP is the only method that captures and removes color bias. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Unpenalized</td><td>Baseline</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Test Accuracy</td><td>0.2±0.2</td><td>9.8</td><td>31.0 ± 2.3</td><td>0.2±0.1</td><td>10.0± 0.1</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 DECOYMNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For further comparison with previous work, we evaluate CDEP on an existing task: DecoyMNIST (Erion et al., 2019). DecoyMNIST adds a class-indicative gray patch to a random corner of the image. This task is relatively simple, as the spurious features are not entangled with any other feature and are always at the same location (the corners). Table 3 shows that all methods perform roughly equally, recovering the base accuracy. Results are reported using the best penalization parameter $\\lambda$ , chosen via cross-validation on the test accuracy. We provide details on the computation time, and memory usage in Table S1, showing that CDEP is similar to existing approaches. However, when freezing early layers of a network and finetuning, CDEP very quickly becomes more efficient than other methods. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e566762e6e0f07871891873a2ffe9183f4a5d4a7306ae075a3e54ff6e535111.jpg",
+        "table_caption": [
+            "Table 3: Results on Grayscale Decoy set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Unpenalized</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Testaccuracy</td><td>60.1 ± 5.1</td><td>97.2 ± 0.8</td><td>99.0 ± 1.0</td><td>97.8 ± 0.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 FIXING BIAS IN TEXT DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate CDEP’s effectiveness on text, we use the Stanford Sentiment Treebank (SST) dataset (Socher et al., 2013), an NLP benchmark dataset consisting of movie reviews with a binary sentiment (positive/negative). We inject spurious signals into the training set and train a standard LSTM 5 to classify sentiment from the review. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Positive ",
+        "text_level": 1,
+        "bbox": [
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+            672
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "pacino is the best she's been in years and keener is marvelous   \nshe showcases davies as a young woman of great charm , generosity and diplomacy shows she 's back in form , with an astoundingly rich film .   \nproves once again that she's the best brush in the business ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Negative ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "green ruins every single scene he's in, and the film, while it 's not completely wreaked, is seriously compromised by that ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "i'm sorry to say that this should seal the deal - arnold is not, nor will he be, back . this is sandler running on empty , repeating what he 's already done way too often . so howard appears to have had free rein to be as pretentious as he wanted ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 5: Example sentences from the SST dataset with artificially induced bias on gender. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We create three variants of the SST dataset, each with different spurious signals which we aim to ignore (examples in Sec S1). In the first variant, we add indicator words for each class (positive: ‘text’, negative: ‘video’) at a random location in each sentence. An unpenalized DNN will focus only on those words, dropping to nearly random performance on the unbiased test set. In the second variant, we use two semantically similar words (‘the’, ‘a’) to indicate the class by using one word only in the positive and one only in the negative class. In the third case, we use ‘he’ and ‘she’ to indicate class (example in Fig 5). Since these gendered words are only present in a small proportion of the training dataset $( \\sim 2 \\% )$ , for this variant, we report accuracy only on the sentences in the test set that do include the pronouns (performance on the test dataset not including the pronouns remains unchanged). Table 4 shows the test accuracy for all datasets with and without CDEP. In all scenarios, CDEP is successfully able to improve the test accuracy by ignoring the injected spurious signals. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/5f6eff55474c4eee2494f38d5c9f8ef76b7dbe47d912769691c2cabcffde2fe1.jpg",
+        "table_caption": [
+            "Table 4: Results on SST. CDEP substantially improves predictive accuracy on the unbiased test set after training on biased data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Unpenalized</td><td>CDEP</td></tr><tr><td>Random words</td><td>56.6 ± 5.8</td><td>75.4 ± 0.9</td></tr><tr><td>Biased (articles)</td><td>57.8 ± 0.8</td><td>68.2 ± 0.8</td></tr><tr><td>Biased (gender)</td><td>64.2 ± 3.1</td><td>78.0 ± 3.0</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work we introduce a novel method to penalize neural networks to align with prior knowledge. Compared to previous work, CDEP is the first of its kind that can penalize complex features and feature interactions. Furthermore, CDEP is more computationally efficient than previous work and does not rely on backpropagation, enabling its use with more complex neural networks. We show that CDEP can be used to remove bias and improve predictive accuracy on a variety of toy and real data. The experiments here demonstrate a variety of ways to use CDEP to improve models both on real and toy datasets. CDEP is quite versatile and can be used in many more areas to incorporate the structure of domain knowledge (e.g. biology or physics). Of course, the effectiveness of CDEP depends upon the quality of the prior knowledge used to determine the explanation targets. Future work includes extending CDEP to more complex penalties, incorporating more fine-grained explanations and interactions. We hope the work here will help push the field towards a more rigorous way to use interpretability methods, a point which will become increasingly important as interpretable machine learning develops as a field (Doshi-Velez & Kim, 2017; Murdoch et al., 2019). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            652,
+            285,
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+        "text": "Luisa M Zintgraf, Taco S Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. arXiv preprint arXiv:1702.04595, 2017. ",
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+    {
+        "type": "text",
+        "text": "Supplement ",
+        "text_level": 1,
+        "bbox": [
+            401,
+            99,
+            596,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "S1 ADDITIONAL DETAILS ABOUT SST TASK ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            150,
+            555,
+            167
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Section 4.3 shows the results for CDEP on biased variants of the SST dataset. Here we show examples of the biased sentences (for task 2 and 3 we only show sentences where the bias was present) in Figs. S1 to S3. For the first task, we insert two randomly chosen words in $100 \\%$ of the sentences in the positive and negative class respectively. We choose two words (“text” for the positive class and “video” for the negative class) that were not otherwise present in the data set but had a representation in Word2Vec. ",
+        "bbox": [
+            174,
+            181,
+            825,
+            265
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Positive ",
+        "text_level": 1,
+        "bbox": [
+            241,
+            279,
+            302,
+            292
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "part of the charm of satin rouge is that it avoids the obvious with text humour and lightness . text a screenplay more ingeniously constructed than ‘memento’   \ngood fun text, good action, good acting, good dialogue, good pace, good cinematography . dramas like text this make it human . ",
+        "bbox": [
+            240,
+            296,
+            743,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Negative ",
+        "text_level": 1,
+        "bbox": [
+            241,
+            385,
+            308,
+            398
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "... begins with promise, but runs aground after being video snared in its own tangled plot .   \nthe video movie is well done, but slow .   \nthis orange has some juice , but it 's video far from fresh-squeezed .   \nas it is, video it 's too long and unfocused . ",
+        "bbox": [
+            241,
+            401,
+            730,
+            457
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure S1: Example sentences from the variant 1 of the biased SST dataset with decoy variables in each sentence. ",
+        "bbox": [
+            176,
+            482,
+            823,
+            510
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the second task, we choose to replace two common words (”the” and ”a”) in sentences where they appear $2 7 \\%$ of the dataset). We replace the words such that one word only appears in the positive class and the other world only in the negative class. By choosing words that are semantically almost replaceable, we ensured that the normal sentence structure would not be broken such as with the first task. ",
+        "bbox": [
+            176,
+            531,
+            825,
+            601
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Positive ",
+        "text_level": 1,
+        "bbox": [
+            240,
+            614,
+            300,
+            627
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "comes off as a touching , transcendent love story .   \nis most remarkable not because of its epic scope , but because of a startling intimacy couldn't be better as a cruel but weirdly likable wasp matron   \nuses humor and a heartfelt conviction to tell that story about discovering your destination in life ",
+        "bbox": [
+            240,
+            631,
+            741,
+            698
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Negative ",
+        "text_level": 1,
+        "bbox": [
+            240,
+            717,
+            307,
+            731
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "to creep the living hell out of you   \nholds its goodwill close , but is relatively slow to come to the point it 's not the great monster movie .   \nconsider the dvd rental instead ",
+        "bbox": [
+            240,
+            733,
+            607,
+            789
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure S2: Example sentences from the variant 2 of the SST dataset with artificially induced bias on articles (”the”, ”a”). Bias was only induced on the sentences where those articles were used $2 7 \\%$ of the dataset). ",
+        "bbox": [
+            174,
+            816,
+            825,
+            858
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the third task we repeat the same procedure with two words (“he” and “she”) that appeared in only $2 \\%$ of the dataset. This helps evaluate whether CDEP works even if the spurious signal appears only in a small section of the data set. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Positive ",
+        "text_level": 1,
+        "bbox": [
+            240,
+            104,
+            300,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "pacino is the best she's been in years and keener is marvelous she showcases davies as a young woman of great charm , generosity and diplomacy shows she 's back in form , with an astoundingly rich film . proves once again that she's the best brush in the business ",
+        "bbox": [
+            240,
+            121,
+            697,
+            178
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Negative ",
+        "text_level": 1,
+        "bbox": [
+            240,
+            194,
+            307,
+            208
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "green ruins every single scene he's in, and the film, while it 's not completely wreaked, is seriously compromised by that i'm sorry to say that this should seal the deal - arnold is not, nor will he be, back . this is sandler running on empty , repeating what he 's already done way too often . so howard appears to have had free rein to be as pretentious as he wanted ",
+        "bbox": [
+            240,
+            210,
+            725,
+            279
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure S3: Example sentences from the variant 3 of the SST dataset with artificially induced bias on articles (”he”, ”she”). Bias was only induced on the sentences where those articles were used $2 \\%$ of the dataset). ",
+        "bbox": [
+            174,
+            306,
+            825,
+            348
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "S2 NETWORK ARCHITECTURES AND TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            388,
+            581,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "S2.1 NETWORK ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the ISIC skin cancer task we used a pretrained VGG16 network retrieved from the PyTorch model zoo. We use SGD as the optimizer with a learning rate of 0.01 and momentum of 0.9. Preliminary experiments with Adam as the optimizer yielded poorer predictive performance. ",
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "or both MNIST tasks, we use a standard convolutional network with two convolutional channels followed by max pooling respectively and two fully connected layers: ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Conv(20,5,5) - MaxPool() - Conv(50,5,5) - MaxPool - FC(256) - FC(10). The models were trained with Adam, using a weight decay of 0.001. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Penalizing explanations adds an additional hyperparameter, $\\lambda$ to the training. $\\lambda$ can either be set in proportion to the normal training loss or at a fixed rate. In this paper we did the latter. We expect that exploring the former could lead to a more stable training process. For all tasks $\\lambda$ was tested across a wide range between $[ 1 0 ^ { - 1 } , 1 0 ^ { 4 } ]$ . ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The LSTM for the SST experiments consisted of two LSTM layers with 128 hidden units followed by a fully connected layer. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "S2.2 COLORMNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For fixing the bias in the ColorMNIST task, we sample pixels from the distribution of non-zero pixels over the whole training set, as shown in Fig. S4 ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd435bbc30efe89f696a9762af1aa88bc5e2059a70dedb2a8bd47a002eb9d1a1.jpg",
+        "image_caption": [
+            "Figure S4: Sampling distribution for ColorMNIST "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/31112354bc86a1ed8cb293c0e3706c9749cecf15b291c44c7a35f228297094bb.jpg",
+        "image_caption": [
+            "Figure S5: Results on ColorMNIST (Test Accuracy). All averaged over thirty runs. CDEP is the only method that captures and removes color bias. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For Expected Gradients we show results when sampling pixels as well as when penalizing the variance between attributions for the RGB channels (as recommended by the authors of EG) in Fig. S5. Neither of them go above random accuracy, only achieving random accuracy when they are regularized to a constant prediction. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "S3 RUNTIME AND MEMORY REQUIREMENTS OF DIFFERENT ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This section provides further details on runtime and memory requirements reported in Table S1. We compared the runtime and memory requirements of the available regularization schemes when implemented in Pytorch. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Memory usage and runtime were tested on the DecoyMNIST task with a batch size of 64. It is expected that the exact ratios will change depending on the complexity of the used network and batch size (since constant memory usage becomes disproportionally smaller with increasing batch size). ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The memory usage was read by recording the memory allocated by PyTorch. Since Expected Gradients and RRR require two forward and backward passes, we only record the maximum memory usage. We ran experiments on a single Titan X. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/d235c6702e3f3788a01b05a20b6f7401e703e8400883fcea7824599b62571b70.jpg",
+        "table_caption": [
+            "Table S1: Memory usage and run time were recorded for the DecoyMNIST task. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Unpenalized</td><td>CDEP</td><td>RRR</td><td>Expected Gradients</td></tr><tr><td>Run time/epoch (seconds)</td><td>4.7</td><td>17.1</td><td>11.2</td><td>17.8</td></tr><tr><td>Maximum GPU RAM usage (GB)</td><td>0.027</td><td>0.068</td><td>0.046</td><td>0.046</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "S4 IMAGE SEGMENTATION FOR ISIC SKIN CANCER ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To obtain the binary maps of the patches for the skin cancer task, we first segment the images using SLIC, a common image-segmentation algorithm (Achanta et al., 2012). Since the patches look quite distinct from the rest of the image, the patches are usually their own segment. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Subsequently we take the mean RGB and HSV values for all segments and filtered for segments which the mean was substantially different from the typical caucasian skin tone. Since different images were different from the typical skin color in different attributes, we filtered for those images ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "recursively. As an example, in the image shown in Fig. S6, the patch has a much higher saturation than the rest of the image. For each image we exported a map as seen in Fig. S6. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/0dfb79dd67bc8fc5f51dae08a2f16a404c88adc6fe72b90cc852fd2b5ded03d4.jpg",
+        "image_caption": [
+            "Figure S6: Sample segmentation for the ISIC task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "S5 ADDITIONAL HEATMAP EXAMPLES FOR ISIC ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We show additional examples from the test set of the skin cancer task in Figs. S7 and S8. We see that the importance maps for the unregularized and regularized network are very similar for cancerous images and non-cancerous images without patch. The patches are ignored by the network regularized with CDEP. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/535e5e0e5266fe5e4376cfbc030329bc433f930b6b247d6fde3dd2c00d42bdbc.jpg",
+        "image_caption": [
+            "Figure S7: Heatmaps for benign samples from ISIC "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            289,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/6aaa8ddbf450d4c9d0668084563bf7d71217d11feded933bcd841f43dc60c25d.jpg",
+        "image_caption": [
+            "Figure S8: Heatmaps for cancerous samples from ISIC "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            276,
+            143,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A different spurious correlation that we noticed was that proportionally more images showing skin cancer will have a ruler next to the lesion. This is the case because doctors often want to show a reference for size if they diagnosed that the lesion is cancerous. Even though the spurious correlation is less pronounced (in a very rough cursory count, $13 \\%$ of the cancerous and $5 \\%$ of the benign images contain some sort of measure), the networks learnt to recognize and exploit this spurious correlation. This further highlights the need for CDEP, especially in medical settings. ",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f08ac868aadbe50cbcbd0424d32f3c70d2aeabb154af0d6b0c7d242740f0490.jpg",
+        "image_caption": [
+            "Figure S9: Both networks learnt that proportionally more images with malignant lesions feature a ruler next to the lesion. To make comparison easier, we visualize the heatmap by multiplying it with the image. Visible regions are important for classification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    }
+]