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+# Learning Causal Semantic Representation for Out-of-Distribution Prediction
+
+Chang Liu1∗, Xinwei Sun1, Jindong Wang1, Haoyue Tang2†, Tao Li3†, Tao Qin1, Wei Chen1, Tie-Yan Liu1 1 Microsoft Research Asia, Beijing, 100080. 2 Tsinghua University, Beijing, 100084. 3 Peking University, Beijing, 100871.
+
+# Abstract
+
+Conventional supervised learning methods, especially deep ones, are found to be sensitive to out-of-distribution (OOD) examples, largely because the learned representation mixes the semantic factor with the variation factor due to their domain-specific correlation, while only the semantic factor causes the output. To address the problem, we propose a Causal Semantic Generative model (CSG) based on a causal reasoning so that the two factors are modeled separately, and develop methods for OOD prediction from a single training domain, which is common and challenging. The methods are based on the causal invariance principle, with a novel design in variational Bayes for both efficient learning and easy prediction. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor by fitting training data, and this semantic-identification guarantees the boundedness of OOD generalization error and the success of adaptation. Empirical study shows improved OOD performance over prevailing baselines.
+
+# 1 Introduction
+
+Deep learning has initiated a new era of artificial intelligence where the potential of machine learning models is greatly unleashed. Despite the great success, these methods heavily rely on the assumption that data from training and test domains follow the same distribution (i.e., the IID assumption), while in practice the test domain is often out-of-distribution (OOD), meaning that the test data distribute differently from the training data. Popular models for predicting the output (or label, response, outcome) $y$ from the input (or covariate) $x$ have been found erroneous when confronted with a distribution change, even from an essentially irrelevant perturbation like a position shift or background change for images [91, 6, 102, 41, 2, 27]. These phenomena pose serious concerns on the robustness and trustworthiness of machine learning methods and severely impede them from risk-sensitive scenarios.
+
+Looking into the problem, although deep learning models allow extracting abstract representation for prediction with their powerful approximation capacity, the representation may unconsciously mix up semantic factors $s$ (e.g., shape of an object) and variation factors $v$ (e.g., background, object position) due to a correlation between them (e.g., desks often appear in a workspace background and beds in bedrooms), so the model also relies on the variation factors $v$ for prediction via this correlation. However, this correlation tends to be superficial and spurious (e.g., a desk can also appear in a bedroom, but this does not make it a bed), and may change drastically in a new domain, making the effect from $v$ misleading. So it is desired to learn a representation that identifies $s$ against $v$ .
+
+Formally, the essence of this goal is to leverage causal relations for prediction, since the fundamental distinction between $s$ and $v$ is that only $s$ is the cause of $y$ . Causal relations better reflect basic mechanisms of nature. They bring the merit to machine learning that they tend to be universal and invariant across domains [97, 87, 93, 77, 16, 96, 98], thus provide the most transferable and reliable information to unseen domains. This causal invariance has been shown to lead to proper domain adaptation [97, 123], lower adaptation cost and lighter catastrophic forgetting [87, 9, 56].
+
+In this work, we propose a Causal Semantic Generative model (CSG) following a causal consideration to separately model the semantic (cause of prediction) and variation latent factors, and develop OOD prediction methods with theoretical guarantees on identifiability and the boundedness of OOD prediction error. Addressing the complaint that OOD prediction and causality methods often require multi-domain or intervention data, we focus on the most common and also challenging tasks where only one single training domain is available, including $O O D$ generalization and domain adaptation, where in the latter, unsupervised test-domain data are additionally available for training. The methods and theory are based on the causal invariance principle, which suggests to share generative mechanisms across domains, while the latent factor distribution (i.e., the prior $p ( s , v ) _ { , }$ ) changes. We argue that this causal invariance is more reliable than inference invariance in the other direction adopted by many existing methods [33, 101, 2, 66, 79]. For our method, we design novel and delicate reformulations of the ELBO objective so that we avoid the cost to build and learn two inference models. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor on the single training domain, even in presence of an s-v correlation. We further prove the merits from this identification: prediction error is bounded for OOD generalization, and for domain adaptation, the test-domain prior is identifiable which leads to an accurate prediction. To sum up our contributions, • Up to our knowledge, we are the first to show a theoretical guarantee (under appropriate conditions) to identify the latent cause of prediction (i.e., the semantic factor) on a single training domain, and also the first to show the theoretical benefits of this identification for OOD prediction. The results also contribute to generative representation learning for revealing what is learned. We develop effective methods for OOD generalization and domain adaptation, and achieve mostly better performance than prevailing methods on real-world image classification tasks.
+
+# 2 Related Work
+
+OOD generalization with causality. There are trials that ameliorate discriminative models towards a causal behavior. Bahadori et al. [4] introduce a regularizer that reweights input dimensions based on their approximated causal effects to the output, and Shen et al. [102] reweight training samples by amortizing causal effects among input samples. Their linear input-output assumption is then extended [4, 41] by learning a representation. Some recent works require identity data (finer than label) and enforce inference invariance via variance minimization [42], or leverage a strong domain knowledge to augment images as an independent intervention on variation factors [79]. These methods introduce no additional generative modeling efforts, at the cost of limited capacity for invariant causal mechanisms.
+
+Domain adaptation/generalization with causality. There are methods developed under various causal assumptions [97, 123] or using learned causal relations [93, 77]. Zhang et al. [123], Gong et al. [35, 36] also consider certain ways of mechanism change. The considered causality is among directly observed variables, which may not well suit general data like image pixels where causality rather lies in the conceptual latent level [75, 10, 59].
+
+To consider latent factors, there are domain adaptation [83, 5, 33, 73, 74] and generalization methods [80, 101, 113] that learn a representation with a domain-invariant marginal distribution. Remarkable results have been achieved. Nevertheless, it is found that this invariance is neither sufficient nor necessary to identify the true semantics or lower the adaptation error ([54, 125]; see also Appx. E). Moreover, these methods are based on inference invariance, which may not be as reliable as causal invariance (see Sec. 3.2).
+
+There are also generative methods for domain adaptation/generalization that model latent factors. Cai et al. [18] and Ilse et al. [49] introduce a semantic factor and a domain-feature factor. They assume the two latent factors are independent in both generative and inference models, which is unrealistic. Correlated factors are then considered [3]. But all these works do not adapt the prior for domain change thus resort to inference invariance. Zhang et al. [121] consider a partially observed manipulation variable, while still assuming its independence from the output in both the joint and posterior, and the adaptation is inconsistent with causal invariance. The above methods also do not show guarantees to identify their latent factors. Teshima et al. [108] leverage causal invariance and adapt the prior, yet also assume latent independence and do not separate the semantic factor. They require some supervised test-domain data, and their deterministic and invertible mechanism also indicates inference invariance. In addition, most domain generalization methods require multiple training domains, with exceptions [89] that still seek to augment domains. In contrast, CSG leverages causal invariance, and has guarantee to identify the semantic factor from a single training domain, even with a correlation to the variation factor.
+
+![](images/a8dcff6383a147a171346e43a11b5ec6548372c11164969c3e96cffe2a0d1d25.jpg)  
+Figure 1: (a) Graphical structure of the proposed CSG. Solid arrows represent causal mechanisms $p ( \bar { \boldsymbol { x } } | \boldsymbol { s } , \boldsymbol { v } )$ and $p ( y | s )$ , the undirected $s { - } v$ clique represents a domain-specific prior $p ( s , v )$ , and the dashed bended arrows represent the inference model $q ( s , v | x )$ for learning. $( \mathbf { b } , \mathbf { c } )$ Graphical structures of CSG-ind and CSG-DA for prediction on the test domain. An independent prior $p ^ { \underline { { \parallel } } } ( s , v )$ (constructed from $p ( s , v ) \mathrm { , }$ ) and a new prior $\tilde { p } ( s , v )$ (the dotted $_ { s - v }$ clique) are introduced reflecting the intervention on the test domain. Respective inference models $q ^ { \perp } ( s , v | x )$ and $\tilde { q } ( s , v | x )$ are also shown. All three models share the same causal mechanisms $p ( x | s , v )$ and $p ( y | s )$ .
+
+Disentangled latent representations is also of interest in unsupervised learning. Despite empirical success [22, 43, 21], Locatello et al. [70] conclude that it is impossible to guarantee the disentanglement in unsupervised settings. Subsequent works then introduce ways of supervision like a few latent variable observations [71] or sample similarity [20, 72, 104]. Identifiable VAE [57] and extensions [58, 117] leverage the data of a cause variable of the latent variables and have established theoretical guarantees under a diversity condition. But the works do not depict domain change thus not suitable for OOD prediction. Instead of disentangling latent factors, we focus on identifying the semantic factor $s$ (Sec. 5.1) and its benefit for OOD prediction. Appx. D shows more related work.
+
+# 3 The Causal Semantic Generative Model
+
+To develop the model soberly based on causality, we require its formal definition: two variables have a causal relation, denoted as “cause effect”, if intervening the cause (by changing external variables out of the considered system) may change the effect, but not vice versa [85, 88]. We follow this definition to build our model (Fig. 1a) by analyzing the example that an photographer takes a photo in a scene as $x$ and labels it as $y$ . Appx. C provides more explanations under other perspectives.
+
+(1) It is likely that neither $y  x$ (e.g., intervening the label with noise by distracting the photographer does not change the image) nor $x  y$ holds (e.g., intervening an image by breaking a camera sensor unit does not change how the photographer labels it), as also argued in [88, Sec. 1.4; 59]. So we introduce a latent variable $z$ to capture factors with causal relations. Also for this reason, we need a generative model (vs. discriminative model that only learns $x  y$ ).
+
+(2) The latent variable $z$ as underlying generating factors (e.g., object shape and texture, background and illumination during imaging) is plausible to cause both $x$ (e.g., changing object shape or background makes a different image, but breaking the camera does not change the shape or background) and $y$ (e.g., the photographer would give a different label if the object shape had been different, but noise-corrupting the label does not change the shape). So we orient the edges in the generative direction $z  ( x , y )$ , as also adopted in [78, 88, 108]. This is in contrast to prior works [18, 49, 48, 19] that treat $y$ as the cause of a semantic factor, which, when $y$ is also a noisy observation, makes unreasonable implications (e.g., adding noise to the labels in a dataset automatically changes object features and consequently the images, and changing the object features does not change the label). This difference is also discussed in [88, Sec. 1.4; 59].
+
+(3) We attribute all $x { - } y$ relation to the existence of some latent factor [68, “purely common cause”; 51] and exclude $x - y$ edges. This can be achieved as long as $z$ holds sufficient information of data (e.g., with shape, background etc. fixed, breaking the camera does not change the label, and noisecorrupting the label does not change the image). Promoting this structure reduces arbitrariness in explaining $x { - } y$ relation thus helps identify (part of) $z$ . This is in contrast to prior works [63, 121, 19] that treat $y$ as a cause of $x$ as no latent variable is introduced between.
+
+(4) Not all latent factors are the causes of $y$ (e.g., changing the shape may alter the label, while changing the background does not). We thus split the latent variable as $\boldsymbol { z } = \left( s , v \right)$ and remove the $v  y$ edge, where $s$ represents the semantic factor that causes $y$ , and $v$ describes the variation or diversity in generating $x$ . This formalizes the intuition on the concepts in Introduction (Sec. 1).
+
+(5) The two factors $s$ and $v$ often have a relation (e.g., a desk/bed shape tends to appear with a workspace/bedroom background), but it is usually a spurious correlation (e.g., putting a desk in a bedroom does not automatically change the room as a workspace, nor does it turn the desk into a bed). So we keep the undirected $s$ -v edge. This is in contrast to prior works [18, 49, 121, 108, 79] which assume independent latent variables. Although $v$ is not a cause of $y$ , modeling it explicitly is worth the effort since otherwise it would still be implicitly mixed into $s$ anyway through the $_ { s - v }$ correlation. We summarize these conclusions in the following definition.
+
+Definition 1 (CSG). A Causal Semantic Generative Model (CSG), $p : = \langle p ( s , v ) , p ( x | s , v ) , p ( y | s ) \rangle$ is a generative model on data variables $x \in \mathcal { X } \subseteq \mathbb { R } ^ { d _ { \mathcal { X } } }$ and $y \in \mathcal { V }$ with semantic $s \in \mathcal { S } \subseteq \mathbb { R } ^ { d _ { \mathcal { S } } }$ and variation $v \in \mathcal { V } \subseteq \mathbb { R } ^ { d _ { \mathcal { V } } }$ latent variables, following the graphical structure shown in Fig. 1a.
+
+# 3.1 The Causal Invariance Principle
+
+Through the above process, we see that the s-v correlation embodied in the prior $p ( s , v )$ tends to change across domains. Under a causal view, this means that the domain change comes from a (soft) intervention on $s$ or $v$ or both, leading to a different prior. On the other hand, the generative processes are likely causal mechanisms, so they enjoy the celebrated Independent Causal Mechanisms principle [88, 98] indicating that they are unaffected under the intervention on prior. This leads to the following causal invariance principle for CSG.
+
+Principle 2 (causal invariance). The causal generative mechanisms $p ( x | s , v )$ and $p ( y | s )$ in CSG are invariant across domains, and the change of prior $p ( s , v )$ is the only source of domain change.
+
+This invariance reflects the universality of basic laws of nature and is considered in some prior works [97, 88, 10, 16]. Other works instead introduce domain index [18, 49, 48, 19] or manipulation variables [121, 57, 58] to model distribution change explicitly. They then require multiple training domains or additional observations, while such changes can also be explained under causal invariance as long as the latent variables include all changing factors.
+
+# 3.2 Comparison with Inference Invariance
+
+Most domain adaptation and generalization methods (incl. domain-invariant-representation based [33, 101], invariantlatent-predictor based [2, 66, 79]) use a shared representation extractor across domains. This effectively assumes the invariance in the other direction, i.e. inferring latent factors $z$ from observed data $x$ . We note in its supportive examples (e.g., inferring object position from image, extracting the fundamental frequency from audio), the causal mechanism $p ( x | z )$ is nearly deterministic and invertible such that it preserves the information of $z$ . Formally, for a given $x$ , only one single $z$ value achieves a positive $p ( x | z )$ while all other values lead to zero. The inferred representation given by the posterior via the Bayes rule $p ( z | x ) \propto p ( z ) p ( x | z )$ then concentrates on this $z$ value, which is determined by the causal mechanism $p ( x | z )$ alone, regardless of the domain-specific prior $p ( z )$ . Causal invariance then implies inference invariance.
+
+![](images/35ca21b02faf8e1f334ba2ef6f1e5d815eeae23b16577148ad4a2320126c74dd.jpg)  
+Figure 2: Examples of noisy (left) or degenerate (right) generating mechanisms that lead to ambiguity in inference. Left: handwritten digit that may be generated as either $\mathbf { \ddot { \delta } } ^ { 6 } 3 ^ { \mathit { * } }$ or $\mathbf { \Delta } ^ { 6 6 } 5 ^ { , 9 }$ . Right: Schröder’s stairs that may be generated with either A or B being the nearer surface. Inference results notably rely on the prior on the digits/surfaces, which is domain-specific.
+
+In more general cases, the causal mechanism may be noisy or degenerate (Fig. 2), such that there are multiple $z$ values that give a positive $p ( x | z )$ , i.e. they all could generate the same $x$ . Inference is then ambiguous, and the posterior relies on the prior to choose from these $z$ values. Since the prior changes across domains (e.g., different labelers have different mindset), the inference rule then changes by nature and is not invariant,3 while the causal invariance is rather more fundamental and reliable. To leverage causal invariance, we use a different prior for the test domain (CSG-ind and CSG-DA), which gives a different and more reliable prediction than following inference invariance.
+
+# 4 Method
+
+We now develop methods based on variational Bayes [55, 62] for OOD generalization and domain adaptation using CSG. Appx. F.1 shows all details.
+
+# 4.1 Method for OOD Generalization
+
+For OOD generalization, one only has supervised data from the underlying data distribution $p ^ { * } ( x , y )$ on the training domain. Fitting a CSG $p : = \langle p ( s , v ) , p ( x | s , v ) , p ( y | s ) \rangle$ to data by maximizing likelihood $\mathbb { E } _ { p ^ { * } ( x , y ) } [ \log p ( x , y ) ]$ is intractable, since $\begin{array} { r } { p ( x , y ) : = \int p ( s , v , x , y ) \mathrm { d } s \mathrm { d } \iota } \end{array}$ where $p ( s , v , x , y ) : = p ( s , v ) p ( x | s , v ) p ( y | s )$ d to estimate. The Evidence Lower BOund (ELBO)[55, 112] is a tractable surrogate with the help of an $\begin{array} { r } { \mathcal { L } _ { p , q _ { s , v | x , y } } ( x , y ) : = \mathbb { E } _ { q ( s , v | x , y ) } [ \log \frac { p ( s , v , x , y ) } { q ( s , v | x , y ) } ] } \end{array}$ inference model $q ( s , v | x , y )$ that enjoys easy sampling and density evaluation. It is known that maxqs,v|x,y Lp, qs,v|x,y (x, y) drives q(s, v|x, y) towards the posterior p(s, v|x, y) := p(s,v,x,y)p(x,y) , meanwhile makes $\mathcal { L } _ { p , q _ { s , v \left| x , y \right. } } ( x , y )$ a tighter lower bound of $\log p ( x , y )$ for optimizing CSG $p$ .
+
+However, the subtlety with supervised learning is that prediction is still hard, as the introduced model $q ( s , v | x , y )$ does not help estimate $p ( y | x )$ . To address this, we propose to employ an auxiliary model $q ( s , v , y | x )$ targeting $p ( s , v , y | x )$ . It allows easy sampling of $y$ given $x$ for prediction, and can also serve as the required inference model: $\begin{array} { r } { q ( s , v | x , y ) = \frac { q ( s , v , y | x ) } { q ( y | x ) } } \end{array}$ , where $\begin{array} { r } { q ( y | x ) : = \int q ( s , v , y | x ) \mathrm { d } s \mathrm { d } v } \end{array}$ is also determined by $q ( s , v , y | x )$ . The ELBO objective $\mathbb { E } _ { p ^ { * } ( x , y ) } [ \mathcal { L } _ { p , q _ { s , v | x , y } } ( x , y ) ]$ then becomes:
+
+$$
+\mathbb { E } _ { p ^ { * } ( x ) } \mathbb { E } _ { p ^ { * } ( y | x ) } [ \log q ( y | x ) ] + \mathbb { E } _ { p ^ { * } ( x ) } \mathbb { E } _ { q ( s , v , y | x ) } [ \frac { p ^ { * } ( y | x ) } { q ( y | x ) } \mathrm { l o g } \frac { p ( s , v , x , y ) } { q ( s , v , y | x ) } ] .
+$$
+
+As a functional of $q ( s , v , y | x )$ (instead of $q ( s , v | x , y ) )$ and the $\mathbf { \boldsymbol { C } } \mathbf { \boldsymbol { S } } \mathbf { \boldsymbol { G } } ~ p$ , this objective also drives them towards their targets: the first term is the negative of the standard cross entropy (CE) loss which drives $q ( y | x )$ towards $p ^ { * } ( y | x )$ , and once this is achieved, the second term becomes the expected ELBO $\mathbb { E } _ { p ^ { * } ( x ) } [ \mathcal { L } _ { p , q _ { s , v , y | x } } ( x ) ]$ that drives $q ( s , v , y | x )$ towards $p ( s , v , y | x )$ and $p ( x )$ towards $p ^ { * } ( x )$ . Furthermore, as the target of $q ( s , v , y | x )$ factorizes as $p ( s , v , y | x ) = p ( s , v | x ) p ( y | s )$ (due to Fig. 1a) where $p ( y | s )$ is already known (part of the CSG), we can instead employ a lighter inference model $q ( s , v | x )$ for the minimally intractable component $p ( s , v | x )$ therein, and use $\bar { \boldsymbol { q } } ( s , v | x ) p ( y | s )$ as $q ( s , v , y | x )$ . This turns the objective Eq. (1) to:
+
+$$
+\operatorname* { m a x } _ { p , q _ { s , v \mid x } } \mathbb { E } _ { p ^ { * } ( x , y ) } \Big [ \log q ( y | x ) + \frac { 1 } { q ( y | x ) } \mathbb { E } _ { q ( s , v | x ) } \Big [ p ( y | s ) \log \frac { p ( s , v ) p ( x | s , v ) } { q ( s , v | x ) } \Big ] \Big ] ,
+$$
+
+where $q ( y | x ) : = \mathbb { E } _ { q ( s , v | x ) } [ p ( y | s ) ]$ . The expectations can be estimated by Monte Carlo after applying the reparameterization trick [62]. This is the basic CSG method.
+
+CSG-ind To actively improve OOD generalization performance, we consider using an independent prior $p ^ { \underline { { \parallel } } } ( s , v ) : = p ( s ) \overline { { p ( v ) } }$ for prediction in the test domain (Fig. 1b), where $p ( s )$ and $p ( v )$ are the marginals of the training-domain prior $p ( s , v )$ . Intuitively, $p ^ { \underline { { \parallel } } } ( s , v )$ discards the spurious correlation between $s$ and $v$ on the training domain (e.g., the “desk-workspace”, “bed-bedroom” association), and promotes a cautious neutral belief on the unknown test-domain correlation in defence against all possibilities (e.g., a “desk-bedroom”, “bed-workspace” association). Formally, $p ^ { \underline { { \parallel } } } ( s , \bar { v } )$ has a larger entropy than $p ( s , v )$ [24, Thm. 2.6.6], so it reduces training-domain-specific information and encourages reliance on the causal mechanisms for better generalization. It also amounts to applying the do-operator [85] to Fig. 1a, representing a randomized experiment by independently soft-intervening $s$ or $v$ . In this way, causal invariance is properly leveraged, making a different and more reliable prediction than following inference invariance. Our theory below also shows that $p ^ { \underline { { \parallel } } } ( s , v )$ leads to a smaller generalization error bound (Thm. 6 Remark).
+
+Methodologically, we need the test-domain inference model $q ^ { \perp } ( s , v | x )$ for prediction $p ^ { \underline { { \parallel } } } ( y | x ) \approx$ $\mathbb { E } _ { q ^ { \perp } ( s , v \mid x ) } [ p ( y \mid s ) ]$ , but also need $q ( s , v | x )$ for learning on the training domain. To save the cost of building and learning two inference models, we propose to use $q ^ { \perp } ( s , v | x )$ to represent $q ( s , v | x )$ . Noting that their targets are related by $\begin{array} { r } { p ( s , v | x ) = \frac { p ( s , v ) } { p ^ { \perp } ( s , v ) } \frac { p ^ { \perp } ( x ) } { p ( x ) } p ^ { \perp } ( s , v | x ) } \end{array}$ , we formulate $q ( s , v | x ) =$ $\begin{array} { r } { \frac { p ( s , v ) } { p ^ { \perp } ( s , v ) } \frac { p ^ { \perp } ( x ) } { p ( x ) } q ^ { \perp } ( s , v | x ) } \end{array}$ accordingly, so that this $q ( s , v | x )$ achieves its target if and only if $q ^ { \perp } ( s , v | x )$
+
+does. The objective Eq. (1) then becomes:
+
+$$
+\operatorname* { m a x } _ { p , q _ { s , v | x } ^ { \mathrm { A } } } \mathbb { E } _ { p ^ { s } ( x , y ) } \Big [ \log \pi ( y | x ) + \frac { 1 } { \pi ( y | x ) } \mathbb { E } _ { q ^ { \perp } ( s , v | x ) } \Big [ \frac { p ( s , v ) } { p ^ { \perp } ( s , v ) } p ( y | s ) \log \frac { p ^ { \perp } ( s , v ) p ( x | s , v ) } { q ^ { \perp } ( s , v | x ) } \Big ] \Big ] ,
+$$
+
+where $\begin{array} { r } { \pi ( y | x ) : = \mathbb { E } _ { q ^ { \perp } ( s , v | x ) } \big [ \frac { p ( s , v ) } { p ^ { \perp } ( s , v ) } p ( y | s ) \big ] } \end{array}$ . (Note $p ^ { \underline { { \parallel } } } ( s , v )$ is determined by $p ( s , v )$ in the $\operatorname { C S G } p$
+
+# 4.2 Method for Domain Adaptation
+
+In domain adaptation, one also has unsupervised data from the underlying data distribution $\tilde { p } ^ { * } ( x )$ on the test domain. We can leverage them for better prediction. According to the causal invariance principle (2), we only need a new prior $\tilde { p } ( s , v )$ for the test-domain CSG $\bar { p } : = \langle \tilde { p } ( s , v ) , p ( x | s , v ) , p ( \bar { y } | s ) \rangle$ (Fig. 1c). Fitting test-domain data can be done through the standard ELBO objective with the test-domain inference model $\tilde { q } ( s , v | x )$ :
+
+$$
+\operatorname* { m a x } _ { \tilde { p } , \tilde { q } _ { s , v \mid x } } \mathbb { E } _ { \tilde { p } ^ { * } ( x ) } [ \mathcal { L } _ { \tilde { p } , \tilde { q } _ { s , v \mid x } } ( x ) ] , \mathrm { w h e r e } \ \mathcal { L } _ { \tilde { p } , \tilde { q } _ { s , v \mid x } } ( x ) = \mathbb { E } _ { \tilde { q } ( s , v \mid x ) } \Bigl [ \log \frac { \tilde { p } ( s , v ) p ( x \mid s , v ) } { \tilde { q } ( s , v \mid x ) } \Bigr ] .
+$$
+
+Prediction is given by $\tilde { p } ( y | x ) \approx \mathbb { E } _ { \tilde { q } ( s , v | x ) } [ p ( y | s ) ]$ . Similar to the CSG-ind case, we still need $q ( s , v | x )$ for fitting training-domain data, and we can also avoid a separate $q ( s , v | x )$ model by representing it using $\tilde { q } ( s , v | x )$ . Following the same relation between their targets, we let $q ( s , v | x ) =$ $\begin{array} { r } { \frac { \tilde { p } ( x ) } { p ( x ) } \frac { p ( s , v ) } { \tilde { p } ( s , v ) } \tilde { q } ( s , v | x ) } \end{array}$ , which reformulates the same training-domain objective Eq. (1) as:
+
+$$
+\operatorname* { m a x } _ { p , \tilde { q } _ { s , v \mid x } } \mathbb { E } _ { p ^ { * } ( x , y ) } \Big [ \log \pi ( y | x ) + \frac { 1 } { \pi ( y | x ) } \mathbb { E } _ { \bar { q } ( s , v | x ) } \Big [ \frac { p ( s , v ) } { \tilde { p } ( s , v ) } p ( y | s ) \log \frac { \tilde { p } ( s , v ) p ( x | s , v ) } { \tilde { q } ( s , v | x ) } \Big ] \Big ] ,
+$$
+
+where $\pi ( \boldsymbol { y } | \boldsymbol { x } ) : = \mathbb { E } _ { \boldsymbol { \tilde { q } } ( \boldsymbol { s } , \boldsymbol { v } | \boldsymbol { x } ) } \left[ \frac { p ( \boldsymbol { s } , \boldsymbol { v } ) } { \tilde { p } ( \boldsymbol { s } , \boldsymbol { v } ) } p ( \boldsymbol { y } | \boldsymbol { s } ) \right]$ . The resulting method, termed CSG-DA, solves both optimization problems Eqs. (4, 5) simultaneously.
+
+# 4.3 Implementation and Model Selection
+
+To implement the three CSG methods, we only need one inference model in each. Appx. F.2 shows its construction from a general discriminative model (e.g., how to select its hidden nodes as $s$ and $v$ ). In practice $x$ often has a much larger dimension than $y$ , making the first supervision term overwhelmed by the second unsupervised term in Eqs. (2,3,5). So we downscale the second term.
+
+As recently emphasized [39], an OOD method should include a model selection method, since it is nontrivial and significantly affects performance [95, 120]. For our methods, we use a validation set from the training domain for model selection. This complies with the OOD setup, and is also suggested by our theory below which gives guarantees based on a good fit to the training-domain data distribution. For CSG-ind/DA, the learned predictor targets the test domain, so we do not use it directly for evaluating validation accuracy, but by normalizing $\pi ( \boldsymbol { y } | \boldsymbol { x } )$ . Appx. F.3 shows details.
+
+# 5 Theory
+
+We now establish theory for the identification of the semantic factor (cause of prediction) and subsequent merits for OOD generalization and domain adaptation. We focus on the distribution-level generalization instead of from finite samples to unseen samples under the same distribution, so we only consider the infinite-data regime. Appx. A shows all the proofs and auxiliary theory.
+
+Latent variable identification is hard [65, 81, 116, 70] as it is beyond observational relations [51, 88].   
+Assumptions are thus required to draw definite conclusions.
+
+Assumption 3. (Additive noise) There exist nonlinear functions $f$ and $g$ with bounded derivatives up to the third-order, and independent random variables $\mu$ and $\nu$ , such that $p ( x | s , v ) = p _ { \mu } ( x - f ( s , v ) { \bar { ) } }$ , and $p ( y | s ) = p _ { \nu } ( y - g ( s ) )$ for continuous $y$ or $p ( y | s ) = \mathrm { C a t } ( y | g ( s ) )$ for categorical $y$ .
+
+(Bijectivity) Assume $f$ is bijective and $g$ is injective.
+
+The additive noise assumption is widely adopted in causal discovery [51, 17]. It disables expressing the same joint in the other direction [122, Thm. 8; 86, Prop. 23] so that CSG unnecessarily indicates inference invariance. For this reason, we exclude GAN [37] and flow-based [61] implementations. Bijectivity is a common assumption for identifiability [51, 100, 57, 68]. It is sufficient [86, Prop. 17; 88, Prop. 7.4] for the more fundamental [86, Prop. 7; 88, p.109] requirement of causal minimality [86, p.2012; 88, Def. 6.33]. Particularly, $s$ and $v$ may otherwise have dummy dimensions that $f$ and $g$ simply ignore, raising another ambiguity against identifiability. On the other hand, according to the commonly acknowledged manifold hypothesis [115, 31], we can take $\mathcal { X }$ as the lower-dimensional data manifold and such a bijection exists as a coordinate map, which is an injection to the original data space and also allows $d _ { S } + d _ { \mathcal { V } } < d _ { \mathcal { X } }$ .
+
+# 5.1 Identifiability Theory
+
+We first formalize the goal of identifying the semantic factor.
+
+Definition 4 (semantic-identification). We say a learned CSG $p$ is semantic-identified, if there exists a homeomorphism4 $\Phi$ on $s \times \nu$ , such that (i) its output dimensions in $s$ is constant of $v$ : $\Phi ^ { S } ( s , v ) =$ $\Phi ^ { S } ( s , v ^ { \prime } ) , \forall \bar { v } , v ^ { \prime } \in \mathcal { V }$ (hence denote $\Phi ^ { S } ( s , v )$ as $\bar { \Phi ^ { S } } ( s ) )$ , and (ii) it is a reparameterization of the ground-truth CSG $p ^ { * }$ $\boldsymbol { \mathbf { \rho } } ) ^ { * } \colon \Phi _ { \# } [ p _ { s , v } ^ { * } ] = p _ { s , v }$ , $p ^ { * } ( x | s , v ) = p ( x | \Phi ( s , v ) )$ and $p ^ { * } ( \dot { y } | s ) = p ( y | \Phi ^ { S } ( s ) )$ .
+
+Here, $\Phi _ { \# } [ p _ { s , v } ^ { * } ]$ denotes the pushed-forward distribution5 of $p _ { s , v } ^ { * }$ by $\Phi$ , i.e. the distribution of $\Phi ( s , v )$ when $( s , v ) \sim p _ { s , v } ^ { * }$ . As the ground-truth CSG could at most provide its information via the data distribution $p ^ { * } ( x , y )$ , a well-learned CSG that achieves $p ( x , y ) = p ^ { * } ( x , y )$ still has the degree of freedom in parameterizing $( s , v )$ . This is described by this reparameterization $\Phi$ (Appx. Lemma 9). At the heart of the definition, the $v$ -constancy of $\Phi ^ { S }$ implies that $\Phi$ is semantic-preserving: the learned model does not mix the ground-truth $v$ into its $s$ , so that the learned $s$ holds equivalent information to the ground-truth $s$ . The definition can thus be seen as the semantic equivalence (Appx. Def. 10, Prop. 14) to the ground-truth $\mathbf { C S G } p ^ { * }$ .
+
+For related concepts, this identification cannot be characterized by the statistical independence between $s$ and $v$ (vs. [18, 49, 121]), which is not sufficient [70] nor necessary (due to the existence of spurious correlation). It is also weaker than disentanglement [44, 11], which additionally requires the learned $v$ to be constant of the ground-truth $s$ . The following theorem shows that semanticidentification can be achieved on a single domain under certain conditions.
+
+Theorem 5 (semantic-identifiability). With Assumption 3, a CSG $p$ is semantic-identified, if it is well-learned such that $p ( x , y ) = p ^ { * } ( x , y )$ , under the conditions that $\log p ( s , v )$ and $\log p ^ { * } ( s , v )$ are bounded up to the second-order, and that6 (i) $1 / \sigma _ { \mu } ^ { 2 }  \infty$ where $\sigma _ { \mu } ^ { 2 } : = \mathbb { E } [ \mu ^ { \top } \mu ]$ , or (ii) $p _ { \mu }$ (e.g., $a$ Gaussian) has an a.e. non-zero characteristic function.
+
+Remarks. (1) (Condition and Intuition) Compared with the multi-domain case [87, 93, 2], identifiability on a single training domain comes at a cost and requires certain conditions. One may imagine that in some extreme cases e.g., all desks appear in workspace and all beds in bedrooms, it is impossible to distinguish whether $y$ labels the object or the background (unlearnable OOD problem [119]). The theorem finds an appropriate condition that excludes such cases: when $\log p ^ { * } ( s , v )$ is bounded, deterministic $_ { s - v }$ relations are not allowed as they concentrate $\boldsymbol { p } ^ { * } ( s , v )$ on a lower-dimensional subspace in $s \times \nu$ thus make it unbounded.
+
+It also leads to the intuition of identifiability: a bounded $\log p ^ { * } ( s , v )$ indicates a stochastic s-v relation, so mixing the ground-truth $v$ into the learned $s$ makes the inference of $s$ more noisy due to the intrinsic diversity/uncertainty of this $v$ . As prediction is made via the inferred $s$ , this worsens prediction accuracy thus violates the “well-learned” requirement. Compared with discriminative models, CSG makes more faithful inference, and its causal structure leads to a proper description of domain change.
+
+(2) In condition (i), $1 / \sigma _ { \mu } ^ { 2 }$ measures the intensity of the causal mechanism $p ( x | s , v )$ . When it is large, the “strong” $p ( x | s , v )$ helps disambiguating values of $( s , v )$ in generating a given $x$ . The formal version in Appx. Thm. $5 '$ shows a quantitative reference for large enough intensity, and Appx. B gives a non-asymptotic extension showing how the intensity trades-off the tolerance of equalities in Def. 4. Condition (ii) goes beyond inference invariance. It roughly implies that different $( s , v )$ values a.s. produce different $p ( x | s , v )$ , so their roles in generating $x$ become clear which helps identification.
+
+(3) The theorem does not contradict the impossibility result by Locatello et al. [70], which considers disentangling each latent dimension with an unconstrained $( \dot { s } , v )  ( x , y )$ , while we only identify $s$ as a whole, with the $v  y$ edge removed which breaks the s-v symmetry.
+
+# 5.2 OOD Generalization Theory
+
+Now we show the benefit of semantic-identification for OOD generalization that the prediction error is bounded. Note the optimal predictor $\tilde { \mathbb { E } } ^ { * } [ y | x ] ^ { \eta }$ on the test domain is defined by the corresponding ground-truth CSG $\tilde { p } ^ { * }$ , which differs from $p ^ { * }$ only in the test-domain prior $\tilde { p } ^ { * } ( s , v )$ (Principle 2).
+
+Theorem 6 (OOD generalization error). 8 With Assumption 3, for a semantic-identified CSG $p$ on the training domain with semantic-preserving reparameterization $\Phi$ , we have up to $O ( \sigma _ { \mu } ^ { 4 } )$ ,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \tilde { p } ^ { * } ( x ) } \| \mathbb { E } [ y | x ] - \tilde { \mathbb { E } } ^ { * } [ y | x ] \| _ { 2 } ^ { 2 } \leqslant \sigma _ { \mu } ^ { 4 } B _ { f ^ { - 1 } } ^ { \prime 4 } B _ { g } ^ { \prime 2 } \mathbb { E } _ { \tilde { p } _ { s , v } } \big \| \nabla \log ( \tilde { p } _ { s , v } / p _ { s , v } ) \big \| _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+where $B _ { f ^ { - 1 } } ^ { \prime }$ and $B _ { g } ^ { \prime }$ bound the 2-norms9 of the Jacobians of $f ^ { - 1 }$ and $g$ , respectively, and $\widetilde { p } _ { s , v } : =$ $\Phi _ { \# } [ \tilde { p } _ { s , v } ^ { * } ]$ is the test-domain prior under the parameterization of the $C S G p$ .
+
+In the bound, the term $\mathbb { E } _ { \tilde { p } _ { s , v } } \left\| \nabla \log ( \tilde { p } _ { s , v } / p _ { s , v } ) \right\| _ { 2 } ^ { 2 }$ is the Fisher divergence measuring the difference between the two priors. As the prior change is the only source of domain change, this term also measures the “OODness” in terms of the effect on prediction. The bound also shows that when the causal mechanism $p ( x | s , v )$ is strong (small $\sigma _ { \mu }$ ), it dominates prediction over the prior change, as the generalization error becomes small. Compared with other methods, using a CSG enforces causal invariance, so the boundedness of OOD generalization error becomes more plausible in practice.
+
+Remark. The bound also shows the advantage of CSG-ind (Sec. 4.1). The Fisher divergence is revealed [28] to have a similar behavior as the forward KL divergence $p _ { s , v } \mapsto \mathrm { K L } ( \tilde { p } _ { s , v } \Vert p _ { s , v } )$ that it is very sensitive to the insufficient coverage of $p _ { s , v }$ on the support of $\tilde { p } _ { s , v }$ [46, 109], since $\log ( \tilde { p } _ { s , v } / p _ { s , v } )$ is infinitely large on the uncovered region. As the independent prior $p _ { s , v } ^ { \perp }$ has a larger support than $p _ { s , v }$ , it is less likely to miss the support of $\tilde { p } _ { s , v }$ , so it induces a generally smaller Fisher divergence. CSG-ind thus generally has a smaller OOD generalization error bound than CSG.
+
+# 5.3 Domain Adaptation Theory
+
+CSG-DA (Sec. 4.2) learns a new prior $\tilde { p } _ { s , v }$ by fitting unsupervised test-domain data, with causal mechanisms shared. If the mechanisms are semantic-identified, the ground-truth test-domain prior $\tilde { p } _ { s , v } ^ { * }$ can also be identified under the learned parameterization, and prediction is made precise.
+
+Theorem 7 (domain adaptation error). With conditions of Thm. 5, for a semantic-identified $C S G p$ on the training domain with semantic-preserving reparameterization $\Phi$ , if its new prior $\tilde { p } _ { s , v }$ is well
+
+learned such that $\tilde { p } ( x ) = \tilde { p } ^ { * } ( x )$ , then $\tilde { p } _ { s , v } = \Phi _ { \# } [ \tilde { p } _ { s , v } ^ { * } ]$ , and $\tilde { \mathbb { E } } [ y | x ] = \tilde { \mathbb { E } } ^ { * } [ y | x ]$ for any $x \in \mathrm { s u p p } ( \tilde { p } _ { x } ^ { * } )$
+
+Different from existing domain adaptation bounds (Appx. E), Theorems 6,7 allow different inference models in the two domains, thus go beyond inference invariance.
+
+# 6 Experiments
+
+For OOD generalization baselines, there is not much choice beyond the standard CE loss optimization, as domain adaptation methods require test-domain data and most domain generalization methods degenerate to CE with one training domain. The exception within our scope is a causal discriminative method CNBB [41]. For domain adaptation, we consider well-acknowledged methods DANN [33], DAN [73], CDAN [74] and recent compelling methods MDD [124] and BNM [25] (shown in Appx. Tables 2,3). Appx. G shows more details, results, and discussions. 10
+
+Shifted-MNIST. We first consider an OOD prediction task on MNIST to classify digits $\mathbf { \bar { \theta } } ^ { 6 6 }$ ”s and “1”s. To make a spurious correlation, in the training data, we horizontally shift each $ { { } ^ { 6 } } 0 ^ { 9 }$ at random by $\delta _ { 0 } \sim \mathcal { N } ( - 5 , 1 ^ { 2 } )$ pixels, while each “1” by $\delta _ { 1 } \sim \bar { \mathcal { N } } ( 5 , 1 ^ { 2 } )$ pixels. We consider two test domains with different digit-position distributions: each digit is not moved $\delta _ { 0 } = \delta _ { 1 } = 0$ in the first, and is shifted at random by $\bar { \delta _ { 0 } } , \delta _ { 1 } \sim \mathcal { N } ( 0 , 2 ^ { 2 } )$ pixels in the second. We implement all methods using a multilayer perceptron which is not naturally shift invariant. We use a larger architecture for non-generative methods to compensate the additional generative component of generative methods.
+
+The performance is shown in Table 1(top 2 rows). For OOD generalization, CE is misled by the more noticeable position factor due to the spurious correlation to digits, and resorts to random guess (even worse) when position is not informative for prediction. CNBB ameliorates the position confusion, but not as thoroughly without modeling causal mechanisms. In contrast, our CSG gives more genuine predictions in unseen domains, thanks to the identification of the semantic factor. CSG-ind performs even better, justifying the merit of using an independent prior for prediction. For domain adaptation, CSG-DA achieves the best results. Existing adaptation methods even worsen the result (negative transfer), as the misleading position representation gets strengthened on the unsupervised test data. CSG is benefited from adaptation in a proper way that identifies the semantic factor.
+
+Table 1: Test accuracy $( \% )$ by various methods (ours in bold) for OOD generalization (left 4 cols) and domain adaptation (right 5 cols) on Shifted-MNIST (top 2 rows), ImageCLEF-DA (middle 4 rows) and PACS (bottom 4 rows) datasets. Averaged over 10 runs. Appx. Tables 2,3 show more results.   
+
+<table><tr><td>task</td><td>CE</td><td>CNBB</td><td>CSG</td><td>CSG-ind</td><td>DANN</td><td>DAN</td><td>CDAN</td><td>MDD</td><td>CSG-DA</td></tr><tr><td>8=δ1=0</td><td>42.9±3.1</td><td>54.7±3.3</td><td>81.4±7.4</td><td>82.6±4.0</td><td>40.9±3.0</td><td>40.4±2.0</td><td>41.0±0.5</td><td>41.9±0.8</td><td>97.6±4.0</td></tr><tr><td>80,~N(0,2²）</td><td>47.8±1.5</td><td>59.2±2.4</td><td>61.7±3.6</td><td>62.3±2.2</td><td>46.2±0.7</td><td>45.6±0.7</td><td>46.3±0.6</td><td>45.8±0.3</td><td>72.0±9.2</td></tr><tr><td>C→P</td><td>65.5±0.3</td><td>72.7±1.1</td><td>73.6±0.6</td><td>74.0±1.3</td><td>74.3±0.5</td><td>69.2±0.4</td><td>74.5±0.3</td><td>74.1±0.7</td><td>75.1±0.5</td></tr><tr><td>P→C</td><td>91.2±0.3</td><td>91.7±0.2</td><td>92.3±0.4</td><td>92.7±0.2</td><td>91.5±0.6</td><td>89.8±0.4</td><td>93.5±0.4</td><td>92.1±0.6</td><td>93.4±0.3</td></tr><tr><td>I→P</td><td>74.8±0.3</td><td>75.4±0.6</td><td>76.9±0.3</td><td>77.2±0.2</td><td>75.0±0.6</td><td>74.5±0.4</td><td>76.7±0.3</td><td>76.8±0.4</td><td>77.4±0.3</td></tr><tr><td>P→I</td><td>83.9±0.1</td><td>88.7±0.5</td><td>90.4±0.3</td><td>90.9±0.2</td><td>86.0±0.3</td><td>82.2±0.2</td><td>90.6±0.3</td><td>90.2±1.1</td><td>91.1±0.5</td></tr><tr><td>others→P</td><td>97.8±0.0</td><td>96.9±0.2 97.7±0.2</td><td></td><td>97.8±0.2</td><td>[97.6±0.2</td><td>97.6±0.4</td><td>97.0±0.4</td><td>97.6±0.3</td><td>97.9±0.2</td></tr><tr><td>others→A</td><td>88.1±0.1</td><td>73.1±0.3</td><td>88.5±0.6</td><td>88.6±0.6</td><td>85.9±0.5</td><td>84.5±1.2</td><td>84.0±0.9</td><td>88.1±0.8</td><td>88.8±0.7</td></tr><tr><td>others-C</td><td>77.9±1.3</td><td>50.2±1.2 84.4±0.9</td><td></td><td>84.6±0.8</td><td>79.9±1.4 81.9±1.9</td><td></td><td>78.5±1.5 83.2±1.1</td><td></td><td>84.7±0.8</td></tr><tr><td>others-→S</td><td>79.1±0.9</td><td>43.3±1.2 80.7±1.0</td><td></td><td>81.1±1.2</td><td>75.2±2.8 77.4±3.1</td><td></td><td>71.8±3.9 80.2±2.2</td><td></td><td>81.4±0.8</td></tr></table>
+
+ImageCLEF-DA is a standard benchmark for domain adaptation [1]. It has 12 classes and three domains of real-world images: Caltech-256, ImageNet, Pascal VOC 2012. We select four OOD prediction tasks $\mathbf { C } {  } \mathbf { P }$ , $\mathbf { I } {  } \mathbf { P }$ that have not seen good enough results. We adopt the same setup as [74]. As shown in Table 1(middle 4 rows), CSG-ind again achieves the best OOD generalization results, and even outperforms some domain adaptation methods. Our CSG also outperforms the baselines mostly. For domain adaptation, CSG-DA is the best in most cases and on par with the best in others.
+
+PACS is a more recent benchmark dataset [69]. It has 7 classes and is named after its four domains: Photo, Art, Cartoon, Sketch; each contains images of a certain style. We follow the same setup as [39]; particularly, we pool together all domains but the test one as the single training domain. Results in Table 1(bottom 4 rows) show the same trend. CSG-DA even outperforms most domain generalization methods reported in [39], which are fed with more information. Appx. Tables 2,3 also show the results on an even larger dataset VLCS [30], which present a similar observation.
+
+Visualization. Appx. Fig. 5 visualizes the learned models using LIME [91]. The results show our methods focus more on the semantic regions and shapes, indicating a causal representation is learned.
+
+Dataset analysis. The results indicate our methods are more powerful on shifted-MNIST and PACS (and VLCS) than ImageCLEF-DA. This meets the intuition of identifiability (Thm. 5 Remark (1)): the random position or pooled training domain shows a diverse $v$ for each $s$ (while with a misleading spurious correlation), so identification is better guaranteed to overcome the spurious correlation.
+
+Ablation study. To show the benefit of modeling $s$ and $v$ separately, we compare with a counterpart of CSG that treats $s$ and $v$ as a whole (equivalently, $v  y$ is kept; see Appx. F.1.4 for method details). Appx. Tables 2,3 show that our methods outperform this baseline in all cases. This shows the separate modeling makes CSG consciously drive semantic representation into the dedicated variable $s$ .
+
+# 7 Conclusion and Discussion
+
+We propose a Causal Semantic Generative model for single-domain OOD prediction tasks, which builds upon a causal reasoning, and models the semantic (cause of prediction) and variation factors separately. By the causal invariance principle, we develop novel and efficient learning and prediction methods, and prove the semantic-identifiability and the subsequent bounded generalization error and the success of adaptation. Experiments show the improved performance over prevailing baselines.
+
+Notably, we answered the questions in the recent farseeing paper [98] on causal representation learning: we found an appropriate condition under which “causal variables can be recovered”, and provided “compelling evidence on the advantages (of causal modeling) in terms of generalization”. Also, separating semantics from variation extends to broader examples. Neural nets are found to change their prediction under a different texture [34, 15]. Adversarial vulnerability [107, 38, 67] extends variation factors to human-imperceptible features, i.e. adversarial noise, which is found to have a strong correlation to the semantics [50]. The separation also matters for fairness when a sensitive variation factor may affect prediction. This work also inspires the dual connection between causal representation learning (“fill in the blanks” given a graph) and causal discovery (“link the nodes” given observed variables). Our theory shows the identifiability condition for causal discovery (the additive noise assumption) also makes causal representation identifiable. Studying the general connection between the two tasks is an interesting future work.
+
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+        "text": "Learning Causal Semantic Representation for Out-of-Distribution Prediction ",
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+        "text": "Chang Liu1∗, Xinwei Sun1, Jindong Wang1, Haoyue Tang2†, Tao Li3†, Tao Qin1, Wei Chen1, Tie-Yan Liu1 1 Microsoft Research Asia, Beijing, 100080. 2 Tsinghua University, Beijing, 100084. 3 Peking University, Beijing, 100871. ",
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+        "text": "Abstract ",
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+        "text": "Conventional supervised learning methods, especially deep ones, are found to be sensitive to out-of-distribution (OOD) examples, largely because the learned representation mixes the semantic factor with the variation factor due to their domain-specific correlation, while only the semantic factor causes the output. To address the problem, we propose a Causal Semantic Generative model (CSG) based on a causal reasoning so that the two factors are modeled separately, and develop methods for OOD prediction from a single training domain, which is common and challenging. The methods are based on the causal invariance principle, with a novel design in variational Bayes for both efficient learning and easy prediction. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor by fitting training data, and this semantic-identification guarantees the boundedness of OOD generalization error and the success of adaptation. Empirical study shows improved OOD performance over prevailing baselines. ",
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+        "text": "1 Introduction ",
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+        "text": "Deep learning has initiated a new era of artificial intelligence where the potential of machine learning models is greatly unleashed. Despite the great success, these methods heavily rely on the assumption that data from training and test domains follow the same distribution (i.e., the IID assumption), while in practice the test domain is often out-of-distribution (OOD), meaning that the test data distribute differently from the training data. Popular models for predicting the output (or label, response, outcome) $y$ from the input (or covariate) $x$ have been found erroneous when confronted with a distribution change, even from an essentially irrelevant perturbation like a position shift or background change for images [91, 6, 102, 41, 2, 27]. These phenomena pose serious concerns on the robustness and trustworthiness of machine learning methods and severely impede them from risk-sensitive scenarios. ",
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+        "type": "text",
+        "text": "Looking into the problem, although deep learning models allow extracting abstract representation for prediction with their powerful approximation capacity, the representation may unconsciously mix up semantic factors $s$ (e.g., shape of an object) and variation factors $v$ (e.g., background, object position) due to a correlation between them (e.g., desks often appear in a workspace background and beds in bedrooms), so the model also relies on the variation factors $v$ for prediction via this correlation. However, this correlation tends to be superficial and spurious (e.g., a desk can also appear in a bedroom, but this does not make it a bed), and may change drastically in a new domain, making the effect from $v$ misleading. So it is desired to learn a representation that identifies $s$ against $v$ . ",
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+        "text": "Formally, the essence of this goal is to leverage causal relations for prediction, since the fundamental distinction between $s$ and $v$ is that only $s$ is the cause of $y$ . Causal relations better reflect basic mechanisms of nature. They bring the merit to machine learning that they tend to be universal and invariant across domains [97, 87, 93, 77, 16, 96, 98], thus provide the most transferable and reliable information to unseen domains. This causal invariance has been shown to lead to proper domain adaptation [97, 123], lower adaptation cost and lighter catastrophic forgetting [87, 9, 56]. ",
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+    {
+        "type": "text",
+        "text": "In this work, we propose a Causal Semantic Generative model (CSG) following a causal consideration to separately model the semantic (cause of prediction) and variation latent factors, and develop OOD prediction methods with theoretical guarantees on identifiability and the boundedness of OOD prediction error. Addressing the complaint that OOD prediction and causality methods often require multi-domain or intervention data, we focus on the most common and also challenging tasks where only one single training domain is available, including $O O D$ generalization and domain adaptation, where in the latter, unsupervised test-domain data are additionally available for training. The methods and theory are based on the causal invariance principle, which suggests to share generative mechanisms across domains, while the latent factor distribution (i.e., the prior $p ( s , v ) _ { , }$ ) changes. We argue that this causal invariance is more reliable than inference invariance in the other direction adopted by many existing methods [33, 101, 2, 66, 79]. For our method, we design novel and delicate reformulations of the ELBO objective so that we avoid the cost to build and learn two inference models. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor on the single training domain, even in presence of an s-v correlation. We further prove the merits from this identification: prediction error is bounded for OOD generalization, and for domain adaptation, the test-domain prior is identifiable which leads to an accurate prediction. To sum up our contributions, • Up to our knowledge, we are the first to show a theoretical guarantee (under appropriate conditions) to identify the latent cause of prediction (i.e., the semantic factor) on a single training domain, and also the first to show the theoretical benefits of this identification for OOD prediction. The results also contribute to generative representation learning for revealing what is learned. We develop effective methods for OOD generalization and domain adaptation, and achieve mostly better performance than prevailing methods on real-world image classification tasks. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "OOD generalization with causality. There are trials that ameliorate discriminative models towards a causal behavior. Bahadori et al. [4] introduce a regularizer that reweights input dimensions based on their approximated causal effects to the output, and Shen et al. [102] reweight training samples by amortizing causal effects among input samples. Their linear input-output assumption is then extended [4, 41] by learning a representation. Some recent works require identity data (finer than label) and enforce inference invariance via variance minimization [42], or leverage a strong domain knowledge to augment images as an independent intervention on variation factors [79]. These methods introduce no additional generative modeling efforts, at the cost of limited capacity for invariant causal mechanisms. ",
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+    {
+        "type": "text",
+        "text": "Domain adaptation/generalization with causality. There are methods developed under various causal assumptions [97, 123] or using learned causal relations [93, 77]. Zhang et al. [123], Gong et al. [35, 36] also consider certain ways of mechanism change. The considered causality is among directly observed variables, which may not well suit general data like image pixels where causality rather lies in the conceptual latent level [75, 10, 59]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To consider latent factors, there are domain adaptation [83, 5, 33, 73, 74] and generalization methods [80, 101, 113] that learn a representation with a domain-invariant marginal distribution. Remarkable results have been achieved. Nevertheless, it is found that this invariance is neither sufficient nor necessary to identify the true semantics or lower the adaptation error ([54, 125]; see also Appx. E). Moreover, these methods are based on inference invariance, which may not be as reliable as causal invariance (see Sec. 3.2). ",
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+    {
+        "type": "text",
+        "text": "There are also generative methods for domain adaptation/generalization that model latent factors. Cai et al. [18] and Ilse et al. [49] introduce a semantic factor and a domain-feature factor. They assume the two latent factors are independent in both generative and inference models, which is unrealistic. Correlated factors are then considered [3]. But all these works do not adapt the prior for domain change thus resort to inference invariance. Zhang et al. [121] consider a partially observed manipulation variable, while still assuming its independence from the output in both the joint and posterior, and the adaptation is inconsistent with causal invariance. The above methods also do not show guarantees to identify their latent factors. Teshima et al. [108] leverage causal invariance and adapt the prior, yet also assume latent independence and do not separate the semantic factor. They require some supervised test-domain data, and their deterministic and invertible mechanism also indicates inference invariance. In addition, most domain generalization methods require multiple training domains, with exceptions [89] that still seek to augment domains. In contrast, CSG leverages causal invariance, and has guarantee to identify the semantic factor from a single training domain, even with a correlation to the variation factor. ",
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+        "img_path": "images/a8dcff6383a147a171346e43a11b5ec6548372c11164969c3e96cffe2a0d1d25.jpg",
+        "image_caption": [
+            "Figure 1: (a) Graphical structure of the proposed CSG. Solid arrows represent causal mechanisms $p ( \\bar { \\boldsymbol { x } } | \\boldsymbol { s } , \\boldsymbol { v } )$ and $p ( y | s )$ , the undirected $s { - } v$ clique represents a domain-specific prior $p ( s , v )$ , and the dashed bended arrows represent the inference model $q ( s , v | x )$ for learning. $( \\mathbf { b } , \\mathbf { c } )$ Graphical structures of CSG-ind and CSG-DA for prediction on the test domain. An independent prior $p ^ { \\underline { { \\parallel } } } ( s , v )$ (constructed from $p ( s , v ) \\mathrm { , }$ ) and a new prior $\\tilde { p } ( s , v )$ (the dotted $_ { s - v }$ clique) are introduced reflecting the intervention on the test domain. Respective inference models $q ^ { \\perp } ( s , v | x )$ and $\\tilde { q } ( s , v | x )$ are also shown. All three models share the same causal mechanisms $p ( x | s , v )$ and $p ( y | s )$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "Disentangled latent representations is also of interest in unsupervised learning. Despite empirical success [22, 43, 21], Locatello et al. [70] conclude that it is impossible to guarantee the disentanglement in unsupervised settings. Subsequent works then introduce ways of supervision like a few latent variable observations [71] or sample similarity [20, 72, 104]. Identifiable VAE [57] and extensions [58, 117] leverage the data of a cause variable of the latent variables and have established theoretical guarantees under a diversity condition. But the works do not depict domain change thus not suitable for OOD prediction. Instead of disentangling latent factors, we focus on identifying the semantic factor $s$ (Sec. 5.1) and its benefit for OOD prediction. Appx. D shows more related work. ",
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+    {
+        "type": "text",
+        "text": "3 The Causal Semantic Generative Model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To develop the model soberly based on causality, we require its formal definition: two variables have a causal relation, denoted as “cause effect”, if intervening the cause (by changing external variables out of the considered system) may change the effect, but not vice versa [85, 88]. We follow this definition to build our model (Fig. 1a) by analyzing the example that an photographer takes a photo in a scene as $x$ and labels it as $y$ . Appx. C provides more explanations under other perspectives. ",
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+        "type": "text",
+        "text": "(1) It is likely that neither $y  x$ (e.g., intervening the label with noise by distracting the photographer does not change the image) nor $x  y$ holds (e.g., intervening an image by breaking a camera sensor unit does not change how the photographer labels it), as also argued in [88, Sec. 1.4; 59]. So we introduce a latent variable $z$ to capture factors with causal relations. Also for this reason, we need a generative model (vs. discriminative model that only learns $x  y$ ). ",
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+    {
+        "type": "text",
+        "text": "(2) The latent variable $z$ as underlying generating factors (e.g., object shape and texture, background and illumination during imaging) is plausible to cause both $x$ (e.g., changing object shape or background makes a different image, but breaking the camera does not change the shape or background) and $y$ (e.g., the photographer would give a different label if the object shape had been different, but noise-corrupting the label does not change the shape). So we orient the edges in the generative direction $z  ( x , y )$ , as also adopted in [78, 88, 108]. This is in contrast to prior works [18, 49, 48, 19] that treat $y$ as the cause of a semantic factor, which, when $y$ is also a noisy observation, makes unreasonable implications (e.g., adding noise to the labels in a dataset automatically changes object features and consequently the images, and changing the object features does not change the label). This difference is also discussed in [88, Sec. 1.4; 59]. ",
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+    {
+        "type": "text",
+        "text": "(3) We attribute all $x { - } y$ relation to the existence of some latent factor [68, “purely common cause”; 51] and exclude $x - y$ edges. This can be achieved as long as $z$ holds sufficient information of data (e.g., with shape, background etc. fixed, breaking the camera does not change the label, and noisecorrupting the label does not change the image). Promoting this structure reduces arbitrariness in explaining $x { - } y$ relation thus helps identify (part of) $z$ . This is in contrast to prior works [63, 121, 19] that treat $y$ as a cause of $x$ as no latent variable is introduced between. ",
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+    {
+        "type": "text",
+        "text": "(4) Not all latent factors are the causes of $y$ (e.g., changing the shape may alter the label, while changing the background does not). We thus split the latent variable as $\\boldsymbol { z } = \\left( s , v \\right)$ and remove the $v  y$ edge, where $s$ represents the semantic factor that causes $y$ , and $v$ describes the variation or diversity in generating $x$ . This formalizes the intuition on the concepts in Introduction (Sec. 1). ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "(5) The two factors $s$ and $v$ often have a relation (e.g., a desk/bed shape tends to appear with a workspace/bedroom background), but it is usually a spurious correlation (e.g., putting a desk in a bedroom does not automatically change the room as a workspace, nor does it turn the desk into a bed). So we keep the undirected $s$ -v edge. This is in contrast to prior works [18, 49, 121, 108, 79] which assume independent latent variables. Although $v$ is not a cause of $y$ , modeling it explicitly is worth the effort since otherwise it would still be implicitly mixed into $s$ anyway through the $_ { s - v }$ correlation. We summarize these conclusions in the following definition. ",
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+    {
+        "type": "text",
+        "text": "Definition 1 (CSG). A Causal Semantic Generative Model (CSG), $p : = \\langle p ( s , v ) , p ( x | s , v ) , p ( y | s ) \\rangle$ is a generative model on data variables $x \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { d _ { \\mathcal { X } } }$ and $y \\in \\mathcal { V }$ with semantic $s \\in \\mathcal { S } \\subseteq \\mathbb { R } ^ { d _ { \\mathcal { S } } }$ and variation $v \\in \\mathcal { V } \\subseteq \\mathbb { R } ^ { d _ { \\mathcal { V } } }$ latent variables, following the graphical structure shown in Fig. 1a. ",
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+    {
+        "type": "text",
+        "text": "3.1 The Causal Invariance Principle ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Through the above process, we see that the s-v correlation embodied in the prior $p ( s , v )$ tends to change across domains. Under a causal view, this means that the domain change comes from a (soft) intervention on $s$ or $v$ or both, leading to a different prior. On the other hand, the generative processes are likely causal mechanisms, so they enjoy the celebrated Independent Causal Mechanisms principle [88, 98] indicating that they are unaffected under the intervention on prior. This leads to the following causal invariance principle for CSG. ",
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+    {
+        "type": "text",
+        "text": "Principle 2 (causal invariance). The causal generative mechanisms $p ( x | s , v )$ and $p ( y | s )$ in CSG are invariant across domains, and the change of prior $p ( s , v )$ is the only source of domain change. ",
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+    {
+        "type": "text",
+        "text": "This invariance reflects the universality of basic laws of nature and is considered in some prior works [97, 88, 10, 16]. Other works instead introduce domain index [18, 49, 48, 19] or manipulation variables [121, 57, 58] to model distribution change explicitly. They then require multiple training domains or additional observations, while such changes can also be explained under causal invariance as long as the latent variables include all changing factors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 Comparison with Inference Invariance ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Most domain adaptation and generalization methods (incl. domain-invariant-representation based [33, 101], invariantlatent-predictor based [2, 66, 79]) use a shared representation extractor across domains. This effectively assumes the invariance in the other direction, i.e. inferring latent factors $z$ from observed data $x$ . We note in its supportive examples (e.g., inferring object position from image, extracting the fundamental frequency from audio), the causal mechanism $p ( x | z )$ is nearly deterministic and invertible such that it preserves the information of $z$ . Formally, for a given $x$ , only one single $z$ value achieves a positive $p ( x | z )$ while all other values lead to zero. The inferred representation given by the posterior via the Bayes rule $p ( z | x ) \\propto p ( z ) p ( x | z )$ then concentrates on this $z$ value, which is determined by the causal mechanism $p ( x | z )$ alone, regardless of the domain-specific prior $p ( z )$ . Causal invariance then implies inference invariance. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/35ca21b02faf8e1f334ba2ef6f1e5d815eeae23b16577148ad4a2320126c74dd.jpg",
+        "image_caption": [
+            "Figure 2: Examples of noisy (left) or degenerate (right) generating mechanisms that lead to ambiguity in inference. Left: handwritten digit that may be generated as either $\\mathbf { \\ddot { \\delta } } ^ { 6 } 3 ^ { \\mathit { * } }$ or $\\mathbf { \\Delta } ^ { 6 6 } 5 ^ { , 9 }$ . Right: Schröder’s stairs that may be generated with either A or B being the nearer surface. Inference results notably rely on the prior on the digits/surfaces, which is domain-specific. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In more general cases, the causal mechanism may be noisy or degenerate (Fig. 2), such that there are multiple $z$ values that give a positive $p ( x | z )$ , i.e. they all could generate the same $x$ . Inference is then ambiguous, and the posterior relies on the prior to choose from these $z$ values. Since the prior changes across domains (e.g., different labelers have different mindset), the inference rule then changes by nature and is not invariant,3 while the causal invariance is rather more fundamental and reliable. To leverage causal invariance, we use a different prior for the test domain (CSG-ind and CSG-DA), which gives a different and more reliable prediction than following inference invariance. ",
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+    {
+        "type": "text",
+        "text": "4 Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We now develop methods based on variational Bayes [55, 62] for OOD generalization and domain adaptation using CSG. Appx. F.1 shows all details. ",
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+    {
+        "type": "text",
+        "text": "4.1 Method for OOD Generalization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For OOD generalization, one only has supervised data from the underlying data distribution $p ^ { * } ( x , y )$ on the training domain. Fitting a CSG $p : = \\langle p ( s , v ) , p ( x | s , v ) , p ( y | s ) \\rangle$ to data by maximizing likelihood $\\mathbb { E } _ { p ^ { * } ( x , y ) } [ \\log p ( x , y ) ]$ is intractable, since $\\begin{array} { r } { p ( x , y ) : = \\int p ( s , v , x , y ) \\mathrm { d } s \\mathrm { d } \\iota } \\end{array}$ where $p ( s , v , x , y ) : = p ( s , v ) p ( x | s , v ) p ( y | s )$ d to estimate. The Evidence Lower BOund (ELBO)[55, 112] is a tractable surrogate with the help of an $\\begin{array} { r } { \\mathcal { L } _ { p , q _ { s , v | x , y } } ( x , y ) : = \\mathbb { E } _ { q ( s , v | x , y ) } [ \\log \\frac { p ( s , v , x , y ) } { q ( s , v | x , y ) } ] } \\end{array}$ inference model $q ( s , v | x , y )$ that enjoys easy sampling and density evaluation. It is known that maxqs,v|x,y Lp, qs,v|x,y (x, y) drives q(s, v|x, y) towards the posterior p(s, v|x, y) := p(s,v,x,y)p(x,y) , meanwhile makes $\\mathcal { L } _ { p , q _ { s , v \\left| x , y \\right. } } ( x , y )$ a tighter lower bound of $\\log p ( x , y )$ for optimizing CSG $p$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, the subtlety with supervised learning is that prediction is still hard, as the introduced model $q ( s , v | x , y )$ does not help estimate $p ( y | x )$ . To address this, we propose to employ an auxiliary model $q ( s , v , y | x )$ targeting $p ( s , v , y | x )$ . It allows easy sampling of $y$ given $x$ for prediction, and can also serve as the required inference model: $\\begin{array} { r } { q ( s , v | x , y ) = \\frac { q ( s , v , y | x ) } { q ( y | x ) } } \\end{array}$ , where $\\begin{array} { r } { q ( y | x ) : = \\int q ( s , v , y | x ) \\mathrm { d } s \\mathrm { d } v } \\end{array}$ is also determined by $q ( s , v , y | x )$ . The ELBO objective $\\mathbb { E } _ { p ^ { * } ( x , y ) } [ \\mathcal { L } _ { p , q _ { s , v | x , y } } ( x , y ) ]$ then becomes: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f80f50143e6c02ae0219955eaf44b0f7bdc5f71d59d71590fcb994062c3f688e.jpg",
+        "text": "$$\n\\mathbb { E } _ { p ^ { * } ( x ) } \\mathbb { E } _ { p ^ { * } ( y | x ) } [ \\log q ( y | x ) ] + \\mathbb { E } _ { p ^ { * } ( x ) } \\mathbb { E } _ { q ( s , v , y | x ) } [ \\frac { p ^ { * } ( y | x ) } { q ( y | x ) } \\mathrm { l o g } \\frac { p ( s , v , x , y ) } { q ( s , v , y | x ) } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As a functional of $q ( s , v , y | x )$ (instead of $q ( s , v | x , y ) )$ and the $\\mathbf { \\boldsymbol { C } } \\mathbf { \\boldsymbol { S } } \\mathbf { \\boldsymbol { G } } ~ p$ , this objective also drives them towards their targets: the first term is the negative of the standard cross entropy (CE) loss which drives $q ( y | x )$ towards $p ^ { * } ( y | x )$ , and once this is achieved, the second term becomes the expected ELBO $\\mathbb { E } _ { p ^ { * } ( x ) } [ \\mathcal { L } _ { p , q _ { s , v , y | x } } ( x ) ]$ that drives $q ( s , v , y | x )$ towards $p ( s , v , y | x )$ and $p ( x )$ towards $p ^ { * } ( x )$ . Furthermore, as the target of $q ( s , v , y | x )$ factorizes as $p ( s , v , y | x ) = p ( s , v | x ) p ( y | s )$ (due to Fig. 1a) where $p ( y | s )$ is already known (part of the CSG), we can instead employ a lighter inference model $q ( s , v | x )$ for the minimally intractable component $p ( s , v | x )$ therein, and use $\\bar { \\boldsymbol { q } } ( s , v | x ) p ( y | s )$ as $q ( s , v , y | x )$ . This turns the objective Eq. (1) to: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06acdf312bf6be6024b4c365537ab00d73f5f58da4fd021b62d36112285fb6a2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { p , q _ { s , v \\mid x } } \\mathbb { E } _ { p ^ { * } ( x , y ) } \\Big [ \\log q ( y | x ) + \\frac { 1 } { q ( y | x ) } \\mathbb { E } _ { q ( s , v | x ) } \\Big [ p ( y | s ) \\log \\frac { p ( s , v ) p ( x | s , v ) } { q ( s , v | x ) } \\Big ] \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $q ( y | x ) : = \\mathbb { E } _ { q ( s , v | x ) } [ p ( y | s ) ]$ . The expectations can be estimated by Monte Carlo after applying the reparameterization trick [62]. This is the basic CSG method. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "CSG-ind To actively improve OOD generalization performance, we consider using an independent prior $p ^ { \\underline { { \\parallel } } } ( s , v ) : = p ( s ) \\overline { { p ( v ) } }$ for prediction in the test domain (Fig. 1b), where $p ( s )$ and $p ( v )$ are the marginals of the training-domain prior $p ( s , v )$ . Intuitively, $p ^ { \\underline { { \\parallel } } } ( s , v )$ discards the spurious correlation between $s$ and $v$ on the training domain (e.g., the “desk-workspace”, “bed-bedroom” association), and promotes a cautious neutral belief on the unknown test-domain correlation in defence against all possibilities (e.g., a “desk-bedroom”, “bed-workspace” association). Formally, $p ^ { \\underline { { \\parallel } } } ( s , \\bar { v } )$ has a larger entropy than $p ( s , v )$ [24, Thm. 2.6.6], so it reduces training-domain-specific information and encourages reliance on the causal mechanisms for better generalization. It also amounts to applying the do-operator [85] to Fig. 1a, representing a randomized experiment by independently soft-intervening $s$ or $v$ . In this way, causal invariance is properly leveraged, making a different and more reliable prediction than following inference invariance. Our theory below also shows that $p ^ { \\underline { { \\parallel } } } ( s , v )$ leads to a smaller generalization error bound (Thm. 6 Remark). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Methodologically, we need the test-domain inference model $q ^ { \\perp } ( s , v | x )$ for prediction $p ^ { \\underline { { \\parallel } } } ( y | x ) \\approx$ $\\mathbb { E } _ { q ^ { \\perp } ( s , v \\mid x ) } [ p ( y \\mid s ) ]$ , but also need $q ( s , v | x )$ for learning on the training domain. To save the cost of building and learning two inference models, we propose to use $q ^ { \\perp } ( s , v | x )$ to represent $q ( s , v | x )$ . Noting that their targets are related by $\\begin{array} { r } { p ( s , v | x ) = \\frac { p ( s , v ) } { p ^ { \\perp } ( s , v ) } \\frac { p ^ { \\perp } ( x ) } { p ( x ) } p ^ { \\perp } ( s , v | x ) } \\end{array}$ , we formulate $q ( s , v | x ) =$ $\\begin{array} { r } { \\frac { p ( s , v ) } { p ^ { \\perp } ( s , v ) } \\frac { p ^ { \\perp } ( x ) } { p ( x ) } q ^ { \\perp } ( s , v | x ) } \\end{array}$ accordingly, so that this $q ( s , v | x )$ achieves its target if and only if $q ^ { \\perp } ( s , v | x )$ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "does. The objective Eq. (1) then becomes: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf86af16044fdfe947ed0077dfa8005c3dee5fc2dbdb7228821daf15fee52c79.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { p , q _ { s , v | x } ^ { \\mathrm { A } } } \\mathbb { E } _ { p ^ { s } ( x , y ) } \\Big [ \\log \\pi ( y | x ) + \\frac { 1 } { \\pi ( y | x ) } \\mathbb { E } _ { q ^ { \\perp } ( s , v | x ) } \\Big [ \\frac { p ( s , v ) } { p ^ { \\perp } ( s , v ) } p ( y | s ) \\log \\frac { p ^ { \\perp } ( s , v ) p ( x | s , v ) } { q ^ { \\perp } ( s , v | x ) } \\Big ] \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\pi ( y | x ) : = \\mathbb { E } _ { q ^ { \\perp } ( s , v | x ) } \\big [ \\frac { p ( s , v ) } { p ^ { \\perp } ( s , v ) } p ( y | s ) \\big ] } \\end{array}$ . (Note $p ^ { \\underline { { \\parallel } } } ( s , v )$ is determined by $p ( s , v )$ in the $\\operatorname { C S G } p$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 Method for Domain Adaptation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In domain adaptation, one also has unsupervised data from the underlying data distribution $\\tilde { p } ^ { * } ( x )$ on the test domain. We can leverage them for better prediction. According to the causal invariance principle (2), we only need a new prior $\\tilde { p } ( s , v )$ for the test-domain CSG $\\bar { p } : = \\langle \\tilde { p } ( s , v ) , p ( x | s , v ) , p ( \\bar { y } | s ) \\rangle$ (Fig. 1c). Fitting test-domain data can be done through the standard ELBO objective with the test-domain inference model $\\tilde { q } ( s , v | x )$ : ",
+        "bbox": [
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+            198,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61217e73a7c0dc4b970a4b367c6e16303b5de8744e4747822624005fc4df2b24.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\tilde { p } , \\tilde { q } _ { s , v \\mid x } } \\mathbb { E } _ { \\tilde { p } ^ { * } ( x ) } [ \\mathcal { L } _ { \\tilde { p } , \\tilde { q } _ { s , v \\mid x } } ( x ) ] , \\mathrm { w h e r e } \\ \\mathcal { L } _ { \\tilde { p } , \\tilde { q } _ { s , v \\mid x } } ( x ) = \\mathbb { E } _ { \\tilde { q } ( s , v \\mid x ) } \\Bigl [ \\log \\frac { \\tilde { p } ( s , v ) p ( x \\mid s , v ) } { \\tilde { q } ( s , v \\mid x ) } \\Bigr ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Prediction is given by $\\tilde { p } ( y | x ) \\approx \\mathbb { E } _ { \\tilde { q } ( s , v | x ) } [ p ( y | s ) ]$ . Similar to the CSG-ind case, we still need $q ( s , v | x )$ for fitting training-domain data, and we can also avoid a separate $q ( s , v | x )$ model by representing it using $\\tilde { q } ( s , v | x )$ . Following the same relation between their targets, we let $q ( s , v | x ) =$ $\\begin{array} { r } { \\frac { \\tilde { p } ( x ) } { p ( x ) } \\frac { p ( s , v ) } { \\tilde { p } ( s , v ) } \\tilde { q } ( s , v | x ) } \\end{array}$ , which reformulates the same training-domain objective Eq. (1) as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cecee319ddfe39d7c66232a609c60d9cb15e7a167947fe7647c32f19c7f2958.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { p , \\tilde { q } _ { s , v \\mid x } } \\mathbb { E } _ { p ^ { * } ( x , y ) } \\Big [ \\log \\pi ( y | x ) + \\frac { 1 } { \\pi ( y | x ) } \\mathbb { E } _ { \\bar { q } ( s , v | x ) } \\Big [ \\frac { p ( s , v ) } { \\tilde { p } ( s , v ) } p ( y | s ) \\log \\frac { \\tilde { p } ( s , v ) p ( x | s , v ) } { \\tilde { q } ( s , v | x ) } \\Big ] \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            367,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\pi ( \\boldsymbol { y } | \\boldsymbol { x } ) : = \\mathbb { E } _ { \\boldsymbol { \\tilde { q } } ( \\boldsymbol { s } , \\boldsymbol { v } | \\boldsymbol { x } ) } \\left[ \\frac { p ( \\boldsymbol { s } , \\boldsymbol { v } ) } { \\tilde { p } ( \\boldsymbol { s } , \\boldsymbol { v } ) } p ( \\boldsymbol { y } | \\boldsymbol { s } ) \\right]$ . The resulting method, termed CSG-DA, solves both optimization problems Eqs. (4, 5) simultaneously. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Implementation and Model Selection ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To implement the three CSG methods, we only need one inference model in each. Appx. F.2 shows its construction from a general discriminative model (e.g., how to select its hidden nodes as $s$ and $v$ ). In practice $x$ often has a much larger dimension than $y$ , making the first supervision term overwhelmed by the second unsupervised term in Eqs. (2,3,5). So we downscale the second term. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As recently emphasized [39], an OOD method should include a model selection method, since it is nontrivial and significantly affects performance [95, 120]. For our methods, we use a validation set from the training domain for model selection. This complies with the OOD setup, and is also suggested by our theory below which gives guarantees based on a good fit to the training-domain data distribution. For CSG-ind/DA, the learned predictor targets the test domain, so we do not use it directly for evaluating validation accuracy, but by normalizing $\\pi ( \\boldsymbol { y } | \\boldsymbol { x } )$ . Appx. F.3 shows details. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Theory ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now establish theory for the identification of the semantic factor (cause of prediction) and subsequent merits for OOD generalization and domain adaptation. We focus on the distribution-level generalization instead of from finite samples to unseen samples under the same distribution, so we only consider the infinite-data regime. Appx. A shows all the proofs and auxiliary theory. ",
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+    },
+    {
+        "type": "text",
+        "text": "Latent variable identification is hard [65, 81, 116, 70] as it is beyond observational relations [51, 88].   \nAssumptions are thus required to draw definite conclusions. ",
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+    {
+        "type": "text",
+        "text": "Assumption 3. (Additive noise) There exist nonlinear functions $f$ and $g$ with bounded derivatives up to the third-order, and independent random variables $\\mu$ and $\\nu$ , such that $p ( x | s , v ) = p _ { \\mu } ( x - f ( s , v ) { \\bar { ) } }$ , and $p ( y | s ) = p _ { \\nu } ( y - g ( s ) )$ for continuous $y$ or $p ( y | s ) = \\mathrm { C a t } ( y | g ( s ) )$ for categorical $y$ . ",
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+    {
+        "type": "text",
+        "text": "(Bijectivity) Assume $f$ is bijective and $g$ is injective. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The additive noise assumption is widely adopted in causal discovery [51, 17]. It disables expressing the same joint in the other direction [122, Thm. 8; 86, Prop. 23] so that CSG unnecessarily indicates inference invariance. For this reason, we exclude GAN [37] and flow-based [61] implementations. Bijectivity is a common assumption for identifiability [51, 100, 57, 68]. It is sufficient [86, Prop. 17; 88, Prop. 7.4] for the more fundamental [86, Prop. 7; 88, p.109] requirement of causal minimality [86, p.2012; 88, Def. 6.33]. Particularly, $s$ and $v$ may otherwise have dummy dimensions that $f$ and $g$ simply ignore, raising another ambiguity against identifiability. On the other hand, according to the commonly acknowledged manifold hypothesis [115, 31], we can take $\\mathcal { X }$ as the lower-dimensional data manifold and such a bijection exists as a coordinate map, which is an injection to the original data space and also allows $d _ { S } + d _ { \\mathcal { V } } < d _ { \\mathcal { X } }$ . ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Identifiability Theory ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We first formalize the goal of identifying the semantic factor. ",
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+    },
+    {
+        "type": "text",
+        "text": "Definition 4 (semantic-identification). We say a learned CSG $p$ is semantic-identified, if there exists a homeomorphism4 $\\Phi$ on $s \\times \\nu$ , such that (i) its output dimensions in $s$ is constant of $v$ : $\\Phi ^ { S } ( s , v ) =$ $\\Phi ^ { S } ( s , v ^ { \\prime } ) , \\forall \\bar { v } , v ^ { \\prime } \\in \\mathcal { V }$ (hence denote $\\Phi ^ { S } ( s , v )$ as $\\bar { \\Phi ^ { S } } ( s ) )$ , and (ii) it is a reparameterization of the ground-truth CSG $p ^ { * }$ $\\boldsymbol { \\mathbf { \\rho } } ) ^ { * } \\colon \\Phi _ { \\# } [ p _ { s , v } ^ { * } ] = p _ { s , v }$ , $p ^ { * } ( x | s , v ) = p ( x | \\Phi ( s , v ) )$ and $p ^ { * } ( \\dot { y } | s ) = p ( y | \\Phi ^ { S } ( s ) )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\Phi _ { \\# } [ p _ { s , v } ^ { * } ]$ denotes the pushed-forward distribution5 of $p _ { s , v } ^ { * }$ by $\\Phi$ , i.e. the distribution of $\\Phi ( s , v )$ when $( s , v ) \\sim p _ { s , v } ^ { * }$ . As the ground-truth CSG could at most provide its information via the data distribution $p ^ { * } ( x , y )$ , a well-learned CSG that achieves $p ( x , y ) = p ^ { * } ( x , y )$ still has the degree of freedom in parameterizing $( s , v )$ . This is described by this reparameterization $\\Phi$ (Appx. Lemma 9). At the heart of the definition, the $v$ -constancy of $\\Phi ^ { S }$ implies that $\\Phi$ is semantic-preserving: the learned model does not mix the ground-truth $v$ into its $s$ , so that the learned $s$ holds equivalent information to the ground-truth $s$ . The definition can thus be seen as the semantic equivalence (Appx. Def. 10, Prop. 14) to the ground-truth $\\mathbf { C S G } p ^ { * }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For related concepts, this identification cannot be characterized by the statistical independence between $s$ and $v$ (vs. [18, 49, 121]), which is not sufficient [70] nor necessary (due to the existence of spurious correlation). It is also weaker than disentanglement [44, 11], which additionally requires the learned $v$ to be constant of the ground-truth $s$ . The following theorem shows that semanticidentification can be achieved on a single domain under certain conditions. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (semantic-identifiability). With Assumption 3, a CSG $p$ is semantic-identified, if it is well-learned such that $p ( x , y ) = p ^ { * } ( x , y )$ , under the conditions that $\\log p ( s , v )$ and $\\log p ^ { * } ( s , v )$ are bounded up to the second-order, and that6 (i) $1 / \\sigma _ { \\mu } ^ { 2 }  \\infty$ where $\\sigma _ { \\mu } ^ { 2 } : = \\mathbb { E } [ \\mu ^ { \\top } \\mu ]$ , or (ii) $p _ { \\mu }$ (e.g., $a$ Gaussian) has an a.e. non-zero characteristic function. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Remarks. (1) (Condition and Intuition) Compared with the multi-domain case [87, 93, 2], identifiability on a single training domain comes at a cost and requires certain conditions. One may imagine that in some extreme cases e.g., all desks appear in workspace and all beds in bedrooms, it is impossible to distinguish whether $y$ labels the object or the background (unlearnable OOD problem [119]). The theorem finds an appropriate condition that excludes such cases: when $\\log p ^ { * } ( s , v )$ is bounded, deterministic $_ { s - v }$ relations are not allowed as they concentrate $\\boldsymbol { p } ^ { * } ( s , v )$ on a lower-dimensional subspace in $s \\times \\nu$ thus make it unbounded. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "It also leads to the intuition of identifiability: a bounded $\\log p ^ { * } ( s , v )$ indicates a stochastic s-v relation, so mixing the ground-truth $v$ into the learned $s$ makes the inference of $s$ more noisy due to the intrinsic diversity/uncertainty of this $v$ . As prediction is made via the inferred $s$ , this worsens prediction accuracy thus violates the “well-learned” requirement. Compared with discriminative models, CSG makes more faithful inference, and its causal structure leads to a proper description of domain change. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(2) In condition (i), $1 / \\sigma _ { \\mu } ^ { 2 }$ measures the intensity of the causal mechanism $p ( x | s , v )$ . When it is large, the “strong” $p ( x | s , v )$ helps disambiguating values of $( s , v )$ in generating a given $x$ . The formal version in Appx. Thm. $5 '$ shows a quantitative reference for large enough intensity, and Appx. B gives a non-asymptotic extension showing how the intensity trades-off the tolerance of equalities in Def. 4. Condition (ii) goes beyond inference invariance. It roughly implies that different $( s , v )$ values a.s. produce different $p ( x | s , v )$ , so their roles in generating $x$ become clear which helps identification. ",
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+    {
+        "type": "text",
+        "text": "(3) The theorem does not contradict the impossibility result by Locatello et al. [70], which considers disentangling each latent dimension with an unconstrained $( \\dot { s } , v )  ( x , y )$ , while we only identify $s$ as a whole, with the $v  y$ edge removed which breaks the s-v symmetry. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 OOD Generalization Theory ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Now we show the benefit of semantic-identification for OOD generalization that the prediction error is bounded. Note the optimal predictor $\\tilde { \\mathbb { E } } ^ { * } [ y | x ] ^ { \\eta }$ on the test domain is defined by the corresponding ground-truth CSG $\\tilde { p } ^ { * }$ , which differs from $p ^ { * }$ only in the test-domain prior $\\tilde { p } ^ { * } ( s , v )$ (Principle 2). ",
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 6 (OOD generalization error). 8 With Assumption 3, for a semantic-identified CSG $p$ on the training domain with semantic-preserving reparameterization $\\Phi$ , we have up to $O ( \\sigma _ { \\mu } ^ { 4 } )$ , ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5489ef8f7d75f51f5b3f70fec914818abe57784aecddf73e715c06c09d0aed4b.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\tilde { p } ^ { * } ( x ) } \\| \\mathbb { E } [ y | x ] - \\tilde { \\mathbb { E } } ^ { * } [ y | x ] \\| _ { 2 } ^ { 2 } \\leqslant \\sigma _ { \\mu } ^ { 4 } B _ { f ^ { - 1 } } ^ { \\prime 4 } B _ { g } ^ { \\prime 2 } \\mathbb { E } _ { \\tilde { p } _ { s , v } } \\big \\| \\nabla \\log ( \\tilde { p } _ { s , v } / p _ { s , v } ) \\big \\| _ { 2 } ^ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            212
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $B _ { f ^ { - 1 } } ^ { \\prime }$ and $B _ { g } ^ { \\prime }$ bound the 2-norms9 of the Jacobians of $f ^ { - 1 }$ and $g$ , respectively, and $\\widetilde { p } _ { s , v } : =$ $\\Phi _ { \\# } [ \\tilde { p } _ { s , v } ^ { * } ]$ is the test-domain prior under the parameterization of the $C S G p$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the bound, the term $\\mathbb { E } _ { \\tilde { p } _ { s , v } } \\left\\| \\nabla \\log ( \\tilde { p } _ { s , v } / p _ { s , v } ) \\right\\| _ { 2 } ^ { 2 }$ is the Fisher divergence measuring the difference between the two priors. As the prior change is the only source of domain change, this term also measures the “OODness” in terms of the effect on prediction. The bound also shows that when the causal mechanism $p ( x | s , v )$ is strong (small $\\sigma _ { \\mu }$ ), it dominates prediction over the prior change, as the generalization error becomes small. Compared with other methods, using a CSG enforces causal invariance, so the boundedness of OOD generalization error becomes more plausible in practice. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Remark. The bound also shows the advantage of CSG-ind (Sec. 4.1). The Fisher divergence is revealed [28] to have a similar behavior as the forward KL divergence $p _ { s , v } \\mapsto \\mathrm { K L } ( \\tilde { p } _ { s , v } \\Vert p _ { s , v } )$ that it is very sensitive to the insufficient coverage of $p _ { s , v }$ on the support of $\\tilde { p } _ { s , v }$ [46, 109], since $\\log ( \\tilde { p } _ { s , v } / p _ { s , v } )$ is infinitely large on the uncovered region. As the independent prior $p _ { s , v } ^ { \\perp }$ has a larger support than $p _ { s , v }$ , it is less likely to miss the support of $\\tilde { p } _ { s , v }$ , so it induces a generally smaller Fisher divergence. CSG-ind thus generally has a smaller OOD generalization error bound than CSG. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 Domain Adaptation Theory ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "CSG-DA (Sec. 4.2) learns a new prior $\\tilde { p } _ { s , v }$ by fitting unsupervised test-domain data, with causal mechanisms shared. If the mechanisms are semantic-identified, the ground-truth test-domain prior $\\tilde { p } _ { s , v } ^ { * }$ can also be identified under the learned parameterization, and prediction is made precise. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 7 (domain adaptation error). With conditions of Thm. 5, for a semantic-identified $C S G p$ on the training domain with semantic-preserving reparameterization $\\Phi$ , if its new prior $\\tilde { p } _ { s , v }$ is well",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "learned such that $\\tilde { p } ( x ) = \\tilde { p } ^ { * } ( x )$ , then $\\tilde { p } _ { s , v } = \\Phi _ { \\# } [ \\tilde { p } _ { s , v } ^ { * } ]$ , and $\\tilde { \\mathbb { E } } [ y | x ] = \\tilde { \\mathbb { E } } ^ { * } [ y | x ]$ for any $x \\in \\mathrm { s u p p } ( \\tilde { p } _ { x } ^ { * } )$ ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Different from existing domain adaptation bounds (Appx. E), Theorems 6,7 allow different inference models in the two domains, thus go beyond inference invariance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For OOD generalization baselines, there is not much choice beyond the standard CE loss optimization, as domain adaptation methods require test-domain data and most domain generalization methods degenerate to CE with one training domain. The exception within our scope is a causal discriminative method CNBB [41]. For domain adaptation, we consider well-acknowledged methods DANN [33], DAN [73], CDAN [74] and recent compelling methods MDD [124] and BNM [25] (shown in Appx. Tables 2,3). Appx. G shows more details, results, and discussions. 10 ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Shifted-MNIST. We first consider an OOD prediction task on MNIST to classify digits $\\mathbf { \\bar { \\theta } } ^ { 6 6 }$ ”s and “1”s. To make a spurious correlation, in the training data, we horizontally shift each $ { { } ^ { 6 } } 0 ^ { 9 }$ at random by $\\delta _ { 0 } \\sim \\mathcal { N } ( - 5 , 1 ^ { 2 } )$ pixels, while each “1” by $\\delta _ { 1 } \\sim \\bar { \\mathcal { N } } ( 5 , 1 ^ { 2 } )$ pixels. We consider two test domains with different digit-position distributions: each digit is not moved $\\delta _ { 0 } = \\delta _ { 1 } = 0$ in the first, and is shifted at random by $\\bar { \\delta _ { 0 } } , \\delta _ { 1 } \\sim \\mathcal { N } ( 0 , 2 ^ { 2 } )$ pixels in the second. We implement all methods using a multilayer perceptron which is not naturally shift invariant. We use a larger architecture for non-generative methods to compensate the additional generative component of generative methods. ",
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+        "type": "text",
+        "text": "The performance is shown in Table 1(top 2 rows). For OOD generalization, CE is misled by the more noticeable position factor due to the spurious correlation to digits, and resorts to random guess (even worse) when position is not informative for prediction. CNBB ameliorates the position confusion, but not as thoroughly without modeling causal mechanisms. In contrast, our CSG gives more genuine predictions in unseen domains, thanks to the identification of the semantic factor. CSG-ind performs even better, justifying the merit of using an independent prior for prediction. For domain adaptation, CSG-DA achieves the best results. Existing adaptation methods even worsen the result (negative transfer), as the misleading position representation gets strengthened on the unsupervised test data. CSG is benefited from adaptation in a proper way that identifies the semantic factor. ",
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+            "Table 1: Test accuracy $( \\% )$ by various methods (ours in bold) for OOD generalization (left 4 cols) and domain adaptation (right 5 cols) on Shifted-MNIST (top 2 rows), ImageCLEF-DA (middle 4 rows) and PACS (bottom 4 rows) datasets. Averaged over 10 runs. Appx. Tables 2,3 show more results. "
+        ],
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+        "table_body": "<table><tr><td>task</td><td>CE</td><td>CNBB</td><td>CSG</td><td>CSG-ind</td><td>DANN</td><td>DAN</td><td>CDAN</td><td>MDD</td><td>CSG-DA</td></tr><tr><td>8=δ1=0</td><td>42.9±3.1</td><td>54.7±3.3</td><td>81.4±7.4</td><td>82.6±4.0</td><td>40.9±3.0</td><td>40.4±2.0</td><td>41.0±0.5</td><td>41.9±0.8</td><td>97.6±4.0</td></tr><tr><td>80,~N(0,2²）</td><td>47.8±1.5</td><td>59.2±2.4</td><td>61.7±3.6</td><td>62.3±2.2</td><td>46.2±0.7</td><td>45.6±0.7</td><td>46.3±0.6</td><td>45.8±0.3</td><td>72.0±9.2</td></tr><tr><td>C→P</td><td>65.5±0.3</td><td>72.7±1.1</td><td>73.6±0.6</td><td>74.0±1.3</td><td>74.3±0.5</td><td>69.2±0.4</td><td>74.5±0.3</td><td>74.1±0.7</td><td>75.1±0.5</td></tr><tr><td>P→C</td><td>91.2±0.3</td><td>91.7±0.2</td><td>92.3±0.4</td><td>92.7±0.2</td><td>91.5±0.6</td><td>89.8±0.4</td><td>93.5±0.4</td><td>92.1±0.6</td><td>93.4±0.3</td></tr><tr><td>I→P</td><td>74.8±0.3</td><td>75.4±0.6</td><td>76.9±0.3</td><td>77.2±0.2</td><td>75.0±0.6</td><td>74.5±0.4</td><td>76.7±0.3</td><td>76.8±0.4</td><td>77.4±0.3</td></tr><tr><td>P→I</td><td>83.9±0.1</td><td>88.7±0.5</td><td>90.4±0.3</td><td>90.9±0.2</td><td>86.0±0.3</td><td>82.2±0.2</td><td>90.6±0.3</td><td>90.2±1.1</td><td>91.1±0.5</td></tr><tr><td>others→P</td><td>97.8±0.0</td><td>96.9±0.2 97.7±0.2</td><td></td><td>97.8±0.2</td><td>[97.6±0.2</td><td>97.6±0.4</td><td>97.0±0.4</td><td>97.6±0.3</td><td>97.9±0.2</td></tr><tr><td>others→A</td><td>88.1±0.1</td><td>73.1±0.3</td><td>88.5±0.6</td><td>88.6±0.6</td><td>85.9±0.5</td><td>84.5±1.2</td><td>84.0±0.9</td><td>88.1±0.8</td><td>88.8±0.7</td></tr><tr><td>others-C</td><td>77.9±1.3</td><td>50.2±1.2 84.4±0.9</td><td></td><td>84.6±0.8</td><td>79.9±1.4 81.9±1.9</td><td></td><td>78.5±1.5 83.2±1.1</td><td></td><td>84.7±0.8</td></tr><tr><td>others-→S</td><td>79.1±0.9</td><td>43.3±1.2 80.7±1.0</td><td></td><td>81.1±1.2</td><td>75.2±2.8 77.4±3.1</td><td></td><td>71.8±3.9 80.2±2.2</td><td></td><td>81.4±0.8</td></tr></table>",
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+        "text": "ImageCLEF-DA is a standard benchmark for domain adaptation [1]. It has 12 classes and three domains of real-world images: Caltech-256, ImageNet, Pascal VOC 2012. We select four OOD prediction tasks $\\mathbf { C } {  } \\mathbf { P }$ , $\\mathbf { I } {  } \\mathbf { P }$ that have not seen good enough results. We adopt the same setup as [74]. As shown in Table 1(middle 4 rows), CSG-ind again achieves the best OOD generalization results, and even outperforms some domain adaptation methods. Our CSG also outperforms the baselines mostly. For domain adaptation, CSG-DA is the best in most cases and on par with the best in others. ",
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+        "text": "PACS is a more recent benchmark dataset [69]. It has 7 classes and is named after its four domains: Photo, Art, Cartoon, Sketch; each contains images of a certain style. We follow the same setup as [39]; particularly, we pool together all domains but the test one as the single training domain. Results in Table 1(bottom 4 rows) show the same trend. CSG-DA even outperforms most domain generalization methods reported in [39], which are fed with more information. Appx. Tables 2,3 also show the results on an even larger dataset VLCS [30], which present a similar observation. ",
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+        "type": "text",
+        "text": "Visualization. Appx. Fig. 5 visualizes the learned models using LIME [91]. The results show our methods focus more on the semantic regions and shapes, indicating a causal representation is learned. ",
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+        "text": "Dataset analysis. The results indicate our methods are more powerful on shifted-MNIST and PACS (and VLCS) than ImageCLEF-DA. This meets the intuition of identifiability (Thm. 5 Remark (1)): the random position or pooled training domain shows a diverse $v$ for each $s$ (while with a misleading spurious correlation), so identification is better guaranteed to overcome the spurious correlation. ",
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+        "text": "Ablation study. To show the benefit of modeling $s$ and $v$ separately, we compare with a counterpart of CSG that treats $s$ and $v$ as a whole (equivalently, $v  y$ is kept; see Appx. F.1.4 for method details). Appx. Tables 2,3 show that our methods outperform this baseline in all cases. This shows the separate modeling makes CSG consciously drive semantic representation into the dedicated variable $s$ . ",
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+        "text": "7 Conclusion and Discussion ",
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+        "text": "We propose a Causal Semantic Generative model for single-domain OOD prediction tasks, which builds upon a causal reasoning, and models the semantic (cause of prediction) and variation factors separately. By the causal invariance principle, we develop novel and efficient learning and prediction methods, and prove the semantic-identifiability and the subsequent bounded generalization error and the success of adaptation. Experiments show the improved performance over prevailing baselines. ",
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+        "text": "Notably, we answered the questions in the recent farseeing paper [98] on causal representation learning: we found an appropriate condition under which “causal variables can be recovered”, and provided “compelling evidence on the advantages (of causal modeling) in terms of generalization”. Also, separating semantics from variation extends to broader examples. Neural nets are found to change their prediction under a different texture [34, 15]. Adversarial vulnerability [107, 38, 67] extends variation factors to human-imperceptible features, i.e. adversarial noise, which is found to have a strong correlation to the semantics [50]. The separation also matters for fairness when a sensitive variation factor may affect prediction. This work also inspires the dual connection between causal representation learning (“fill in the blanks” given a graph) and causal discovery (“link the nodes” given observed variables). Our theory shows the identifiability condition for causal discovery (the additive noise assumption) also makes causal representation identifiable. Studying the general connection between the two tasks is an interesting future work. ",
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+    {
+        "type": "text",
+        "text": "References ",
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parse/train/B1YfAfcgl/B1YfAfcgl.md

@@ -0,0 +1,468 @@
+# ENTROPY-SGD: BIASING GRADIENT DESCENT INTO WIDE VALLEYS
+
+Pratik Chaudhari1, Anna Choromanska2, Stefano Soatto1, Yann LeCun3,4, Carlo Baldassi5, Christian Borgs6, Jennifer Chayes6, Levent Sagun3, Riccardo Zecchina5
+
+1 Computer Science Department, University of California, Los Angeles   
+2 Department of Electrical and Computer Engineering, New York University   
+3 Courant Institute of Mathematical Sciences, New York University   
+4 Facebook AI Research, New York   
+5 Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino   
+6 Microsoft Research New England, Cambridge
+
+Email: pratikac $@$ ucla.edu, $\operatorname { a c } 5 4 5 5 @$ nyu.edu, soatto $@$ ucla.edu, yann $@$ cs.nyu.edu, carlo.baldassi $@$ polito.it, borgs $@$ microsoft.com, jchayes $@$ microsoft.com, sagun@cims.nyu.edu, riccardo.zecchina $@$ polito.it
+
+# ABSTRACT
+
+This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent neural networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.
+
+# 1 INTRODUCTION
+
+This paper presents a new optimization tool for deep learning designed to exploit the local geometric properties of the objective function. Consider the histogram we obtained in Fig. 1 showing the spectrum of the Hessian at an extremum discovered by Adam (Kingma & Ba, 2014) for a convolutional neural network on MNIST (LeCun et al., 1998) $( \approx 4 7 , 0 0 0$ weights, cf. Sec. 5.1). It is evident that:
+
+(i) a large number of directions $( \approx 9 4 \%$ ) have near-zero eigenvalues (magnitude less than $1 0 ^ { - 4 }$ ), (ii) positive eigenvalues (right inset) have a long tail with the largest one being almost 40, (iii) negative eigenvalues (left inset), which are directions of descent that the optimizer missed, have a much faster decay (the largest negative eigenvalue is only $- 0 . 4 6 )$ .
+
+Interestingly, this trend is not unique to this particular network. Rather, its qualitative properties are shared across a variety of network architectures, network sizes, datasets or optimization algorithms (refer to Sec. 5 for more experiments). Local minima that generalize well and are discovered by gradient descent lie in “wide valleys” of the energy landscape, rather than in sharp, isolated minima. For an intuitive understanding of this phenomenon, imagine a Bayesian prior concentrated at the minimizer of the expected loss, the marginal likelihood of wide valleys under this prior is much higher than narrow, sharp valleys even if the latter are close to the global minimum in training loss. Almost-flat regions of the energy landscape are robust to data perturbations, noise in the activations, as well as perturbations of the parameters, all of which are widely-used techniques to achieve good generalization. This suggests that wide valleys should result in better generalization and, indeed, standard optimization algorithms in deep learning seem to discover exactly that — without being explicitly tailored to do so. For another recent analysis of the Hessian, see the parallel work of Sagun et al. (2016).
+
+![](images/fe6f472c542ef09360b26afca748671709746b760967305acfe6f04eaaa0004e.jpg)  
+small-LeNet: Eigenspectrum of the Hessian at local minimum   
+Eigenvalues   
+Figure 1: Eigenspectrum of the Hessian at a local minimum of a CNN on MNIST (two independent runs). Remark: The central plot shows the eigenvalues in a small neighborhood of zero whereas the left and right insets show the entire tails of the eigenspectrum.
+
+Based on this understanding of how the local geometry looks at the end of optimization, can we modify SGD to actively seek such regions? Motivated by the work of Baldassi et al. (2015) on shallow networks, instead of minimizing the original loss $f ( x )$ , we propose to maximize
+
+$$
+F ( x , \gamma ) = \log \int _ { x ^ { \prime } \in { \mathbb { R } ^ { n } } } \exp \left( - f ( x ^ { \prime } ) - \frac { \gamma } { 2 } \left\| x - x ^ { \prime } \right\| _ { 2 } ^ { 2 } \right) d x ^ { \prime } .
+$$
+
+The above is a log-partition function that measures both the depth of a valley at a location $x \in$ $\mathbb { R } ^ { n }$ , and its flatness through the entropy of $f ( x ^ { \prime } )$ ; we call it “local entropy” in analogy to the free entropy used in statistical physics. The Entropy-SGD algorithm presented in this paper employs stochastic gradient Langevin dynamics (SGLD) to approximate the gradient of local entropy. Our algorithm resembles two nested loops of SGD: the inner loop consists of SGLD iterations while the outer loop updates the parameters. We show that the above modified loss function results in a smoother energy landscape defined by the hyper-parameter $\gamma$ which we can think of as a “scope” that seeks out valleys of specific widths. Actively biasing the optimization towards wide valleys in the energy landscape results in better generalization error. We present experimental results on fullyconnected and convolutional neural networks (CNNs) on the MNIST and CIFAR-10 (Krizhevsky, 2009) datasets and recurrent neural networks (RNNs) on the Penn Tree Bank dataset (PTB) (Marcus et al., 1993) and character-level text prediction. Our experiments show that Entropy-SGD scales to deep networks used in practice, obtains comparable generalization error as competitive baselines and also trains much more quickly than SGD (we get a $2 \mathbf { x }$ speed-up over SGD on RNNs).
+
+# 2 RELATED WORK
+
+Our above observation about the spectrum of Hessian (further discussed in Sec. 5) is similar to results on a perceptron model in Dauphin et al. (2014) where the authors connect the loss function of a deep network to a high-dimensional Gaussian random field. They also relate to earlier studies such as Baldi & Hornik (1989); Fyodorov & Williams (2007); Bray & Dean (2007) which show that critical points with high training error are exponentially likely to be saddle points with many negative directions and all local minima are likely to have error that is very close to that of the global minimum. The authors also argue that convergence of gradient descent is affected by the proliferation of saddle points surrounded by high error plateaus — as opposed to multiple local minima. One can also see this via an application of Kramer’s law: the time spent by diffusion is inversely proportional to the smallest negative eigenvalue of the Hessian at a saddle point (Bovier & den Hollander, 2006).
+
+The existence of multiple — almost equivalent — local minima in deep networks has been predicted using a wide variety of theoretical analyses and empirical observations, e.g., papers such as Choromanska et al. (2015a;b); Chaudhari & Soatto (2015) that build upon results from statistical physics as also others such as Haeffele & Vidal (2015) and Janzamin et al. (2015) that obtain similar results for matrix and tensor factorization problems. Although assumptions in these works are somewhat unrealistic in the context of deep networks used in practice, similar results are also true for linear networks which afford a more thorough analytical treatment (Saxe et al., 2014). For instance, Soudry & Carmon (2016) show that with mild over-parameterization and dropout-like noise, training error for a neural network with one hidden layer and piece-wise linear activation is zero at every local minimum. All these results suggest that the energy landscape of deep neural networks should be easy to optimize and they more or less hold in practice — it is easy to optimize a prototypical deep network to near-zero loss on the training set (Hardt et al., 2015; Goodfellow & Vinyals, 2015).
+
+Obtaining good generalization error, however, is challenging: complex architectures are sensitive to initial conditions and learning rates (Sutskever et al., 2013) and even linear networks (Kawaguchi, 2016) may have degenerate and hard to escape saddle points (Ge et al., 2015; Anandkumar & Ge, 2016). Techniques such as adaptive (Duchi et al., 2011) and annealed learning rates, momentum (Tieleman & Hinton, 2012), as well as architectural modifications like dropout (Srivastava et al., 2014), batch-normalization (Ioffe & Szegedy, 2015; Cooijmans et al., 2016), weight scaling (Salimans & Kingma, 2016) etc. are different ways of tackling this issue by making the underlying landscape more amenable to first-order algorithms. However, the training process often requires a combination of such techniques and it is unclear beforehand to what extent each one of them helps.
+
+Closer to the subject of this paper are results by Baldassi et al. (2015; 2016a;b) who show that the energy landscape of shallow networks with discrete weights is characterized by an exponential number of isolated minima and few very dense regions with lots of local minima close to each other. These dense local minima can be shown to generalize well for random input data; more importantly, they are also accessible by efficient algorithms using a novel measure called “robust ensemble” that amplifies the weight of such dense regions. The authors use belief propagation to estimate local entropy for simpler models such as committee machines considered there. A related work in this context is EASGD (Zhang et al., 2015) which trains multiple deep networks in parallel and modulates the distance of each worker from the ensemble average. Such an ensemble training procedure enables improved generalization by ensuring that different workers land in the same wide valley and indeed, it turns out to be closely related to the replica theoretic analysis of Baldassi et al. (2016a).
+
+Our work generalizes the local entropy approach above to modern deep networks with continuous weights. It exploits the same phenomenon of wide valleys in the energy landscape but does so without incurring the hardware and communication complexity of replicated training or being limited to models where one can estimate local entropy using belief propagation. The enabling technique in our case is using Langevin dynamics for estimating the gradient of local entropy, which can be done efficiently even for large deep networks using mini-batch updates.
+
+Motivated by the same final goal, viz. flat local minima, the authors in Hochreiter & Schmidhuber (1997b) introduce hard constraints on the training loss and the width of local minima and show using the Gibbs formalism (Haussler & Opper, 1997) that this leads to improved generalization. As the authors discuss, the effect of hyper-parameters for the constraints is intricately tied together and they are difficult to choose even for small problems. Our local entropy based objective instead naturally balances the energetic term (training loss) and the entropic term (width of the valley). The role of γ is clear as a focusing parameter (cf. Sec. 4.3) and effectively exploiting this provides significant computational advantages. Among other conceptual similarities with our work, let us note that local entropy in a flat valley is a direct measure of the width of the valley which is similar to their usage of Hessian, while the Gibbs variant to averaging in weight space (Eqn. 33 of Hochreiter & Schmidhuber (1997b)) is similar to our Eqn. (7). Indeed, Gibbs formalism used in their analysis is a very promising direction to understanding generalization in deep networks (Zhang et al., 2016).
+
+Homotopy continuation methods convolve the loss function to solve sequentially refined optimization problems (Allgower & Georg, 2012; Mobahi & Fisher III, 2015), similarly, methods that perturb the weights or activations to average the gradient (Gulcehre et al., 2016) do so with an aim to smooth the rugged energy landscape. Such smoothing is however very different from local entropy. For instance, the latter places more weight on wide local minima even if they are much shallower than the global minimum (cf. Fig. 2); this effect cannot be obtained by smoothing. In fact, smoothing can introduce an artificial minimum between two nearby sharp valleys which is detrimental to generalization. In order to be effective, continuation techniques also require that minimizers of successively smaller convolutions of the loss function lie close to each other (Hazan et al., 2016); it is not clear whether this is true for deep networks. Local entropy, on the other hand, exploits wide minima which have been shown to exist in a variety of learning problems (Monasson & Zecchina, 1995; Cocco et al., 1996). Please refer to Appendix C for a more elaborate discussion as well as possible connections to stochastic variational inference (Blei et al., 2016).
+
+# 3 LOCAL ENTROPY
+
+We first provide a simple intuition for the concept of local entropy of an energy landscape. The discussion in this section builds upon the results of Baldassi et al. (2016a) and extends it for the case of continuous variables. Consider a cartoon energy landscape in Fig. 2 where the $\mathbf { X }$ -axis denotes the configuration space of the parameters. We have constructed two local minima: a shallower although wider one at $x _ { \mathrm { r o b u s t } }$ and a very sharp global minimum at xnon robust. Under a Bayesian prior on the parameters, say a Gaussian of a fixed variance at locations $x _ { \mathrm { r o b u s t } }$ and $x _ { \mathrm { n o n - r o b u s t } }$ respectively, the wider local minimum has a higher marginalized likelihood than the sharp valley on the right.
+
+The above discussion suggests that parameters that lie in wider local minima like $x _ { \mathrm { r o b u s t } }$ , which may possibly have a higher loss than the global minimum, should generalize better than the ones that are simply at the global minimum. In a neighborhood of $x _ { \mathrm { r o b u s t } }$ , “local entropy” as introduced in Sec. 1 is large because it includes the contributions from a large region of good parameters; conversely, near $x _ { \mathrm { n o n - r o b u s t } }$ , there are fewer such contributions and the resulting local entropy is low. The local entropy thus provides a way of picking large, approximately flat, regions of the landscape over sharp, narrow valleys in spite of the latter possibly having a lower loss. Quite conveniently, the local entropy is also computed from the partition function with a local re-weighting term.
+
+Formally, for a parameter vector $x \in \mathbb { R } ^ { n }$ , consider a Gibbs distribution corresponding to a given energy landscape $f ( x )$ :
+
+![](images/f9080b5c33ff0c459ae76b79b64ba2c0a37369c205de046f172d7251db689ca8.jpg)  
+Figure 2: Local entropy concentrates on wide valleys in the energy landscape.
+
+$$
+\mathrm { P } ( x ; \beta ) = Z _ { \beta } ^ { - 1 } \ \mathrm { e x p } \ ( - \beta \ f ( x ) ) ;
+$$
+
+where $\beta$ is known as the inverse temperature and $Z _ { \beta }$ is a normalizing constant, also known as the partition function. As $\beta \to \infty$ , the probability distribution above concentrates on the global minimum of $f ( x )$ (assuming it is unique) given as:
+
+$$
+x ^ { * } = \underset { x } { \operatorname { a r g m i n } } \ f ( x ) ,
+$$
+
+which establishes the link between the Gibbs distribution and a generic optimization problem (2). We would instead like the probability distribution — and therefore the underlying optimization problem — to focus on flat regions such as $x _ { \mathrm { r o b u s t } }$ in Fig. 2. With this in mind, let us construct a modified Gibbs distribution:
+
+$$
+\mathbf { P } ( x ^ { \prime } ; x , \beta , \gamma ) = Z _ { x , \beta , \gamma } ^ { - 1 } \exp \left( - \beta f ( x ^ { \prime } ) - \beta \frac { \gamma } { 2 } \| x - x ^ { \prime } \| _ { 2 } ^ { 2 } \right) .
+$$
+
+The distribution in (3) is a function of a dummy variable $x ^ { \prime }$ and is parameterized by the original location $x$ . The parameter $\gamma$ biases the modified distribution (3) towards $x$ ; a large $\gamma$ results in a
+
+$P ( x ^ { \prime } ; x , \beta , \gamma )$ with all its mass near $x$ irrespective of the energy term $f ( x ^ { \prime } )$ . For small values of $\gamma _ { : }$ , the term $f ( x ^ { \prime } )$ in the exponent dominates and the modified distribution is similar to the original Gibbs distribution in (1). We will set the inverse temperature $\beta$ to 1 because $\gamma$ affords us similar control on the Gibbs distribution.
+
+Definition 1 (Local entropy). The local free entropy of the Gibbs distribution in (1), colloquially called “local entropy” in the sequel and denoted by $F ( x , \gamma )$ , is defined as the log-partition function of modified Gibbs distribution (3), i.e.,
+
+$$
+\begin{array} { l } { \displaystyle { F ( x , \gamma ) = \log Z _ { x , 1 , \gamma } } } \\ { \displaystyle { \phantom { \sum _ { i } \log Z _ { x , 1 } \exp Z _ { x } } = \log \int _ { x ^ { \prime } } \exp \left( - f ( x ^ { \prime } ) - \frac { \gamma } { 2 } \ : \| x - x ^ { \prime } \| _ { 2 } ^ { 2 } \right) d x ^ { \prime } } . } \end{array}
+$$
+
+The parameter $\gamma$ is used to focus the modified Gibbs distribution upon a local neighborhood of $x$ and we call it a “scope” henceforth.
+
+Effect on the energy landscape: Fig. 2 shows the negative local entropy $- F ( x , \gamma )$ for two different values of $\gamma .$ . We expect $x _ { \mathrm { r o b u s t } }$ to be more robust than $x _ { \mathrm { { n o n } } }$ -robust to perturbations of data or parameters and thus generalize well and indeed, the negative local entropy in Fig. 2 has a global minimum near $x _ { \mathrm { r o b u s t } }$ . For low values of $\gamma ,$ , the energy landscape is significantly smoother than the original landscape and still maintains our desired characteristic, viz. global minimum at a wide valley. As $\gamma$ increases, the local entropy energy landscape gets closer to the original energy landscape and they become equivalent at $\gamma \to \infty$ . On the other hand, for very small values of $\gamma ,$ the local entropy energy landscape is almost uniform.
+
+Connection to classical entropy: The quantity we have defined as local entropy in Def. 1 is different from classical entropy which counts the number of likely configurations under a given distribution. For a continuous parameter space, this is given by
+
+$$
+S ( x , \beta , \gamma ) = - \int _ { x ^ { \prime } } \log \mathrm { P } ( x ^ { \prime } ; x , \beta , \gamma ) d \mathrm { P } ( x ^ { \prime } ; x , \beta , \gamma )
+$$
+
+for the Gibbs distribution in (3). Minimizing classical entropy however does not differentiate between flat regions that have very high loss versus flat regions that lie deeper in the energy landscape. For instance in Fig. 2, classical entropy is smallest in the neighborhood of xcandidate which is a large region with very high loss on the training dataset and is unlikely to generalize well.
+
+# 4 ENTROPY-GUIDED SGD
+
+Simply speaking, our Entropy-SGD algorithm minimizes the negative local entropy from Sec. 3. This section discusses how the gradient of local entropy can be computed via Langevin dynamics. The reader will see that the resulting algorithm has a strong flavor of “SGD-inside-SGD”: the outer SGD updates the parameters, while an inner SGD estimates the gradient of local entropy.
+
+Consider a typical classification setting, let $x \in \mathbb { R } ^ { n }$ be the weights of a deep neural network and $\xi _ { k } \in \Xi$ be samples from a dataset $\Xi$ of size $N$ . Let $f ( x ; \xi _ { k } )$ be the loss function, e.g., cross-entropy of the classifier on a sample $\xi _ { k }$ . The original optimization problem is:
+
+$$
+x ^ { * } = \underset { x } { \mathrm { a r g m i n } } \ \frac { 1 } { N } \ \sum _ { k = 1 } ^ { N } \ f ( x ; \ \xi _ { k } ) ;
+$$
+
+where the objective $f ( x , \xi _ { k } )$ is typically a non-convex function in both the weights $x$ and the samples $\xi _ { k }$ . The Entropy-SGD algorithm instead solves the problem
+
+$$
+x _ { \mathrm { E n t r o p y - S G D } } ^ { * } = \underset { x } { \mathrm { a r g m i n } } \ - F ( x , \gamma ; \ \Xi ) ;
+$$
+
+where we have made the dependence of local entropy $F ( x , \gamma )$ on the dataset $\Xi$ explicit.
+
+The gradient of local entropy over a randomly sampled mini-batch of $m$ samples denoted by $\boldsymbol { \xi } _ { \ell _ { i } } \in \Xi ^ { \ell }$ for $i \leq m$ is easy to derive and is given by
+
+$$
+- \nabla _ { x } F \left( x , \gamma ; \Xi ^ { \ell } \right) = \gamma \left( x - \left. x ^ { \prime } ; \Xi ^ { \ell } \right. \right) ;
+$$
+
+where the notation $\langle \cdot \rangle$ denotes an expectation of its arguments (we have again made the dependence on the data explicit) over a Gibbs distribution of the original optimization problem modified to focus on the neighborhood of $x$ ; this is given by
+
+$$
+\mathrm { P } ( x ^ { \prime } ; x , \gamma ) \propto \exp \left[ - \left( \frac { 1 } { m } \sum _ { i = 1 } ^ { m } f \left( x ^ { \prime } ; \xi _ { \ell _ { i } } \right) \right) - \frac { \gamma } { 2 } \left. x - x ^ { \prime } \right. _ { 2 } ^ { 2 } \right] .
+$$
+
+Computationally, the gradient in (7) involves estimating $\left. x ^ { \prime } ; \Xi ^ { \ell } \right.$ with the current weights fixed to $x$ . This is an expectation over a Gibbs distribution and is hard to compute. We can however approximate it using Markov chain Monte-Carlo (MCMC) techniques. In this paper, we use stochastic gradient Langevin dynamics (SGLD) (Welling & Teh, 2011) that is an MCMC algorithm for drawing samples from a Bayesian posterior and scales to large datasets using mini-batch updates. Please see Appendix A for a brief overview of SGLD. For our application, as lines 3-6 of Alg. 1 show, SGLD resembles a few iterations of SGD with a forcing term $- \gamma ( x - x ^ { \prime } )$ and additive gradient noise.
+
+We can obtain some intuition on how Entropy-SGD works using the expression for the gradient: the term $\langle x ^ { \prime } ; \cdot \rangle$ is the average over a locally focused Gibbs distribution and for two local minima in the neighborhood of $x$ roughly equivalent in loss, this term points towards the wider one because $\langle x ^ { \prime } ; \cdot \rangle$ is closer to it. This results in a net gradient that takes SGD towards wider valleys. Moreover, if we unroll the SGLD steps used to compute $\left( x - \langle { x ^ { \prime } ; \cdot } \rangle \right)$ (cf. line 5 in Alg. 1), it resembles one large step in the direction of the (noisy) average gradient around the current weights $x$ and Entropy-SGD thus looks similar to averaged SGD in the literature (Polyak & Juditsky, 1992; Bottou, 2012). These two phenomena intuitively explain the improved generalization performance of Entropy-SGD.
+
+# 4.2 ALGORITHM AND IMPLEMENTATION DETAILS
+
+Alg. 1 provides the pseudo-code for one iteration of the Entropy-SGD algorithm. At each iteration, lines 3-6 perform $L$ iterations of Langevin dynamics to estimate $\pmb { \mu } = \langle \bar { x } ^ { \prime } ; \Xi ^ { \ell } \rangle$ . The weights $x$ are updated with the modified gradient on line 7.
+
+<table><tr><td colspan="2">Algorithm1: Entropy-SGD algorithm</td></tr><tr><td colspan="2">Input ：current weights x,Langevin iterations L</td></tr><tr><td colspan="2">Hyper-parameters:scope γ,learning rate n,SGLD step size n&#x27;</td></tr><tr><td colspan="2"></td></tr><tr><td colspan="2">// SGLD iterations;</td></tr><tr><td colspan="2">1x,μ←x;</td></tr><tr><td colspan="2">2 forl≤Ldo El← sample mini-batch;</td></tr><tr><td>3</td><td></td></tr><tr><td>4</td><td>dx&#x27;←m∑m=Vxf(x²;5e)-γ (x-x);</td></tr><tr><td>5 6</td><td>x&#x27;← x&#x27;-n&#x27;dx&#x27;+√n&#x27;εN(0,1); μ←(1-α)μ+αx&#x27;;</td></tr><tr><td colspan="2"></td></tr><tr><td colspan="2">// Update weights; 7 x←x-nx(x-μ)</td></tr></table>
+
+Let us now discuss a few implementation details. Although we have written Alg. 1 in the classical SGD setup, we can easily modify it to include techniques such as momentum and gradient preconditioning (Duchi et al., 2011) by changing lines 5 and 7. In our experiments, we have used SGD with Nesterov’s momentum (Sutskever et al., 2013) and Adam for outer and inner loops with similar qualitative results. We use exponential averaging to estimate $\mu$ in the SGLD loop (line 6)
+
+with $\alpha = 0 . 7 5$ so as to put more weight on the later samples, this is akin to a burn-in in standard SGLD.
+
+We set the number of SGLD iterations to $L = [ 5 , 2 0 ]$ depending upon the complexity of the dataset. The learning rate $\eta ^ { \prime }$ is fixed for all our experiments to values between $\eta ^ { \prime } \in \ [ 0 . 1 , 1 ]$ . We found that annealing $\eta ^ { \prime }$ (for instance, setting it to be the same as the outer learning rate $\eta$ ) is detrimental; indeed a small learning rate leads to poor estimates of local entropy towards the end of training where they are most needed. The parameter $\varepsilon$ in SGLD on line 5 is the thermal noise and we fix this to $\varepsilon \in [ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } ]$ . Having thus fixed $L , \varepsilon$ and $\eta ^ { \prime }$ , an effective heuristic to tune the remaining parameter $\gamma$ is to match the magnitude of the gradient of the local entropy term, viz. $\gamma \left( x - \mu \right)$ , to the gradient for vanilla SGD, viz. $m ^ { - 1 } \sum _ { i = 1 } ^ { m } \bar { \nabla _ { x } } f ( x ; \xi _ { \ell _ { i } } )$ .
+
+# 4.3 SCOPING OF γ
+
+The scope $\gamma$ is fixed in Alg. 1. For large values of $\gamma ,$ the SGLD updates happen in a small neighborhood of the current parameters $x$ while low values of $\gamma$ allow the inner SGLD to explore further away from $x$ . In the context of the discussion in Sec. 3, a “reverse-annealing” of the scope $\gamma ,$ i.e. increasing γ as training progresses has the effect of exploring the energy landscape on progressively finer scales. We call this process “scoping” which is similar to that of Baldassi et al. (2016a) and use a simple exponential schedule given by
+
+$$
+\gamma ( t ) = \gamma _ { 0 } \ ( 1 + \gamma _ { 1 } ) ^ { t } ;
+$$
+
+for the $t ^ { \mathrm { { t h } } }$ parameter update. We have experimented with linear, quadratic and bounded exponential $( \gamma _ { 0 } \left( 1 - e ^ { - \tau t } \right) )$ ) scoping schedules and obtained qualitatively similar results.
+
+Scoping of $\gamma$ unfortunately interferes with the learning rate annealing that is popular in deep learning, this is a direct consequence of the update step on line 7 of Alg. 1. In practice, we therefore scale down the local entropy gradient by $\gamma$ before the weight update and modify the line to read
+
+$$
+x  x - \eta ( x - \mu ) .
+$$
+
+Our general strategy during hyper-parameter tuning is to set the initial scope $\%$ to be very small, pick a large value of $\eta$ and set $\gamma _ { 1 }$ to be such that the magnitude of the local entropy gradient is comparable to that of SGD. We can use much larger learning rates than SGD in our experiments because the local entropy gradient is less noisy than the original back-propagated gradient. This also enables very fast progress in the beginning with a smooth landscape of a small $\gamma$ .
+
+# 4.4 THEORETICAL PROPERTIES
+
+We can show that Entropy-SGD results in a smoother loss function and obtains better generalization error than the original objective (5). With some overload of notation, we assume that the original loss $f ( x )$ is $\beta$ -smooth, i.e., for all $x , y \in \mathbb { R } ^ { n }$ , we have $\| \nabla f ( x ) - \nabla f ( y ) \| \leq \beta \ \| x - y \|$ . We additionally assume for the purpose of analysis that no eigenvalue of the Hessian $\nabla ^ { 2 } f ( x )$ lies in the set $[ - 2 \gamma - c , c ]$ for some small $c > 0$ .
+
+Lemma 2. The objective $F ( x , \gamma ; \Xi )$ in (6) is $\frac { \alpha } { 1 + \gamma ^ { - 1 } \ c }$ -Lipschitz and $\frac { \beta } { 1 + \gamma ^ { - 1 } ~ c }$ -smooth.
+
+Please see Appendix B for the proof. The local entropy objective is thus smoother than the original objective. Let us now obtain a bound on the improvement in generalization error. We denote an optimization algorithm, viz., SGD or Entropy-SGD by $A ( \Xi )$ , it is a function of the dataset $\Xi$ and outputs the parameters $x ^ { * }$ upon termination. Stability of the algorithm (Bousquet & Elisseeff, 2002) is then a notion of how much its output differs in loss upon being presented with two datasets, Ξ and $\Xi ^ { \prime }$ , that differ in at most one sample:
+
+$$
+\operatorname* { s u p } _ { \xi \in \Xi \cup \Xi ^ { \prime } } \left[ f ( A ( \Xi ) , \xi ) - f \left( A ( \Xi ^ { \prime } ) , \xi \right) \right] \leq \varepsilon .
+$$
+
+Hardt et al. (2015) connect uniform stability to generalization error and show that an $\varepsilon$ -stable algorithm $A ( \Xi )$ has generalization error bounded by $\varepsilon$ , i.e., if $A ( \Xi )$ terminates with parameters $x ^ { * }$ ,
+
+$$
+| \mathbb { E } _ { \Xi } \left( R _ { \Xi } ( x ^ { * } ) - R ( x ^ { * } ) \right) | \le \varepsilon ;
+$$
+
+where the left hand side is the generalization error: it is the difference between the empirical loss $\begin{array} { r } { R _ { \Xi } ( x ) : = N ^ { - 1 } \sum _ { k = 1 } ^ { N } \ f ( x , \xi _ { k } ) } \end{array}$ and the population loss $R ( x ) : = \mathbb { E } _ { \xi } ~ f ( x , \xi )$ . We now employ the following theorem that bounds the stability of an optimization algorithm through the smoothness of its loss function and the number of iterations on the training set.
+
+Theorem 3 (Hardt et al. (2015)). For an $\alpha$ -Lipschitz and $\beta$ -smooth loss function, if SGD converges in $T$ iterations on $N$ samples with decreasing learning rate $\eta _ { t } \leq 1 / t$ the stability is bounded by
+
+$$
+\varepsilon \stackrel { < } { \approx } \frac { 1 } { N } \alpha ^ { 1 / ( 1 + \beta ) } T ^ { 1 - 1 / ( 1 + \beta ) } .
+$$
+
+Using Lemma 2 and Theorem 3 we have
+
+$$
+\varepsilon _ { \mathrm { E n t r o p y - S G D } } \lesssim \left( \alpha T ^ { - 1 } \right) ^ { \left( 1 - \frac { 1 } { 1 + \gamma ^ { - 1 } c } \right) \beta } \varepsilon _ { \mathrm { S G D } } ,
+$$
+
+which shows that Entropy-SGD generalizes better than SGD for all $T > \alpha$ if they both converge after $T$ passes over the samples.
+
+Let us note that while the number of passes over the dataset for Entropy-SGD and SGD are similar for our experiments on CNNs, Entropy-SGD makes only half as many passes as SGD for our experiments on RNNs. As an aside, it is easy to see from the proof of Lemma 2 that for a convex loss function $f ( x )$ , the local entropy objective does not change the minimizer of the original problem.
+
+Remark 4. The above analysis hinges upon an assumption that the Hessian $\nabla ^ { 2 } f ( x )$ does not have eigenvalues in the set $[ - 2 \gamma - c , c ]$ for a constant $c > 0$ . This is admittedly unrealistic, for instance, the eigenspectrum of the Hessian at a local minimum in Fig. 1 has a large fraction of its eigenvalues almost zero. Let us however remark that the result in Thm. 3 by Hardt et al. (2015) assumes global conditions on the smoothness of the loss function; one imagines that Eqn. 9 remains qualitatively the same (with respect to $T$ in particular) even if this assumption is violated to an extent before convergence happens. Obtaining a rigorous generalization bound without this assumption would require a dynamical analysis of SGD and seems out of reach currently.
+
+# 5 EXPERIMENTS
+
+In Sec. 5.1, we discuss experiments that suggest that the characteristics of the energy landscape around local minimal accessible by SGD are universal to deep architectures. We then present experimental results on two standard image classification datasets, viz. MNIST and CIFAR-10 and two datasets for text prediction, viz. PTB and the text of War and Peace. Table 1 summarizes the results of these experiments on deep networks.
+
+# 5.1 UNIVERSALITY OF THE HESSIAN AT LOCAL MINIMA
+
+We use automatic differentiation1 to compute the Hessian at a local minimum obtained at the end of training for the following networks:
+
+(i) small-LeNet on MNIST: This network has 47, 658 parameters and is similar to LeNet but with 10 and 20 channels respectively in the first two convolutional layers and 128 hidden units in the fully-connected layer. We train this with Adam to obtain a test error of $2 . 4 \%$ .   
+(ii) small-mnistfc on MNIST: A fully-connected network (50, 890 parameters) with one layer of 32 hidden units, ReLU non-linearities and cross-entropy loss; it converges to a test error of $2 . 5 \%$ with momentum-based SGD.   
+(iii) char-lstm for text generation: This is a recurrent network with 48 hidden units and Long Short-Term Memory (LSTM) architecture (Hochreiter & Schmidhuber, 1997a). It has 32, 640 parameters and we train it with Adam to re-generate a small piece of text consisting of 256 lines of length 32 each and 96-bit one-hot encoded characters.
+
+(iv) All-CNN-BN on CIFAR-10: This is similar to the All-CNN-C network (Springenberg et al., 2014) with $\approx 1 . 6$ million weights (cf. Sec. 5.3) which we train using Adam to obtain an error of $1 1 . 2 \%$ . Exact Hessian computation is in this case expensive and thus we instead compute the diagonal of the Fisher information matrix (Wasserman, 2013) using the element-wise first and second moments of the gradients that Adam maintains, i.e., $\mathrm { d i a g } ( I ) = \mathbb { E } ( g ^ { 2 } ) - ( \mathbb { E } ~ g ) ^ { 2 }$ where $g$ is the back-propagated gradient. Fisher information measures the sensitivity of the log-likelihood of data given parameters in a neighborhood of a local minimum and thus is exactly equal to the Hessian of the negative log-likelihood. We will consider the diagonal of the empirical Fisher information matrix as a proxy for the eigenvalues of the Hessian, as is common in the literature.
+
+We choose to compute the exact Hessian and to keep the computational and memory requirements manageable, the first three networks considered above are smaller than standard deep networks used in practice. For the last network, we sacrifice the exact computation and instead approximate the Hessian of a large deep network. We note that recovering an approximate Hessian from Hessianvector products (Pearlmutter, 1994) could be a viable strategy for large networks.
+
+Fig. 1 in the introductory Sec. 1 shows the eigenspectrum of the Hessian for small-LeNet while Fig. 3 shows the eigenspectra for the other three networks. A large proportion of eigenvalues of the Hessian are very close to zero or positive with a very small (relative) magnitude. This suggests that the local geometry of the energy landscape is almost flat at local minima discovered by gradient descent. This agrees with theoretical results such as Baldassi et al. (2016c) where the authors predict that flat regions of the landscape generalize better. Standard regularizers in deep learning such as convolutions, max-pooling and dropout seem to bias SGD towards flatter regions in the energy landscape. Away from the origin, the right tails of the eigenspectra are much longer than the left tails. Indeed, as discussed in numerous places in literature (Bray & Dean, 2007; Dauphin et al., 2014; Choromanska et al., 2015a), SGD finds low-index critical points, i.e., optimizers with few negative eigenvalues of the Hessian. What is interesting and novel is that the directions of descent that SGD misses do not have a large curvature.
+
+# 5.2 MNIST
+
+We consider two prototypical networks: the first, called mnistfc, is a fully-connected network with two hidden layers of 1024 units each and the second is a convolutional neural network with the same size as LeNet but with batch-normalization (Ioffe & Szegedy, 2015); both use a dropout of probability 0.5 after each layer. As a baseline, we train for 100 epochs with Adam and a learning rate of $1 0 ^ { - 3 }$ that drops by a factor of 5 after every 30 epochs to obtain an average error of $1 . 3 9 \pm 0 . 0 3 \%$ and $0 . 5 1 \pm 0 . 0 1 \%$ for mnistfc and LeNet respectively, over 5 independent runs.
+
+For both these networks, we train Entropy-SGD for 5 epochs with $L = 2 0$ and reduce the dropout probability (0.15 for mnistfc and 0.25 for LeNet). The learning rate of the SGLD updates is fixed to $\eta ^ { \prime } = 0 . 1$ while the outer loop’s learning rate is set to $\eta = 1$ and drops by a factor of 10 after the second epoch; we use Nesterov’s momentum for both loops. The thermal noise in SGLD updates (line 5 of Alg. 1) is set to $1 0 ^ { - 3 }$ . We use an exponentially increasing value of $\gamma$ for scoping, the initial value of the scope is set to $\gamma = 1 0 ^ { - 4 }$ and this increases by a factor of 1.001 after each parameter update. The results in Fig. 4a and Fig. 4b show that Entropy-SGD obtains a comparable generalization error: $1 . 3 7 \pm 0 . 0 3 \%$ and $0 . 5 0 \pm 0 . 0 1 \%$ , for mnistfc and LeNet respectively. While Entropy-SGD trains slightly faster in wall-clock time for LeNet; it is marginally slower for mnistfc which is a small network and trains in about two minutes anyway.
+
+Remark on the computational complexity: Since Entropy-SGD runs $L$ steps of SGLD before each parameter update, the effective number of passes over the dataset is $L$ times that of SGD or Adam for the same number of parameter updates. We therefore plot the error curves of Entropy-SGD in Figs. 4, 5, and 6 against the “effective number of epochs”, i.e. by multiplying the abscissae by a factor of $L$ . (we define $L = 1$ for SGD or Adam). Modulo the time required for the actual parameter updates (which are fewer for Entropy-SGD), this is a direct measure of wall-clock time, agnostic to the underlying hardware and implementation.
+
+char-lstm: Hessian eigenspectrum at local minimum (a) small-mnistfc (2 runs): Peak (clipped here) at zero $( | \lambda | \leq 1 0 ^ { - 2 } )$ accounts for $90 \%$ of the entries.
+
+![](images/336104945b2ccebddd4ac5db7ee5eb2d35a1c83e82765d6c182dc5a24f7bd6af.jpg)  
+Eigenvalues
+
+![](images/3ca7f5dcd4ecab6ad8e76a11626cd2899437ceee8e9af938df6da67976093a32.jpg)  
+Eigenvalues
+
+(b) char-LSTM (5 runs): Almost $9 5 \%$ eigenvalues have absolute value below $1 0 ^ { - 5 }$ .
+
+![](images/a7f5a2dee37afd08233cfc30b8033d88abfc5efc20b2568c5fd8a33d1e6cbb44.jpg)  
+(c) Negative and positive eigenvalues of the Fisher information matrix of All-CNN-BN at a local minimum (4 independent runs). The origin has a large peak with $\approx 9 5 \%$ near-zero $( | \lambda | \leq 1 0 ^ { - 5 } )$ eigenvalues (clipped here).
+
+Figure 3: Universality of the Hessian: for a wide variety of network architectures, sizes and datasets, optima obtained by SGD are mostly flat (large peak near zero), they always have a few directions with large positive curvature (long positive tails). A very small fraction of directions have negative curvature, and the magnitude of this curvature is extremely small (short negative tails).
+
+![](images/c5d0293acd789944f02d935e05d3ea8cc35794d3f246deaf7b1f263d3d3d8a2b.jpg)  
+Figure 4: Comparison of Entropy-SGD vs. Adam on MNIST
+
+# 5.3 CIFAR-10
+
+We train on CIFAR-10 without data augmentation after performing global contrast normalization and ZCA whitening (Goodfellow et al., 2013). We consider the All-CNN-C network of Springenberg et al. (2014) with the only difference that a batch normalization layer is added after each convolutional layer; all other architecture and hyper-parameters are kept the same. We train for 200 epochs with SGD and Nesterov’s momentum during which the initial learning rate of 0.1 decreases by a factor of 5 after every 60 epochs. We obtain an average error of $7 . 7 1 \pm 0 . 1 9 \%$ in 200 epochs vs. $9 . 0 8 \%$ error in 350 epochs that the authors in Springenberg et al. (2014) report and this is thus a very competitive baseline for this network. Let us note that the best result in the literature on non-augmented CIFAR-10 is the ELU-network by Clevert et al. (2015) with $6 . 5 5 \%$ test error.
+
+We train Entropy-SGD with $L = 2 0$ for 10 epochs with the original dropout of 0.5. The initial learning rate of the outer loop is set to $\eta = 1$ and drops by a factor of 5 every 4 epochs, while the learning rate of the SGLD updates is fixed to $\eta ^ { \prime } = 0 . 1$ with thermal noise $\varepsilon = 1 0 ^ { - 4 }$ . As the scoping scheme, we set the initial value of the scope to $\gamma _ { 0 } = 0 . 0 3$ which increases by a factor of 1.001 after each parameter update. Fig. 5 shows the training and validation error curves for Entropy-SGD compared with SGD. It shows that local entropy performs as well as SGD on a large CNN; we obtain a validation error of $7 . 8 1 \pm 0 . 0 9 \%$ in about 160 effective epochs.
+
+We see almost no plateauing of training loss or validation error for Entropy-SGD in Fig. 5a; this trait is shared across different networks and datasets in our experiments and is an indicator of the additional smoothness of the local entropy landscape coupled with a good scoping schedule for $\gamma .$
+
+![](images/ac3566167ec08871865ea227af1e776bc23d239d4cd1ed5fba2ec2eb18151904.jpg)  
+Figure 5: Comparison of Entropy-SGD vs. SGD on CIFAR-10
+
+# 5.4 RECURRENT NEURAL NETWORKS
+
+# 5.4.1 PENN TREE BANK
+
+We train an LSTM network on the Penn Tree Bank (PTB) dataset for word-level text prediction. This dataset contains about one million words divided into a training set of about 930, 000 words, a validation set of 74, 000 words and 82, 000 words with a vocabulary of size 10, 000. Our network called PTB-LSTM consists of two layers with 1500 hidden units, each unrolled for 35 time steps; note that this is a large network with about 66 million weights. We recreated the training pipeline of Zaremba et al. (2014) for this network (SGD without momentum) and obtained a word perplexity of $8 1 . 4 3 \pm 0 . 2$ on the validation set and $7 8 . 6 \pm 0 . 2 6$ on the test set with this setup; these numbers closely match the results of the original authors.
+
+We run Entropy-SGD on PTB-LSTM for 5 epochs with $L = 5$ , note that this results in only 25 effective epochs. We do not use scoping for this network and instead fix $\gamma = 1 0 ^ { - 3 }$ . The initial learning rate of the outer loop is $\eta = 1$ which reduces by a factor of 10 at each epoch after the third epoch. The SGLD learning rate is fixed to $\eta ^ { \prime } = 1$ with $\varepsilon = 1 0 ^ { - 4 }$ . We obtain a word perplexity of $8 0 . 1 1 6 \pm 0 . 0 6 9$ on the validation set and $7 7 . 6 5 6 \pm 0 . 1 7 1$ on the test set. As Fig. 6a shows, Entropy-SGD trains significantly faster than SGD (25 effective epochs vs. 55 epochs of SGD) and also achieves a slightly better generalization perplexity.
+
+# 5.4.2 CHAR-LSTM ON WAR AND PEACE
+
+Next, we train an LSTM to perform character-level text-prediction. As a dataset, following the experiments of Karpathy et al. (2015), we use the text of War and Peace by Leo Tolstoy which contains about 3 million characters divided into training $( 8 0 \% )$ , validation $( 1 0 \% )$ and test $( 1 0 \% )$ sets. We use an LSTM consisting of two layers of 128 hidden units unrolled for 50 time steps and a vocabulary of size 87. We train the baseline with Adam for 50 epochs with an initial learning rate of 0.002 that decays by a factor of 2 after every 5 epochs to obtain a validation perplexity of $1 . 2 2 4 \pm 0 . 0 0 8$ and a test perplexity of $1 . 2 2 6 \pm 0 . 0 1$ .
+
+As noted in Sec. 4.2, we can easily wrap Alg. 1 inside other variants of SGD such as Adam; this involves simply substituting the local entropy gradient in place of the usual back-propagated gradient. For this experiment, we constructed Entropy-Adam which is equivalent to Adam with the local entropy gradient (which is computed via SGLD). We run Entropy-Adam for 5 epochs with $L = 5$ and a fixed $\gamma { = } 0 . 0 1$ with an initial learning rate of 0.01 that decreases by a factor of 2 at each epoch. Note that this again results in only 25 effective epochs, i.e. half as much wall-clock time as SGD or Adam. We obtain a validation perplexity of $1 . 2 1 3 \pm 0 . 0 0 7$ and a test perplexity of $1 . 2 1 7 { \pm } 0 . 0 0 5$ over 4 independent runs which is better than the baseline. Fig. 6b shows the error curves for this experiment.
+
+![](images/efaf369d187c7cde4890cecd6930a9f730ce4c1554e9e750a51f2550ef7f596c.jpg)  
+Figure 6: Comparison of Entropy-SGD vs. SGD / Adam on RNNs
+
+Table 1: Experimental results: Entropy-SGD vs. SGD / Adam   
+
+<table><tr><td> Model</td><td>Entropy-SGD</td><td colspan="3"> SGD / Adam</td></tr><tr><td></td><td>Error(%)/PerplexityEpochs|Error(%)/PerplexityEpochs</td><td></td><td></td><td></td></tr><tr><td> mnistfc</td><td>1.37 ±0.03</td><td>120</td><td>1.39 ± 0.03</td><td>66</td></tr><tr><td>LeNet</td><td>0.5±0.01</td><td>80</td><td>0.51±0.01</td><td>100</td></tr><tr><td> All-CNN-BN</td><td>7.81±0.09</td><td>160</td><td>7.71±0.19</td><td>180</td></tr><tr><td>PTB-LSTM</td><td>77.656±0.171</td><td>25</td><td>78.6±0.26</td><td>55</td></tr><tr><td> char-LSTM</td><td>1.217 ± 0.005</td><td>25</td><td>1.226 ± 0.01</td><td>40</td></tr></table>
+
+Tuning the momentum in Entropy-SGD was crucial to getting good results on RNNs. While the SGD baseline on PTB-LSTM does not use momentum (and in fact, does not train well with momentum) we used a momentum of 0.5 for Entropy-SGD. On the other hand, the baseline for char-LSTM was trained with Adam with $\beta _ { 1 } = 0 . 9$ $\lvert \beta _ { 1 }$ in Adam controls the moving average of the gradient) while we set $\beta _ { 1 } = 0 . 5$ for Entropy-Adam. In contrast to this observation about RNNs, all our experiments on CNNs used a momentum of 0.9. In order to investigate this difficulty, we monitored the angle between the local entropy gradient and the vanilla SGD gradient during training. This angle changes much more rapidly for RNNs than for CNNs which suggests a more rugged energy landscape for the former; local entropy gradient is highly uncorrelated with the SGD gradient in this case.
+
+# 6 DISCUSSION
+
+In our experiments, Entropy-SGD results in generalization error comparable to SGD, but always with lower cross-entropy loss on the training set. This suggests the following in the context of energy landscapes of deep networks. Roughly, wide valleys favored by Entropy-SGD are located deeper in the landscape with a lower empirical loss than local minima discovered by SGD where it presumably gets stuck. Such an interpretation is in contrast to theoretical models of deep networks (cf. Sec. 2) which predict multiple equivalent local minima with the same loss. Our work suggests that geometric properties of the energy landscape are crucial to generalize well and provides algorithmic approaches to exploit them. However, the literature lacks general results about the geometry of the loss functions of deep networks — convolutional neural networks in particular — and this is a promising direction for future work.
+
+If we focus on the inner loop of the algorithm, SGLD updates compute the average gradient (with Langevin noise) in a neighborhood of the parameters while maintaining the Polyak average of the new parameters. Such an interpretation is very close to averaged SGD of Polyak & Juditsky (1992) and Bottou (2012) and worth further study. Our experiments show that while Entropy-SGD trains significantly faster than SGD for recurrent networks, it gets relatively minor gains in terms of wallclock time for CNNs. Estimating the gradient of local entropy cheaply with few SGLD iterations, or by using a smaller network to estimate it in a teacher-student framework (Balan et al., 2015) is another avenue for extensions to this work.
+
+# 7 CONCLUSIONS
+
+We introduced an algorithm named Entropy-SGD for optimization of deep networks. This was motivated from the observation that the energy landscape near a local minimum discovered by SGD is almost flat for a wide variety of deep networks irrespective of their architecture, input data or training methods. We connected this observation to the concept of local entropy which we used to bias the optimization towards flat regions that have low generalization error. Our experiments showed that Entropy-SGD is applicable to large convolutional and recurrent deep networks used in practice.
+
+# 8 ACKNOWLEDGMENTS
+
+This work was supported by ONR N00014-13-1-034, AFOSR F9550-15-1-0229 and ARO W911NF-15-1-0564/66731-CS.
+
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+
+# A STOCHASTIC GRADIENT LANGEVIN DYNAMICS (SGLD)
+
+Local entropy in Def. (1) is an expectation over the entire configuration space $x \in \mathbb { R } ^ { n }$ and is hard to compute; we can however approximate its gradient using Markov chain Monte-Carlo (MCMC) techniques. In this section, we briefly review stochastic gradient Langevin dynamics (Welling & Teh, 2011) that is an MCMC algorithm designed to draw samples from a Bayesian posterior and scales to large datasets using mini-batch updates.
+
+For a parameter vector $x \in \mathbb { R } ^ { n }$ with a prior distribution $p ( x )$ and if the probability of generating a data item $\xi _ { k }$ given a model parameterized by $x$ is $p ( \xi _ { k } | x )$ , the posterior distribution of the parameters based on $N$ data items can be written as
+
+$$
+p \left( x | \xi _ { k \leq N } \right) \propto p ( x ) \prod _ { k = 1 } ^ { N } p \left( \xi _ { k } | x \right) .
+$$
+
+Langevin dynamics (Neal, 2011) injects Gaussian noise into maximum-a-posteriori (MAP) updates to prevent over-fitting the solution $x ^ { * }$ of the above equation. The updates can be written as
+
+$$
+\Delta x _ { t } = \frac { \eta } { 2 } \ \left( \nabla \log p ( x _ { t } ) + \sum _ { k = 1 } ^ { N } \nabla p ( \xi _ { k } | x _ { t } ) \right) + \sqrt { \eta } \ \varepsilon _ { t } ;
+$$
+
+where $\mathbf { \mathcal { E } } _ { t } \sim \mathrm { N } ( 0 , \varepsilon ^ { 2 } )$ is Gaussian noise and $\eta$ is the learning rate. In this form, Langevin dynamics faces two major hurdles for applications to large datasets. First, computing the gradient $\begin{array} { r l } { \sum _ { k = 1 } ^ { N } } & { { } \nabla p ( \xi _ { k } | x _ { t } ) } \end{array}$ over all samples for each update $\Delta x _ { t }$ becomes prohibitive. However, as Welling & Teh (2011) show, one can instead simply use the average gradient over $m$ data samples (mini-batch) as follows:
+
+$$
+\Delta x _ { t } = \frac { \eta _ { t } } { 2 } \left( \nabla \log p ( x _ { t } ) + \frac { N } { m } \sum _ { k = 1 } ^ { m } \nabla p ( \xi _ { k } | x _ { t } ) \right) + \sqrt { \eta _ { t } } \ \varepsilon _ { t } .
+$$
+
+Secondly, Langevin dynamics in (11) is the discrete-time approximation of a continuous-time stochastic differential equation (Mandt et al., 2016) thereby necessitating a Metropolis-Hastings (MH) rejection step (Roberts & Stramer, 2002) which again requires computing $p ( \xi _ { k } | x )$ over the entire dataset. However, if the learning rate $\eta _ { t } \to 0$ , we can also forgo the MH step (Chen et al., 2014). Welling & Teh (2011) also argue that the sequence of samples $x _ { t }$ generated by updating (12) converges to the correct posterior (10) and one can hence compute the statistics of any function $g ( x )$ of the parameters using these samples. Concretely, the posterior expectation $\mathbb { E } \left[ g ( x ) \right]$ is given by E [g(x)] ≈ ∑ts=1 ηt g(xt )t ; which is the average computed by weighing each sample by the corresponding learning rate in (12). In this paper, we will consider a uniform prior on the parameters $x$ and hence the first term in (12), viz., $\nabla { \log } p ( x _ { t } )$ vanishes.
+
+Let us note that there is a variety of increasingly sophisticated MCMC algorithms applicable to our problem, e.g., Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) by Chen et al. (2014) based on volume preserving flows in the “parameter-momentum” space, stochastic annealing thermostats (Santa) by Chen et al. (2015) etc. We can also employ these techniques, although we use SGLD for ease of implementation; the authors in Ma et al. (2015) provide an elaborate overview.
+
+# B PROOFS
+
+Proof of Lemma 2. The gradient $- \boldsymbol { \nabla } F ( \boldsymbol { x } )$ is computed in Sec. 4.1 to be γ $\left( x - \left. { x ^ { \prime } ; \Xi ^ { \ell } } \right. \right)$ . Consider the term
+
+$$
+\begin{array} { l } { { \displaystyle - \left. x ^ { \prime } ; x \right. = x - Z _ { x , \gamma } ^ { - 1 } \displaystyle \int _ { x } x ^ { \prime } e ^ { - f \left( x ^ { \prime } \right) - \frac { \gamma } { 2 } } \left. x - x ^ { \prime } \right. ^ { 2 } d x ^ { \prime } } } \\ { { \displaystyle \approx x - Z _ { x , \gamma } ^ { - 1 } \displaystyle \int _ { s } \left( x + s \right) e ^ { - f \left( x \right) - \nabla f \left( x \right) ^ { \top } s - \frac { 1 } { 2 } s ^ { \top } \left( \gamma + \nabla ^ { 2 } f \left( x \right) \right) s } d s } } \\ { { \displaystyle = x \left( 1 - Z _ { x , \gamma } ^ { - 1 } \displaystyle \int _ { s } e ^ { - f \left( x \right) - \nabla f \left( x \right) ^ { \top } s - \frac { 1 } { 2 } s ^ { \top } \left( \gamma + \nabla ^ { 2 } f \left( x \right) \right) s } d s \right) - Z _ { x , \gamma } ^ { - 1 } \displaystyle \int _ { s } s e ^ { - f \left( x \right) - \nabla f \left( x \right) ^ { \top } s - \frac { 1 } { 2 } s ^ { \top } \left( \gamma + \nabla ^ { 2 } f \left( x \right) \right) s } d s } } \\ { { \displaystyle = - Z _ { x , \gamma } ^ { - 1 } e ^ { - f \left( x \right) } \displaystyle \int _ { s } s e ^ { - \nabla f \left( x \right) ^ { \top } s - \frac { 1 } { 2 } s ^ { \top } \left( \gamma + \nabla ^ { 2 } f \left( x \right) \right) s } d s } . } \end{array}
+$$
+
+The above expression is the mean of a distribution $\propto e ^ { - \nabla f ( x ) ^ { \top } s - \frac { 1 } { 2 } ~ s ^ { \top } \left( \gamma + \nabla ^ { 2 } f ( x ) \right) s }$ . We can approximate it using the saddle point method as the value of $s$ that minimizes the exponent to get
+
+$$
+x - \left. x ^ { \prime } ; x \right. \approx \left( \nabla ^ { 2 } f ( x ) + \gamma I \right) ^ { - 1 } \nabla f ( x ) .
+$$
+
+Let us denote $A ( x ) : = \left( I + \gamma ^ { - 1 } \nabla ^ { 2 } f ( x ) \right) ^ { - 1 }$ . Plugging this into the condition for smoothness, we have
+
+$$
+\begin{array} { r l } & { \| \nabla F ( x , \gamma ) - \nabla F ( y , \gamma ) \| = \| A ( x ) \nabla f ( x ) - A ( y ) \nabla f ( y ) \| } \\ & { \qquad \le \left( \underset { x } { \operatorname* { s u p } } \| A ( x ) \| \right) \beta \| x - y \| . } \end{array}
+$$
+
+Unfortunately, we can only get a uniform bound if we assume that for a small constant $c > 0$ , no eigenvalue of $\nabla ^ { 2 } f ( x )$ lies in the set $[ - 2 \gamma - c , c ]$ . This gives
+
+$$
+\left( \operatorname* { s u p } _ { x } \left\| A ( x ) \right\| \right) \leq \frac { 1 } { 1 + \gamma ^ { - 1 } c } .
+$$
+
+This shows that a smaller value of $\gamma$ results in a smoother energy landscape, except at places with very flat directions. The Lipschitz constant also decreases by the same factor.
+
+# C CONNECTION TO VARIATIONAL INFERENCE
+
+The fundamental motivations of (stochastic) variational inference (SVI) and local entropy are similar: they both aim to generalize well by constructing a distribution on the weight space. In this section, we explore whether they are related and how one might reconcile the theoretical and algorithmic implications of the local entropy objective with that of SVI.
+
+Let $\Xi$ denote the entire dataset, $z$ denote the weights of a deep neural network and $x$ be the parameters of a variational distribution $q _ { x } ( z )$ . The Evidence Lower Bound (ELBO) can be then be written as
+
+$$
+\log p ( \Xi ) \geq \mathbb { E } _ { z \sim q _ { x } ( z ) } \ [ \log p ( \Xi \mid z ) ] - \mathbf { K L } \left( q _ { x } ( z ) \mid \mid p ( z ) \right) ;
+$$
+
+where $p ( z )$ denotes a parameter-free prior on the weights and controls, through their KL-divergence, how well the posited posterior $q _ { x } ( z )$ fits the data. Stochastic variational inference involves maximizing the right hand side of the above equation with respect to $x$ after choosing a suitable prior $p ( z )$ and a family of distributions $q _ { x } ( z )$ . These choices are typically dictated by the ease of sampling $z \sim q _ { x } ( z )$ , e.g. a mean-field model where $q _ { x } ( z )$ factorizes over $z$ , and being able to compute the KL-divergence term, e.g. a mixture of Gaussians.
+
+On the other hand, if we define the loss as the log-likelihood of data, viz. $f ( z ) : = - \log p ( \Xi | z )$ , we can write the logarithm of the local entropy in Eqn. (4) as
+
+$$
+\begin{array} { l } { \displaystyle \log F ( \boldsymbol { x } , \boldsymbol { \gamma } ) = \log \int _ { z \in \mathbb { R } ^ { n } } \exp \left[ - f ( z ; \Xi ) - \frac { \gamma } { 2 } \left. \boldsymbol { x } - \boldsymbol { z } \right. ^ { 2 } \right] d z , } \\ { \displaystyle \geq \int _ { z \in \mathbb { R } ^ { n } } \left[ \log p ( \Xi \mid z ) - \frac { \gamma } { 2 } \left. \boldsymbol { x } - \boldsymbol { z } \right. ^ { 2 } \right] d z ; } \end{array}
+$$
+
+by an application of Jensen’s inequality. It is thus clear that Eqn. (13) and (14) are very different in general and one cannot choose a prior, or a variational family, that makes them equivalent and interpret local entropy as ELBO.
+
+Eschewing rigor, formally, if we modify Eqn. (13) to allow the prior $p ( z )$ to depend upon $x$ , we can see that the two lower bounds above are equivalent iff $q _ { x } ( z )$ belongs to a “flat variational family”, i.e. uniform distributions with $x$ as the mean and $\begin{array} { r } { p _ { x } ( z ) \propto \exp \left( - \frac { \gamma } { 2 } \parallel x - z \parallel ^ { 2 } \right) } \end{array}$ . We emphasize that the distribution $p _ { x } ( z )$ depends on the parameters $x$ themselves and is thus, not really a prior, or one that can be derived using the ELBO.
+
+This “moving prior” is absent in variational inference and indeed, a crucial feature of the local entropy objective. The gradient of local entropy in Eqn. (7) clarifies this point:
+
+$$
+\nabla F ( x , \gamma ) = - \gamma ( x - \langle z ; \Xi \rangle ) = - \gamma \mathbb { E } _ { z } \sim _ { r ( z ; x ) } \ [ z ] ;
+$$
+
+where the distribution $r ( z ; x )$ is given by
+
+$$
+r ( z ; x ) \propto \ p ( \Xi \mid z ) \ \exp \left( - \frac \gamma 2 \ \| x - z \| ^ { 2 } \right) ;
+$$
+
+it thus contains a data likelihood term along with a prior that “moves” along with the current iterate $x$ .
+
+Let us remark that methods in the deep learning literature that average the gradient through perturbations in the neighborhood of $x$ (Mobahi, 2016) or noisy activation functions (Gulcehre et al., 2016) can be interpreted as computing the data likelihood in ELBO (without the KL-term); such an averaging is thus different from local entropy.
+
+# C.1 COMPARISON WITH SGLD
+
+We use stochastic gradient Langevin dynamics (cf. Appendix A) to estimate the gradient of local entropy in Alg. 1. It is natural then, to ask the question whether vanilla SGLD performs as well as local entropy. To this end, we compare the performance of SGLD on two prototypical networks: LeNet on MNIST and All-CNN-BN on CIFAR-10. We follow the experiments in Welling & Teh (2011) and Chen et al. (2015) and set the learning rate schedule to be $\overline { { \eta / ( 1 + t ) } } ^ { b }$ where the initial learning rate $\eta$ and $b$ are hyper-parameters. We make sure that other architectural aspects (dropout, batch-normalization) and regularization (weight decay) are consistent with the experiments in Sec. 5.
+
+After a hyper-parameter search, we obtained a test error on LeNet of $0 . 6 3 \pm 0 . 1 \%$ after 300 epochs and $9 . 8 9 \pm 0 . 1 1 \%$ on All-CNN-BN after 500 epochs. Even if one were to disregard the slow convergence of SGLD, its generalization error is much worse than our experimental results; we get $0 . 5 0 \pm 0 . 0 1 \%$ on LeNet and $7 . 8 1 \pm 0 . 0 9 \%$ on All-CNN-BN with Entropy-SGD. For comparison, the authors in Chen et al. (2015) report $0 . 7 1 \%$ error on MNIST on a slightly larger network. Our results with local entropy on RNNs are much better than those reported in Gan et al. (2016) for SGLD. On the PTB dataset, we obtain a test perplexity of $7 7 . 6 5 6 \pm 0 . 1 7 1$ vs. 94.03 for the same model whereas we obtain a test perplexity of $1 . 2 1 3 \pm 0 . 0 0 7$ vs. 1.3375 for char-LSTM on the War and Peace dataset.
+
+Training deep networks with SGLD, or other more sophisticated MCMC algorithms such as SGHMC, SGNHT etc. (Chen et al., 2014; Neal, 2011) to errors similar to those of SGD is difficult, and the lack of such results in the literature corroborates our experimental experience. Roughly speaking, local entropy is so effective because it operates on a transformation of the energy landscape that exploits entropic effects. Conventional MCMC techniques such as SGLD or Nose’-Hoover thermostats (Ding et al., 2014) can only trade energy for entropy via the temperature parameter which does not allow the direct use of the geometric information of the energy landscape and does not help with narrow minima.

parse/train/B1hcZZ-AW/B1hcZZ-AW.md

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+# N2N LEARNING: NETWORK TO NETWORK COMPRESSION VIA POLICY GRADIENT REINFORCEMENT LEARNING
+
+Anubhav Ashok Robotics Institute Carnegie Mellon University bhav@cmu.edu
+
+Nicholas Rhinehart Robotics Institute Carnegie Mellon University nrhineha@cs.cmu.edu
+
+Fares Beainy   
+Volvo Construction Equipment Volvo Group   
+fares.beainy@volvo.com
+
+Kris M. Kitani Robotics Institute Carnegie Mellon University kkitani@cs.cmu.edu
+
+# ABSTRACT
+
+While wider and deeper neural network architectures continue to advance the state-of-the-art for many computer vision tasks, real-world adoption of these networks is impeded by hardware and speed constraints. Conventional model compression methods attempt to address this problem by modifying the architecture manually or using pre-defined heuristics. Since the space of all reduced architectures is very large, modifying the architecture of a deep neural network in this way is a difficult task. In this paper, we tackle this issue by introducing a principled method for learning reduced network architectures in a data-driven way using reinforcement learning. Our approach takes a larger ‘teacher’ network as input and outputs a compressed ‘student’ network derived from the ‘teacher’ network. In the first stage of our method, a recurrent policy network aggressively removes layers from the large ‘teacher’ model. In the second stage, another recurrent policy network carefully reduces the size of each remaining layer. The resulting network is then evaluated to obtain a reward – a score based on the accuracy and compression of the network. Our approach uses this reward signal with policy gradients to train the policies to find a locally optimal student network. Our experiments show that we can achieve compression rates of more than $1 0 \times$ for models such as ResNet34 while maintaining similar performance to the input ‘teacher’ network. We also present a valuable transfer learning result which shows that policies which are pre-trained on smaller ‘teacher’ networks can be used to rapidly speed up training on larger ‘teacher’ networks.
+
+# 1 INTRODUCTION
+
+While carefully hand-designed deep convolutional networks continue to increase in size and in performance, they also require significant power, memory and computational resources, often to the point of prohibiting their deployment on smaller devices. As a result, researchers have developed model compression techniques based on Knowledge Distillation to compress a large (teacher) network to a smaller (student) network using various training techniques (e.g., soft output matching, hint layer matching, uncertainty modeling). Unfortunately, state-of-the-art knowledge distillation methods share a common feature: they require carefully hand-designed architectures for the student model. Hand-designing networks is a tedious sequential process, often loosely guided by a sequence of trial-and-error based decisions to identify a smaller network architecture. This process makes it very difficult to know if the resulting network is optimal. Clearly, there is a need to develop more principled methods of identifying optimal student architectures.
+
+![](images/e47dd5c04d343b25d0b7ab3408802f24cfe8610069d8d2668c8fd923f7c0279c.jpg)  
+Figure 1: Layer Removal Policy removes layers of Teacher network architecture (stage-1 candidates) then Layer Shrinkage Policy reduces parameters (stage-2 candidates).
+
+Towards a more principled approach to network architecture compression, we present a reinforcement learning approach to identify a compressed high-performance architecture (student) given knowledge distilled from a larger high-performing model (teacher). We make a key conceptual assumption that formulates the sequential process of converting a teacher network to a student network as a Markov Decision Process (MDP). Under this model, a state $s$ represents the network architecture. Clearly, the domain of the state $s$ is very large since it contains every possible reduced architecture of the teacher network. A deterministic transition in this state space, $T ( s ^ { \prime } | s , a )$ , is determined by selecting the action $a$ , e.g., removing a convolutional filter or reducing the size of a fully connected layer. Each action will transform one architecture $s$ to another architecture $s ^ { \prime }$ . Under the MDP, the strategy for selecting an action given a certain state is represented by the policy $\pi ( a | s )$ , which stochastically maps a state to an action. The process of reinforcement learning is used to learn an optimal policy based on a reward function $r ( s )$ defined over the state space. In our work, we define the reward function based on the accuracy and the compression rate of the specified architecture $s$ .
+
+A straightforward application of reinforcement learning to this problem can be very slow depending on the definition of the action space. For example, an action could be defined as removing a single filter from every layer of a convolutional neural network. Since the search space is exponential in the size of the action space and sequence length, it certainly does not scale to modern networks that have hundreds of layers.
+
+Our proposed approach addresses the problem of scalability in part, by introducing a two-stage action selection mechanism which first selects a macro-scale “layer removal” action, followed by a micro-scale “layer shrinkage” action. In this way we enable our reinforcement learning process to efficiently explore the space of reduced networks. Each network architecture that is generated by our policy is then trained with Knowledge Distillation (Hinton et al., 2015). Figure 1 illustrates our proposed approach.
+
+To the best of our knowledge, this is the first paper to provide a principled approach to the task of network compression, where the architecture of the student network is obtained via reinforcement learning. To facilitate reinforcement learning, we propose a reward function that encodes both the compression rate and the accuracy of the student model. In particular, we propose a novel formulation of the compression reward term based on a relaxation of a constrained optimization problem, which encodes the hardware-based computational budget items in the form of linear constraints.
+
+We demonstrate the effectiveness of our approach over several network architectures and several visual learning tasks of varying difficulty (MNIST, SVHN, CIFAR-10, CIFAR-100, Caltech-256). We also demonstrate that the compression policies exhibit generalization across networks with similar architectures. In particular, we use a policy trained on a ResNet-18 model on a ResNet-34 model and show that it greatly accelerates the reinforcement learning process.
+
+# 2 RELATED WORK
+
+We first discuss methods for compressing models to a manually designed network (pruning and distillation). Towards automation, we discuss methods for automatically constructing highperformance networks, orthogonal to the task of compression.
+
+Pruning: Pruning-based methods preserve the weights that matter most and remove the redundant weights LeCun et al. (1989), Hassibi et al. (1993), Srinivas & Babu (2015), Han et al. (2015b), Han et al. (2015a), Mariet & Sra (2015), Anwar et al. (2015), Guo et al. (2016). While pruning-based approaches typically operate on the weights of the teacher model, our approach operates on a much larger search space over both model weights and model architecture. Additionally, our method offers greater flexibility as it allows the enforcement of memory, inference time, power, or other hardware constraints. This allows our approach to find the optimal architecture for the given dataset and constraints instead of being limited to that of the original model.
+
+Knowledge Distillation: Knowledge distillation is the task of training a smaller network (a “student”) to mimic a “teacher” network, performing comparably to the input network (a “teacher”) Bucilu et al. (2006), Ba & Caruana (2014), Hinton et al. (2015), Romero et al. (2014), Urban et al. (2016). The work of Hinton et al. (2015) generalized this idea by training the student to learn from both the teacher and from the training data, demonstrating that this approach outperforms models trained using only training data. In Romero et al. (2014), the approach uses Knowledge Distillation with an intermediate hint layer to train a thinner but deeper student network containing fewer parameters to outperform the teacher network. In previous Knowledge Distillation approaches, the networks are hand designed, possibly after many rounds of trial-and-error. In this paper, we train a policy to learn the optimal student architecture, instead of hand-designing one. In a sense, we automate Knowledge Distillation, employing the distillation method of Ba & Caruana (2014) as a component of our learning process. In the experiments section we show that our learned architectures outperform those described in Romero et al. (2014) and Hinton et al. (2015).
+
+Architecture Search: There has been much work on exploring the design space of neural networks Saxe et al. (2011), Zoph & Le (2016), Baker et al. (2016), Ludermir et al. (2006), Miikkulainen et al. (2017), Real et al. (2017), Snoek et al. (2012), Snoek et al. (2015), Stanley & Miikkulainen (2002), Jozefowicz et al. (2015), Murdock et al. (2016), Feng & Darrell (2015), Warde-Farley et al. (2014), Iandola et al. (2016). The principal aim of previous work in architecture search has been to build models that maximize performance on a given dataset. On the other hand, our goal is to find a compressed architecture while maintaining reasonable performance on a given dataset. Our approach also differs from existing architecture search method since we use the teacher model as the search space for our architecture instead of constructing networks from scratch. Current methods that construct networks from scratch either operate on a very large search space, making it computationally expensive Zoph & Le (2016), Real et al. (2017), Miikkulainen et al. (2017), Jozefowicz et al. (2015) or operate on a highly restricted search space Baker et al. (2016), Snoek et al. (2015). Our approach instead leverages the idea that since the teacher model is able to achieve high accuracy on the dataset, it already contains the components required to solve the task well and therefore is a suitable search space for the compressed architecture.
+
+# 3 APPROACH
+
+Our goal is to learn an optimal compression strategy (policy) via reinforcement learning, that takes a Teacher network as input and systematically reduces it to output a small Student network.
+
+# 3.1 MARKOV DECISION PROCESS
+
+We formulate the sequential process of finding a reduced architecture as a sequential decision making problem. The decision process is modeled as a Markov Decision Process (MDP). Formally, the MDP is defined as the tuple $\mathcal { M } = \{ \boldsymbol { S } , \mathcal { A } , T , r , \gamma \}$ .
+
+States: $s$ is the state space, a finite set consisting of all possible reduced network architectures that can be derived from the Teacher model. For example, a VGG network (Simonyan & Zisserman,
+
+![](images/2d429d21620b19a0dca8de598f32786e8a492c5c2a28119a919b35e7f793643f.jpg)  
+Figure 2: a) Layer removal policy network, b) Layer shrinkage policy network
+
+2014) represents the state $s \in S$ (the initial state) and by removing one convolutional filter from the first layer we obtain a new network architecture $s ^ { \prime }$ .
+
+Actions: $\mathcal { A }$ is a finite set of actions that can transform one network architecture into another network architecture. In our approach there are two classes of action types: layer removal actions and layer parameter reduction actions. The definition of these actions are further described in Section 3.2.1 and 3.2.2.
+
+Transition Function: $T : S \times A  S$ is the state transition dynamic. Here, $T$ is deterministic since an action $a$ always transforms a network architecture $s$ to the resulting network architecture $s ^ { \prime }$ with probability one.
+
+Discount Factor: $\gamma$ is the discount factor. We use $\gamma = 1$ so that all rewards contribute equally to the final return.
+
+Reward: $r : \mathcal { S }  \mathbb { R }$ is the reward function. The rewards of network architecture $r ( s )$ can be interpreted to be a score associated with a given network architecture s. Note that we define the reward to be 0 for intermediate states, which represent “incomplete” networks, and only compute a non-trivial reward for the final state. The reward function is described in detail in Section 3.4.
+
+# 3.2 STUDENT-TEACHER REINFORCEMENT LEARNING
+
+Under this MDP, the task of reinforcement learning is to learn an optimal policy $\pi : { \mathcal { S } }  A$ , such that it maximizes the expected total reward, with the total reward given by:
+
+$$
+R ( \vec { s } ) = \sum _ { i = 0 } ^ { L = | \vec { s } | } r ( s _ { i } ) = r ( s _ { L } ) .
+$$
+
+We take a policy gradient reinforcement learning approach and iteratively update the policy based on sampled estimates of the reward. The design of the action space is critical for allowing the policy gradient method to effectively search the state space. If the actions are selected to be very incremental, a long sequence of actions would be needed to make a significant change to the network architecture, making credit assignment difficult. To address this issue, we propose a two stage reinforcement learning procedure. In the first stage a policy selects a sequence of actions deciding whether to keep or remove each layer of the teacher architecture. In the second stage, a different policy selects a sequence of discrete actions corresponding to the magnitude by which to attenuate configuration variables of each remaining layer. In this way, we are able to efficiently explore the state space to find the optimal student network.
+
+# Algorithm 1 Student-Teacher Reinforcement Learning
+
+<table><tr><td colspan="2"></td></tr><tr><td colspan="2">1: procedure STUDENT-TEACHER RL(S,A,T,r,γ)</td></tr><tr><td>2:</td><td>So←Teacher</td></tr><tr><td>3: 4:</td><td>fori=1 to N do</td></tr><tr><td></td><td>for t = 1 to Li do</td></tr><tr><td>5:</td><td>at ~ Tremove(St-1; 0remove,i-1)</td></tr><tr><td>6:</td><td>St←T(St-1,at)</td></tr><tr><td>7:</td><td>end for</td></tr><tr><td>8:</td><td>R←r(sL1)</td></tr><tr><td>9:</td><td>0remove,i ←θremove,i-1J(0remove,i-1)</td></tr><tr><td>10:</td><td>end for</td></tr><tr><td>11:</td><td>So ← Stage-1 Candidate</td></tr><tr><td>12:</td><td>fori=1 to N2 do</td></tr><tr><td>13:</td><td>for t = 1 to L2 do</td></tr><tr><td>14: 15:</td><td>at ~ Tshrink(St-1; 0shrink,i-1)</td></tr><tr><td>16:</td><td>St←T(St-1,at)</td></tr><tr><td></td><td>end for</td></tr><tr><td>17:</td><td>R←r(sL2)</td></tr><tr><td>18: 19:</td><td>shrink,i←sriniJ（shink)</td></tr><tr><td>20:</td><td>end for</td></tr><tr><td></td><td>Output: Compressed model</td></tr><tr><td>21: end procedure</td><td></td></tr></table>
+
+A sketch of the algorithm is given in Algorithm 3.2. For both layer removal and shrinkage policies, we repeatedly sample architectures and update the policies based on the reward achieved by the architectures. We now describe the details of the two stages of student-teacher reinforcement learning.
+
+# 3.2.1 LAYER REMOVAL
+
+In the layer removal stage, actions $a _ { t }$ correspond to the binary decision to keep or remove a layer. The length of the trajectory for layer removal is $T = L$ , the number of layers in the network. At each step $t$ of layer removal, the Bidirectional LSTM policy (See Figure 2a) observes the hidden states, $h _ { t - 1 } , h _ { t + 1 }$ , as well as information $x _ { t }$ about the current layer: $\pi _ { \mathrm { r e m o v e } } ( a _ { t } | h _ { t - 1 } , h _ { t + 1 } , x _ { t } )$ . Information about the current layer $l$ is given as
+
+$$
+x _ { t } = ( l , k , s , p , n , s _ { \mathrm { s t a r t } } , s _ { \mathrm { e n d } } ) ,
+$$
+
+where $l$ is the layer type, $k$ kernel size, $s$ stride, $p$ padding and $n$ number of outputs (filters or connections). To model more complex architectures, such as ResNet, $s _ { \mathrm { s t a r t } }$ and $s _ { \mathrm { e n d } }$ are used to inform the policy network about skip connections. For a layer inside a block containing a skip connection, $s _ { \mathrm { s t a r t } }$ is the number of layers prior to which the skip connection began and $s _ { \mathrm { e n d } }$ is the number of layers remaining until the end of the block. Additionally it is to be noted that although actions are stochastically sampled from the outputs at each time step, the hidden states that are passed on serve as a sufficient statistic for $x _ { 0 } , a _ { 0 } . . . x _ { t - 1 } , a _ { t - 1 }$ (Wierstra et al., 2010).
+
+# 3.2.2 LAYER SHRINKAGE
+
+The length of the trajectory for layer shrinkage is $\begin{array} { r } { T \ = \ \sum _ { l = 1 } ^ { L } H _ { l } } \end{array}$ , where $H$ is the number of configuration variables for each layer. At each step $t$ of layer shrinkage, the policy observes the hidden state $h _ { t - 1 }$ , the previously sampled action $a _ { t - 1 }$ and current layer information $x _ { t }$ : $\pi _ { \mathrm { s h r i n k } } \big ( a _ { t } \big | a _ { t - 1 } , h _ { t - 1 } , x _ { t } \big )$ . The parameterization of $x _ { t }$ is similar to layer removal except that the previous action is appended to the representation in an autoregressive manner (See Figure 2b). The action space for layer shrinkage is defined as $a _ { t } \in [ 0 . 1 , 0 . 2 , \ldots , 1 ]$ (each action corresponds to how much to shrink a layer parameter) and an action is produced for each configurable variable for each layer. Examples include kernel size, padding, and number of output filters or connections.
+
+# 3.3 REWARD FUNCTION
+
+The design of the reward function plays a critical role in learning the policies. A poorly designed reward that provides no discrimination between good and bad student architectures prevents policies from learning the trade-offs in architecture space. The objective of model compression is to maximize compression while maintaining a high accuracy. Since there is no benefit in producing highly compressed models which have bad performance, we want to provide a harsher penalty for a model with high compression $^ +$ low accuracy than one with low compression $^ +$ high accuracy. Furthermore we would also like to define a general reward function that does not depend on dataset/model specific hyperparameters. Additional discussion on the design of the reward function is provided in the appendix.
+
+In our approach, we define the reward function as follows:
+
+$$
+\begin{array} { l } { R = R _ { c } \cdot R _ { a } } \\ { \quad = C ( 2 - C ) \cdot \frac { A } { A _ { \mathrm { t e a c h e r } } } } \end{array}
+$$
+
+Where $C$ is the relative compression ratio of the student model, $A$ is the validation accuracy of the student model and $A _ { \mathrm { t e a c h e r } }$ is the validation accuracy of the teacher model provided defined as a constant. $R _ { c }$ and $R _ { a }$ refer to the compression and accuracy reward respectively. We compute the reward as a product of the compression and accuracy reward since we want the reward to scale with both quantities dependently. The compression reward, $R _ { c } = C ( 2 - C )$ , is computed using a non-linear function that biases the policy towards producing models that maintain accuracy while optimizing for compression. The relative compression $C \in [ 0 , 1 )$ is defined in terms of the ratio of trainable parameters of each model: C = 1 − #params(student)# ( ) . It is noted here that other compression methods that use quantization or coding define compression ratio in terms of number of bits instead of parameters. The accuracy reward, Ra, is defined with respect to the teacher model as Ra = AAteacher , where $A \in [ 0 , 1 ]$ refers to the validation accuracy of the student model and $A _ { \mathrm { t e a c h e r } }$ refers to the validation accuracy of the teacher model. We note that both accuracy and compression rewards are normalized with respect to the teacher and thus do not require additional hyperparameters to perform task-specific weighting. Lastly, it is possible that the policies may produce degenerate architectures in such cases, a reward if $^ { - 1 }$ is assigned (details in appendix).
+
+# 3.3.1 CONSTRAINTS AS REWARDS
+
+Our approach allows us to incorporate pre-defined hardware or resource budget constraints by rewarding architectures that meet the constraints and discouraging those that do not. Formally, our constrained optimization problem is
+
+$$
+\begin{array} { r } { \operatorname* { m a x } E _ { a _ { 1 : T } } [ R ] } \\ { \mathrm { s u b j e c t ~ t o ~ } A x \leq b , } \end{array}
+$$
+
+where $A$ and $b$ form our constraints, and $x$ is vector of constrained variables. We relax these hard constraints by redefining our reward function as:
+
+$$
+R = { \left\{ \begin{array} { l l } { R _ { a } \cdot R _ { c } } & { { \mathrm { i f ~ } } A x \leq b } \\ { - 1 } & { { \mathrm { o t h e r w i s e . } } } \end{array} \right. }
+$$
+
+The introduction of the non-smooth penalty may result in a reduced exploration of the search space and hence convergence to a worse local minimum. To encourage early exploration gradually incorporate constraints over time:
+
+$$
+R = { \left\{ \begin{array} { l l } { R _ { a } \cdot R _ { c } } & { { \mathrm { i f ~ } } A x \leq b } \\ { \epsilon _ { t } ( R _ { a } \cdot R _ { c } + 1 ) - 1 } & { { \mathrm { o t h e r w i s e } } , } \end{array} \right. }
+$$
+
+where $\epsilon _ { t } \in [ 0 , 1 ]$ monotonically decreases with $t$ and $\epsilon _ { 0 } = 1$ . As it is possible to incorporate a variety of constraints such as memory, time, power, accuracy, label-wise accuracy, our method is flexible enough to produce models practically viable in a diversity of settings. This is in contrast to conventional model compression techniques which require many manual repetitions of the algorithm in order to find networks that meet the constraints as well as optimally balance the accuracy-size tradeoff.
+
+# 3.4 OPTIMIZATION
+
+We now describe the optimization procedure for each our stochastic policies, $\pi _ { \mathrm { r e m o v e } }$ and $\pi _ { \mathrm { s h r i n k } }$ . The procedure is the same for each policy, thus we use $\pi$ in what follows. Each policy network is parameterized by its own $\theta$ .
+
+Our objective function is the expected reward over all sequences of actions $a _ { 1 : T }$ , i.e.:
+
+We use the REINFORCE policy gradient algorithm from Williams (1992) to train both of our policy networks.
+
+$$
+\begin{array} { l } { { \displaystyle \nabla _ { \theta } J ( \theta ) = \nabla _ { \theta } E _ { a _ { 1 : T } \sim P _ { \theta } } ( R ) } } \\ { ~ } \\ { { \displaystyle = \sum _ { t = 1 } ^ { T } E _ { a _ { 1 : T } \sim P _ { \theta } } [ \nabla _ { \theta } \log P _ { \theta } ( a _ { t } | a _ { 1 : ( t - 1 ) } ) R ] } } \\ { { \displaystyle ~ \approx \frac { 1 } { m } \sum _ { k = 1 } ^ { m } \sum _ { t = 1 } ^ { T } [ \nabla _ { \theta } \log P _ { \theta } ( a _ { t } | h _ { t } ) R _ { k } ] } } \end{array}
+$$
+
+where $m$ is the number of rollouts for a single gradient update, $T$ is the length of the trajectory, $P _ { \theta } { \left( { a _ { t } } | { h _ { t } } \right) }$ is the probability of selecting action $a _ { t }$ given the hidden state $h _ { t }$ , generated by the current stochastic policy parameterized by $\theta$ and $R _ { k }$ is the reward of the $k ^ { \mathrm { { t h } } }$ rollout.
+
+The above is an unbiased estimate of our gradient, but has high variance. A common trick is to use a state-independent baseline function to reduce the variance:
+
+$$
+\nabla _ { \theta } J ( \theta ) \approx \frac { 1 } { m } \sum _ { k = 1 } ^ { m } \sum _ { t = 1 } ^ { T } [ \nabla _ { \theta } \log P _ { \theta } ( a _ { t } | h _ { t } ) ( R _ { k } - b ) ]
+$$
+
+We use an exponential moving average of the previous rewards as the baseline $b$ . An Actor-Critic policy was also tested. While there was a minor improvement in stability, it failed to explore as effectively in some cases, resulting in a locally optimal solution. Details are in the appendix.
+
+# 3.5 KNOWLEDGE DISTILLATION
+
+Student models are trained using data labelled by a teacher model. Instead of using hard labels, we use the un-normalized log probability values (the logits) of the teacher model. Training using the logits helps to incorporate dark knowledge (Hinton et al., 2015) that regularizes students by placing emphasis on the relationships learned by the teacher model across all of the outputs.
+
+As in Ba & Caruana (2014), the student is trained to minimize the mean $L _ { 2 }$ loss on the training data $\left\{ \left( x ^ { i } , z ^ { i } \right) \right\} _ { i = 1 } ^ { N }$ . Where $z ^ { i }$ are the logits of the teacher model.
+
+$$
+\mathcal { L } _ { \mathrm { K D } } ( f ( x ; W ) , z ) = \frac { 1 } { N } \sum _ { i } | | f ( x ^ { ( i ) } ; W ) - z ^ { ( i ) } | | _ { 2 } ^ { 2 }
+$$
+
+where W represents the weights of the student network and $f ( x ^ { ( i ) } ; W )$ is the model prediction on the $i ^ { t h }$ training data sample.
+
+Final student models were trained to convergence with hard and soft labels using the following loss function.
+
+$$
+\mathcal { L } ( \mathcal { W } ) = \mathcal { L } _ { \mathrm { h a r d } } ( f ( x ; W ) , y _ { \mathrm { t r u e } } ) + \lambda * \mathcal { L } _ { \mathrm { K D } } ( f ( x ; W ) , z )
+$$
+
+Where $\mathcal { L } _ { \mathrm { h a r d } }$ is the loss function used for training with hard labels (in our case cross-entropy) and $y _ { \mathrm { t r u e } }$ are the ground truth labels.
+
+# 4 EXPERIMENTS
+
+In the following experiments, we first show that our method is able to find highly compressed student architectures with high performance on multiple datasets and teacher architectures, often exceeding performance of the teacher model. We compare the results obtained to current baseline methods of model compression, showing competitive performance. Then we demonstrate the viability of our method in highly resource constrained conditions by running experiments with strong model size constraints. Finally, we show that it is possible to rapidly speed up training when using larger teacher models by reusing policies that are pretrained on smaller teacher models.
+
+Table 1: Summary of Compression results.   
+
+<table><tr><td colspan="6">MNIST</td></tr><tr><td colspan="6">Architecture</td></tr><tr><td rowspan="2">VGG-13</td><td>Teacher</td><td>Acc. 99.54%</td><td>#Params △Acc.Compr. 9.4M</td><td></td><td></td></tr><tr><td>Student (Stage1)</td><td>99.55%</td><td>73K</td><td>+0.01% 127x</td><td></td></tr><tr><td colspan="6">CIFAR-10</td></tr><tr><td rowspan="3">VGG-19</td><td></td><td>91.97%</td><td>20.2M</td><td></td><td></td></tr><tr><td>Teacher Student (Stage1)</td><td>92.05%</td><td>1.7M</td><td>+0.08% 11.8x</td><td></td></tr><tr><td>Student (Stagel+Stage2)</td><td>91.64%</td><td>984K</td><td>-0.33%</td><td>20.53x</td></tr><tr><td rowspan="3">ResNet-18</td><td></td><td>92.01%</td><td>11.17M</td><td></td><td></td></tr><tr><td>Teacher Student (Stage1)</td><td>91.97%</td><td>2.12M</td><td>-0.04%</td><td>5.26x</td></tr><tr><td>Student (Stagel+Stage2)</td><td>91.81%</td><td>1.00M</td><td>-0.2%</td><td>11.10x</td></tr><tr><td rowspan="3">ResNet-34</td><td>Teacher</td><td>92.05%</td><td>21.28M</td><td></td><td></td></tr><tr><td>Student (Stage1)</td><td>93.54%</td><td>63.87M</td><td>+1.49%</td><td>5.5x</td></tr><tr><td>Student (Stage1+Stage2) 92.35% 2.07M</td><td></td><td></td><td>+0.30%</td><td>10.2x</td></tr><tr><td colspan="6">SVHN</td></tr><tr><td rowspan="3">ResNet-18</td><td>Teacher</td><td>95.24% 11.17M</td><td></td><td></td><td></td></tr><tr><td>Student (Stage1) Student (Stage1+Stage2) 95.38% 564K</td><td></td><td>95.66% 2.24M</td><td>+0.42%</td><td>4.97x</td></tr><tr><td>CIFAR-100</td><td></td><td></td><td>+0.18%</td><td>19.8x</td></tr><tr><td colspan="6"></td></tr><tr><td rowspan="3">ResNet-18</td><td>Teacher</td><td>72.22% 11.22M</td><td></td><td></td><td></td></tr><tr><td>Student (Stage1)</td><td>69.64% 4.76M</td><td></td><td>-2.58%</td><td>2.35x</td></tr><tr><td>Student (Stage1+Stage2)</td><td>68.01%</td><td>2.42M</td><td>-4.21%</td><td>4.64x</td></tr><tr><td rowspan="2">ResNet-34</td><td>Teacher</td><td>72.86%</td><td>21.33M</td><td></td><td></td></tr><tr><td>Student (Stage1)</td><td>70.11%</td><td>4.25M</td><td>-2.75%</td><td>5.02x</td></tr><tr><td colspan="6">Caltech256</td></tr><tr><td rowspan="3">ResNet-18</td><td>Teacher</td><td>47.65%</td><td>11.31M</td><td></td><td></td></tr><tr><td>Student (Stage1)</td><td>44.71% 3.62M</td><td></td><td></td><td>-2.94% 3.12x</td></tr><tr><td>Student (Stage1+Stage2) 44.63% 2.45M</td><td></td><td></td><td>-3.02%</td><td>64.61x</td></tr><tr><td colspan="6">ImageNet32x32</td></tr><tr><td>ResNet-34</td><td>Teacher</td><td>30.87% 21.79M</td><td></td><td></td><td></td></tr><tr><td></td><td>Student (Stage1)</td><td>30.22% 3.34M</td><td></td><td>-0.65%</td><td>6.51x</td></tr></table>
+
+# 4.1 DATASETS
+
+MNIST The MNIST (LeCun et al., 1998) dataset consists of $2 8 \times 2 8$ pixel grey-scale images depicting handwritten digits. We use the standard 60,000 training images and 10,000 test images for experiments. Although MNIST is easily solved with smaller networks, we used a high capacity models (e.g., VGG-13) to show that the policies learned by our approach are able to effectively and aggressively remove redundancies from large network architectures.
+
+CIFAR-10 The CIFAR-10 (Krizhevsky & Hinton, 2009) dataset consists of 10 classes of objects and is divided into 50,000 train and 10,000 test images $3 2 \mathrm { x } 3 2$ pixels). This dataset provides an incremental level of difficulty over the MNIST dataset, using multi-channel inputs to perform model compression.
+
+SVHN The Street View House Numbers (Netzer et al., 2011) dataset contains 3232 colored digit images with 73257 digits for training, 26032 digits for testing. This dataset is slightly larger that CIFAR-10 and allows us to observe the performance on a wider breadth of visual tasks.
+
+CIFAR-100 To further test the robustness of our approach, we evaluated it on the CIFAR-100 dataset. CIFAR-100 is a harder dataset with 100 classes instead of 10, but the same amount of data, 50,000 train and 10,000 test images (32x32). Since there is less data per class, there is a steeper size-accuracy tradeoff. We show that our approach is able to produce solid results despite these limitations.
+
+Caltech-256 To test the effectiveness of our approach in circumstances where data is sparse, we run experiments on the Caltech-256 dataset (Griffin et al., 2007). This dataset contains more classes and less data per class than CIFAR-100, containing 256 classes and a total of 30607 images $( 2 2 4 \mathbf { x } 2 2 4 )$ . We trained the networks from scratch instead of using pretraining in order to standardize our comparisons across datasets.
+
+ImageNet32x32 To test the efficiency of our approach, an experiment was conducted on a large scale dataset, ImageNet32x32 (Chrabaszcz et al., 2017). This dataset contains the same training/validation splits as the original ImageNet (Krizhevsky et al., 2012) dataset. It consists of 1.28 million training images and 50,000 validation images with 1000 object classes. However, unlike the original ImageNet dataset which uses $2 2 4 { \mathrm { x } } 2 2 4 { \mathrm { R G B } }$ images, ImageNet32x32 uses 32x32 RGB images, which reduces training time while increasing the difficulty of the task.
+
+# 4.2 TRAINING DETAILS
+
+In the following experiments, student models were trained as described in Section 3.5. We observed heuristically that 5 epochs was sufficient to compare performance.
+
+The layer removal and layer shrinkage policy networks were trained using the Adam optimizer with a learning rate of 0.003 and 0.01 respectively. Both recurrent policy networks were trained using the REINFORCE algorithm (batch size $^ { : = 5 }$ ) with standard backpropagation through time. A grid search was done to determine the ideal learning rate and batch size (details in appendix).
+
+# 4.3 COMPRESSION EXPERIMENTS
+
+In this section we evaluate the ability of our approach to learn policies to find compressed architectures without any constraints. In the following experiments, we expect that the policies learned by our approach will initially start out as random and eventually tend towards an optimal size-accuracy trade-off which results in a higher reward. Definitions of architectures are available in the appendix.
+
+![](images/b13694a26e9e9e4bc04ca2da035b1dcf2b44e404aacfb104fe819ee81a5bb61e.jpg)  
+Figure 3: Student learning on MNIST. Reward, Accuracy, Compression vs Iteration (Top: Stage 1, Bottom: Stage 2)
+
+MNIST To evaluate the compression performance we use (1) a Conv4 network consisting of 4 convolutional layers and (2) a high capacity VGG-13 network.
+
+Figure 3 shows the results of our compression approach for each teacher network. The lines represent the compression (blue), accuracy (green) and reward (orange). The y-axis represents the score of those quantities, between 0 and 1. The $\mathbf { X }$ -axis is the iteration number. We also highlight the largest and smallest models with red circles to give a sense of the magnitude of compression. This experiment appears to confirm our original expectation that the policies would improve over time.
+
+![](images/156459651a49b48038906e448cb2020faf5fe1cbb170594b9fa9f387594ff757.jpg)  
+Figure 4: Student learning on CIFAR-10. Reward, Accuracy, Compression vs Iteration (Top: Stage 1, Bottom: Stage 2)
+
+CIFAR-10 On the CIFAR-10 dataset we ran experiments using the following teacher networks: (1) VGG-19, (2) ResNet-18 and (3) ResNet-34 networks. The experimental results are shown in Figure 4. It is interesting to note that on CIFAR-10, our learned student networks perform almost as well or better the teacher networks despite a $1 0 \mathrm { x }$ compression rate.
+
+SVHN On the SVHN dataset, we ran experiments using ResNet-18 network as the teacher model. We observed that the reward and compression steadily increased while the accuracy remained stable, confirming similar results to that of CIFAR-10. This is a promising indication that our approach works for a breadth of tasks and isn’t dataset specific. Results are in the appendix.
+
+CIFAR-100 We also verified our approach on a harder dataset, CIFAR-100 to show how our approach performs with less data per class (Figure 5). Considering the largely reduced number of parameters, the compressed network achieves reasonably high accuracy. A notable aspect of many of the final compressed models is that ReLU layers within residual blocks were removed. Another interesting result is that the compressed ResNet-34 student model outperforms the ResNet-18 model despite having fewer parameters. This can likely be explained by the increased number of residual blocks in the ResNet-34 model.
+
+Caltech-256 The Caltech-256 experiments (appendix) show the performance of our approach when training data is scarce. We would like to verify that our approach does not overly compress the network by overfitting to the small number of training examples. As with the other experiments, the policies appears to learn to maximize reward over time, although the positive trend is not as pronounced due to the lack of training data. This is expected since less data means the reward signal is less robust to sources of noise, which in turn affects training of the policy.
+
+ImageNet32x32 We conducted an experiment on the ImageNet32x32 dataset to test the performance of our approach on a large scale dataset. Due to the increased difficulty of this dataset, the teacher model (ResNet-34) achieved a top-1 accuracy of $3 0 . 8 7 \%$ after training for 40 epochs. Despite the difficulty of the dataset, our approach was still able to find a compressed model with similar performance (- $. 0 . 6 5 \%$ drop). The runtime for 100 iterations of the layer removal policy on a ResNet-34 teacher and a batch size of 3 was approximately 272 hours. More details regarding the runtime can be found in Section 12.
+
+![](images/73ae4251ed68d46b78b8fafe7597ee6574aab78f4e72be975e2302560700a117.jpg)  
+Figure 5: Student learning on CIFAR-100.
+
+# 4.4 BASELINES
+
+We compare the performance of our approach to current model compression methods, namely pruning and Knowledge Distillation (with hand-designed model). We note here that compression rate is defined as the ratio of number of parameters instead of number of bits, which some other compression methods (quantization, coding) use. To provide a fair comparison with our method, the same trained teacher models used in our method were used.
+
+# 4.4.1 PRUNING
+
+Table 2: Pruning (Baseline)   
+
+<table><tr><td>Model</td><td>Acc.</td><td>#Params</td><td>Compr. △ Acc.</td><td></td></tr><tr><td>Teacher (MNIST/VGG-13)</td><td>99.54%</td><td>9.4M</td><td></td><td></td></tr><tr><td>Pruning</td><td>99.12%</td><td>162K</td><td>58x</td><td>-0.42%</td></tr><tr><td>Ours</td><td>99.55%</td><td>73K</td><td>127x</td><td>+0.01%</td></tr><tr><td>Teacher (CIFAR-10/VGG-19)</td><td>91.97%</td><td>20.2M</td><td></td><td></td></tr><tr><td>Pruning</td><td>91.06%</td><td>2.3M</td><td>8.7x</td><td>-0.91%</td></tr><tr><td>Ours</td><td>92.05%</td><td>1.7M</td><td>11.8x</td><td>+0.08%</td></tr></table>
+
+We compare our method to pruning, which is a model compression approach that operates directly on the weight space of a network, removing redundant weights or filters. We perform pruning based on Molchanov et al. (2016), which removes filters using a greedy criteria based approach and then finetunes the network. With pruning, the performance of the final model can vary depending on the degree to which it was pruned. To ensure a fair comparison, we stop pruning when 1. accuracy drops below $1 \%$ of the student model obtained by our method or 2. the number of parameters is less than our method. Pruning is done 5 times to control for variance and the best performing model is reported.
+
+The results of this experiment, reported in Table 2, show that while the pruned models show good compression rates, our approach outperforms this baseline on both datasets. These results could indicate that operating on the architecture space of the model might result in more consistent results than using heuristics to operate on the weight space directly.
+
+# 4.4.2 KNOWLEDGE DISTILLATION
+
+Table 3: Knowledge distillation with hand designed models (Baseline)   
+
+<table><tr><td>Model</td><td>Acc.</td><td>#Params</td><td>Compr. △ Acc.</td><td></td></tr><tr><td>Teacher (SVHN/ResNet-18)</td><td>95.24%</td><td>11.17M</td><td></td><td></td></tr><tr><td>SqueezeNet1.1</td><td>89.34%</td><td>727K</td><td>15x</td><td>-5.90%</td></tr><tr><td>Ours</td><td>95.38%</td><td>564K</td><td>19.8x</td><td>+0.18%</td></tr><tr><td>Teacher (CIFAR-10/ResNet-18)</td><td>92.01%</td><td>11.17M</td><td></td><td></td></tr><tr><td>FitNet-4</td><td>91.33%</td><td>1.2M</td><td>9.3x</td><td>-0.63%</td></tr><tr><td>VGG-small</td><td>83.93%</td><td>1.06M</td><td>10.5x</td><td>-8.08%</td></tr><tr><td>Ours</td><td>91.81%</td><td>1.00M</td><td>11.0x</td><td>-0.20%</td></tr></table>
+
+We also tested the validity of our hypothesis that hand designed models may not be optimal for Knowledge distillation. We compare models generated by our method to hand designed models that contain a similar number of parameters. We perform experiments with 3 hand designed model architectures, FitNet-4, SqueezeNet and a reduced network based on VGG, (VGG-small) which contains 10 layers. These networks were then trained to convergence with Knowledge Distillation on the CIFAR-10 dataset and the SVHN datasets.
+
+For the implementation of FitNet-4 (17 layers), we used the same model architecture described in Mishkin & Matas (2015) with the ReLU activation and Xavier initialization. The paper reported a baseline accuracy of 90.63 when trained from scratch and $1 . 2 \mathbf { M }$ parameters (Table 3 in Mishkin & Matas (2015)). For SqueezeNet, we implemented the 1.1 version described in Iandola et al. (2016), which contained 727K parameters after adapting it to CIFAR-10. We benchmarked VGG-small and FitNet on the CIFAR-10 dataset and SqueezeNet on the SVHN dataset in order to provide a fair comparison with our best models in terms of the number of parameters.
+
+From the results reported in Table 3, we observe that our method performs better than the handdesigned models on both datasets despite containing fewer parameters. The CIFAR-10 results seem to indicate that model selection is an important factor in Knowledge Distillation. Our model and the FitNet-4 model both outperform the VGG-small model, further confirming our hypothesis that hand-designing models may not be the optimal approach for use with Knowledge Distillation.
+
+# 4.5 COMPRESSION WITH SIZE CONSTRAINTS
+
+Table 4: Model Compression with Size Constraints   
+
+<table><tr><td>Model</td><td>Acc.</td><td>#Params</td><td>Compr.</td><td>Constr.</td></tr><tr><td>Teacher (MNIST/VGG-13)</td><td>99.54%</td><td>9.4M</td><td>1x</td><td>N/A</td></tr><tr><td>Student (Stage 1 &amp; 2)</td><td>98.91%</td><td>17K</td><td>553x</td><td>20K</td></tr><tr><td>Teacher (CIFAR-10/VGG-19)</td><td>91.97%</td><td>20.2M</td><td>1x</td><td>N/A</td></tr><tr><td>Student (Stage 1 &amp; 2)</td><td>90.8%</td><td>573K</td><td>35x</td><td>1M</td></tr></table>
+
+While the experiments to this point used no explicit constraints, in this experiment, we add a size constraint in terms of the number of parameters via the reward function as in Section 3.3.1. We expect the optimization to be harder because the range of acceptable architectures is reduced.
+
+Results are summarized in Table 4. These promising results suggest that the compression policies are able to produce sensible results despite being heavily constrained, thus demonstrating the viability of the approach in practice.
+
+# 4.6 TRANSFER LEARNING
+
+Naively applying our approach to a new teacher network means that the compression policies must be learned from scratch for each new problem. We would like to know if layer removal and shrinkage policy networks can be reused to accelerate compression for new teacher architectures. In the following experiments, we train a policy on an initial teacher model and then apply it to another teacher model to test whether the policy has learned a general strategy for compressing a network. Since both a pretrained policy and a randomly initialized policy is expected to eventually converge to a locally optimal policy given enough iterations, we provide performance measures over the the first 10 policy update iterations.
+
+Table 5: Transfer Learning Performance during first 10 iterations.   
+
+<table><tr><td></td><td colspan="3">ResNet18→ResNet34</td><td colspan="3">ResNet34→ResNet18</td><td colspan="3">VGG11→VGG19</td></tr><tr><td></td><td></td><td>Reward Comp.</td><td>. Acc.</td><td>Reward Comp. Acc.</td><td></td><td></td><td>Reward Comp. Acc.</td><td></td><td></td></tr><tr><td>Pre-trained</td><td>d 0.81</td><td>78.1%</td><td>79.5%</td><td>0.76</td><td>65.5%</td><td>82.3%</td><td>0.52</td><td>46.0%</td><td>71.7%</td></tr><tr><td>Scratch</td><td>0.50</td><td>34.8%</td><td>82.4%</td><td>0.53</td><td>39.7%</td><td>82.8%</td><td>-0.07</td><td>20.2%</td><td>42.5 %</td></tr></table>
+
+Results are summarized in Table 5. The slight drop in accuracy (third subcolumn) in models produced by the pretrained policy is expected due to the tradeoff between compression and accuracy. However, the average reward (first subcolumn) is always higher when we use a pretrained policy. Note that in the VGG experiment, the reward is negative since the non-pretrained policy starts off by producing degenerate models. However, the pretrained policy starts off from a different initialization that does not.
+
+This is an important result as it shows promising evidence that we can even transfer learned knowledge from a smaller model to a larger model, rapidly accelerating the policy search procedure on very deep networks.
+
+# 5 CONCLUSION
+
+We introduced a novel method for compressing neural networks. Our approach employs a two-stage layer removal and layer shrinkage procedure to learn how to compress large neural networks. By leveraging signals for accuracy and compression as supervision, our method efficiently learns to search the space of model architectures. We show that our method performs well over a variety of datasets and architectures. We also observe generalization capabilities of our method through transfer learning, allowing our procedure to be made even more efficient. Our method is also able to incorporate other practical constraints, such as power or inference time, thus showing potential for application in a real world setting.
+
+# ACKNOWLEDGEMENTS
+
+This work was sponsored in part by IARPA (D17PC00340).
+
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+
+# APPENDIX
+
+# 6 ACTOR-CRITIC
+
+Policy gradient based Actor-Critic algorithms have been shown to improve the stability of the policy search. This is achieved by replacing the baseline with a learned estimate of the value function at each time step.
+
+Formally, with vanilla REINFORCE we have,
+
+$$
+\nabla _ { \boldsymbol { \theta } } J ( \boldsymbol { \theta } ) \approx \frac { 1 } { m } \sum _ { k = 1 } ^ { m } \sum _ { t = 1 } ^ { T } [ \nabla _ { \boldsymbol { \theta } } \log P _ { \boldsymbol { \theta } } ( a _ { t } | h _ { t } ) ( R _ { k } - b _ { k } ) ]
+$$
+
+In the Actor-Critic algorithm we replace $b _ { k }$ with $V _ { k } ^ { \theta }$ , resulting in a new gradient estimate,
+
+$$
+\nabla _ { \boldsymbol { \theta } } J ( \boldsymbol { \theta } ) \approx \frac { 1 } { m } \sum _ { k = 1 } ^ { m } \sum _ { t = 1 } ^ { T } [ \nabla _ { \boldsymbol { \theta } } \log P _ { \boldsymbol { \theta } } ( a _ { t } | h _ { t } ) ( R _ { k } - V _ { k } ^ { \boldsymbol { \theta } } ) ]
+$$
+
+We implement the Critic network by adding an additional fully-connected layer that takes as input the hidden state of the LSTM and outputs a single scalar value. Figures 6-7 the results of the experiments performed.
+
+![](images/7074e9df4c49ee87b50190d86386f2f25c211b10553f94d82662a3cee13f27d7.jpg)  
+Figure 6: MNIST Left: Actor-critic Right: REINFORCE, averaged over 3 runs
+
+![](images/b17d932b16361feddd051f9d170a3c75befff3442b3af0bd3dd7c10ac1c222de.jpg)  
+Figure 7: CIFAR-10 Left: Actor-critic Right: REINFORCE, averaged over 3 runs
+
+For the MNIST dataset, our results show that there is a slight improvement in stability, although they both converge at a similar rate.
+
+For the CIFAR-10 dataset, although the Actor-critic version was more stable, it did not perform as well as the vanilla REINFORCE algorithm.
+
+# 7 LEARNING RATE AND BATCH SIZE
+
+The learning rate and batch size were selected via a grid search. The following graphs show the rate of convergence for different learning rates and batch sizes.
+
+# 7.1 LEARNING RATE
+
+In order to determine the learning rate, we performed a grid search over 0.03, 0.003, 0.0003. We performed this grid search on the MNIST dataset using the VGG-13 network to save time. For the stage-1 policy, it was observed that $_ { \mathrm { l r = 0 . 0 3 } }$ did not converge while $\scriptstyle 1 \mathrm { r } = 0 . 0 0 0 3$ converged too slowly. Thus we used $\scriptstyle 1 \mathrm { r } = 0 . 0 0 3$ as the learning rate.
+
+![](images/e81925cf3b855c3b9ada0a3a5ad9f6cae13ebcfe98c7827cd16194e6507441cd.jpg)  
+Figure 8: Average reward over 3 runs for various learning rates on the MNIST dataset
+
+# 7.2 BATCH SIZE
+
+Similarly we performed a grid search to determine the optimal batch size over 1, 5, 10. A batch size of 1 was too unstable while a batch size of 10 offered no substantial improvements to justify the additional computation. Thus we observed that a batch size of 5 worked the best.
+
+![](images/6a61abd58bafa44e0225d7dc414bdef0ba2cb96614f4c8b311425c5dfc740967.jpg)  
+Figure 9: Average reward over 3 runs for batch sizes Left: 1, Middle: 5, Right: 10 on the MNIST dataset
+
+# 8 TRANSFER LEARNING EXPERIMENTS
+
+Below are the results of the transfer learning experiments, as observed, the pretrained policies start off with a high reward unlike the policies trained from scratch.
+
+![](images/a8fd4d223af6985eeadb5db2e039d3df78532441e4fdd269239a927d56b53397.jpg)  
+Figure 10: Transfer learning experiments
+
+# 9 ADDITIONAL EXPERIMENTS
+
+The following section contains results about additional compression experiments that were conducted.
+
+![](images/a6a110550cfe01d050c64692cd3201d96cbc3d80e316b03ed85ed32df937b3c3.jpg)  
+Figure 11: ResNet-18 experiments on SVHN, (Left: Stage 1, Right: Stage 2)
+
+![](images/941c13e1e544894790036473e20166058c2c7db0052bd427d393e0138ba6948e.jpg)  
+Figure 12: ResNet-18 experiments on Caltech, (Left: Stage 1, Right: Stage 2)
+
+# 10 IMPLEMENTATION DETAILS
+
+The following section contains the implementation details required to replicate the experiments. All of the experiments were implemented in PyTorch with 1 NVIDIA TitanX GPU.
+
+![](images/93e3983f27fea08cd4906a6875de05052ee41021acf95f66aa3ae5ebfbf2bd1e.jpg)  
+Figure 13: Stage1 ResNet-34 experiments on ImageNet32x32
+
+# 10.1 POLICIES
+
+Removal policy The removal policy was implemented with 2 hidden layers and 30 hidden units and trained with the Adam optimizer and a learning rate of 0.003. The shrinkage policy was implemented with 2 hidden layers and 50 hidden units and trained with the Adam optimizer and with a learning rate of 0.1. These policies were each trained for at least 100 epochs for each experiment. Batch size of 5 rollouts was used.
+
+# 10.2 TEACHER MODELS
+
+MNIST Teacher models for MNIST were trained for 50 epochs with a starting learning rate of 0.01.   
+The learning rate is reduced by a factor of 10 in the 30th epoch. A batch size of 64 was used.
+
+CIFAR-10/100 Teacher models for CIFAR-10/100 were trained for 150 epochs with a starting learning rate of 0.001. The learning rate is decreased by a factor of 10 in the 80th and 120th epochs. Standard data augmentation with horizontal mirroring $\scriptstyle ( \mathrm { p } = 0 . 5 )$ , random cropping with padding of 4 pixels and mean subtraction of (0.5, 0.5, 0.5). A batch size of 128 was used.
+
+SVHN Teacher models for SVHN were trained for 150 epochs with a starting learning rate of 0.001. The learning rate is decreased by a factor of 10 in the 80th and 120th epochs. Mean subtraction of (0.5, 0.5, 0.5) and a batch size of 128 was used.
+
+Caltech256 To make the experiments controlled over all datasets the Caltech256 models were trained from scratch. It is to be noted that Caltech256 models are usually initialized with pre-trained ImageNet weights since data is sparse. The training procedure consisted of 50 epochs with an initial learning rate of 0.01. It was reduced to 0.001 after the 50th epoch. Data augmentation such as horizontal flipping and random cropping alongside mean subtraction was used. ImageNet32x32 The ResNet-34 teacher model for the ImageNet32x32 experiment was trained using a method similar to that described in Chrabaszcz et al. (2017). It was trained for 40 epochs with a starting learning rate of 0.01. The learning rate was reduced by a factor of 5.0 every 10 epochs. Mean subtraction was used with a batch size of 128.
+
+# 11 REWARD DESIGN
+
+In this section we go into greater detail regarding the design of the chosen reward function compared to a naive reward. For our objective of model compression, we want the reward to reflect the following qualitative heuristics.
+
+1. A model with $\uparrow$ compression but $\downarrow$ accuracy should be penalized more than a model with ↓ compression and $\uparrow$ accuracy. Since we do not want to produce highly compressed models which do not perform well on the task, we do not want to let the compression score dominate the reward.
+
+2. The reward function should montonically increase with both compression and accuracy.
+
+Defining a naive, symmetrical reward function results in the following failure case. Suppose we define our reward as:
+
+$$
+R = A * C
+$$
+
+where $A , C$ are the relative validation accuracy and compression achieved by the student model. Let us consider the following 2 cases:
+
+1. $\uparrow$ accuracy, $\downarrow$ compression. $\mathbf A = 1$ , $C = 0 . 2 5$   
+2. ↓ accuracy, $\uparrow$ compression. $\mathbf { A } = 0 . 2 5 , \mathbf { C } = 1$
+
+In both cases $R = A * C = 0 . 2 5$ , which we do not want. If we use the reward function defined in the paper we get a reward of 0.25 and 0.4375 for each of the cases, which is closer to our true objective. In our empirical experiments, the non-linear reward outperformed the naive one. Other more complex reward functions that respect the above criteria may also work well.
+
+The visualization of the reward manifold in Figure 14 better illustrates the difference. As observed, a naive reward function is symmetric while our reward function returns a lower reward for low accuracy, high compression models compared to high accuracy, low compression models. Both functions are monotonically increasing.
+
+![](images/d0a6d064f55e93c61fd3dc020caadddaac9649e52cad65213e39d264c5ede48e.jpg)  
+Figure 14: Reward manifold of naive reward vs. our reward
+
+# 11.2 DEGENERATE CASES
+
+The following section outlines a few of the cases which are considered degenerate and for which a fixed reward of -1 is assigned.
+
+1. Empty architecture - Depending on how it is implemented, the policies could possibly output ”remove” actions for each layer during the layer removal stage. In this case, the output would be an empty architecture with no trainable parameters. 2. Large FC layer - If too many layers are removed in the feature extraction portion of the convolutional neural network, the size of the feature map before the fully connected layers would be large. In this case, although we have a well defined reward, training the network could be impractical 3. Specialized architectures - When dealing with more complex architectures, there may be inter-layer dependencies which impose certain requirements. For example, in a ResNet, the dimensionality of the feature maps at the start and end of each residual block has to match.
+
+# 12 TOTAL TRAINING TIME
+
+To give the reader an approximate estimate of the time taken to train the policies, we have included Table 6 which shows the time taken to train a layer removal policy for 100 iterations. These experiments were done in PyTorch with a single NVIDIA TitanX GPU and an Intel Xeon E5-2660
+
+Table 6: Training time of Layer Removal policy (100 iterations)   
+
+<table><tr><td rowspan=1 colspan=1>Architecture               Time (hrs)</td></tr><tr><td rowspan=1 colspan=1>MNIST</td></tr><tr><td rowspan=1 colspan=1>VGG-13                   4</td></tr><tr><td rowspan=1 colspan=1>CIFAR-10</td></tr><tr><td rowspan=1 colspan=1>VGG-19                   17ResNet-18                 17ResNet-34                 54</td></tr><tr><td rowspan=1 colspan=1>SVHN</td></tr><tr><td rowspan=1 colspan=1>ResNet-18                22</td></tr><tr><td rowspan=1 colspan=1>CIFAR-100</td></tr><tr><td rowspan=1 colspan=1>ResNet-18                 20ResNet-34                 55</td></tr><tr><td rowspan=1 colspan=1>Caltech256</td></tr><tr><td rowspan=1 colspan=1>ResNet-18                 175</td></tr><tr><td rowspan=1 colspan=1>ImageNet32x32</td></tr><tr><td rowspan=1 colspan=1>ResNet-34 (batch_size=3) 272</td></tr></table>
+
+CPU. We note that runtime varies based on many factors such as hardware, machine usage and the inherent stochasticity in the approach. The times listed are simply an approximate estimate to how long the method takes on average.
+
+# 13 FUTURE DIRECTIONS
+
+This paper introduces a general method to generate an architecture that optimizes the size-capacity trade-off with respect to a particular task. The current limitation with this method is that we need to train each student model for a few epochs to determine a reward for it. This step can be computationally expensive depending on the dataset. Results from Saxe et al. (2011), Jarrett et al. (2009) and Cox & Pinto (2011) seem to suggest that initializing models with random weights could be an efficient way to evaluate architectures provided the right non-linearities and pooling are used. Another way to provide a better initialization could be to use a hypernetwork which takes the student model architecture as input and produces weights for the model. Other methods that select an informative subset of the training and test dataset to efficiently evaluate the network could also be interesting to explore. Another interesting direction would be to use the pretrained policies for transfer learning on different architecture search problems (apart from compression) to see if any generalizable information about deep architectures is being learned.

parse/train/B1l2bp4YwS/B1l2bp4YwS.md

@@ -0,0 +1,339 @@
+# WHAT GRAPH NEURAL NETWORKS CANNOT LEARN: DEPTH VS WIDTH
+
+Andreas Loukas Ecole Polytechnique F ´ ed´ erale Lausanne ´ andreas.loukas@epfl.ch
+
+# ABSTRACT
+
+This paper studies the expressive power of graph neural networks falling within the message-passing framework $( G N N _ { \mathsf { m p } } )$ . Two results are presented. First, $\mathsf { G N N } _ { \mathsf { m p } }$ are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that $\mathsf { G N N } _ { \mathsf { m p } }$ can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the repurposing of seminal results from distributed computing and leads to lower bounds for an array of decision, optimization, and estimation problems involving graphs. Strikingly, several of these problems are deemed impossible unless the product of a $\mathsf { G N N } _ { \mathsf { m p } }$ ’s depth and width exceeds a polynomial of the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation.
+
+# 1 INTRODUCTION
+
+A fundamental question in machine learning is to determine what a model can and cannot learn. In deep learning, there has been significant research effort in establishing expressivity results for feedforward (Cybenko, 1989; Hornik et al., 1989; Lu et al., 2017) and recurrent neural networks (Neto et al., 1997), as well as more recently for Transformers and Neural GPUs (Perez et al., 2019). ´ We have also seen the first results studying the universality of graph neural networks, i.e., neural networks that take graphs as input. Maron et al. (2019b) derived a universal approximation theorem over invariant functions targeted towards deep networks whose layers are linear and equivariant to permutation of their input. Universality was also shown for equivariant functions and a particular shallow architecture (Keriven & Peyre, 2019). ´
+
+Universality statements allow us to grasp the expressive power of models in the limit. In theory, given enough data and the right training procedure, a universal network will be able to solve any task that it is presented with. Nevertheless, the insight brought by such results can also be limited. Knowing that a sufficiently large network can be used to solve any problem does not reveal much about how neural networks should be designed in practice. It also cannot guarantee that said network will be able to solve a given task given a particular training procedure, such as stochastic gradient descent.
+
+On the other hand, it might be easier to obtain insights about models by studying their limitations. After all, the knowledge of what cannot be computed (and thus learned) by a network of specific characteristics applies independently of the training procedure. Further, by helping us comprehend the difficulty of a task in relation to a model, impossibility results can yield practical advice on how to select model hyperparameters. Take, for instance, the problem of graph classification. Training a graph classifier entails identifying what constitutes a class, i.e., finding properties shared by graphs in one class but not the other, and then deciding whether new graphs abide to said learned properties. However, if the aforementioned decision problem is shown to be impossible by a graph neural network of certain depth then we can be certain that the same network will not learn how to classify a sufficiently diverse test set correctly, independently of which learning algorithm is employed. We should, therefore, focus on networks deeper that the lower bound when performing experiments.
+
+Table 1: Summary of main results. Subgraph verification\* entails verifying one of the following predicates for a given subgraph: is connected, contains a cycle, forms a spanning tree, is bipartite, is a cut, is an $^ { s - t }$ cut. All problems are defined in Appendix A.   
+
+<table><tr><td>problem</td><td>bound</td><td>problem</td><td>bound</td></tr><tr><td>cycle detection (odd)</td><td>dw = Ω(n/log n)</td><td>shortest path</td><td>dw = Ω(√n/log n)</td></tr><tr><td>cycle detection (even)</td><td>dw = Ω(√n/log n)</td><td>max. indep.set</td><td>dw = Ω(n²/log² n) for w = O(1)</td></tr><tr><td>subgraph verification*</td><td>d√w = Ω(√n/ log n)</td><td>min.vertex cover</td><td>dw = Ω(n²/log²n) for w = O(1)</td></tr><tr><td>min. spanning tree</td><td>d√w = Ω(√n/log n)</td><td>perfect coloring</td><td>dw = Ω(n²/log²n) for w = O(1)</td></tr><tr><td>min. cut</td><td>d√w = Ω(√n/log n)</td><td>girth 2-approx.</td><td>dw = Ω(√n/log n)</td></tr><tr><td>diam.computation</td><td>dw = Ω(n/log n)</td><td>diam. 3/2-approx.</td><td>dw = Ω(√n/log n)</td></tr></table>
+
+# 1.1 MAIN RESULTS
+
+This paper studies the expressive power of message-passing graph neural networks $( G N N _ { \mathsf { m p } } )$ ) (Gilmer et al., 2017). This model encompasses several state-of-the-art networks, including GCN (Kipf & Welling, 2016), gated graph neural networks (Li et al., 2015), molecular fingerprints (Duvenaud et al., 2015), interaction networks (Battaglia et al., 2016), molecular convolutions (Kearnes et al., 2016), among many others. Networks using a global state (Battaglia et al., 2018) or looking at multiple hops per layer (Morris et al., 2019; Liao et al., 2019; Isufi et al., 2020) are not directly $\mathsf { G N N } _ { \mathsf { m p } }$ , but they can often be re-expressed as such. The provided contributions are two-fold:
+
+I. What $\mathsf { G N N } _ { \mathsf { m p } }$ can compute. Section 3 derives sufficient conditions such that a $G N N _ { \mathsf { m p } }$ can compute any function on its input that is computable by a Turing machine. This result compliments recent universality results (Maron et al., 2019b; Keriven & Peyre, 2019) that considered approximation ´ (rather than computability) over specific classes of functions (permutation invariant and equivariant) and particular architectures. The claim follows in a straightforward manner by establishing the equivalence of $\mathsf { G N N } _ { \mathsf { m p } }$ with LOCAL (Angluin, 1980; Linial, 1992; Naor & Stockmeyer, 1993), a classical model in distributed computing that is itself Turing universal. In a nutshell, $\mathsf { G N N } _ { \mathsf { m p } }$ are shown to be universal if four strong conditions are met: there are enough layers of sufficient expressiveness and width, and nodes can uniquely distinguish each other. Since Turing universality is a strictly stronger property than universal approximation, Chen et al. (2019)’s argument further implies that a Turing universal $\mathsf { G N N } _ { \mathsf { m p } }$ can solve the graph isomorphism problem (a sufficiently deep and wide network can compute the isomorphism class of its input).
+
+II. What $\mathsf { G N N } _ { \mathsf { m p } }$ cannot compute (and thus learn). Section 4 analyses the implications of restricting the depth $d$ and width $w$ of $\mathsf { G N N } _ { \mathsf { m p } }$ that do not use a readout function. Specifically, it is proven that $G N N _ { \mathsf { m p } }$ lose a significant portion of their power when the product $d w$ , which I call capacity, is restricted. The analysis relies on a new technique that enables repurposing impossibility results from the context of distributed computing to the graph neural network setting. Specifically, lower bounds for the following problems are presented: (i) detecting whether a graph contains a cycle of specific length; (ii) verifying whether a given subgraph is connected, contains a cycle, is a spanning tree, is bipartite, is a simple path, corresponds to a cut or Hamiltonial cycle; (iii) approximating the shortest path between two nodes, the minimum cut, and the minimum spanning tree; (iv) finding a maximum independent set, a minimum vertex cover, or a perfect coloring; (v) computing or approximating the diameter and girth. The bounds are summarized in Table 1 and the problem definitions can be found in Appendix A. Section 5 presents some empirical evidence of the theory.
+
+Though formulated in a graph-theoretic sense, the above problems are intimately linked to machine learning on graphs. Detection, verification, and computation problems are relevant to classification: knowing what properties of a graph a $G N N _ { \mathsf { m p } }$ cannot see informs us also about which features of a graph can it extract. Further, there have been attempts to use $\mathsf { G N N } _ { \mathsf { m p } }$ to devise heuristics for graph-based optimization problems (Khalil et al., 2017; Battaglia et al., 2018; Li et al., 2018; Joshi et al., 2019; Bianchi et al., 2019), such as the ones discussed above. The presented results can then be taken as a worst-case analysis for the efficiency of $G N N _ { \mathsf { m p } }$ in such endeavors.
+
+# 1.2 DISCUSSION
+
+The results of this paper carry several intriguing implications. To start with, it is shown that the capacity dw of a $G N N _ { m p }$ plays a significant role in determining its power. Solving many problems is shown to be impossible unless $d w = \tilde { \Omega } ( n ^ { \delta } )$ , where $\delta \in [ 1 / 2 , 2 ]$ , $n$ is the number of nodes of the graph, and $f ( n ) = { \tilde { \Omega } } ( g ( n ) )$ is interpreted as $f ( n )$ being, up to logarithmic factors, larger than $g ( n )$ as $n$ grows. This reveals a direct trade-off between the depth and width of a graph neural network. Counter-intuitively, the dependence on n can be significant even if the problem appears local in nature or one only looks for approximate solutions. For example, detecting whether $G$ contains a short cycle of odd length cannot be done unless $d w = \tilde { \Omega } ( n )$ . Approximation helps, but only to a limited extent; computing the graph diameter requires $\Dot { d w } = \tilde { \Omega } ( n )$ and this reduces to $d w = \tilde { \Omega } ( \sqrt { n } )$ for any $^ 3 / 2$ -factor approximation. Further, it is impossible to approximate within any constant factor the shortest path, the minimum cut, and the minimum spanning tree, all three of which have polynomial-time solutions, unless $d \sqrt { w } = \tilde { \Omega } ( \sqrt { n } )$ . Finally, for truly hard problems, the capacity may even need to be super-linear on $n$ . Specifically, it is shown that, even if the layers of the $G N N _ { \mathsf { m p } }$ are allowed to take exponential time, solving certain NP-hard problems necessitates $d = \tilde { \Omega } ( n ^ { 2 } )$ depth for any constant-width network.
+
+Relation to previous impossibility results. In contrast to universality (Maron et al., 2019b; Keriven & Peyre, 2019), the limitations of ´ $\mathsf { G N N } _ { \mathsf { m p } }$ have been much less studied. In particular, the bounds presented here are the first impossibility results that (i) explicitly connect $\mathsf { G N N } _ { \mathsf { m p } }$ properties (depth and width) with graph properties and that (ii) go beyond isomorphism by addressing decision, optimization, and estimation graph problems. Three main directions of related work can be distinguished. First, Dehmamy et al. (2019) bounded the ability of graph convolutional networks (i.e., $\mathsf { G N N } _ { \mathsf { m p } }$ w/o messaging functions) to compute specific polynomial functions of the adjacency matrix, referred to as graph moments by the authors. Second, Xu et al. (2018) and Morris et al. (2019) established the equivalence of anonymous $\mathsf { G N N } _ { \mathsf { m p } }$ (those that do not rely on node identification) to the Weisfeiler-Lehman (WL) graph isomorphism test. The equivalence implies that anonymous networks are blind to the many graph properties that WL cannot see: e.g., any two regular graphs with the same number of nodes are identical from the perspective of the WL test (Arvind et al., 2015; Kiefer et al., 2015). Third, in parallel to this work, Sato et al. (2019) utilized a connection to LOCAL to derive impossibility results for the ability of a class of novel partially-labeled $G N N _ { \mathsf { m p } }$ to find good approximations for three NP-hard optimization problems. Almost all of the above negative results occur due to nodes being unable to distinguish between neighbors at multiple hops (see Appendix D). With discriminative attributes $\mathsf { G N N } _ { \mathsf { m p } }$ become significantly more powerful (without necessarily sacrificing permutation in/equivariance). Still, as this work shows, even in this setting certain problems remain impossible when the depth and width of the $\mathsf { G N N } _ { \mathsf { m p } }$ is restricted. For instance, though cycles can be detected (something impossible in anonymous networks ( $\mathrm { { X u } }$ et al., 2018; Morris et al., 2019)), even for short cycles one now needs $d w = \tilde { \Omega } ( n )$ . Further, in contrast to Sato et al. (2019), an approximation ratio below 2 for the minimum vertex cover is not impossible, but necessitates $d w = \tilde { \Omega } ( n ^ { 2 } )$ .
+
+Limitations. First, all lower bounds are of a worst-case nature: a problem is deemed impossible if there exists a graph for which it cannot be solved. The discovery of non worst-case capacity bounds remains an open problem. Second, rather than taking into account specific parametric functions, each layer is assumed to be sufficiently powerful to compute any function of its input. This strong assumption does not significantly limit the applicability of the results, simply because all lower bounds that hold with universal layers also apply to those that are limited computationally. Lastly, it will be assumed that nodes can uniquely identify each other. Node identification is compatible with permutation invariance/equivariance as long as the network output is asked to be invariant to the particular way the ids have been assigned. In the literature, one-hot encoded node ids are occasionally useful (Kipf & Welling, 2016; Berg et al., 2017). When attempting to learn functions across multiple graphs, ids should be ideally substituted by sufficiently discriminative node attributes (attributes that uniquely identify each node within each receptive field it belongs to can serve as ids). Nevertheless, similar to the unbounded computation assumption, if a problem cannot be solved by a graph neural network in the studied setting, it also cannot be solved without identifiers and discriminative attributes. Thus, the presented lower bounds also apply to partially and fully anonymous networks.
+
+Notation. I consider connected graphs $G = ( \nu , \mathcal { E } )$ consisting of $n = | \nu |$ nodes. The edge going from $v _ { j }$ to $v _ { i }$ is written as $e _ { i  j }$ and it is asserted that if $e _ { i  j } \in \mathcal { E }$ then also $e _ { j  i } \in \mathcal { E }$ . The neighborhood ${ \mathcal { N } } _ { i }$ of a node $v _ { i } \in \mathcal V$ consists of all nodes $v _ { j }$ for which $e _ { i  j } \in \mathcal { E }$ . The degree of $v _ { i }$ is denoted by $\deg _ { i }$ , $\Delta$ is the maximum degree of all nodes and the graph diameter $\delta _ { G }$ is the length of the longest shortest path between any two nodes. In the self-loop graph $G ^ { * } = ( \nu , { \mathcal { E } } ^ { * } )$ , the neighborhood set of $v _ { i }$ is given by ${ \mathcal { N } } _ { i } ^ { * } = { \mathcal { N } } _ { i } \cup v _ { i }$ .
+
+# 2 THE GRAPH NEURAL NETWORK COMPUTATIONAL MODEL
+
+Graph neural networks are parametric and differentiable learning machines. Their input is usually an attributed graph $G _ { a } = ( G , ( a _ { i } : v _ { i } \in \mathcal { V } )$ , $( a _ { i  j } : e _ { i  j } \in \mathcal { E } )$ ), where vectors $a _ { 1 } , \ldots , a _ { n }$ encode relevant node attributes and $a _ { i  j }$ are edge attributes, e.g., encoding edge direction.
+
+Model 1 formalizes the graph neural network operation by placing it in the message passing model (Gilmer et al., 2017). The computation proceeds in layers, within which a message $m _ { i  j }$ is passed along each directed edge $e _ { i  j } \in \mathcal { E }$ going from $v _ { j }$ to $v _ { i }$ and each node updates its internal representation by aggregating its state with the messages sent by its incoming neighbors $v _ { j } \in \mathcal N _ { i }$ . The network output can be either of two things: a vector $x _ { i }$ for each node $v _ { i }$ or a single vector $x _ { G }$ obtained by combining the representations of all nodes using a readout function. Vectors $x _ { i } / x _ { G }$ could be scalars (node/graph regression), binary variables (node/graph classification) or multi-dimensional (node/graph embedding). I use the symbols $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ and $G N N _ { \mathrm { m p } } ^ { \mathrm { g } }$ to distinguish between models that return a vector per node and one per graph, respectively.
+
+<table><tr><td>Computational model 1 Message passing graph neural network (GNNmp)</td><td></td></tr><tr><td></td><td rowspan="3"></td></tr><tr><td>forlayerl=1,...,d do for every edge ei←j ∈ ε* (in parallel) do</td></tr><tr><td>（e） MsGe xi (l-1) ,x (l-1) mij ,Ui,Uj,ai←j)</td></tr></table>
+
+The operation of a $\mathsf { G N N } _ { \mathsf { m p } }$ is primarily determined by the messaging, update, and readout functions. I assume that $\mathbf { M } \mathbf { S } \mathbf { G } _ { \ell }$ and $\mathrm { U } \mathrm { P } _ { \ell }$ are general functions that act on intermediate node representations and node ids (the notation is overloaded such that $v _ { i }$ refers to both the $i$ -th node as well as its unique id). As is common in the literature (Lu et al., 2017; Battaglia et al., 2018), these functions are instantiated by feed-forward neural networks. Thus, by the universal approximation theorem and its variants (Cybenko, 1989; Hornik et al., 1989), they can approximate any general function that maps vectors onto vectors, given sufficient depth and/or width. Function READ is useful when one needs to retrieve a representation that is invariant of the number of nodes. The function takes as an input a multiset, i.e., a set with possibly repeating elements, and returns a vector. Commonly, READ is chosen to be a dimension squashing operator, such as a sum or a histogram, followed by a feed-forward neural network $\mathrm { { X u } }$ et al., 2018; Seo et al., 2019).
+
+Depth and width. The depth $d$ is equal to the number of layers of the network. Larger depth means that each node has the opportunity to learn more about the rest of the graph (i.e., it has a larger receptive field). The width $w$ of a $G N N _ { \mathsf { m p } }$ is equal to the largest dimension of state $x _ { i } ^ { ( l ) }$ over all layers $l$ and nodes $v _ { i } \in \mathcal V$ . Since nodes need to be able to store their own unique ids, in the following it is assumed that each variable manipulated by the network is represented in finite-precision using $p = \Theta ( \log n )$ bits (though this is not strictly necessary for the analysis).
+
+# 3 SUFFICIENT CONDITIONS FOR TURING UNIVERSALITY
+
+This section studies what graph neural networks can compute. It is demonstrated that, even without readout function, a network is computationally universal1 if it has enough layers of sufficient
+
+width, nodes can uniquely distinguish each other, and the functions computed within each layer are sufficiently expressive. The derivation entails establishing that $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ is equivalent to LOCAL, a classical model used in the study the distributed algorithms that is itself Turing universal.
+
+# 3.1 THE LOCAL COMPUTATIONAL MODEL
+
+A fundamental question in theoretical computer science is determining what can and cannot be computed efficiently by a distributed algorithm. The LOCAL model, initially studied by Angluin (1980), Linial (1992), and Naor & Stockmeyer (1993), provides a common framework for analyzing the effect of local decision. Akin to $G N N _ { \mathsf { m p } }$ , in LOCAL a graph plays a double role: it is both the input of the system and captures the network topology of the distributed system that solves the problem. In this spirit, the nodes of the graph are here both the machines where computation takes place as well as the variables of the graph-theoretic problem we wish to solve—similarly, edges model communication links between machines as well as relations between nodes. Each node $v _ { i } \in \mathcal V$ is given a problem-specific local input and has to produce a local output. The input contains necessary the information that specifies the problem instance. All nodes execute the same algorithm, they are fault-free, and they are provided with unique identifiers.
+
+A pseudo-code description is given in Model 2. Variables $s _ { i } ^ { ( l ) }$ and s(l)i←j refer respectively to the state of $v _ { i }$ in round $l$ and to the message sent by $v _ { j }$ to $v _ { i }$ in the same round. Both are represented as strings. The computation starts simultaneously and unfolds in synchronous rounds $l = 1 , \ldots , d$ Three things can occur within each round: each node receives a string of unbounded size from its incoming neighbors; each node updates its internal state by performing some local computation; and each node sends a string to every one of its outgoing neighbors. Functions $\mathbf { A L G } _ { l } ^ { 1 }$ and $\dot { \mathrm { A L G } } _ { l } ^ { 2 }$ are algorithms computed locally by a Turing machine running on node $v _ { i }$ . Before any computation is done, each node $v _ { i }$ is aware of its own attribute $a _ { i }$ as well as of all edge attributes $\{ \bar { a } _ { i  j } : \overline { { v _ { j } } } \in \mathcal { N } _ { i } ^ { * } \}$
+
+<table><tr><td>Computational model 2 LOCAL (computed distributedly by each node Ui ∈ V).</td></tr><tr><td>Initialization: Set s (0) = (ai, Ui) and sj (0) = (aj,Uj) for all ei←j ∈ε.</td></tr><tr><td>for roundl=1,...,d do Receive S←j (e-1) from Uj ∈N*,compute</td></tr><tr><td>s(=ALl({(s）,i):Nt},u),</td></tr><tr><td>and send s() = ALG² (s,ui) to Uj ∈N.</td></tr></table>
+
+In LOCAL, there are no restrictions on how much information a node can send at every round.   
+Asserting that each message $s _ { i  j } ^ { ( \ell ) }$ is at most $b$ bits yields the CONGEST model (Peleg, 2000).
+
+# 3.2 TURING UNIVERSALITY
+
+The reader might have observed that LOCAL resembles closely $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ in its structure, with only a few minor differences: firstly, whereas a LOCAL algorithm A may utilize messages in any way it chooses, a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ network $\mathsf { N }$ always sums received messages before any local computation. The two models also differ in the arguments of the messaging function and the choice of information representation (string versus vector). Yet, as the following theorem shows, the differences between $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ and LOCAL are inconsequential when seen from the perspective of their expressive power:
+
+Theorem 3.1 (Equivalence). Let $N _ { \ell } ( G _ { a } )$ be the binary representation of the state $( x _ { 1 } ^ { ( \ell ) } , \ldots , x _ { n } ^ { ( \ell ) } )$ of a $G N N _ { m p } ^ { n }$ network N and $\mathbf { \delta A } _ { \ell } ( G _ { a } ) = ( s _ { 1 } ^ { ( \ell ) } , \ldots , s _ { n } ^ { ( \ell ) } )$ that of a LOCAL algorithm A. If $\mathbf { M } \mathbf { S } \mathbf { G } _ { \ell }$ and $\mathrm { U P } _ { \ell }$ are Turing complete functions, then, for any algorithm $A$ there exists $N$ (resp. for any N there exists $A$ ) such that
+
+$\pmb { A } _ { \ell } ( G _ { a } ) = N _ { \ell } ( G _ { a } )$ for every layer $\ell$ and $G _ { a } \in \mathcal { G } _ { a }$ where ${ \mathcal { G } } _ { a }$ is the set of all attributed graphs.
+
+This equivalence enables us to reason about the power of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ by building on the well-studied properties of LOCAL. In particular, it is well known in distributed computing that, as long as the number of rounds $d$ of a distributed algorithm is larger than the graph diameter $\delta _ { G }$ , every node in a LOCAL can effectively make decisions based on the entire graph (Linial, 1992). Together with Theorem 3.1, the above imply that, if computation and memory are not an issue, one may construct a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ that effectively computes any computable function w.r.t. its input.
+
+Corollary 3.1. $G N N _ { m p } ^ { n }$ can compute any Turing computable function over connected attributed graphs if the following conditions are jointly met: each node is uniquely identified; $\mathbf { M } \mathbf { S } \mathbf { G } _ { l }$ and $\mathrm { U } \mathrm { P } _ { l }$ are Turing-complete for every layer $\ell$ ; the depth is at least $d \geq \delta _ { G }$ layers; and the width is unbounded.
+
+Why is this result relevant? From a cursory review, it might seem that universality is an abstract result with little implication to machine learning architects. After all, the utility of a learning machine is usually determined not with regards to its expressive power but with its ability to generalize to unseen examples. Nevertheless, it can be argued that universality is an essential property of a good learning model. This is for two main reasons: First, universality guarantees that the learner does not have blind-spots in its hypothesis space. No matter how good the optimization algorithm is, how rich the dataset, and how overparameterized the network is, there will always be functions which a non universal learner cannot learn. Second, a universality result provides a glimpse on how the size of the learner’s hypothesis space is affected by different design choices. For instance, Corollary 3.1 puts forth four necessary conditions for universality: the $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ should be sufficiently deep and wide, nodes should be able to uniquely and consistently identify each other, and finally, the functions utilized in each layer should be sufficiently complex. The following sectimportance of two of these universality conditions. It will be shown that $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ es further into thelose a significant portion of their power when the depth and width conditions are relaxed.
+
+The universality of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { g } }$ . Though a universality result could also be easily derived for networks with a readout function, the latter is not included as it deviates from how graph neural networks are meant to function: given a sufficiently powerful readout function, a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { g } }$ of $d = 1$ depth and $O ( \Delta )$ width can be used to compute any Turing computable function. The nodes should simply gather one hop information about their neighbors; the readout function can then reconstruct the problem input based on the collective knowledge and apply any computation needed.
+
+# 4 IMPOSSIBILITY RESULTS AS A FUNCTION OF DEPTH AND WIDTH
+
+This section analyzes the effect of depth and width in the expressive power of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ . Specifically, I will consider problems that cannot be solved by a network of a given depth and width.
+
+To be able to reason in terms of width, it will be useful to also enforce that the message size in LOCAL at each round is at most $b$ bits. This model goes by the name CONGEST in the distributed computing literature (Peleg, 2000). In addition, it will be assumed that nodes do not have access to a random generator. With this in place, the following theorem shows us how to translate impossibility results from CONGEST to $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ :
+
+Theorem 4.1. If a problem $P$ cannot be solved in less than d rounds in CONGEST using messages of at most $b$ bits, then $P$ cannot be solved by a $G N N _ { m p } ^ { n }$ of width $w \leq ( b - \log _ { 2 } n ) / p = \bar { O } ( b / \log n )$ and depth $d$ .
+
+The $p = \Theta ( \log n )$ factor corresponds to the length of the binary representation of every variable—the precision needs to depend logarithmically on $n$ for the node ids to be unique. With this result in place, the following sections re-state several known lower bounds in terms of a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ ’s depth and width.
+
+# 4.1 IMPOSSIBILITY RESULTS FOR DECISION PROBLEMS
+
+I first consider problems where one needs to decide whether a given graph satisfies a certain property (Feuilloley $\&$ Fraigniaud, 2016). Concretely, given a decision problem $P$ and a graph $G$ , the $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ should output $x _ { i } \in \{ t r u e , f a l s e \}$ for all $v _ { i } \in \mathcal V$ . The network then accepts the premise if the logical conjunction of $\{ x _ { 1 } , \ldots , x _ { n } \}$ is true and rejects it otherwise. Such problems are intimately connected to graph classification: classifying a graph entails identifying what constitutes a class from some training set and using said learned definition to decide the label of graphs sampled from the test set. Instead, I will suppose that the class definition is available to the classifier and I will focus on the corresponding decision problem. As a consequence, every lower bound presented below for a decision problem must also be respected by a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ classifier that attains zero error on the corresponding graph classification problem.
+
+Subgraph detection. In this type of problems, the objective is to decide whether $G$ contains a subgraph belonging to a given family. I focus specifically on detecting whether $G$ contains a cycle $C _ { k }$ , i.e., a simple undirected graph of $k$ nodes each having exactly two neighbors. As the following result shows, even with ids, cycle detection remains relatively hard:
+
+Corollary 4.1 (Repurposed from (Drucker et al., 2014; Korhonen & Rybicki, 2018)). There exists√ graph $G$ on which every $G N N _ { m p } ^ { n }$ of width w requires depth at least $d \stackrel { . } { = } \Omega ( \sqrt { n } / ( w \log n ) )$ and $d =$ $\Omega ( n / ( w \log n ) )$ ) to detect if $G$ contains a cycle $C _ { k }$ for even $k \geq 4$ and odd $k \geq 5$ , respectively.
+
+Whereas an anonymous $\mathsf { G N N } _ { \mathsf { m p } }$ cannot detect cycles (e.g., distinguish between two $C _ { 3 }$ vs one $C _ { 6 }$ (Maron et al., 2019a)), it seems that with ids the product of depth and width should exhibit an (at least) linear dependence on $n$ . The intuition behind this bound can be found in Appendix C and empirical evidence in support of the theory are presented in Section 5.
+
+Subgraph verification. Suppose that the network is given a subgraph $H = ( \mathcal V _ { H } , \mathcal E _ { H } )$ of $G$ in its input. This could, for instance, be achieved by selecting the attributes of each node and edge to be a one-hot encoding of their membership on $\nu _ { H }$ and ${ \mathcal { E } } _ { H }$ , respectively. The question considered is whether the neural network can verify a certain property of $H$ . More concretely, does a graph neural network exist that can successfully verify $H$ as belonging to a specific family of graphs w.r.t. $G 2$ In contrast to the standard decision paradigm, here every node should reach the same decision—either accepting or rejecting the hypothesis. The following result is a direct consequence of the seminal work by Sarma et al. (2012):
+
+Corollary 4.2 (Repurposed from (Sarma et al., 2012)). There exists a graph $G$ on which every $G N N _ { m p } ^ { n }$ of width w requires depth at least $\begin{array} { r } { d = \Omega ( \sqrt { \frac { n } { w \log ^ { 2 } n } } + \delta _ { G } ) } \end{array}$ to verify if some subgraph $H$ of $G$ is connected, contains a cycle, forms a spanning tree of $G$ , is bipartite, is a cut of $G$ , or is an s-t cut of G. Furthermore, the depth should be at least $\begin{array} { r } { d = \bar { \Omega } \left( \left( \frac { n } { w \log n } \right) ^ { \gamma } + \delta _ { G } \right) } \end{array}$ with γ = 12 − 12(δ 0 −1) to verify if $H$ is a Hamiltonian cycle or a simple path.
+
+Therefore, even if one knows where to look in $G$ , verifying whether a given subgraph meets a given property can be non-trivial, and this holds for several standard graph-theoretic properties. For instance, if we constrain ourselves to networks of constant width, detecting whether a subgraph is√ connected can, up to logarithmic factors, require $\Omega ( { \sqrt { n } } )$ depth in the worst case.
+
+# 4.2 IMPOSSIBILITY RESULTS FOR OPTIMIZATION PROBLEMS
+
+I turn my attention to the problems involving the exact or approximate optimization of some graphtheoretic objective function. From a machine learning perspective, the considered problems can be interpreted as node/edge classification problems: each node/edge is tasked with deciding whether it belongs to the optimal set or not. Take, for instance, the maximum independent set, where one needs to find the largest cardinality node set, such that no two of them are adjacent. Given only information identifying nodes, $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ will be asked to classify each node as being part of the maximum independent set or not.
+
+Polynomial-time problems. Let me first consider three problems that possess known polynomialtime solutions. To make things easier for the $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ , I relax the objective and ask for an approximate solution rather than optimal. An algorithm (or neural network) is said to attain an $\alpha$ -approximation if it produces a feasible output whose utility is within a factor $\alpha$ of the optimal. Let OPT be the utility of the optimal solution and ALG that of the $\alpha$ -approximation algorithm. Depending on whether the problem entails minimization or maximization, the ratio ALG/OPT is at most $\alpha$ and at least $1 / \alpha$ , respectively.
+
+According to the following corollary, it is non-trivial to find good approximate solutions:
+
+Corollary 4.3 (Repurposed from (Sarma et al., 2012; Ghaffari & Kuhn, 2013)). There exists graphs $G$ and $G ^ { \prime }$ of diameter $\delta _ { G } = \Theta ( \log n )$ and $\delta _ { G ^ { \prime } } = { \cal O } ( 1 )$ on which every $G N N _ { m p } ^ { n }$ of width w requires depth at least $\begin{array} { r } { d = \Omega ( \sqrt { \frac { n } { w \log ^ { 2 } n } } ) } \end{array}$ and $\begin{array} { r } { d ^ { \prime } = \Omega ( ( \frac { n } { w \log n } ) ^ { \gamma } ) } \end{array}$ with $\begin{array} { r } { \gamma = \frac { 1 } { 2 } - \frac { 1 } { 2 ( \delta _ { G ^ { \prime } } - 1 ) } } \end{array}$ 12(δ 0 −1) , respectively, to approximate within any constant factor: the minimum cut problem, the shortest s-t path problem, or the minimum spanning tree problem.
+
+Thus, even for simple problems (complexity-wise), in the worst case a constant width $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ should be almost $\Omega ( { \sqrt { n } } )$ deep even if the graph diameter is exponentially smaller than $n$ .
+
+NP-hard problems. So what about truly hard problems? Clearly, one cannot expect a $G N N _ { \mathsf { m p } }$ to solve an NP-hard time in polynomial time2. However, it might be interesting as a thought experiment to consider a network whose layers take exponential time on the input size—e.g., by selecting the $\mathbf { M } \mathbf { S } \mathbf { G } _ { l }$ and $\mathrm { U } \mathrm { P } _ { l }$ functions to be feed-forward networks of exponential depth and width. Could one ever expect such a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ to arrive at the optimal solution?
+
+The following corollary provides necessary conditions for three well-known NP-hard problems:
+
+Corollary 4.4 (Repurposed from (Censor-Hillel et al., 2017)). There exists a graph $G$ on which every $G N N _ { m p } ^ { n }$ of width $w = O ( 1 )$ requires depth at least $d = \Omega ( n ^ { 2 } / \log ^ { 2 } n )$ to solve: the minimum vertex cover problem; the maximum independent set problem; the perfect coloring problem.
+
+Thus, even if each layer is allowed to take exponential time, the depth should be quadratically larger than the graph diameter $\delta _ { G } = { \cal { O } } ( n )$ to have a chance of finding the optimal solution. Perhaps disappointingly, the above result suggests that it may not be always possible to exploit the distributed decision making performed by $\mathsf { G N N } _ { \mathsf { m p } }$ architectures to find solutions faster than classical (centralized) computational paradigms.
+
+# 4.3 IMPOSSIBILITY RESULTS FOR ESTIMATION PROBLEMS
+
+Finally, I will consider problems that involve the computation or estimation of some real function that takes as an input the graph and attributes. The following corollary concerns the computation of two well-known graph invariants: the diameter $\delta _ { G }$ and the girth. The latter is defined as the length of the shortest cycle and is infinity if the graph has no cycles.
+
+Corollary 4.5 (Repurposed from (Frischknecht et al., 2012)). There exists a graph $G$ on which every $G N N _ { m p } ^ { n }$ of width w requires depth at least $d = \Omega ( n / ( w \log n ) + \delta _ { G } )$ to compute the graph diameter $\delta _ { G }$ and $d = \Omega ( \sqrt { n } / ( w \log n ) + \delta _ { G } )$ to approximate the graph diameter and girth within a factor of $^ 3 / 2$ and 2, respectively.
+
+Term $\delta _ { G }$ appears in the lower bounds because both estimation problems require global information. Further, approximating the diameter within a $\%$ factor seems to be simpler than computing it. Yet, in both cases, one cannot achieve this using a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ whose capacity is constant. As a final remark, the graphs giving rise to the lower bounds of Corollary 4.5 have constant diameter and $\Theta ( n ^ { 2 } )$ edges. However, similar bounds can be derived also for graphs with $O ( n \log n )$ edges (Abboud et al., 2016). For the case of exact computation, the lower bound is explained in Appendix C.
+
+# 5 EMPIRICAL EVIDENCE
+
+This section aims to empirically test the connection between the capacity $d w$ of a $\mathsf { G N N } _ { \mathsf { m p } }$ , the number of nodes $n$ of its input, and its ability to solve a given task. In particular, I considered the problem of 4-cycle classification and tasked the neural network with classifying graphs based on whether they contained a cycle of length four. Following the lower bound construction described in Appendix A, I generated five distributions over graphs with $n \in ( 8 , 1 6 , 2 4 , 3 2 , 4 4 )$ nodes and an average diameter of $( 4 , 6 , 8 , 9 , 1 1 )$ , respectively (this was achieved by setting $p \in ( 6 , 8 , 1 0 , 1 2 , 1 4 )$ , see Figure 3a). For each such distribution, I generated a training and test set consisting respectively of 1000 and 200 examples. Both sets were exactly balanced, i.e., any example graph from the training and test set had exactly $5 0 \%$ chance of containing a 4-cycle.
+
+The experiment aimed to evaluate how able were $\mathsf { G N N } _ { \mathsf { m p } }$ of different capacities to attain high accuracy on the test set. To this end, I performed grid search over the hyperparameters $w \in ( 2 , 1 0 , 2 0 )$ and $d \in ( 5 , 1 0 , 2 0 , 1 5 )$ . To reduce the dependence on the initial conditions and training length, for each hyperparameter combination, I trained 4 networks independently (using Adam and learning rate decay) for 4000 epochs. The $\mathsf { G N N } _ { \mathsf { m p } }$ chosen was that proposed by $\mathrm { X u }$ et al. (2018), with the addition of residual connections—this network outperformed all others that I experimented with.
+
+![](images/9c0b68c7d53cfa7504ecdca7b601197823b7b36e49a1c5952c915b0f68b568ce.jpg)  
+Figure 1: Accuracy as a function of $\mathsf { G N N } _ { \mathsf { m p } }$ capacity $d w$ and $n$ . (Best seen in color.)
+
+It is important to stress that empirically verifying lower bounds for neural networks is challenging, because it involves searching over the space of all possible networks in order to find the ones that perform the best. For this reason, an experiment such as the one described above cannot be used to verify3 the tightness of the bounds: we can never be certain whether the results obtained are the best possible or whether the optimization resulted in a local minimum. In that view, the following results should be interpreted in a qualitative sense. The question that I will ask is: to which extend do the trends uncovered match those predicted by the theory? More specifically, does the ability of a network to detect 4-cycles depend on the relation between $d w$ and $n$ ?
+
+To answer this question, Figure 1 depicts the training and test accuracy as a function of the capacity $d w$ for all the 240 networks trained (5 distributions $\times 3$ widths $\times ~ 4$ depths $\times ~ 4$ iterations). The accuracy of the best performing networks with the smallest capacity is shown in Figures 1c and 1d. It is important to stress that, based on Weisfeiler-Lehman analyses, anonymous $\mathsf { G N N } _ { \mathsf { m p } }$ cannot solve the considered task. However, as it seen in the figures, the impossibility is annulled when using node ids4. Indeed, even small neural networks could consistently classify all test examples perfectly (i.e., achieving $100 \%$ test accuracy) when $n \leq 1 6$ . Moreover, as the theoretical results predicted, there is a strong correlation between the test accuracy, $d w$ and $n$ (recall that Corollary 4.1 predicts $d w = \tilde { \Omega } ( \sqrt { n } ) )$ . Figure 1d shows that networks of the same capacity were consistently less accurate on the test set as $n$ increased (even though the cycle length remained 4 in all experiments). It is also striking to observe that even the most powerful networks considered could not achieve a test accuracy above $9 5 \%$ for $n > 1 6$ ; for $n = 4 0$ their best accuracy was below $80 \%$ .
+
+![](images/31952bba08cbf07926ecadfc216461eeb454c8004884b1386ca08f53b2de511f.jpg)  
+Figure 2: (a) GNNs are significantly more powerful when given discriminative node attributes. (b) Test accuracy indicated by color as a function of normalized depth and width. Points in highlighted areas correspond to networks with super-critical capacity, whereas the diagonal line separates networks that more deep than wide. (For improved visibility, points are slightly perturbed. Best seen in color.)
+
+Effect of anonymity. Figure 2a plots example training and test curves for $G N N _ { \mathsf { m p } }$ trained with four different node attributes: no attributes (anonymous), a one-hot encoding of the node degrees (degree), a one-hot encoding of node ids (unique id), and a one-hot encoding of node ids that changed across graphs (random unique id). It can be clearly observed that there is a direct correlation between accuracy and the type of attributes used. With non- or partially-discriminative attributes, the network could not detect cycles even in the training set. The cycle detection problem was solved exactly with unique ids, but when the latter were inconsistently assigned, the network could not learn to generalize.
+
+Exchangeability of depth and width. Figure 2b examines further the relationship between depth, width, and test accuracy. This time, networks were separated depending on their depth and width normalized by the square root of the “critical capacity”. For each $n$ , the critical capacity is the minimum $d w$ of a network that was able to solve the task on a graph of $n$ nodes—here, solving amounts to a test accuracy above $9 5 \%$ . In this way, a network of depth √ $d$ and width $w$ tested on $n$ nodes corresponds to a point positioned at $x = \overset { \cdot } { d } / \sqrt { \mathrm { c r i t i c a l } } , y = \overset { \cdot } { w } / \sqrt { \mathrm { c r i t i c a l } }$ and no network positioned at $x y < 1$ can solve the task (non-highlighted region in the bottom left corner). As seen, there is a crisp phase transition between the regime of under- and super-critical capacity: almost every network meeting the condition $d w \ge$ critical was able to solve the task, irrespective of whether the depth or width was larger. Note that, the exchangeability of depth and width cannot be guaranteed by the proposed theory which asserts that the condition $\hat { d w } = \tilde { \Omega ( \sqrt { n } ) }$ is necessary—but not sufficient. The empirical results however do agree with the hypothesis that, for 4-cycle classification, depth and width are indeed exchangeable.
+
+# 6 CONCLUSION
+
+This work studied the expressive power of graph neural networks falling within the message-passing framework. Two results were derived. First, sufficient conditions were provided such that $G N N _ { \mathsf { m p } }$ can compute any function computable by a Turing machine with the same connected graph as input. Second, it was discovered that the product of a $\mathsf { G N N } _ { \mathsf { m p } }$ ’s depth and width plays a prominent role in determining whether the network can solve various graph-theoretic problems. Specifically, it was shown that $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ with $d w = \tilde { \Omega } ( n ^ { \delta } )$ and $\delta \in [ 0 . 5 , 2 ]$ cannot solve a range of decision, optimization, and estimation problems involving graphs. Overall, the proposed results demonstrate that the power of graph neural networks depends critically on their capacity and illustrate the importance of using discriminative node attributes.
+
+Acknowledgements. I thank the Swiss National Science Foundation for supporting this work in the context of the project “Deep Learning for Graph-Structured Data” (grant number PZ00P2 179981).
+
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+
+# A GRAPH THEORY DEFINITIONS
+
+The main graph-theoretic terms encountered in this work are:
+
+• $k$ -cycle detection: a $k$ -cycle is a subraph of $G$ consisting of $k$ nodes, each with degree two. The $k$ -cycle detection problem entails determining if $G$ contains a $k$ -cycle.   
+• Hamiltonian cycle: a cycle of length $n$   
+• (minimum) spanning tree: a spanning tree is a tree subgraph of $G$ consisting of $n$ nodes. The minimum spanning tree problem entails finding the spanning tree of $G$ of minimum weight (the weight of a tree is equal to the sum of its edge weights).   
+• (minimum) cut: a cut is a subgraph of $G$ that when deleted leaves $G$ disconnected. The minimum cut problem entails finding the cut of minimum weight (the weight of a cut is equal to the sum of its edge weights).   
+• $s$ -t cut: a subgraph of $G$ such that removing all subgraph edges from $G$ will leave the nodes $s$ and $t$ of $G$ disconnected.   
+• (shortest) path: a simple path is subgraph of $G$ where all nodes have degree 2 except from the two endpoint nodes whose degree is one. The shortest path problem entails finding the simple path of minimum weight that connects two given nodes (the weight of a path is equal to the sum of its edge weights).
+
+• (maximum) independent set: an independent set is a set of nodes in a graph no two of which are adjacent. The maximum independent set problem entails finding the independent set of maximum cardinality.
+
+• (minimum) vertex cover: a vertex cover of $G$ is a set of nodes such that each edge of $G$ is incident to at least one node in the set. The minimum vertex cover problem entails finding the vertex cover of minimum cardinality.
+
+• (perfect) coloring: a coloring of $G$ is a labeling of the nodes with distinct colors such that no two adjacent nodes are colored using same color. The perfect coloring problem entails finding a coloring with the smallest number of colors.
+
+• diameter: the diameter $\delta _ { G }$ of $G$ equals the length of the longest shortest path.
+
+• girth: the girth of $G$ equals the length of the shortest cycle. It is infinity if no cycles are present.
+
+# B DEFERRED PROOFS
+
+# B.1 PROOF OF THEOREM 3.1
+
+The claim is proven by expressing the state of node $v _ { i }$ in the two models in the same form. It is not difficult to see that for each layer of the $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ one has
+
+$$
+\begin{array} { r l r } {  { x _ { i } ^ { ( l ) } = \operatorname { U P } _ { \ell } \Big ( \sum _ { v _ { j } \in \mathcal { N } _ { i } ^ { \ast } } m _ { i  j } ^ { ( \ell ) } \Big ) } } \\ & { } & { = \operatorname { U P } _ { \ell } \Bigg ( \sum _ { v _ { j } \in \mathcal { N } _ { i } ^ { \ast } } \operatorname { M S G } _ { \ell } \Big ( x _ { i } ^ { ( \ell - 1 ) } , x _ { j } ^ { ( \ell - 1 ) } , v _ { i } , v _ { j } , a _ { i  j } \Big ) \Big ) } \\ & { } & { = \operatorname { A G G } _ { \ell } ( \Big \{ \Big ( x _ { i } ^ { ( \ell - 1 ) } , x _ { j } ^ { ( \ell - 1 ) } , v _ { i } , v _ { j } , a _ { i  j } \Big ) : v _ { j } \in \mathcal { N } _ { i } ^ { \ast } \Big \} ) , \quad \mathrm { ( f r o m ~ ( X u ~ e t ~ a l . ~ 2 0 1 8 , L e m m a ~ 5 ) ) } } \end{array}
+$$
+
+where $\mathbf { A G G } _ { \ell }$ is an aggregation function, i.e., a map from the set of multisets onto some vector space. In the last step, I used a result of $\mathrm { X u }$ et al. $\mathrm { X u }$ et al. (2018) stating that each aggregation function can be decomposed as an element-wise function over each element of the multiset, followed by summation of all elements, and then a final function.
+
+Similarly, one may write:
+
+(by definition)
+
+$$
+\begin{array} { r l } & { s _ { i } ^ { ( \ell ) } = \mathrm { A L G } _ { \ell } ^ { 1 } ( \{ ( s _ { i  j } ^ { ( \ell - 1 ) } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \} , v _ { i } ) } \\ & { \quad \quad = \mathrm { A L G } _ { \ell } ^ { 1 } ( \{ ( \mathrm { A L G } _ { \ell - 1 } ^ { 2 } ( s _ { j } ^ { ( \ell - 1 ) } , v _ { j } ) , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \} , v _ { i } ) } \\ & { \quad \quad = \mathrm { A L G } _ { \ell } ( \{ ( s _ { j } ^ { ( \ell - 1 ) } , v _ { i } , v _ { j } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \} ) , } \end{array}
+$$
+
+with the last step following by restructuring the input and defining $\mathbf { A L G } _ { \ell }$ as the Turning machine that simulates the action of both $\mathbf { \bar { A } L G } _ { \ell } ^ { 2 }$ and $\mathbf { A L G } _ { \ell - 1 } ^ { 1 }$ .
+
+Since one may encode any vector into a string and vice versa, w.l.o.g. one may assume that the state of each node in LOCAL is encoded as a vector $x _ { i }$ . Then, to complete the proof, one still needs to demonstrate that the functions
+
+$$
+\begin{array} { r } { \mathrm { A G G } ( \{ ( x _ { i } , x _ { j } , v _ { i } , v _ { j } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \} ) \quad \mathrm { a n d } \quad \mathrm { A L G } ( \{ ( x _ { j } , v _ { i } , v _ { j } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \} ) } \end{array}
+$$
+
+are equivalent (in the interest of brevity the layer/round indices have been dropped). If this holds then each layer of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ is equivalent to a round of LOCAL and the claim follows.
+
+I first note that, since its input is a multiset, $\mathbf { A L G } _ { l }$ is also an aggregation function. To demonstrate equivalence, one thus needs to show that, despite not having identical inputs, each of the two aggregation functions can be used to replace the other. For the forward direction, it suffices to show that for every aggregation function AGG there exists ALG with the same output. Indeed, one may always construct $\mathbf { A L G } = \mathbf { A G G } \circ g$ , where $g$ takes as input $\{ ( x _ { j } , v _ { i } , v _ { j } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \}$ , identifies $x _ { i }$ (by searching for $v _ { i } , v _ { i } )$ and appends it to each element of the multiset yielding $\{ ( x _ { i } , x _ { j } , v _ { i } , v _ { j } , a _ { i  j } ) : v _ { j } \in \mathcal { N } _ { i } ^ { * } \}$ . The backward direction can also be proven with an elementary construction: given ALG, one sets $\mathbf { A G G } = \mathbf { A L G } \circ h$ , where $h$ deletes $x _ { i }$ from each element of the multiset.
+
+# B.2 PROOF OF COROLLARY 3.1
+
+In the LOCAL model the reasoning is elementary (Linial, 1992; Fraigniaud et al., 2013; Seidel, 2015): suppose that the graph is represented by a set of edges and further consider that $\mathbf { A L G } _ { l }$ amounts to a union operation. Then in $d = \delta _ { G }$ rounds, the state of each node will contain the entire graph. The function $\mathbf { A L G } _ { d } ^ { 1 }$ can then be used to make the final computation. This argument also trivially holds for node/edge attributes. The universality of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ then follows by the equivalence of LOCAL and GNNnmp.
+
+# B.3 PROOF OF THEOREM 4.1
+
+First note that, since the $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ and LOCAL models are equivalent, if no further memory/width restrictions are placed, an impossibility for one implies also an impossibility for the other. It can also be seen in Theorem 3.1 that there is a one to one mapping between the internal state of each node at each level between the two models (i.e., variables $x _ { i } ^ { \bar { ( l ) } }$ and $s _ { i } ^ { ( l ) }$ ). As such, impossibility results that rely on restrictions w.r.t. state size (in terms of bits) also transfer between the models.
+
+To proceed, I demonstrate that a depth lower bound in the CONGEST model (i.e., in the LOCAL model with bounded message size) also implies the existence of a depth lower bound in the LOCAL model with a bounded state size—with this result in place, the proof of the main claim follows directly. As in the statement of the theorem, one starts by assuming that $P$ cannot be solved in less than $d$ rounds when messages are bounded to be at most $b$ bits. Then, for the sake of contradiction, it is supposed that there exists an algorithm $A \in { \mathsf { L O C A L } }$ that can solve $P$ in less than $d$ rounds with a state of at most $c$ bits, but unbounded message size. I argue that the existence of this algorithm also implies the existence of a second algorithm $A ^ { \prime }$ whose messages are bounded by $c + \log _ { 2 } n$ : since each message s(l)j←i is the output of a universal Turing machine $\mathbf { A L G } _ { l } ^ { 2 }$ that takes as input the tuple $( s _ { i } ^ { ( l ) } , v _ { i } )$ , algorithm $A ^ { \prime }$ directly sends the input and relies on the universality of $\mathbf { A L G } _ { l + 1 } ^ { 1 }$ to simulate the action of $\mathbf { A L G } _ { l } ^ { 2 }$ . The message size bound follows by adding the size $c$ of the state with that of representing the node id $( \log _ { 2 } n$ bits suffice for unique node ids). This line of reasoning leads to a contradiction when $c \leq b - \log _ { 2 } n$ , as it implies that there exists an algorithm (namely $A ^ { \prime }$ ) that can solve $P$ in less than $d$ rounds while using messages of at most $b$ bits. Hence, no algorithm whose state is less than $b - \log _ { 2 } n$ bits can solve $P$ in LOCAL, and the width of $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ has to be at least $( b - \log _ { 2 } n ) / p$ .
+
+# C AN EXPLANATION OF THE LOWER BOUNDS FOR CYCLE DETECTION AND DIAMETER ESTIMATION
+
+A common technique for obtaining lower bounds in the CONGEST model is by reduction to the set-disjointness problem in two-player communication complexity: Suppose that Alice and Bob are each given some secret string ( $. s _ { a }$ and $s _ { b }$ ) of $q$ bits. The two players use the string to construct a set by selecting the elements from the base set $\{ 1 , 2 , \ldots , q \}$ for which the corresponding bit is one. It is known that Alice and Bob cannot determine whether their sets are disjoint or not without exchanging at least $\Omega ( q )$ bits (Kalyanasundaram $\&$ Schintger, 1992; Chor & Goldreich, 1988).
+
+The reduction involves constructing a graph that is partially known by each player. Usually, Alice and Bob start knowing half of the graph (red and green induced subgraphs in Figure 3). The players then use their secret string to control some aspect of their private topology (subgraphs annotated in dark gray). Let the resulting graph be $G ( s _ { a } , s _ { b } )$ and denote by cut the number of edges connecting the subgraphs controlled by Alice and Bob. To derive a lower bound for some problem $P$ , one needs to prove that a solution for $P$ in $G ( s _ { a } , s _ { b } )$ would also reveal whether the two sets are disjoint or not. Since each player can exchange at most ${ \dot { O } } ( b \cdot \operatorname { c u t } )$ bits per round, at least $\Omega ( q / ( b \cdot \mathrm { c u t } ) )$ rounds are needed in total ilower bound for EST. By Theorem 4.1, one then obtains a . $d = \Omega ( q / ( w \log n \cdot \mathbf { c u t } ) )$ ) depth $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$
+
+The two examples in Figure 3 illustrate the graphs $G ( s _ { a } , s _ { b } )$ giving rise to the lower bounds for even $k$ -cycle detection and diameter estimation. To reduce occlusion, only a subset of the edges are shown.
+
+![](images/334052ccd1ea255b8f322db9884932a7383060b9bf5027131d74c6d05939d850.jpg)  
+Figure 3: Examples of graphs giving rise to lower bounds.
+
+(a) In the construction of Korhonen & Rybicki (2018), each player starts from a complete bipartite graph of $p = { \sqrt { q } }$ nodes (nodes annotated in dark grey) with nodes numbered from 1 to $2 p$ . The nodes with the same id are connected yielding a cut of size $2 p$ . Each player then uses its secret (there are as many bits as bipartite edges) to decide which of the bipartite edges will be deleted (corresponding to zero bits). Remaining edges are substituted by a path of length $k / 2 - 1$ . This happens in a way that ensures that $G ( s _ { a } , s _ { b } )$ contains a cycle of length $k$ (half known by Alice and half by Bob) if and only if the two sets are disjoint: the cycle will pass through nodes $t$ and $p + t$ of each player to signify that the $t { \cdot }$ -th bits of $s _ { a }$ and $s _ { b }$ are both one. It can then be shown that $n \doteq \dot { \Theta } ( p ^ { 2 } )$ from which it follows that:√ CONGEST requires at least $d = \Omega ( q / ( b \cdot \mathbf { c u t } ) ) = \Omega ( n / ( b \cdot p ) ) = \Omega ( \sqrt { n } / b )$ bits to decide if there is a cycle of length $k$ ; and $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ has to have $d = \Omega ( \sqrt { n } / ( w \log n ) )$ depth to do the same.
+
+(b) In the construction of Abboud et al. (2016), each string consists of $q = \Omega ( n )$ bits. The strings are used to encode the connectivity of subgraphs annotated in dark gray: an edge exists between the red nodes $i$ and $q$ if and only if the $i$ -th bit of $s _ { a }$ is one (and similarly for green). Due to the graph construction, the cut between Alice and Bob has $O ( \log { q } )$ edges. Moreover, About et al. proved that $G ( s _ { a } , s _ { b } )$ has diameter at least five if and only if the sets defined by $s _ { a }$ and $s _ { b }$ are disjoint. This implies that $d = \Omega ( n / ( w \log ^ { 2 } n ) )$ depth is necessary to compute the graph diameter in $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ .
+
+# D THE COST OF ANONYMITY
+
+There is a striking difference between the power of anonymous networks and those in which nodes have the ability to uniquely identify each other, e.g., based on ids or discriminative attributes (see the survey by Suomela (2013)).
+
+To illustrate this phenomenon, I consider a thought experiment where a node is tasked with reconstructing the graph topology in the LOCAL model. In the left, Figure 4 depicts the red node’s knowledge after two rounds (equivalent to a $\mathsf { G N N } _ { \mathsf { m p } } ^ { \mathsf { n } }$ having $d = 2$ ) when each node has a unique identifier (color). At the end of the first round, each node is aware of its neighbors and after two rounds the entire graph has been successfully reconstructed in red’s memory.
+
+In the right subfigure, nodes do not possess ids (as in the analysis of (Xu et al., 2018; Morris et al., 2019)) and thus cannot distinguish which of their neighbors are themselves adjacent. As such, the red node cannot tell whether the graph contains cycles: after two rounds there are at least two plausible topologies that could explain its observations.
+
+![](images/ad2fa39385f3ec73579ebc093a2176cac62316873a1efcbbbdbecc3e9a632134.jpg)  
+Figure 4: Toy example of message exchange from the perspective of the red node. The arrows show where each received message comes from and the message content is shown in light gray boxes. Red’s knowledge of the graph topology is depicted at the bottom.

parse/train/BJfvknCqFQ/BJfvknCqFQ.md

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+# A ROTATION AND A TRANSLATION SUFFICE: FOOLING CNNS WITH SIMPLE TRANSFORMATIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We show that simple spatial transformations, namely translations and rotations alone, suffice to fool neural networks on a significant fraction of their inputs in multiple image classification tasks. Our results are in sharp contrast to previous work in adversarial robustness that relied on more complicated optimization approaches unlikely to appear outside a truly adversarial context. Moreover, the misclassifying rotations and translations are easy to find and require only a few black-box queries to the target model. Overall, our findings emphasize the need to design robust classifiers even for natural input transformations in benign settings.
+
+# 1 INTRODUCTION
+
+Neural networks are now widely embraced as dominant solutions in computer vision (Krizhevsky et al., 2012; He et al., 2016), speech recognition (Graves et al., 2013), and natural language processing (Collobert & Weston, 2008). While their accuracy scores often match (and sometimes go beyond) human-level performance on key benchmarks (He et al., 2015; Taigman et al., 2014), we still do not understand how robust neural networks are. A prominent issue in this context is the existence of so-called adversarial examples, i.e., inputs that are almost indistinguishable from natural data to a human but cause state-of-the-art classifiers to make incorrect predictions with high confidence (Szegedy et al., 2013; Goodfellow et al., 2014). This raises concerns about the use of neural networks in contexts where reliability, dependability, and security are important desiderata.
+
+There is a long line of work on methods for constructing adversarial perturbations in various settings (Szegedy et al., 2013; Goodfellow et al., 2014; Kurakin et al., 2016a;b; Sharif et al., 2016; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2016; Papernot et al., 2017; Madry et al., 2017; Athalye et al., 2017). However, these methods are quite sophisticated and the resulting perturbations tend to be fairly contrived since they often rely on fine-tuned control over a large number of input pixels or audio samples. So one may suspect that adversarial examples constitute a problem only in the presence of a truly malicious attacker and are unlikely to arise in more benign environments. In particular, the focus on intricate worst-case attacks so far raises a natural question:
+
+Are neural networks robust to simple, naturally-occurring transformations of their input?
+
+We address this question by studying two basic image transformations: translations and rotations. While these transformations appear natural to a human, we show that small rotations and translations alone (i.e., without any additional fine-tuned perturbation) can cause a significant drop in the model’s performance. This holds even when the model has been trained using appropriate data augmentation and no visual information is lost due to these transformations (e.g. due to cropping, see Figure 1).
+
+# 1.1 OUR METHODOLOGY AND RESULTS
+
+We start with standard image classifiers for the MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009), and ImageNet (Russakovsky et al., 2015) datasets. The classifiers achieve close to state-of-the-art performance on the respective benchmarks. Nevertheless, we demonstrate that small transformations can cause a significant drop in classification accuracy for these models. Depending on dataset and model, this drop ranges from $34 \%$ to as high as $90 \%$ for the worst combination of rotation angle and translation shift. Even for a small random transformation, the accuracy can drop by up to $30 \%$ . These results demonstrate that robustness to rotations and translations should also be a concern in standard classification problems outside an adversarial security context.
+
+![](images/ea20825e83c337f09b98fee548cdb575b880d5b0d9764434ef6dcaa1ce2b19a0.jpg)  
+Figure 1: Examples of adversarial transformations and their predictions in the standard, ”black canvas”, and reflection padding setting.
+
+Moreover, we show that direct access to the model (or a surrogate) is not necessary to find such misclassifying transformations. Choosing the worst out of 10 random transformations suffices to reduce the accuracy of these models by $26 \%$ on MNIST, $72 \%$ on CIFAR10, and $28 \%$ on ImageNet (top 1 accuracy). Hence our results also give a strong baseline for fooling classifiers with a small number of non-adaptive queries.
+
+Finally, we examine possible ways to alleviate these vulnerabilities. A natural first step is to augment the training procedure with rotations and translations. While this does largely mitigate the problem on MNIST, the models trained on CIFAR10 and ImageNet are still far from robust. We thus propose two natural methods for further increasing the robustness of these models. These methods are based on robust optimization and aggregation of random input transformations. They offer significant improvements in classification accuracy but also come with considerable computational overhead. Even then, they are still not sufficient to completely mitigate the vulnerability. This suggests that obtaining models robust to spatial transformations of their inputs remains a challenge.
+
+Finally, we examine the interplay between rotations / translations and the widely used $\ell _ { \infty }$ -based adversarial examples. We observe that robustness to these two classes of input perturbations is largely orthogonal to each other. In particular, pixel-based robustness does not imply spatial robustness, while combining spatial and $\ell _ { \infty }$ -bounded transformations seems to have a cumulative effect in reducing classification accuracy. This emphasizes the need to broaden the notions of image similarity in the adversarial examples literature beyond the common $\ell _ { p }$ -balls.
+
+# 1.2 SUMMARY OF CONTRIBUTIONS
+
+We perform extensive experiments that provide a fine-grained understanding of rotation / translation robustness on a wide spectrum of datasets and training regimes. In summary, we show that:
+
+• A simple attack based solely on rotations and translations is effective against state-of-theart neural networks. This holds even when the model has been trained with appropriate data augmentation and no image information is lost during the spatial transformation.   
+• Rotation / translation attacks are easy to execute, requiring only a few black-box queries.   
+• It is possible to increase a model’s robustness to rotations and translations at the cost of increased training and / or inference time. However, these methods are still not sufficient to fully recover the accuracy on unmodified images. Robustness to $\ell _ { \infty }$ -bounded perturbations does not significantly affect spatial robustness. Instead, these two notions appear orthogonal to each other.   
+• First-order methods are significantly less effective for finding adversarial transformations than an exhaustive search over a fine grid of transformations. This is in stark contrast to $\ell _ { p }$ -bounded perturbrbations where first-order methods have been very successful (Carlini & Wagner, 2016; Madry et al., 2017). Hence rigorous evaluation of model robustness in this spatial setting requires techniques that are different from $\ell _ { p }$ -bounded adversarial examples.
+
+# 2 ADVERSARIAL ROTATIONS AND TRANSLATIONS
+
+Recall that in the context of image classification, an adversarial example for a given input image $x$ and a classifier $C$ is an image $x ^ { \bar { \prime } }$ that satisfies two properties: (i) on the one hand, the adversarial example $x ^ { \prime }$ causes the classifier $C$ to output a different label on $x ^ { \prime }$ than on $x$ , i.e., we have $C ( x ) \neq$ $C ( { \boldsymbol { x } } ^ { \prime } )$ . (ii) On the other hand, the adversarial example $x ^ { \prime }$ is “visually similar” to $x$ .
+
+Clearly, the notion of visual similarity is not precisely defined here. In fact, providing a precise and rigorous definition is extraordinarily difficult as it would require formally capturing the notion of human perception. Consequently, previous work largely settled on the assumption that $x ^ { \prime }$ is a valid adversarial example for $x$ if and only if $\| x - x ^ { \prime } \| _ { p } \ \leq \varepsilon$ for some $p \in [ 0 , \infty ]$ and $\varepsilon$ small enough. This convention is based on the fact that two images are indeed visually similar when they are close enough in some $\ell _ { p }$ norm. However, the converse is not necessarily true. A small rotation or translation of an image usually appears visually similar to a human, yet can lead to a large change when measured in an $\ell _ { p }$ norm. We aim to expand the range of similarity measures considered in the adversarial examples literature by investigating robustness to small rotations and translations.
+
+Attack methods. Our first goal is to develop sufficiently strong methods for generating adversarial rotations and translations. In the context of pixel-wise $\ell _ { p }$ perturbations, the most successful approach for constructing adversarial examples so far has been to employ optimization methods on a suitable loss function (Szegedy et al., 2013; Goodfellow et al., 2014; Carlini & Wagner, 2016). Following this approach, we parametrize our attack method with a set of tunable parameters and then optimize over these parameters. We perform this optimization in three distinct ways:
+
+• First-Order Method (FO): Starting from a random choice of parameters, we iteratively take steps in the direction of the gradient of the loss function. This is the direction that locally maximizes the loss of the classifier (as a surrogate for misclassification probability). Note that unlike the $\ell _ { p }$ -norm case, we are not optimizing in the pixel space but in the latent space of rotation and translation parameters. Grid Search: We discretize the parameter space and exhaustively examine every possible parametrization of the attack to find one that causes the classifier to give a wrong prediction (if such a parametrization exists). Since our parameter space is low-dimensional enough, this method is computationally feasible (in contrast to a grid search for $\ell _ { p }$ -based adversaries). Worst-of- $k$ : We randomly sample $k$ different choices of attack parameters and choose the one on which the model performs worst. As we increase $k$ , this attack interpolates between a random choice and grid search.
+
+While a first-order attack requires full knowledge of the model to compute the gradient of the loss with respect to the input, the other two attacks do not. They only require the outputs corresponding to chosen inputs, which can be done witho only query access to the target model.
+
+Next, we need to define the exact range of attacks we want to optimize over. For the case of rotation and translation attacks, we wish to find parameters $( \delta u , \delta v , \theta )$ such that rotating the original image by $\theta$ degrees around the center and then translating it by $( \delta u , \delta v )$ pixels causes the classifier to make a wrong prediction. Formally, the pixel at position $( u , v )$ is moved to the following position (assuming the point $( 0 , 0 )$ is the center of the image):
+
+$$
+\left[ { \begin{array} { c } { u ^ { \prime } } \\ { v ^ { \prime } } \end{array} } \right] = \left[ { \begin{array} { c c } { \cos \theta } & { - \sin \theta } \\ { \sin \theta } & { \cos \theta } \end{array} } \right] \cdot \left[ { \begin{array} { c } { u } \\ { v } \end{array} } \right] + \left[ { \begin{array} { c } { \delta u } \\ { \delta v } \end{array} } \right] .
+$$
+
+We implement this transformation in a differentiable manner using the spatial transformer blocks of (Jaderberg et al., 2015). In order to handle pixels that are mapped to non-integer coordinates, the transformer units include a differentiable bilinear interpolation routine. Since our loss function is differentiable with respect to the input and the transformation is in turn differentiable with respect to its parameters, we can obtain gradients of the model’s loss function w.r.t. the perturbation parameters. This enables us to apply a first-order optimization method to our problem.
+
+By defining the spatial transformation for some $x$ as $T ( x ; \delta u , \delta v , \theta )$ , we construct an adversarial perturbation for $x$ by solving the problem
+
+$$
+\operatorname * { m a x } _ { \delta u , \delta v , \theta } \mathcal { L } ( x ^ { \prime } , y ) , \quad \mathrm { f o r } x ^ { \prime } = T ( x ; \delta u , \delta v , \theta ) \ ,
+$$
+
+where $\mathcal { L }$ is the loss function of the neural network1, and $y$ is the correct label for $x$ . Since this is a non-concave maximization problem, there are no guarantees for the global optimality of a general first order method.
+
+# 3 IMPROVING INVARIANCE TO SPATIAL TRANSFORMATIONS
+
+As we will see in Section 4, augmenting the training set with random rotations and translations does improve the robustness of the model against such random transformations. However, data augmentation does not significantly improve the robustness against worst-case attacks and sometimes leads to a drop in accuracy on unperturbed images. To address these issues, we explore two simple baselines that turn out to be surprisingly effective.
+
+Robust Optimization. Instead of performing standard empirical risk minimization to train the classification model, we utilize ideas from robust optimization. Robust optimization has a rich history (Ben-Tal et al., 2009) and has recently been applied successfully in the context of defending neural networks against adversarial examples (Madry et al., 2017; Sinha et al., 2017; Raghunathan et al., 2018; Kolter & Wong, 2017). The main barrier to applying robust optimization for spatial transformations is the lack of an efficient procedure to compute the worst-case perturbation of a given example. Performing a grid search (as described in Section 2) is prohibitive as this would increase the training time by a factor close to the grid size, which can easily be a factor 100 or 1,000. Moreover, the non-convexity of the loss landscape prevents potentially more efficient first-order methods from discovering (approximately) worst-case transformations (see Section 4 for details).
+
+Given that we cannot fully optimize over the space of translations and rotations, we instead use a coarse approximation provided by the worst-of-10 adversary (as described in Section 2). So each time we use an example during training, we first sample 10 transformations of the example uniformly at random from the space of allowed transformations. We then evaluate the model on each of these transformations and train on the one perturbation with the highest loss. This corresponds to approximately minimizing a min-max formulation of robust accuracy similar to (Madry et al., 2017). Training against such an adversary increases the overall time by a factor of roughly six.2
+
+Aggregating Random Transformations. As Section 4 shows, the accuracy against a random transformation is significantly higher than the accuracy against the worst transformation in the allowed attack space. This motivates the following inference procedure: compute a (tyipcally small) number of random transformations of the input image and output the label that occurs most common in the resulting set of predictions. We constrain these random transformations to be within $5 \%$ of the input image size in each translation direction and up to $1 5 ^ { \circ }$ of rotation. 3 The training procedure and model can remain unchanged while the inference time is increased by a small factor (equal to the number of transformations we evaluate on).
+
+Combining Both Methods. The two methods outlined above are orthogonal and in some sense complementary. We can therefore combine robust training (using a worst-of- $\mathbf { \nabla } \cdot \mathbf { k }$ adversary) and majority inference to further increase the robustness of our models.
+
+# 4 EXPERIMENTS
+
+We evaluate standard image classifiers for the MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009) and ImageNet (Russakovsky et al., 2015) datasets. In order to determine the extent to which misclassification is caused by insufficient data augmentation during training, we examine various data augmentation methods. We begin with a description of our experimental setup.
+
+Model Architecture. For MNIST, we use a convolutional neural network derived from the TensorFlow Tutorial (tft). In order to obtain a fully convolutional version of the network, we replace the fully-connected layer by two convolutional layers with 128 and 256 filters each, followed by a global average pooling. For CIFAR10, we consider a standard ResNet (He et al., 2016) model with 4 groups of residual layers with filter sizes [16, 16, 32, 64] and 5 residual units each. We use standard and $\ell _ { \infty }$ -adversarially trained models similar to those studied by Madry et al. (2017).4,5 For ImageNet, we use a ResNet-50 (He et al., 2016) architecture implemented in the tensorpack repository (Wu et al., 2016). We did not modify the model architectures or training procedures.
+
+Attack Space. In order to maintain the visual similarity of images to the natural ones we restrict the space of allowed perturbations to be relatively small. We consider rotations of at most $3 0 ^ { \circ }$ and translations of at most (roughly) $10 \%$ percent of the image size in each direction. This corresponds to 3 pixels for MNIST (image size $2 8 \times 2 8$ ) and CIFAR10 (image size $3 2 \times 3 2$ ), and 24 pixels for ImageNet (image size $2 9 9 \times 2 9 9$ ). For grid search attacks, we consider 5 values per translation direction and 31 values for rotations, equally spaced. For first-order attacks, we use 200 steps of projected gradient descent of step size 0.01 times the parameter range. When rotating and translating the images, we fill the empty space with zeros (black pixels).
+
+Data Augmentation. We consider five variants of training for our models.
+
+• Standard training: The standard training procedure for the respective model architecture.   
+• $\ell _ { \infty }$ -bounded adversarial training: The classifier is trained on $\ell _ { \infty }$ -bounded adversarial examples that are generated with projected gradient descent.   
+• No random cropping: Standard training for CIFAR-10 and ImageNet includes data augmentation via random crops. We investigate the effect of this data augmentation scheme by also training a model without random crops. Random rotations and translations: At each training step, we perform a uniformly random perturbation from the attack space on each training example.   
+• Random rotations and translations from larger intervals: As before, we perform uniformly random perturbations, but now from a superset of the attack space $( 4 0 ^ { \circ } , \pm 1 3 \%$ pixels).
+
+# 4.1 EVALUATING MODEL ROBUSTNESS
+
+We evaluate all models against random and grid search adversaries with rotations and translations considered both separately and together. We report the results in Table 1. We visualize a random subset of successful attacks in Figures 3, 4, and 5 of Appendix A.
+
+Despite the high accuracy of standard models on unperturbed examples and their reasonable performance on random perturbations, a grid search can significantly lower the classifiers’ accuracy on the test set. For the standard models, accuracy drops from $9 9 \%$ to $26 \%$ on MNIST, $93 \%$ to $3 \%$ on CIFAR10, and $76 \%$ to $31 \%$ on ImageNet (Top 1 accuracy).
+
+The addition of random rotations and translations during training greatly improves both the random and adversarial accuracy of the classifier for MNIST and CIFAR10, but less so for ImageNet. For the first two datasets, data augmentation increases the accuracy against a grid adversary by $60 \%$ to $70 \%$ , while the same data augmentation technique adds less than $3 \%$ accuracy on ImageNet.
+
+In Appendix A, we perform a fine-grained investigation of our findings:
+
+• In Figure 8 we examine how many examples can be fooled by (i) rotations only, (ii) translations only, (iii) neither transformation, or (iv) both. We visualize the set of fooling angles for a random sample of the rotations-only grid in Figure 9. We observe that the set of fooling angles is not contiguous. • To investigate how many transformations are adversarial per image, we analyze the percentage of misclassified grid points for each example in Figure 10. While the majority of images has only a small number of adversarial transformations, a significant fraction of images is fooled by $20 \%$ or more of the transformations.
+
+Table 1: Accuracy of different classifiers against rotation and translation adversaries on MNIST, CIFAR10, and ImageNet. The allowed transformations are translations by (roughly) $10 \%$ of the image size and $\pm 3 0 ^ { \circ }$ rotations. The attack parameters are chosen through random sampling or grid search with rotations and translations considered both together (“Rand.”, “Grid”) and separately (“Rand. T.” and “Grid T.” for transformations, “Rand R.” and “Grid R.” for rotations). We consider networks that are trained with (i) the respective standard setup, (ii) no data augmentation (if data augmentation is present in standard setup), (iii) with an $\ell _ { \infty }$ adversary, (iv) with data augmentation corresponding to the attack space $( \pm \mathrm { { 3 p x } , \pm 3 0 ^ { \circ } ) }$ and an enlarged space $( \pm 4 \mathrm { p x } , \pm 4 0 ^ { \circ } )$ , and (v) with worst-of-10 training for both types of augmentations.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=4>Nat.     Rand.    Grid</td><td rowspan=1 colspan=1>Rand. T.  Grid T.</td><td rowspan=1 colspan=1>Rand. R.  Grid R.</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Standard</td><td rowspan=1 colspan=4>99.31%  94.23%  26.02%</td><td rowspan=1 colspan=1>98.61%  89.80%</td><td rowspan=1 colspan=1>95.68%   70.98%</td></tr><tr><td rowspan=5 colspan=1>LSINN</td><td rowspan=1 colspan=1>loo-Adv</td><td rowspan=1 colspan=4>98.65%  88.02%  1.20%</td><td rowspan=1 colspan=1>93.72%  34.13%</td><td rowspan=1 colspan=1>95.27%   72.03%</td></tr><tr><td rowspan=4 colspan=1>Aug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=1 colspan=4>99.53%  99.35%  95.79%</td><td rowspan=1 colspan=1>99.47%  98.66%</td><td rowspan=1 colspan=1>99.34%  98.23%</td></tr><tr><td rowspan=2 colspan=4>99.34%  99.31%  96.95%99.48%  99.37%  97.32%</td><td rowspan=1 colspan=1>99.39%  98.65%</td><td rowspan=2 colspan=1>99.40%   98.49%99.39%  98.62%</td></tr><tr><td rowspan=1 colspan=1>99.50%  99.01%</td><td rowspan=1 colspan=1>99.39%  98.62%</td></tr><tr><td rowspan=1 colspan=3>99.42%</td><td rowspan=1 colspan=2>99.39%  97.88%</td><td rowspan=1 colspan=1>99.45%  98.89%</td><td rowspan=1 colspan=1>99.36%  98.85%</td></tr><tr><td rowspan=7 colspan=1>CITIII</td><td rowspan=7 colspan=1>StandardNo Croploo-AdvAug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=1 colspan=2>92.62%</td><td rowspan=1 colspan=2>60.93%   2.80%</td><td rowspan=1 colspan=1>88.54%  66.17%</td><td rowspan=1 colspan=1>75.36%   24.71%</td></tr><tr><td rowspan=1 colspan=2>90.34%</td><td rowspan=1 colspan=1>01</td><td rowspan=1 colspan=2>54.64%   1.86%</td><td rowspan=1 colspan=1>81.95%  46.07%</td><td rowspan=1 colspan=1>69.23%   18.34%</td></tr><tr><td rowspan=5 colspan=4>80.21%  58.33%   6.02%90.02%  90.92%  58.90%88.83%  91.18%  61.69%91.34%  92.35%  69.17%91.00%  92.11% 71.15%</td><td rowspan=1 colspan=1>78.15%  59.02%</td><td rowspan=1 colspan=1>62.85%   20.98%</td></tr><tr><td rowspan=1 colspan=1>91.76%  79.01%</td><td rowspan=1 colspan=1>91.14%   76.33%</td></tr><tr><td rowspan=1 colspan=1>91.53%  77.42%</td><td rowspan=1 colspan=1>91.10%  76.80%</td></tr><tr><td rowspan=1 colspan=1>92.43%  83.01%</td><td rowspan=2 colspan=1>92.33%  81.82%92.53%  82.25%</td></tr><tr><td rowspan=1 colspan=1>92.28%  82.15%</td></tr><tr><td rowspan=6 colspan=1>1negee</td><td rowspan=6 colspan=1>StandardNo CropAug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=4 colspan=4>75.96%  63.39%  31.42%70.81%  59.09%  16.52%65.96%  68.60%  32.90%66.19%  67.58%  33.86%</td><td rowspan=1 colspan=1>73.24%  60.42%</td><td rowspan=1 colspan=1>67.90%  44.98%</td></tr><tr><td rowspan=1 colspan=1>66.75%  45.17%</td><td rowspan=1 colspan=1>62.78%  34.17%</td></tr><tr><td rowspan=4 colspan=4>65.96%  68.60%  32.90%66.19%  67.58%  33.86%76.14%  73.19%  52.76%74.64%  71.36%  50.23%</td><td rowspan=1 colspan=1>70.27%  45.72%</td><td rowspan=1 colspan=1>69.28%  47.25%</td></tr><tr><td rowspan=1 colspan=1>69.50%  44.60%</td><td rowspan=1 colspan=1>68.88%   48.72%</td></tr><tr><td rowspan=1 colspan=1>74.42%  61.18%</td><td rowspan=2 colspan=1>73.74%   61.06%71.95%  59.23%</td></tr><tr><td rowspan=1 colspan=1>72.86%  59.34%</td></tr></table>
+
+Padding Experiments. A natural question is whether the reduced accuracy of the models is due to the cropping applied during the transformation. We verify that this is not the case by applying zero and reflection padding to the image datasets. We note that the zero padding creates a “black canvas” version of the dataset, ensuring that no information from the original image is lost after a transformation. We show a random set of adversarial examples in this setting in Figure 6 and a full evaluation in Table 4. We also provide more details regarding reflection padding in Section B and provide an evaluation in Table 6. All of these are in Appendix A.
+
+# 4.2 COMPARING ATTACK METHODS
+
+In Table 2 we compare different attack methods on various classifiers and datasets. We observe that worst-of-10 is a powerful adversary despite its limited interaction with the target classifier. The firstorder adversary performs significantly worse. While it is still better than a random transformation , it fails to approximate the ground-truth accuracy of the models and performs significantly worse than the grid adversary and even the worst-of-10 adversary.
+
+Understanding the Failure of First-Order Methods. The fact that first-order methods fail to reliably find adversarial rotations and translations is in sharp contrast to previous work on $\ell _ { p }$ -robustness (Carlini & Wagner, 2016; Madry et al., 2017). For $\ell _ { p }$ -bounded perturbations parametrized directly in pixel space, prior work found the optimization landscape to be well-behaved which allowed first-order methods to consistently find maxima with high loss. In the case of spatial perturbations, we observe that the non-concavity of the problem is a significant barrier for first-order methods. We investigate this issue by visualizing the loss landscape. For a few random examples from the three datasets, we plot the cross-entropy loss of the examples as a function of translation and rotation. Figure 2 shows one example for each dataset and additional examples are visualized in Figure 11 of the appendix. The plots show that the loss landscape is indeed non-concave and contains many local maxima of low value. The low-dimensional problem structure seems to make non-concavity a crucial obstacle. Even for MNIST, where we observe fewer local maxima, the large flat regions prevent first-order methods from finding transformations of high loss.
+
+Table 2: Comparison of attack methods across datasets and models. Worst-of-10 is very effective and significantly reduces the model accuracy despite the limited interaction. The first-order (FO) adversary performs poorly, despite the large number of steps allowed. We compare standard training to Augmentation $( \pm \mathrm { { 3 p x } , \pm 3 0 ^ { \circ } ) }$ . For the full table, see Figure 3 of Appendix A.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">MNIST</td><td colspan="2">CIFAR-10</td><td colspan="2">ImageNet</td></tr><tr><td>Standard</td><td>Aug.</td><td>Standard</td><td>Aug.</td><td>Standard</td><td>Aug.</td></tr><tr><td>Natural</td><td>99.31%</td><td>99.53%</td><td>92.62%</td><td>90.02%</td><td>75.96%</td><td>65.96%</td></tr><tr><td>Worst-of-10</td><td>73.32%</td><td>98.33%</td><td>20.13%</td><td>79.92%</td><td>47.83%</td><td>50.62%</td></tr><tr><td>First-Order</td><td>79.84%</td><td>98.78%</td><td>62.69%</td><td>85.92%</td><td>63.12%</td><td>66.05%</td></tr><tr><td>Grid</td><td>26.02%</td><td>95.79%</td><td>2.80%</td><td>58.92%</td><td>31.42%</td><td>32.90%</td></tr></table>
+
+![](images/037aebd33bb873a22c486788575345d70cc4fdbdf871cfe483c5e7705c723eb1.jpg)  
+Figure 2: Loss landscape of a random example for each dataset when performing left-right translations and rotations. Translations and rotations are restricted to $10 \%$ of the image pixels and $3 0 ^ { \circ }$ , respectively. We observe that the landscape is significantly non-concave, rendering first-order methods to generate adversarial example ineffective. Figure 11 in the appendix shows additional examples.
+
+Relation to Black-Box Attacks. Given its limited interaction with the model, the worst-of-10 adversary achieves a significant reduction in classification accuracy. It performs only 10 random, non-adaptive queries to the model and is still able to find adversarial examples for a large fraction of the inputs (see Table 2). The low query complexity is an important baseline for black-box attacks on neural networks, which recently gained significant interest (Papernot et al., 2017; Chen et al., 2017; Bhagoji et al., 2017; Ilyas et al., 2017). Black-box attacks rely only function evaluations of the target classifier, without additional information such as gradients. The main challenge is to construct an adversarial example from a small number of queries. Our results show that it is possible to find adversarial rotations and translations for a significant fraction of inputs with very few queries.
+
+Combining Spatial and $\ell _ { \infty }$ -Bounded Perturbations Table 1 shows that models trained to be robust to $\ell _ { \infty }$ perturbations do not achieve higher robustness to spatial perturbations. This provides evidence that the two families of perturbation are orthogonal to each other. We further investigate this possibility by considering a combined adversary that utilizes $\ell _ { \infty }$ bounded perturbations on top of rotations and translations. The results are shown in Figure 12. We indeed observe that these combined attacks reduce classification accuracy in an (approximately) additive manner.
+
+# 4.3 EVALUATING OUR DEFENSE METHODS.
+
+As we see in Table 1, training with a worst-of-10 adversary significantly increases the spatial robustness of the model, also compared to data augmentation with random transformations. We conjecture that using more reliable methods to compute the worst-case transformations will further improve these results. Unfortunately, increasing the number of random transformations per training example quickly becomes computationally expensive. And as pointed out above, current first-order methods also appear to be insufficient for finding worst-case transformations efficiently.
+
+Our results for majority-based inference are presented in Table 5 of Appendix A. By combining these two defense, we improve the worst-case performance of the models from $26 \%$ to $98 \%$ on MNIST, from $3 \%$ to $82 \%$ on CIFAR10, and from $31 \%$ to $56 \%$ on ImageNet (Top 1).
+
+# 5 RELATED WORK
+
+The fact that small rotations and translation can fool neural networks on MNIST and CIFAR10 was first observed in (Fawzi & Frossard, 2015). They compute the minimum transformation required to fool the model and use it as a measure for a quantitative comparison of different architectures and training procedures. The main difference to our work is that we focus on the optimization aspect of the problem . We show that a few random queries usually suffice for a successful attack, while firstorder methods are ineffective. Moreover, we go beyond standard data augmentation and evaluate the effectiveness of natural baseline defenses.
+
+The concurrent work of Kanbak et al. (2017) proposes a different first-order method to evaluate the robustness of classifiers based on geodesic distances on a manifold. This metric is harder to interpret than our parametrized attack space. Moreover, given our findings on the non-concavity of the optimization landscape, it is unclear how close their method is to the ground truth (exhaustive enumeration). While they perform a limited study of defenses (adversarial fine-tuning) using their method, it appears to be less effective than our baseline worst-of-10 training. We attribute this difference to the inherent obstacles first-order methods face in this optimization landscape.
+
+Recently, Xiao et al. (2018) and Tramer & Boneh (2017) observed independently that it is possible to \` use various spatial transformations to construct adversarial examples for naturally and adversarially trained models. The main difference from our work is that we show even very simple transformations (translations and rotations) are sufficient to break a variety of classifiers, while the transformations employed in (Xiao et al., 2018) and (Tramer & Boneh, 2017) are more involved. The transformation \` in (Xiao et al., 2018) is based on performing a displacement of individual pixels in the original image constrained to be globally smooth and then optimized for misclassification probability. Tramer & \` Boneh (2017) consider an $\ell _ { \infty }$ -bounded pixel-wise perturbation of a version of the original image that has been slightly rotated and in which a few random pixels have been flipped. Both of these methods require direct access to the attacked model (or a surrogate) to compute (or at least estimate) the gradient of the loss function with respect to the model’s input. In contrast, our attacks can be implemented using only a small number of random, non-adaptive transformations of the input.
+
+# 6 CONCLUSIONS
+
+We examined the robustness of state-of-the-art image classifiers to translations and rotations. We observed that even a small number of randomly chosen perturbations of the input are sufficient to considerably degrade the classifier’s performance.
+
+The fact that common neural networks are vulnerable to simple and naturally occurring spatial transformations (and that these transformations can be found easily from just a few random tries) indicates that adversarial robustness should be a concern not only in a fully worst-case security setting. We conjecture that additional techniques need to be incorporated in the architecture and training procedures of modern classifiers to achieve worst-case spatial robustness. Also, our results underline the need to consider broader notions of similarity than only pixel-wise distances when studying adversarial misclassification attacks. In particular, we view combining the pixel-wise distances with rotations and translations as a next step towards the “right” notion of similarity in the context of images.
+
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+
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+
+# A OMITTED TABLES AND FIGURES
+
+![](images/6be8b1e90011565dbfe18086124e632ae6c4f7015bb8b51d90e17ae7749163ea.jpg)  
+Figure 3: MNIST. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \circ }$ of the original image and translations up to 3 pixels per direction (image size $2 8 \times 2 8$ ). Each example is visualized along with its predicted label in the original and perturbed versions.
+
+![](images/26b7cdf24cc48d671ac5febb8af59b365b797de29104fd693f5d7858c1d01466.jpg)  
+Figure 4: CIFAR10. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \circ }$ of the original and translations up to 3 pixels per directions (image size $3 2 \times 3 2 ,$ ). Each example is visualized along with its predicted label in the original and perturbed version.
+
+![](images/355d6ece4af015878f75c7904074cb3ce8fc4acf1a2682a83017eb3cdd94580a.jpg)  
+Figure 5: ImageNet. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \circ }$ of the original and translations up to 24 pixels per directions (image size $2 9 9 \times 2 9 9$ ). Each example is visualized along with its predicted label in the original and perturbed version.
+
+![](images/9adc97254f9a49fc289bf63a114c93c63d72ba5fe677cc63dbad9dc88db6a269.jpg)  
+Figure 6: Sample adversarial transformations for the ”black-canvas” setting for the standard models on CIFAR10 and ImageNet.
+
+![](images/34f61e1cb5c8fcd102da1367f3949ca26b3ba5ce8a043f83a1151970aa4d7030.jpg)  
+Figure 7: Sample adversarial transformations for the reflection padding setting for the standard models on CIFAR10.
+
+![](images/af22e88fb367464ae08d512ced58677326738b42ff7c1d32c031f9152e88a8f2.jpg)  
+Figure 8: Fine-grained dataset analysis. For each model, we visualize what percent of the test set can be fooled via various methods. We compute how many examples can be fooled with either translations or rotations (”any”), how many can be fooled only by one of these, and how many require a combination to be fooled (”both”).
+
+![](images/36b377a5f8e6a35b475cb5fcb813f0fc17f6ad24d95ced1b9f5569a67d62fb9e.jpg)  
+Figure 9: Visualizing which angles fool the classifier for 50 random examples. For each dataset and model, we visualize one example per row. Red corresponds to misclassification of the images. We observe that the angles fooling the models form a highly non-convex set.
+
+![](images/d63c0b01d615e7cd366824c1817405e3b76745a1255c2ab4b6273fbdf12922c4.jpg)  
+Figure 10: Cumulative Density Function plots. For each fraction of grid points $p$ , we plot the percentage of correctly classified test set examples that are fooled by at least $p$ of the grid points. For instance, we can see from the first plot, MNIST Translations and Rotations, that approximately $10 \%$ of the correctly classified natural examples are misclassified under $1 / 5$ of the grid points transformations.
+
+Table 3: Comparison of attack methods across datasets and models.   
+
+<table><tr><td></td><td>Model</td><td>Natural</td><td>Worst-of-10</td><td>FO</td><td>Grid</td></tr><tr><td>LSINW</td><td>Standard lo-Adversarially Trained Aug.30 (±3px,±30°) Aug. 40 (±4±,±40°)</td><td>99.31% 98.65% 99.53% 99.34%</td><td>73.32% 51.18% 98.33% 98.49%</td><td>79.84% 81.23% 98.78% 98.74%</td><td>26.02% 1.20% 95.79% 96.95%</td></tr><tr><td>CITIIIT</td><td>Standard No Crop loo-Adversarially Trained Aug.30 (±3px,±30°) Aug. 40 (±4px,±40°)</td><td>92.62% 90.34% 80.21% 90.02% 88.83%</td><td>20.13% 15.04% 19.38% 79.92% 80.47%</td><td>62.69% 52.27% 33.24% 85.92% 85.48%</td><td>2.80% 1.86% 6.02% 58.92% 61.69%</td></tr><tr><td>1eenee</td><td>Standard No Crop Aug.30 (±24px,±30°) Aug.40 (±32px,±40°)</td><td>75.96% 70.81% 65.96% 66.19%</td><td>47.83% 35.52% 50.62% 51.11%</td><td>63.12% 55.93% 66.05% 66.14%</td><td>31.42% 16.52% 32.90% 33.86%</td></tr></table>
+
+Table 4: Evaluation of a subset of Table 1 in the “black-canvas” setting (images are zero-padded to avoid cropping due to rotations and translations). The models are trained on padded images.   
+
+<table><tr><td></td><td></td><td>Natural</td><td>Random</td><td>Worst-of-10</td><td>Grid</td><td>Trans. Grid</td><td>Rot. Grid</td></tr><tr><td rowspan="4">CITIII</td><td>Standard</td><td>91.81%</td><td>70.23%</td><td>25.51%</td><td>6.55%</td><td>83.38%</td><td>12.44%</td></tr><tr><td>No Crop</td><td>89.70%</td><td>52.86%</td><td>14.14%</td><td>1.17%</td><td>47.94%</td><td>9.46%</td></tr><tr><td>Aug.30 (±3px,±30°)</td><td>91.45%</td><td>90.82%</td><td>80.53%</td><td>63.64%</td><td>82.28%</td><td>76.32%</td></tr><tr><td>Aug.40 (±4px,±40°)</td><td>91.24%</td><td>91.00%</td><td>81.81%</td><td>66.64%</td><td>81.75%</td><td>78.57%</td></tr><tr><td rowspan="4">1aeege</td><td>Standard</td><td>73.60%</td><td>46.59%</td><td>29.51%</td><td>15.38%</td><td>28.03%</td><td>23.81%</td></tr><tr><td>No Crop</td><td>66.28%</td><td>38.70%</td><td>14.17%</td><td>3.43%</td><td>8.87%</td><td>10.97%</td></tr><tr><td>Aug.30 (±24px,±30°)</td><td>64.60%</td><td>67.75%</td><td>47.32%</td><td>28.51%</td><td>45.33%</td><td>39.33%</td></tr><tr><td>Aug.40 (±32px,±40°)</td><td>49.20%</td><td>57.69%</td><td>38.36%</td><td>22.10%</td><td>32.84%</td><td>32.95%</td></tr></table>
+
+![](images/61eeaa79b4ebcc0e672828f115ea91e7b4c7ab09d22c040a193ca6de379d6445.jpg)  
+Figure 11: Loss landscape of 4 random examples for each dataset when performing left-right translations and rotations. Translations and rotations are restricted to $10 \%$ of the image pixels and $3 0 ^ { \circ }$ respectively. We observe that the landscape is significantly non-concave, making rendering FO methods for adversarial example generation powerless.
+
+![](images/603dbd3b56ec7b704aaf472369fd8da9b002abacfcff3173c1d2f263c55c6e19.jpg)  
+Figure 12: Accuracy of different classifiers against $\ell _ { \infty }$ -bounded adversaries with various values of $\varepsilon$ and spatial transformations. For each value of $\varepsilon$ , we perform PGD to find the most adversarial $\ell _ { \infty }$ - bounded perturbation. Additionally, we combine PGD with random rotations and translations and with a grid search over rotations and translations in order to find the transformation that combines with PGD in the most adversarial way.
+
+# B MIRROR PADDING
+
+In the experiments of Section 4, we filled the remaining pixels of rotated and translated images with black (also known as zero or constant padding). This is the standard approach used when performing random cropping for data augmentation purposes. We briefly examined the effect of mirror padding, that is replacing empty pixels by reflecting the image around the border6. The results are shown in Table 6. We observed that training with one padding method and evaluating using the other resulted in a significant drop in accuracy. Training using one of these methods randomly for each example resulted in a model which roughly matched the best-case of the two individual cases.
+
+Table 5: Majority Defense. Accuracy of different models on the natural evaluation set and against a combined rotation and translation adversary using aggregation of multiple random transformations.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Natural Acc.Stand.   Vote</td><td rowspan=1 colspan=1>Grid Acc.Stand.   Vote</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Standard</td><td rowspan=1 colspan=1>99.31% 98.71%</td><td rowspan=2 colspan=1>26.02% 18.80%95.79%95.32%96.95% 97.65%97.32% 96.95%97.88% 98.47%</td></tr><tr><td rowspan=1 colspan=1>JSINN</td><td rowspan=1 colspan=1>Aug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>99.53%99.41%99.34%99.25%99.48% 99.40%99.42% 99.41%</td></tr><tr><td rowspan=1 colspan=1>CITIIIO</td><td rowspan=1 colspan=1>StandardAug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>92.62% 80.37%90.02% 92.70%88.83% 92.50%91.34% 93.38%91.00% 93.40%</td><td rowspan=1 colspan=1>2.82%  7.85%58.90% 69.65%61.69% 76.54%69.17% 77.33%71.15% 81.52%</td></tr><tr><td rowspan=1 colspan=1>1eeeeer</td><td rowspan=1 colspan=1>StandardAug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>75.96% 73.19%65.96% 72.44%66.19% 71.46%76.14%74.92%74.64%73.38%</td><td rowspan=1 colspan=1>31.42% 40.21%32.90% 44.46%33.86% 46.98%52.76%56.45%50.23%56.23%</td></tr></table>
+
+Table 6: CIFAR10: The effect of using reflection or zero padding when training a model. The experimental setup matches that of Section 4. Zero padding refers to filling the empty pixels caused by translations and rotations with black. Mirror padding corresponds to using a reflection of the images. ”Both” refers to training using both methods and alternating randomly between them for each training example.   
+
+<table><tr><td></td><td>Natural</td><td>Random (Zero)</td><td>Random (Mirror)</td><td>Grid Search (Zero)</td><td>Grid Search (Mirror)</td></tr><tr><td>Standard Nat</td><td>92.62%</td><td>60.76%</td><td>66.42%</td><td>8.08%</td><td>5.37%</td></tr><tr><td>Standard Adv</td><td>80.21%</td><td>59.79%</td><td>67.12%</td><td>7.20%</td><td>12.89%</td></tr><tr><td>Aug. A, Zero</td><td>90.25%</td><td>91.09%</td><td>87.67%</td><td>59.87%</td><td>40.55%</td></tr><tr><td>Aug. B, Zero</td><td>89.55%</td><td>91.40%</td><td>87.94%</td><td>62.42%</td><td>42.37%</td></tr><tr><td>Aug. A, Mirror</td><td>92.25%</td><td>88.43%</td><td>91.05%</td><td>41.46%</td><td>53.95%</td></tr><tr><td>Aug. B, Mirror</td><td>92.03%</td><td>88.58%</td><td>91.34%</td><td>45.44%</td><td>57.97%</td></tr><tr><td>Aug. A, Both</td><td>91.80%</td><td>90.98%</td><td>91.28%</td><td>56.95%</td><td>52.60%</td></tr><tr><td>Aug. B, Both</td><td>91.57%</td><td>91.87%</td><td>91.11%</td><td>60.46%</td><td>56.13%</td></tr></table>

parse/train/BJfvknCqFQ/BJfvknCqFQ_content_list.json

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+    {
+        "type": "text",
+        "text": "A ROTATION AND A TRANSLATION SUFFICE: FOOLING CNNS WITH SIMPLE TRANSFORMATIONS ",
+        "text_level": 1,
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+    {
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+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We show that simple spatial transformations, namely translations and rotations alone, suffice to fool neural networks on a significant fraction of their inputs in multiple image classification tasks. Our results are in sharp contrast to previous work in adversarial robustness that relied on more complicated optimization approaches unlikely to appear outside a truly adversarial context. Moreover, the misclassifying rotations and translations are easy to find and require only a few black-box queries to the target model. Overall, our findings emphasize the need to design robust classifiers even for natural input transformations in benign settings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            407,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks are now widely embraced as dominant solutions in computer vision (Krizhevsky et al., 2012; He et al., 2016), speech recognition (Graves et al., 2013), and natural language processing (Collobert & Weston, 2008). While their accuracy scores often match (and sometimes go beyond) human-level performance on key benchmarks (He et al., 2015; Taigman et al., 2014), we still do not understand how robust neural networks are. A prominent issue in this context is the existence of so-called adversarial examples, i.e., inputs that are almost indistinguishable from natural data to a human but cause state-of-the-art classifiers to make incorrect predictions with high confidence (Szegedy et al., 2013; Goodfellow et al., 2014). This raises concerns about the use of neural networks in contexts where reliability, dependability, and security are important desiderata. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There is a long line of work on methods for constructing adversarial perturbations in various settings (Szegedy et al., 2013; Goodfellow et al., 2014; Kurakin et al., 2016a;b; Sharif et al., 2016; Moosavi-Dezfooli et al., 2016; Carlini & Wagner, 2016; Papernot et al., 2017; Madry et al., 2017; Athalye et al., 2017). However, these methods are quite sophisticated and the resulting perturbations tend to be fairly contrived since they often rely on fine-tuned control over a large number of input pixels or audio samples. So one may suspect that adversarial examples constitute a problem only in the presence of a truly malicious attacker and are unlikely to arise in more benign environments. In particular, the focus on intricate worst-case attacks so far raises a natural question: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Are neural networks robust to simple, naturally-occurring transformations of their input? ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We address this question by studying two basic image transformations: translations and rotations. While these transformations appear natural to a human, we show that small rotations and translations alone (i.e., without any additional fine-tuned perturbation) can cause a significant drop in the model’s performance. This holds even when the model has been trained using appropriate data augmentation and no visual information is lost due to these transformations (e.g. due to cropping, see Figure 1). ",
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+    {
+        "type": "text",
+        "text": "1.1 OUR METHODOLOGY AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We start with standard image classifiers for the MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009), and ImageNet (Russakovsky et al., 2015) datasets. The classifiers achieve close to state-of-the-art performance on the respective benchmarks. Nevertheless, we demonstrate that small transformations can cause a significant drop in classification accuracy for these models. Depending on dataset and model, this drop ranges from $34 \\%$ to as high as $90 \\%$ for the worst combination of rotation angle and translation shift. Even for a small random transformation, the accuracy can drop by up to $30 \\%$ . These results demonstrate that robustness to rotations and translations should also be a concern in standard classification problems outside an adversarial security context. ",
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+    {
+        "type": "image",
+        "img_path": "images/ea20825e83c337f09b98fee548cdb575b880d5b0d9764434ef6dcaa1ce2b19a0.jpg",
+        "image_caption": [
+            "Figure 1: Examples of adversarial transformations and their predictions in the standard, ”black canvas”, and reflection padding setting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            253,
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Moreover, we show that direct access to the model (or a surrogate) is not necessary to find such misclassifying transformations. Choosing the worst out of 10 random transformations suffices to reduce the accuracy of these models by $26 \\%$ on MNIST, $72 \\%$ on CIFAR10, and $28 \\%$ on ImageNet (top 1 accuracy). Hence our results also give a strong baseline for fooling classifiers with a small number of non-adaptive queries. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, we examine possible ways to alleviate these vulnerabilities. A natural first step is to augment the training procedure with rotations and translations. While this does largely mitigate the problem on MNIST, the models trained on CIFAR10 and ImageNet are still far from robust. We thus propose two natural methods for further increasing the robustness of these models. These methods are based on robust optimization and aggregation of random input transformations. They offer significant improvements in classification accuracy but also come with considerable computational overhead. Even then, they are still not sufficient to completely mitigate the vulnerability. This suggests that obtaining models robust to spatial transformations of their inputs remains a challenge. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, we examine the interplay between rotations / translations and the widely used $\\ell _ { \\infty }$ -based adversarial examples. We observe that robustness to these two classes of input perturbations is largely orthogonal to each other. In particular, pixel-based robustness does not imply spatial robustness, while combining spatial and $\\ell _ { \\infty }$ -bounded transformations seems to have a cumulative effect in reducing classification accuracy. This emphasizes the need to broaden the notions of image similarity in the adversarial examples literature beyond the common $\\ell _ { p }$ -balls. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.2 SUMMARY OF CONTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            609,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We perform extensive experiments that provide a fine-grained understanding of rotation / translation robustness on a wide spectrum of datasets and training regimes. In summary, we show that: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• A simple attack based solely on rotations and translations is effective against state-of-theart neural networks. This holds even when the model has been trained with appropriate data augmentation and no image information is lost during the spatial transformation.   \n• Rotation / translation attacks are easy to execute, requiring only a few black-box queries.   \n• It is possible to increase a model’s robustness to rotations and translations at the cost of increased training and / or inference time. However, these methods are still not sufficient to fully recover the accuracy on unmodified images. Robustness to $\\ell _ { \\infty }$ -bounded perturbations does not significantly affect spatial robustness. Instead, these two notions appear orthogonal to each other.   \n• First-order methods are significantly less effective for finding adversarial transformations than an exhaustive search over a fine grid of transformations. This is in stark contrast to $\\ell _ { p }$ -bounded perturbrbations where first-order methods have been very successful (Carlini & Wagner, 2016; Madry et al., 2017). Hence rigorous evaluation of model robustness in this spatial setting requires techniques that are different from $\\ell _ { p }$ -bounded adversarial examples. ",
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+        "type": "text",
+        "text": "2 ADVERSARIAL ROTATIONS AND TRANSLATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Recall that in the context of image classification, an adversarial example for a given input image $x$ and a classifier $C$ is an image $x ^ { \\bar { \\prime } }$ that satisfies two properties: (i) on the one hand, the adversarial example $x ^ { \\prime }$ causes the classifier $C$ to output a different label on $x ^ { \\prime }$ than on $x$ , i.e., we have $C ( x ) \\neq$ $C ( { \\boldsymbol { x } } ^ { \\prime } )$ . (ii) On the other hand, the adversarial example $x ^ { \\prime }$ is “visually similar” to $x$ . ",
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+        "type": "text",
+        "text": "Clearly, the notion of visual similarity is not precisely defined here. In fact, providing a precise and rigorous definition is extraordinarily difficult as it would require formally capturing the notion of human perception. Consequently, previous work largely settled on the assumption that $x ^ { \\prime }$ is a valid adversarial example for $x$ if and only if $\\| x - x ^ { \\prime } \\| _ { p } \\ \\leq \\varepsilon$ for some $p \\in [ 0 , \\infty ]$ and $\\varepsilon$ small enough. This convention is based on the fact that two images are indeed visually similar when they are close enough in some $\\ell _ { p }$ norm. However, the converse is not necessarily true. A small rotation or translation of an image usually appears visually similar to a human, yet can lead to a large change when measured in an $\\ell _ { p }$ norm. We aim to expand the range of similarity measures considered in the adversarial examples literature by investigating robustness to small rotations and translations. ",
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+        "type": "text",
+        "text": "Attack methods. Our first goal is to develop sufficiently strong methods for generating adversarial rotations and translations. In the context of pixel-wise $\\ell _ { p }$ perturbations, the most successful approach for constructing adversarial examples so far has been to employ optimization methods on a suitable loss function (Szegedy et al., 2013; Goodfellow et al., 2014; Carlini & Wagner, 2016). Following this approach, we parametrize our attack method with a set of tunable parameters and then optimize over these parameters. We perform this optimization in three distinct ways: ",
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+        "type": "text",
+        "text": "• First-Order Method (FO): Starting from a random choice of parameters, we iteratively take steps in the direction of the gradient of the loss function. This is the direction that locally maximizes the loss of the classifier (as a surrogate for misclassification probability). Note that unlike the $\\ell _ { p }$ -norm case, we are not optimizing in the pixel space but in the latent space of rotation and translation parameters. Grid Search: We discretize the parameter space and exhaustively examine every possible parametrization of the attack to find one that causes the classifier to give a wrong prediction (if such a parametrization exists). Since our parameter space is low-dimensional enough, this method is computationally feasible (in contrast to a grid search for $\\ell _ { p }$ -based adversaries). Worst-of- $k$ : We randomly sample $k$ different choices of attack parameters and choose the one on which the model performs worst. As we increase $k$ , this attack interpolates between a random choice and grid search. ",
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+    {
+        "type": "text",
+        "text": "While a first-order attack requires full knowledge of the model to compute the gradient of the loss with respect to the input, the other two attacks do not. They only require the outputs corresponding to chosen inputs, which can be done witho only query access to the target model. ",
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+        "type": "text",
+        "text": "Next, we need to define the exact range of attacks we want to optimize over. For the case of rotation and translation attacks, we wish to find parameters $( \\delta u , \\delta v , \\theta )$ such that rotating the original image by $\\theta$ degrees around the center and then translating it by $( \\delta u , \\delta v )$ pixels causes the classifier to make a wrong prediction. Formally, the pixel at position $( u , v )$ is moved to the following position (assuming the point $( 0 , 0 )$ is the center of the image): ",
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+    {
+        "type": "equation",
+        "img_path": "images/85acaf978ec2be4368437539b02c9ff807108a80b6b40fbaa564d4983681f907.jpg",
+        "text": "$$\n\\left[ { \\begin{array} { c } { u ^ { \\prime } } \\\\ { v ^ { \\prime } } \\end{array} } \\right] = \\left[ { \\begin{array} { c c } { \\cos \\theta } & { - \\sin \\theta } \\\\ { \\sin \\theta } & { \\cos \\theta } \\end{array} } \\right] \\cdot \\left[ { \\begin{array} { c } { u } \\\\ { v } \\end{array} } \\right] + \\left[ { \\begin{array} { c } { \\delta u } \\\\ { \\delta v } \\end{array} } \\right] .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We implement this transformation in a differentiable manner using the spatial transformer blocks of (Jaderberg et al., 2015). In order to handle pixels that are mapped to non-integer coordinates, the transformer units include a differentiable bilinear interpolation routine. Since our loss function is differentiable with respect to the input and the transformation is in turn differentiable with respect to its parameters, we can obtain gradients of the model’s loss function w.r.t. the perturbation parameters. This enables us to apply a first-order optimization method to our problem. ",
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+        "type": "text",
+        "text": "By defining the spatial transformation for some $x$ as $T ( x ; \\delta u , \\delta v , \\theta )$ , we construct an adversarial perturbation for $x$ by solving the problem ",
+        "bbox": [
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+    {
+        "type": "equation",
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+        "text": "$$\n\\operatorname * { m a x } _ { \\delta u , \\delta v , \\theta } \\mathcal { L } ( x ^ { \\prime } , y ) , \\quad \\mathrm { f o r } x ^ { \\prime } = T ( x ; \\delta u , \\delta v , \\theta ) \\ ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the loss function of the neural network1, and $y$ is the correct label for $x$ . Since this is a non-concave maximization problem, there are no guarantees for the global optimality of a general first order method. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3 IMPROVING INVARIANCE TO SPATIAL TRANSFORMATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As we will see in Section 4, augmenting the training set with random rotations and translations does improve the robustness of the model against such random transformations. However, data augmentation does not significantly improve the robustness against worst-case attacks and sometimes leads to a drop in accuracy on unperturbed images. To address these issues, we explore two simple baselines that turn out to be surprisingly effective. ",
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+    {
+        "type": "text",
+        "text": "Robust Optimization. Instead of performing standard empirical risk minimization to train the classification model, we utilize ideas from robust optimization. Robust optimization has a rich history (Ben-Tal et al., 2009) and has recently been applied successfully in the context of defending neural networks against adversarial examples (Madry et al., 2017; Sinha et al., 2017; Raghunathan et al., 2018; Kolter & Wong, 2017). The main barrier to applying robust optimization for spatial transformations is the lack of an efficient procedure to compute the worst-case perturbation of a given example. Performing a grid search (as described in Section 2) is prohibitive as this would increase the training time by a factor close to the grid size, which can easily be a factor 100 or 1,000. Moreover, the non-convexity of the loss landscape prevents potentially more efficient first-order methods from discovering (approximately) worst-case transformations (see Section 4 for details). ",
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+    {
+        "type": "text",
+        "text": "Given that we cannot fully optimize over the space of translations and rotations, we instead use a coarse approximation provided by the worst-of-10 adversary (as described in Section 2). So each time we use an example during training, we first sample 10 transformations of the example uniformly at random from the space of allowed transformations. We then evaluate the model on each of these transformations and train on the one perturbation with the highest loss. This corresponds to approximately minimizing a min-max formulation of robust accuracy similar to (Madry et al., 2017). Training against such an adversary increases the overall time by a factor of roughly six.2 ",
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+        "type": "text",
+        "text": "Aggregating Random Transformations. As Section 4 shows, the accuracy against a random transformation is significantly higher than the accuracy against the worst transformation in the allowed attack space. This motivates the following inference procedure: compute a (tyipcally small) number of random transformations of the input image and output the label that occurs most common in the resulting set of predictions. We constrain these random transformations to be within $5 \\%$ of the input image size in each translation direction and up to $1 5 ^ { \\circ }$ of rotation. 3 The training procedure and model can remain unchanged while the inference time is increased by a small factor (equal to the number of transformations we evaluate on). ",
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+        "type": "text",
+        "text": "Combining Both Methods. The two methods outlined above are orthogonal and in some sense complementary. We can therefore combine robust training (using a worst-of- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ adversary) and majority inference to further increase the robustness of our models. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate standard image classifiers for the MNIST (LeCun et al., 1998), CIFAR10 (Krizhevsky & Hinton, 2009) and ImageNet (Russakovsky et al., 2015) datasets. In order to determine the extent to which misclassification is caused by insufficient data augmentation during training, we examine various data augmentation methods. We begin with a description of our experimental setup. ",
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+        "type": "text",
+        "text": "Model Architecture. For MNIST, we use a convolutional neural network derived from the TensorFlow Tutorial (tft). In order to obtain a fully convolutional version of the network, we replace the fully-connected layer by two convolutional layers with 128 and 256 filters each, followed by a global average pooling. For CIFAR10, we consider a standard ResNet (He et al., 2016) model with 4 groups of residual layers with filter sizes [16, 16, 32, 64] and 5 residual units each. We use standard and $\\ell _ { \\infty }$ -adversarially trained models similar to those studied by Madry et al. (2017).4,5 For ImageNet, we use a ResNet-50 (He et al., 2016) architecture implemented in the tensorpack repository (Wu et al., 2016). We did not modify the model architectures or training procedures. ",
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+        "type": "text",
+        "text": "Attack Space. In order to maintain the visual similarity of images to the natural ones we restrict the space of allowed perturbations to be relatively small. We consider rotations of at most $3 0 ^ { \\circ }$ and translations of at most (roughly) $10 \\%$ percent of the image size in each direction. This corresponds to 3 pixels for MNIST (image size $2 8 \\times 2 8$ ) and CIFAR10 (image size $3 2 \\times 3 2$ ), and 24 pixels for ImageNet (image size $2 9 9 \\times 2 9 9$ ). For grid search attacks, we consider 5 values per translation direction and 31 values for rotations, equally spaced. For first-order attacks, we use 200 steps of projected gradient descent of step size 0.01 times the parameter range. When rotating and translating the images, we fill the empty space with zeros (black pixels). ",
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+    {
+        "type": "text",
+        "text": "Data Augmentation. We consider five variants of training for our models. ",
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+        "type": "text",
+        "text": "• Standard training: The standard training procedure for the respective model architecture.   \n• $\\ell _ { \\infty }$ -bounded adversarial training: The classifier is trained on $\\ell _ { \\infty }$ -bounded adversarial examples that are generated with projected gradient descent.   \n• No random cropping: Standard training for CIFAR-10 and ImageNet includes data augmentation via random crops. We investigate the effect of this data augmentation scheme by also training a model without random crops. Random rotations and translations: At each training step, we perform a uniformly random perturbation from the attack space on each training example.   \n• Random rotations and translations from larger intervals: As before, we perform uniformly random perturbations, but now from a superset of the attack space $( 4 0 ^ { \\circ } , \\pm 1 3 \\%$ pixels). ",
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+    {
+        "type": "text",
+        "text": "4.1 EVALUATING MODEL ROBUSTNESS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate all models against random and grid search adversaries with rotations and translations considered both separately and together. We report the results in Table 1. We visualize a random subset of successful attacks in Figures 3, 4, and 5 of Appendix A. ",
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+        "type": "text",
+        "text": "Despite the high accuracy of standard models on unperturbed examples and their reasonable performance on random perturbations, a grid search can significantly lower the classifiers’ accuracy on the test set. For the standard models, accuracy drops from $9 9 \\%$ to $26 \\%$ on MNIST, $93 \\%$ to $3 \\%$ on CIFAR10, and $76 \\%$ to $31 \\%$ on ImageNet (Top 1 accuracy). ",
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+        "type": "text",
+        "text": "The addition of random rotations and translations during training greatly improves both the random and adversarial accuracy of the classifier for MNIST and CIFAR10, but less so for ImageNet. For the first two datasets, data augmentation increases the accuracy against a grid adversary by $60 \\%$ to $70 \\%$ , while the same data augmentation technique adds less than $3 \\%$ accuracy on ImageNet. ",
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+        "type": "text",
+        "text": "In Appendix A, we perform a fine-grained investigation of our findings: ",
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+        "type": "text",
+        "text": "• In Figure 8 we examine how many examples can be fooled by (i) rotations only, (ii) translations only, (iii) neither transformation, or (iv) both. We visualize the set of fooling angles for a random sample of the rotations-only grid in Figure 9. We observe that the set of fooling angles is not contiguous. • To investigate how many transformations are adversarial per image, we analyze the percentage of misclassified grid points for each example in Figure 10. While the majority of images has only a small number of adversarial transformations, a significant fraction of images is fooled by $20 \\%$ or more of the transformations. ",
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+        "text": "",
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+        "type": "text",
+        "text": "Table 1: Accuracy of different classifiers against rotation and translation adversaries on MNIST, CIFAR10, and ImageNet. The allowed transformations are translations by (roughly) $10 \\%$ of the image size and $\\pm 3 0 ^ { \\circ }$ rotations. The attack parameters are chosen through random sampling or grid search with rotations and translations considered both together (“Rand.”, “Grid”) and separately (“Rand. T.” and “Grid T.” for transformations, “Rand R.” and “Grid R.” for rotations). We consider networks that are trained with (i) the respective standard setup, (ii) no data augmentation (if data augmentation is present in standard setup), (iii) with an $\\ell _ { \\infty }$ adversary, (iv) with data augmentation corresponding to the attack space $( \\pm \\mathrm { { 3 p x } , \\pm 3 0 ^ { \\circ } ) }$ and an enlarged space $( \\pm 4 \\mathrm { p x } , \\pm 4 0 ^ { \\circ } )$ , and (v) with worst-of-10 training for both types of augmentations. ",
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+    {
+        "type": "table",
+        "img_path": "images/3c248e4f134748181385f5a6cad906ff4299d1723b6c56df6c8c5f1a3995d015.jpg",
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+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=4>Nat.     Rand.    Grid</td><td rowspan=1 colspan=1>Rand. T.  Grid T.</td><td rowspan=1 colspan=1>Rand. R.  Grid R.</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Standard</td><td rowspan=1 colspan=4>99.31%  94.23%  26.02%</td><td rowspan=1 colspan=1>98.61%  89.80%</td><td rowspan=1 colspan=1>95.68%   70.98%</td></tr><tr><td rowspan=5 colspan=1>LSINN</td><td rowspan=1 colspan=1>loo-Adv</td><td rowspan=1 colspan=4>98.65%  88.02%  1.20%</td><td rowspan=1 colspan=1>93.72%  34.13%</td><td rowspan=1 colspan=1>95.27%   72.03%</td></tr><tr><td rowspan=4 colspan=1>Aug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=1 colspan=4>99.53%  99.35%  95.79%</td><td rowspan=1 colspan=1>99.47%  98.66%</td><td rowspan=1 colspan=1>99.34%  98.23%</td></tr><tr><td rowspan=2 colspan=4>99.34%  99.31%  96.95%99.48%  99.37%  97.32%</td><td rowspan=1 colspan=1>99.39%  98.65%</td><td rowspan=2 colspan=1>99.40%   98.49%99.39%  98.62%</td></tr><tr><td rowspan=1 colspan=1>99.50%  99.01%</td><td rowspan=1 colspan=1>99.39%  98.62%</td></tr><tr><td rowspan=1 colspan=3>99.42%</td><td rowspan=1 colspan=2>99.39%  97.88%</td><td rowspan=1 colspan=1>99.45%  98.89%</td><td rowspan=1 colspan=1>99.36%  98.85%</td></tr><tr><td rowspan=7 colspan=1>CITIII</td><td rowspan=7 colspan=1>StandardNo Croploo-AdvAug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=1 colspan=2>92.62%</td><td rowspan=1 colspan=2>60.93%   2.80%</td><td rowspan=1 colspan=1>88.54%  66.17%</td><td rowspan=1 colspan=1>75.36%   24.71%</td></tr><tr><td rowspan=1 colspan=2>90.34%</td><td rowspan=1 colspan=1>01</td><td rowspan=1 colspan=2>54.64%   1.86%</td><td rowspan=1 colspan=1>81.95%  46.07%</td><td rowspan=1 colspan=1>69.23%   18.34%</td></tr><tr><td rowspan=5 colspan=4>80.21%  58.33%   6.02%90.02%  90.92%  58.90%88.83%  91.18%  61.69%91.34%  92.35%  69.17%91.00%  92.11% 71.15%</td><td rowspan=1 colspan=1>78.15%  59.02%</td><td rowspan=1 colspan=1>62.85%   20.98%</td></tr><tr><td rowspan=1 colspan=1>91.76%  79.01%</td><td rowspan=1 colspan=1>91.14%   76.33%</td></tr><tr><td rowspan=1 colspan=1>91.53%  77.42%</td><td rowspan=1 colspan=1>91.10%  76.80%</td></tr><tr><td rowspan=1 colspan=1>92.43%  83.01%</td><td rowspan=2 colspan=1>92.33%  81.82%92.53%  82.25%</td></tr><tr><td rowspan=1 colspan=1>92.28%  82.15%</td></tr><tr><td rowspan=6 colspan=1>1negee</td><td rowspan=6 colspan=1>StandardNo CropAug. 30Aug. 40W-10 (30)W-10 (40)</td><td rowspan=4 colspan=4>75.96%  63.39%  31.42%70.81%  59.09%  16.52%65.96%  68.60%  32.90%66.19%  67.58%  33.86%</td><td rowspan=1 colspan=1>73.24%  60.42%</td><td rowspan=1 colspan=1>67.90%  44.98%</td></tr><tr><td rowspan=1 colspan=1>66.75%  45.17%</td><td rowspan=1 colspan=1>62.78%  34.17%</td></tr><tr><td rowspan=4 colspan=4>65.96%  68.60%  32.90%66.19%  67.58%  33.86%76.14%  73.19%  52.76%74.64%  71.36%  50.23%</td><td rowspan=1 colspan=1>70.27%  45.72%</td><td rowspan=1 colspan=1>69.28%  47.25%</td></tr><tr><td rowspan=1 colspan=1>69.50%  44.60%</td><td rowspan=1 colspan=1>68.88%   48.72%</td></tr><tr><td rowspan=1 colspan=1>74.42%  61.18%</td><td rowspan=2 colspan=1>73.74%   61.06%71.95%  59.23%</td></tr><tr><td rowspan=1 colspan=1>72.86%  59.34%</td></tr></table>",
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+        "type": "text",
+        "text": "Padding Experiments. A natural question is whether the reduced accuracy of the models is due to the cropping applied during the transformation. We verify that this is not the case by applying zero and reflection padding to the image datasets. We note that the zero padding creates a “black canvas” version of the dataset, ensuring that no information from the original image is lost after a transformation. We show a random set of adversarial examples in this setting in Figure 6 and a full evaluation in Table 4. We also provide more details regarding reflection padding in Section B and provide an evaluation in Table 6. All of these are in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPARING ATTACK METHODS ",
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+    {
+        "type": "text",
+        "text": "In Table 2 we compare different attack methods on various classifiers and datasets. We observe that worst-of-10 is a powerful adversary despite its limited interaction with the target classifier. The firstorder adversary performs significantly worse. While it is still better than a random transformation , it fails to approximate the ground-truth accuracy of the models and performs significantly worse than the grid adversary and even the worst-of-10 adversary. ",
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+    {
+        "type": "text",
+        "text": "Understanding the Failure of First-Order Methods. The fact that first-order methods fail to reliably find adversarial rotations and translations is in sharp contrast to previous work on $\\ell _ { p }$ -robustness (Carlini & Wagner, 2016; Madry et al., 2017). For $\\ell _ { p }$ -bounded perturbations parametrized directly in pixel space, prior work found the optimization landscape to be well-behaved which allowed first-order methods to consistently find maxima with high loss. In the case of spatial perturbations, we observe that the non-concavity of the problem is a significant barrier for first-order methods. We investigate this issue by visualizing the loss landscape. For a few random examples from the three datasets, we plot the cross-entropy loss of the examples as a function of translation and rotation. Figure 2 shows one example for each dataset and additional examples are visualized in Figure 11 of the appendix. The plots show that the loss landscape is indeed non-concave and contains many local maxima of low value. The low-dimensional problem structure seems to make non-concavity a crucial obstacle. Even for MNIST, where we observe fewer local maxima, the large flat regions prevent first-order methods from finding transformations of high loss. ",
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+    {
+        "type": "table",
+        "img_path": "images/6a8c7952e9cdf04380d6773d540b6d1118e093f320520893e9c7e169614b8806.jpg",
+        "table_caption": [
+            "Table 2: Comparison of attack methods across datasets and models. Worst-of-10 is very effective and significantly reduces the model accuracy despite the limited interaction. The first-order (FO) adversary performs poorly, despite the large number of steps allowed. We compare standard training to Augmentation $( \\pm \\mathrm { { 3 p x } , \\pm 3 0 ^ { \\circ } ) }$ . For the full table, see Figure 3 of Appendix A. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">MNIST</td><td colspan=\"2\">CIFAR-10</td><td colspan=\"2\">ImageNet</td></tr><tr><td>Standard</td><td>Aug.</td><td>Standard</td><td>Aug.</td><td>Standard</td><td>Aug.</td></tr><tr><td>Natural</td><td>99.31%</td><td>99.53%</td><td>92.62%</td><td>90.02%</td><td>75.96%</td><td>65.96%</td></tr><tr><td>Worst-of-10</td><td>73.32%</td><td>98.33%</td><td>20.13%</td><td>79.92%</td><td>47.83%</td><td>50.62%</td></tr><tr><td>First-Order</td><td>79.84%</td><td>98.78%</td><td>62.69%</td><td>85.92%</td><td>63.12%</td><td>66.05%</td></tr><tr><td>Grid</td><td>26.02%</td><td>95.79%</td><td>2.80%</td><td>58.92%</td><td>31.42%</td><td>32.90%</td></tr></table>",
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+        "text": "",
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+        "type": "image",
+        "img_path": "images/037aebd33bb873a22c486788575345d70cc4fdbdf871cfe483c5e7705c723eb1.jpg",
+        "image_caption": [
+            "Figure 2: Loss landscape of a random example for each dataset when performing left-right translations and rotations. Translations and rotations are restricted to $10 \\%$ of the image pixels and $3 0 ^ { \\circ }$ , respectively. We observe that the landscape is significantly non-concave, rendering first-order methods to generate adversarial example ineffective. Figure 11 in the appendix shows additional examples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Relation to Black-Box Attacks. Given its limited interaction with the model, the worst-of-10 adversary achieves a significant reduction in classification accuracy. It performs only 10 random, non-adaptive queries to the model and is still able to find adversarial examples for a large fraction of the inputs (see Table 2). The low query complexity is an important baseline for black-box attacks on neural networks, which recently gained significant interest (Papernot et al., 2017; Chen et al., 2017; Bhagoji et al., 2017; Ilyas et al., 2017). Black-box attacks rely only function evaluations of the target classifier, without additional information such as gradients. The main challenge is to construct an adversarial example from a small number of queries. Our results show that it is possible to find adversarial rotations and translations for a significant fraction of inputs with very few queries. ",
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+    {
+        "type": "text",
+        "text": "Combining Spatial and $\\ell _ { \\infty }$ -Bounded Perturbations Table 1 shows that models trained to be robust to $\\ell _ { \\infty }$ perturbations do not achieve higher robustness to spatial perturbations. This provides evidence that the two families of perturbation are orthogonal to each other. We further investigate this possibility by considering a combined adversary that utilizes $\\ell _ { \\infty }$ bounded perturbations on top of rotations and translations. The results are shown in Figure 12. We indeed observe that these combined attacks reduce classification accuracy in an (approximately) additive manner. ",
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+    {
+        "type": "text",
+        "text": "4.3 EVALUATING OUR DEFENSE METHODS. ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As we see in Table 1, training with a worst-of-10 adversary significantly increases the spatial robustness of the model, also compared to data augmentation with random transformations. We conjecture that using more reliable methods to compute the worst-case transformations will further improve these results. Unfortunately, increasing the number of random transformations per training example quickly becomes computationally expensive. And as pointed out above, current first-order methods also appear to be insufficient for finding worst-case transformations efficiently. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our results for majority-based inference are presented in Table 5 of Appendix A. By combining these two defense, we improve the worst-case performance of the models from $26 \\%$ to $98 \\%$ on MNIST, from $3 \\%$ to $82 \\%$ on CIFAR10, and from $31 \\%$ to $56 \\%$ on ImageNet (Top 1). ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The fact that small rotations and translation can fool neural networks on MNIST and CIFAR10 was first observed in (Fawzi & Frossard, 2015). They compute the minimum transformation required to fool the model and use it as a measure for a quantitative comparison of different architectures and training procedures. The main difference to our work is that we focus on the optimization aspect of the problem . We show that a few random queries usually suffice for a successful attack, while firstorder methods are ineffective. Moreover, we go beyond standard data augmentation and evaluate the effectiveness of natural baseline defenses. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The concurrent work of Kanbak et al. (2017) proposes a different first-order method to evaluate the robustness of classifiers based on geodesic distances on a manifold. This metric is harder to interpret than our parametrized attack space. Moreover, given our findings on the non-concavity of the optimization landscape, it is unclear how close their method is to the ground truth (exhaustive enumeration). While they perform a limited study of defenses (adversarial fine-tuning) using their method, it appears to be less effective than our baseline worst-of-10 training. We attribute this difference to the inherent obstacles first-order methods face in this optimization landscape. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recently, Xiao et al. (2018) and Tramer & Boneh (2017) observed independently that it is possible to \\` use various spatial transformations to construct adversarial examples for naturally and adversarially trained models. The main difference from our work is that we show even very simple transformations (translations and rotations) are sufficient to break a variety of classifiers, while the transformations employed in (Xiao et al., 2018) and (Tramer & Boneh, 2017) are more involved. The transformation \\` in (Xiao et al., 2018) is based on performing a displacement of individual pixels in the original image constrained to be globally smooth and then optimized for misclassification probability. Tramer & \\` Boneh (2017) consider an $\\ell _ { \\infty }$ -bounded pixel-wise perturbation of a version of the original image that has been slightly rotated and in which a few random pixels have been flipped. Both of these methods require direct access to the attacked model (or a surrogate) to compute (or at least estimate) the gradient of the loss function with respect to the model’s input. In contrast, our attacks can be implemented using only a small number of random, non-adaptive transformations of the input. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We examined the robustness of state-of-the-art image classifiers to translations and rotations. We observed that even a small number of randomly chosen perturbations of the input are sufficient to considerably degrade the classifier’s performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The fact that common neural networks are vulnerable to simple and naturally occurring spatial transformations (and that these transformations can be found easily from just a few random tries) indicates that adversarial robustness should be a concern not only in a fully worst-case security setting. We conjecture that additional techniques need to be incorporated in the architecture and training procedures of modern classifiers to achieve worst-case spatial robustness. Also, our results underline the need to consider broader notions of similarity than only pixel-wise distances when studying adversarial misclassification attacks. In particular, we view combining the pixel-wise distances with rotations and translations as a next step towards the “right” notion of similarity in the context of images. ",
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+        "text": "A OMITTED TABLES AND FIGURES ",
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+        "image_caption": [
+            "Figure 3: MNIST. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \\circ }$ of the original image and translations up to 3 pixels per direction (image size $2 8 \\times 2 8$ ). Each example is visualized along with its predicted label in the original and perturbed versions. "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/26b7cdf24cc48d671ac5febb8af59b365b797de29104fd693f5d7858c1d01466.jpg",
+        "image_caption": [
+            "Figure 4: CIFAR10. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \\circ }$ of the original and translations up to 3 pixels per directions (image size $3 2 \\times 3 2 ,$ ). Each example is visualized along with its predicted label in the original and perturbed version. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            165,
+            821,
+            768
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/355d6ece4af015878f75c7904074cb3ce8fc4acf1a2682a83017eb3cdd94580a.jpg",
+        "image_caption": [
+            "Figure 5: ImageNet. Successful adversarial examples for the models studied in Section 4. Rotations are restricted to be within $3 0 ^ { \\circ }$ of the original and translations up to 24 pixels per directions (image size $2 9 9 \\times 2 9 9$ ). Each example is visualized along with its predicted label in the original and perturbed version. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            167,
+            818,
+            766
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/9adc97254f9a49fc289bf63a114c93c63d72ba5fe677cc63dbad9dc88db6a269.jpg",
+        "image_caption": [
+            "Figure 6: Sample adversarial transformations for the ”black-canvas” setting for the standard models on CIFAR10 and ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            284,
+            202,
+            722,
+            767
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/34f61e1cb5c8fcd102da1367f3949ca26b3ba5ce8a043f83a1151970aa4d7030.jpg",
+        "image_caption": [
+            "Figure 7: Sample adversarial transformations for the reflection padding setting for the standard models on CIFAR10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            411,
+            119,
+            593,
+            613
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/af22e88fb367464ae08d512ced58677326738b42ff7c1d32c031f9152e88a8f2.jpg",
+        "image_caption": [
+            "Figure 8: Fine-grained dataset analysis. For each model, we visualize what percent of the test set can be fooled via various methods. We compute how many examples can be fooled with either translations or rotations (”any”), how many can be fooled only by one of these, and how many require a combination to be fooled (”both”). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            708,
+            826,
+            827
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/36b377a5f8e6a35b475cb5fcb813f0fc17f6ad24d95ced1b9f5569a67d62fb9e.jpg",
+        "image_caption": [
+            "Figure 9: Visualizing which angles fool the classifier for 50 random examples. For each dataset and model, we visualize one example per row. Red corresponds to misclassification of the images. We observe that the angles fooling the models form a highly non-convex set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            94,
+            877,
+            851
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/d63c0b01d615e7cd366824c1817405e3b76745a1255c2ab4b6273fbdf12922c4.jpg",
+        "image_caption": [
+            "Figure 10: Cumulative Density Function plots. For each fraction of grid points $p$ , we plot the percentage of correctly classified test set examples that are fooled by at least $p$ of the grid points. For instance, we can see from the first plot, MNIST Translations and Rotations, that approximately $10 \\%$ of the correctly classified natural examples are misclassified under $1 / 5$ of the grid points transformations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            196,
+            875,
+            743
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/6728833b7dfea7fcbb46d86864a491b9f6e96bc1daa19e73ee8986a9c97cd894.jpg",
+        "table_caption": [
+            "Table 3: Comparison of attack methods across datasets and models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Model</td><td>Natural</td><td>Worst-of-10</td><td>FO</td><td>Grid</td></tr><tr><td>LSINW</td><td>Standard lo-Adversarially Trained Aug.30 (±3px,±30°) Aug. 40 (±4±,±40°)</td><td>99.31% 98.65% 99.53% 99.34%</td><td>73.32% 51.18% 98.33% 98.49%</td><td>79.84% 81.23% 98.78% 98.74%</td><td>26.02% 1.20% 95.79% 96.95%</td></tr><tr><td>CITIIIT</td><td>Standard No Crop loo-Adversarially Trained Aug.30 (±3px,±30°) Aug. 40 (±4px,±40°)</td><td>92.62% 90.34% 80.21% 90.02% 88.83%</td><td>20.13% 15.04% 19.38% 79.92% 80.47%</td><td>62.69% 52.27% 33.24% 85.92% 85.48%</td><td>2.80% 1.86% 6.02% 58.92% 61.69%</td></tr><tr><td>1eenee</td><td>Standard No Crop Aug.30 (±24px,±30°) Aug.40 (±32px,±40°)</td><td>75.96% 70.81% 65.96% 66.19%</td><td>47.83% 35.52% 50.62% 51.11%</td><td>63.12% 55.93% 66.05% 66.14%</td><td>31.42% 16.52% 32.90% 33.86%</td></tr></table>",
+        "bbox": [
+            232,
+            223,
+            766,
+            443
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/5d0779a84efb1eda24246488ffbd0392b341f5761e6075098e4ad3a14ae37725.jpg",
+        "table_caption": [
+            "Table 4: Evaluation of a subset of Table 1 in the “black-canvas” setting (images are zero-padded to avoid cropping due to rotations and translations). The models are trained on padded images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>Natural</td><td>Random</td><td>Worst-of-10</td><td>Grid</td><td>Trans. Grid</td><td>Rot. Grid</td></tr><tr><td rowspan=\"4\">CITIII</td><td>Standard</td><td>91.81%</td><td>70.23%</td><td>25.51%</td><td>6.55%</td><td>83.38%</td><td>12.44%</td></tr><tr><td>No Crop</td><td>89.70%</td><td>52.86%</td><td>14.14%</td><td>1.17%</td><td>47.94%</td><td>9.46%</td></tr><tr><td>Aug.30 (±3px,±30°)</td><td>91.45%</td><td>90.82%</td><td>80.53%</td><td>63.64%</td><td>82.28%</td><td>76.32%</td></tr><tr><td>Aug.40 (±4px,±40°)</td><td>91.24%</td><td>91.00%</td><td>81.81%</td><td>66.64%</td><td>81.75%</td><td>78.57%</td></tr><tr><td rowspan=\"4\">1aeege</td><td>Standard</td><td>73.60%</td><td>46.59%</td><td>29.51%</td><td>15.38%</td><td>28.03%</td><td>23.81%</td></tr><tr><td>No Crop</td><td>66.28%</td><td>38.70%</td><td>14.17%</td><td>3.43%</td><td>8.87%</td><td>10.97%</td></tr><tr><td>Aug.30 (±24px,±30°)</td><td>64.60%</td><td>67.75%</td><td>47.32%</td><td>28.51%</td><td>45.33%</td><td>39.33%</td></tr><tr><td>Aug.40 (±32px,±40°)</td><td>49.20%</td><td>57.69%</td><td>38.36%</td><td>22.10%</td><td>32.84%</td><td>32.95%</td></tr></table>",
+        "bbox": [
+            174,
+            684,
+            879,
+            825
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/61eeaa79b4ebcc0e672828f115ea91e7b4c7ab09d22c040a193ca6de379d6445.jpg",
+        "image_caption": [
+            "Figure 11: Loss landscape of 4 random examples for each dataset when performing left-right translations and rotations. Translations and rotations are restricted to $10 \\%$ of the image pixels and $3 0 ^ { \\circ }$ respectively. We observe that the landscape is significantly non-concave, making rendering FO methods for adversarial example generation powerless. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            162,
+            797,
+            795
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/603dbd3b56ec7b704aaf472369fd8da9b002abacfcff3173c1d2f263c55c6e19.jpg",
+        "image_caption": [
+            "Figure 12: Accuracy of different classifiers against $\\ell _ { \\infty }$ -bounded adversaries with various values of $\\varepsilon$ and spatial transformations. For each value of $\\varepsilon$ , we perform PGD to find the most adversarial $\\ell _ { \\infty }$ - bounded perturbation. Additionally, we combine PGD with random rotations and translations and with a grid search over rotations and translations in order to find the transformation that combines with PGD in the most adversarial way. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            103,
+            857,
+            484
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "B MIRROR PADDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            606,
+            362,
+            622
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "In the experiments of Section 4, we filled the remaining pixels of rotated and translated images with black (also known as zero or constant padding). This is the standard approach used when performing random cropping for data augmentation purposes. We briefly examined the effect of mirror padding, that is replacing empty pixels by reflecting the image around the border6. The results are shown in Table 6. We observed that training with one padding method and evaluating using the other resulted in a significant drop in accuracy. Training using one of these methods randomly for each example resulted in a model which roughly matched the best-case of the two individual cases. ",
+        "bbox": [
+            173,
+            637,
+            825,
+            726
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/01c187664e018099f57e622d78df3302dc83b5c026f6e5fd7f042be9c26b702b.jpg",
+        "table_caption": [
+            "Table 5: Majority Defense. Accuracy of different models on the natural evaluation set and against a combined rotation and translation adversary using aggregation of multiple random transformations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Natural Acc.Stand.   Vote</td><td rowspan=1 colspan=1>Grid Acc.Stand.   Vote</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Standard</td><td rowspan=1 colspan=1>99.31% 98.71%</td><td rowspan=2 colspan=1>26.02% 18.80%95.79%95.32%96.95% 97.65%97.32% 96.95%97.88% 98.47%</td></tr><tr><td rowspan=1 colspan=1>JSINN</td><td rowspan=1 colspan=1>Aug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>99.53%99.41%99.34%99.25%99.48% 99.40%99.42% 99.41%</td></tr><tr><td rowspan=1 colspan=1>CITIIIO</td><td rowspan=1 colspan=1>StandardAug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>92.62% 80.37%90.02% 92.70%88.83% 92.50%91.34% 93.38%91.00% 93.40%</td><td rowspan=1 colspan=1>2.82%  7.85%58.90% 69.65%61.69% 76.54%69.17% 77.33%71.15% 81.52%</td></tr><tr><td rowspan=1 colspan=1>1eeeeer</td><td rowspan=1 colspan=1>StandardAug 30.Aug 40.W-10 (30)W-10 (40)</td><td rowspan=1 colspan=1>75.96% 73.19%65.96% 72.44%66.19% 71.46%76.14%74.92%74.64%73.38%</td><td rowspan=1 colspan=1>31.42% 40.21%32.90% 44.46%33.86% 46.98%52.76%56.45%50.23%56.23%</td></tr></table>",
+        "bbox": [
+            310,
+            200,
+            687,
+            463
+        ],
+        "page_idx": 20
+    },
+    {
+        "type": "table",
+        "img_path": "images/580a52a53c4d40eecf986e1032e360c6f6a7d1cbffcf7d3f36bde3f59fd5453f.jpg",
+        "table_caption": [
+            "Table 6: CIFAR10: The effect of using reflection or zero padding when training a model. The experimental setup matches that of Section 4. Zero padding refers to filling the empty pixels caused by translations and rotations with black. Mirror padding corresponds to using a reflection of the images. ”Both” refers to training using both methods and alternating randomly between them for each training example. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Natural</td><td>Random (Zero)</td><td>Random (Mirror)</td><td>Grid Search (Zero)</td><td>Grid Search (Mirror)</td></tr><tr><td>Standard Nat</td><td>92.62%</td><td>60.76%</td><td>66.42%</td><td>8.08%</td><td>5.37%</td></tr><tr><td>Standard Adv</td><td>80.21%</td><td>59.79%</td><td>67.12%</td><td>7.20%</td><td>12.89%</td></tr><tr><td>Aug. A, Zero</td><td>90.25%</td><td>91.09%</td><td>87.67%</td><td>59.87%</td><td>40.55%</td></tr><tr><td>Aug. B, Zero</td><td>89.55%</td><td>91.40%</td><td>87.94%</td><td>62.42%</td><td>42.37%</td></tr><tr><td>Aug. A, Mirror</td><td>92.25%</td><td>88.43%</td><td>91.05%</td><td>41.46%</td><td>53.95%</td></tr><tr><td>Aug. B, Mirror</td><td>92.03%</td><td>88.58%</td><td>91.34%</td><td>45.44%</td><td>57.97%</td></tr><tr><td>Aug. A, Both</td><td>91.80%</td><td>90.98%</td><td>91.28%</td><td>56.95%</td><td>52.60%</td></tr><tr><td>Aug. B, Both</td><td>91.57%</td><td>91.87%</td><td>91.11%</td><td>60.46%</td><td>56.13%</td></tr></table>",
+        "bbox": [
+            191,
+            584,
+            808,
+            780
+        ],
+        "page_idx": 20
+    }
+]

parse/train/H1lBj2VFPS/H1lBj2VFPS.md

@@ -0,0 +1,343 @@
+# LINEAR SYMMETRIC QUANTIZATION OF NEURAL NETWORKS FOR LOW-PRECISION INTEGER HARDWARE
+
+Xiandong $\mathbf { Z } \mathbf { h } \mathbf { a } \mathbf { 0 } ^ { 1 , 2 }$ , Ying Wang1,3∗ , Xuyi Cai1,2, Cheng Liu1, Lei Zhang1
+
+Institute of Computing Technology, Chinese Academy of Sciences1   
+University of Chinese Academy of Sciences2   
+State Key Laboratory of Computer Architecture3   
+{zhaoxiandong,wangying2009,caixuyi18s,liucheng,zlei}@ict.ac.cn
+
+# ABSTRACT
+
+With the proliferation of specialized neural network processors that operate on low-precision integers, the performance of Deep Neural Network inference becomes increasingly dependent on the result of quantization. Despite plenty of prior work on the quantization of weights or activations for neural networks, there is still a wide gap between the software quantizers and the low-precision accelerator implementation, which degrades either the efficiency of networks or that of the hardware for the lack of software and hardware coordination at designphase. In this paper, we propose a learned linear symmetric quantizer for integer neural network processors, which not only quantizes neural parameters and activations to low-bit integer but also accelerates hardware inference by using batch normalization fusion and low-precision accumulators (e.g., 16-bit) and multipliers (e.g., 4-bit). We use a unified way to quantize weights and activations, and the results outperform many previous approaches for various networks such as AlexNet, ResNet, and lightweight models like MobileNet while keeping friendly to the accelerator architecture. Additional, we also apply the method to object detection models and witness high performance and accuracy in YOLO-v2. Finally, we deploy the quantized models on our specialized integer-arithmetic-only DNN accelerator to show the effectiveness of the proposed quantizer. We show that even with linear symmetric quantization, the results can be better than asymmetric or non-linear methods in 4-bit networks. In evaluation, the proposed quantizer induces less than $0 . 4 \%$ accuracy drop in ResNet18, ResNet34, and AlexNet when quantizing the whole network as required by the integer processors.
+
+# 1 INTRODUCTION
+
+Deep neural networks have shown excellent performance on various computer vision and natural language processing tasks, such as classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), object detection (Girshick, 2015; Redmon et al., 2016; He et al., 2017), segmentation (Long et al., 2015; Noh et al., 2015), machine translation (Zhang et al., 2018b), speech recognition (Nassif et al., 2019), etc. While the past few years witnessed the success of DNNs on cloud and server-end computers, neural networks have been recently pushed to embedded and mobile areas to enable edge intelligence. For these scenarios, the power provision and computational strength on the edge computing devices are limited. As a result, it is essential to have more efficient network architectures and less expensive inference overhead. Therefore, there is increasing attention from the research community to study the compression of modern deep neural networks that are typically over-parameterized and computationally costly.
+
+Several categories of approaches are proposed to decrease the computational overhead of neural networks, such as lightweight neural network architectures (Howard et al., 2017), neural architecture search (NAS) (Elsken et al., 2018), and network pruning (Han et al., 2015; 2016; Wen et al., 2016;
+
+Table 1: Comparison between different quantizers: All-Layer $( A L )$ denotes quantizing all the parameters of all the operators in networks, including weights, bias, activations, and the scaling factor for low-precision networks; $\pmb { B N }$ donates that the BN operation is only invoked in training but merged into weights and induces no overhead in integer inference; Linear-Symmetric $( L S )$ denotes linear symmetric quantization; Activation Functions $( A F )$ donates the support of Leaky $R e L U$ and activation functions besides ReLU. Structure-Intact $( S I )$ indicates the network structure is unmodified.   
+
+<table><tr><td>Method</td><td>AL</td><td>BN</td><td>LS</td><td>AFe</td><td>SI</td></tr><tr><td>Deep Compression (Han et al.,2016)b</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>WQ (Park et al., 2017)b</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>LQ-Nets (Zhang et al., 2018a)</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>Min-Max Linear Quantizationa</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoReFa (Zhou et al., 2016)c</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RQ (Louizos et al., 2019)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>WRPN (Mishra et al., 2018)</td><td></td><td></td><td>√</td><td></td><td></td></tr><tr><td>PACT (Choi, 2018)d</td><td></td><td></td><td></td><td>√</td><td></td></tr><tr><td>LLSQ(ours)</td><td>厂</td><td></td><td>√</td><td>1</td><td></td></tr></table>
+
+a Naive linear quantization, which finds min-max value at runtime. b Clustering-based approaches to quantize weights. c DoReFa falls into linear asymmetric quantizer due to the need for offset. d In Choi (2018), they use PACT to quantize activations, and DoReFa to quantize weights. e DoReFa, RQ, WRPN, and PACT are designed for $R e L U$ , but they can be extended to support other activation functions in theory.
+
+Molchanov et al., 2017). Besides these techniques, quantizing high-precision floating-point networks to lower bitwidth representation can also drastically decrease both the static parameters and the intermediate data generated during the network inference, resulting in reduced memory footprint and also computational intensity. And this paper focuses on the quantization of neural networks.
+
+Quantization technique is also closely related to the implementation of specialized hardware that maps the procedure of network inference onto the energy-efficient low-precision integer or fixedpoint arithmetic circuits. In the hardware perspective, low-precision integer accelerators or processors are dominating the solutions targeted on neural network inference, especially for mobile and embedded scenarios. Google’s Tensor Processing Unit 1.0 (TPU) (Jouppi et al., 2017), Unified Deep Neural Network Accelerator (UNPU) (Lee et al., 2018), Eyeriss (Chen et al., 2018), Stripes (Judd et al., 2016), Pragmatic(Albericio et al., 2017) and many other newly proposed hardware implementations are generally reliant on the effectiveness of the underlying quantization techniques, which are especially crucial for the low-precision integer hardware designed to process binary, ternary, 4-bit or 8-bit networks. In other words, quantization is not only a method to reduce the memory footprint as in traditional work, but also a mandatory step to make the network deployable on integer hardware.
+
+Though there is a lot of prior work that investigates low-precision quantization, they mainly target on reducing the memory overhead caused by floating or high precision data representation in the networks, but not focus on specialized integer hardware for network inference. To enable the neural network processors to work with low-precision integer operands and minimize the accuracy losses, a good network quantizer must satisfy the constraints as enlisted in Table 1.
+
+First, all the parameters, including weights, bias, activations, partial results that eventually accumulate to an activation, and even the scaling factors, which are indispensable for low-precision networks like binary and ternary representation, must be quantized into low bitwidth integers as required by the underlying specialized hardware. In some prior work (Zhou et al., 2016; Zhu et al., 2017; Zhang et al., 2018a; Mishra et al., 2018; Choi, 2018), they either leave bias and scaling factors unquantized or keep the first and last layer in full or high precision. Besides, some designs rely on high-precision internal register or ALUs to support high-precision partial results that are generated during computation before the final output of activations or features. For example, Krishnamoorthi (2018), which quantizes the weights and activations to 8-bit, directly use 32-bit accumulators to cache the intermediate values or partial results to avoid overflows. However, for 4-bit and lower bitwidth, the integer accelerators cannot afford high bitwidth accumulators, which indicates higher silicon area and power cost. For integer-only-arithmetic, we quantize the bias to fixed-point numbers by using a straight-forward method. The value range of these numbers is wide, resulting in overflows of the low bitwidth accumulators. To overcome this problem, we quantize the bias to 8-bit and finetune the bias of the model. As shown in Figure 1, the bitwidth of accumulators can be reduced to 16-bit.
+
+Second, the BatchNorm (BN) layer does not necessarily need to be processed during inference for the reduction of computation and memory cost. For most of the convolutional neural networks, BN layers are often after the Conv or $F C$ layers. In these situations, $B N$ can be merged into the weights and biases of the corresponding Conv or $F C$ layers. However, in Zhou et al. (2016); Zhang et al. (2018a), they use asymmetric or non-linear quantization, causing barriers to $B N$ fusion. There are two ways to overcome this obstacle. One is $\mathbf { \bar { \Sigma } } ^ { 6 6 }$ folded training”(Krishnamoorthi, 2018), which adopts BN fusion before weights quantization in every training step; the other is to use symmetric linear quantization. However, the first method doubles the training time, while the second one has no additional computational overhead, which will be introduced in Section 3.4.
+
+Third, linear quantization is necessary for state-of-the-art accelerators. There are many non-linear quantization methods which achieve excellent bitwidth reduction efficacy and accuracy tradeoffs. In these cases, it requires additional transformation to have correct arithmetic results after quantizing the value into non-linear distribution. For example, as in Han et al. (2016); Park et al. (2017), it necessitates the operation of table lookup to have correct multiplication between quantized values. However, the linear quantization can make full use of the low-precision arithmetic components in off-the-shelf accelerators. Further, linear quantization can be divided into symmetric mode and asymmetric mode. Asymmetric quantization has one more parameter (e.g., zero-point (Krishnamoorthi, 2018)) than symmetric quantization, and it requires additional subtraction or linearoperation before multiplication. As a result, the symmetrical mode is compatible with the mainstream integer accelerator chip design and do not require the redesign of datapath in these hardware.
+
+Fourth, different CNNs or applications usually use a variety of activation functions. For instance, the object detection model Redmon et al. (2016) typically uses Leaky ReLU. And the bottleneck of ResNet block does not use any activation function. The quantization methods are expected to be adapted to these situations. However, Zhang et al. (2018a); Park et al. (2017) only focus on the quantization of activations after ReLU. In this paper, we demonstrate our method is friendly to different activation methods such as Leaky ReLU.
+
+Some of the previous researches change the network structure for better quantization performance, e.g., Mishra et al. (2018) double or even triple the convolutional filters to reduce accuracy degradation. For the energy-efficient integer neural network chips, it needs to remap the changed network architecture to hardware and adds to computational and memory access overhead due to the increased filters and parameters. As a result, keeping the network structure intact is important.
+
+Concerning all the factors above, in this paper, we present a learned linear symmetric quantization (LLSQ) method and also evaluate it on a low-precision neural network accelerator through hardware-software co-design. Specifically, our mainly contributions are:
+
+• Unlike most of other quantization methods, we quantize the whole network including the first and last layers. We also quantize bias and scaling factors, in support of the low bitwidth integer arithmetic units and accumulators on the accelerator.   
+We adopt learned linear symmetric quantization schemes which are hardware friendly (such as the convenience of $B N$ fusion implementation) while achieving state-of-the-art prediction accuracy. We design a specialized low-precision CNN inference accelerator to validate the methodology, which supports 2/4/8 integer operating and work with high efficiency. We then deploy our quantization model on the accelerator to illustrate the efficacy of the workflow.
+
+# 2 MOTIVATION
+
+Edge or embedded neural network accelerators generally have three primary design goals— smallfootprint, high-throughput/low-latency, and low-power. For different applications and scenarios, the prior researches on specialized deep learning processors are often falling into different categories: cloud-oriented hardware for warehouse machines, low power mobile processors and ultra-low power accelerators for IoT or cyber-physical devices.
+
+For mobile and embedded usage, specialized neural network processors are becoming increasingly popular as an efficient hardware solution of inference. DianNao (Chen et al., 2014) is proposed for fast inference of DNNs and it uses 16-bit fixed-point multipliers for small silicon area and lowenergy. Later, ShiDianNao (Du et al., 2015) is introduced and it burns extremely low energy consumption by putting all weights onto the SRAM to eliminate considerable DRAM accesses. Besides, DeepBurning (Wang et al., 2016) simplifies the design flow of accelerator for different NN models. Eyeriss (Chen et al., 2018) is also another representative of low-power accelerators. And it presents a row-stationary (RS) dataflow to minimize data movement energy consumption on a spatial architecture. To further reduce computation overhead, EIE (Han et al., 2016) exploits the sparsity and low-bit compression of the NNs and achieves better throughput, energy and area efficiency. These typical edge neural network processors are accepting fixed-point data input and using fixed-point processing elements to reduce the power and chip area overhead caused by floating-point arithmetic components and memory. For the cloud scenarios, specialized architectures like TPU (Jouppi et al., 2017) and FPGA-based accelerator cards are also replacing conventional GPGPU and CPU for highthroughput inference tasks. Even for cloud-oriented inference architectures, fixed-point processing architectures like TPU are favored because they are able to deliver much higher throughput for the given power budget and silicon area overhead.
+
+However, for the fixed-point or integer hardware targeted on neural network acceleration, quantization is prerequisite to convert the floating-point network model into the fixed-point format compatible with the specialized hardware, and it is also a critical step to ensure the accuracy of the network after conversion. Many prior quantization methods are intended to reduce the running overhead of networks but ignore the architecture and working mechanism of integer neural network processors, as illustrated in Table 1, and they sometimes face considerable accuracy losses, or performance penalty or even fail to be supported on the realistic integer datapath due to the unconsciousness of the underlying hardware. This problem becomes particularly important for the hardware that is designed to run low bitwidth networks such as binary, ternary, and 2/4-bit models. For instance, Deep compression and WQ are clustering-based quantization methods, and they still need high-precision values to represent the weights, bias, and activations. As a result, they are not compatible with the hardware that only supports low-precision computing. LQ-Nets uses non-linear quantization based on the binary code and basis vector, and it can theoretically calculate the inner products between quantized weights and activations by bitwise operations only. However, it requires intensive modifications to the design of current processors by adding a lot of look-up tables in the datapath. Further, bias and scaling factors are not quantized in PACT and WRPN, resulting in performance penalty when employing additional high-precision or float-point ALUs to deal with them. In contrast, our LLSQ is designed to ease the model quantization flow for the specialized integer neural network processors by conforming to the constraints specified in Table 1. To validate the importance of hardware-aware quantizer and software/hardware co-design, we also design a specialized CNN accelerator for wearable applications. And the specialized accelerator supports 2/4/8 integer operation and adopts the dataflow of low latency and energy design.
+
+# 3 NETWORKS WITH LEARNED LINEAR SYMMETRIC QUANTIZATION
+
+In this section, we firstly give the overview of the proposed quantization scheme. Then we detail the scheme including low-precision representation, quantized network training, and the deployment of quantization model on our specialized integer-only CNN accelerator for fast inference.
+
+# 3.1 OVERVIEW OF LLSQ
+
+Many of the previous researches focus on the quantization-aware training in GPU, showing the potential of low-bit quantization on CNNs. Han et al. (2016); Park et al. (2017); Zhang et al. (2018a) propose non-linear quantization methods but lacks of a detailed description of the hardware feasibility. Krishnamoorthi (2018) provides a quantization scheme that quantizes weights and activations into 8-bit integers and integer-arithmetic-only implementation on ARM CPUs. The method achieves evident hardware acceleration effects, but does not fully exploit lower-precision quantization. Based on the researches, we propose a quantization scheme for state-of-the-art specialized accelerators operating on low-precision integers only. Figure 1 shows an overview of the proposed scheme.
+
+![](images/c9cfdd978cc908a0f869aa6ced258efd9a3f69064b2bf31b5fca1215f8c520d7.jpg)  
+Figure 1: An overview of LLSQ: using pre-trained weights for fast convergence; Retraining of the network with quantized weights and activations; BN fusion for efficient inference; Quantization of bias and scaling factors; Deployment of the quantized model to our accelerator. As shown in this figure, weights, activations, bias, and scaling factors are quantized to low-bit integers. And the bandwidth of accumulator can be set to lower (e.g., 16-bit in our experiments).
+
+Compared with prior work, our proposed quantization scheme pays more attention to the constraints imposed by real hardware. We use a unified learned linear symmetric quantizer to quantize weights and activations. And the quantizer has only one parameter, known as the scaling factor. Linear symmetric quantization consumes little additional resources based on the mainstream integer accelerator designs while achieving state-of-the-art accuracy in various networks. After that, we adopt BN fusion for fast inference on hardware. As for bias and scaling factors, we also quantize them to low-bitwidth integers. The integer accelerator illustrated in Figure 1 is an illustrative case of 4-bit quantization and hardware acceleration.
+
+# 3.2 MAKING FULL USE OF THE PRE-TRAINED PARAMETERS
+
+In experiments, we find that it is more efficient to start with the pre-trained full-precision parameters before quantization. Louizos et al. (2019); Zhou et al. (2017) use pre-trained weights for fast convergence and deployment, while Zhang et al. (2018a); Choi (2018); Cai et al. (2017) train quantized network from scratch to show the robustness of the algorithm. However, for some object detection models, the backbone models and pre-trained weights are essential to the detection performance. Redmon et al. (2016) shows that the pre-trained high-resolution classification network gives an increase of almost $4 \% m A P$ . To have better performance in classification, object detection, and other CNN based tasks, in this paper, we use pre-trained parameters to initialize the networks.
+
+# 3.3 LOW-PRECISION REPRESENTATION AND QUANTIZATION ALGORITHM
+
+We use channel-wise quantization for $C o n \nu$ layers and layer-wise quantization for $F C$ layers and activations. And we adopt the symmetric linear quantization to quantize weights or activations into $k$ bits words(e.g., 4-bit), which can be defined as
+
+$$
+\begin{array} { c } { { { \pmb x } ^ { q } = Q u a n t i z e _ { k } ( { \pmb x } ^ { r } \mid \alpha ) } } \\ { { { \pmb q } = \displaystyle \frac { { \pmb x } ^ { q } } { \alpha } = c l a m p ( \lfloor \displaystyle \frac { { \pmb x } ^ { r } } { \alpha } \rceil , - 2 ^ { k - 1 } , 2 ^ { k - 1 } - 1 ) } } \end{array}
+$$
+
+where $\pmb { x } ^ { r } \in \mathbb { R }$ is one kernel of weights or one layer of activations, the variable $\alpha \in \mathbb { R } ^ { + }$ is the quantization parameter, known as the scaling factor, while $\pmb { q } \in \{ - 2 ^ { k - 1 } , \dotsc , 0 , 1 , \dotsc , 2 ^ { k - 1 } - 1 \}$ is the integer values flowing in the integer accelerator and $\pmb { x } ^ { q } \in \{ - 2 ^ { k - 1 } \alpha , \dotsc , 0 , \alpha , \dotsc , ( 2 ^ { k - 1 } - 1 ) \alpha \}$ is the quantized weights or activations. Note that for activations, which are non-negative values if the ActFun is $R e L U$ , we clamp them to $[ 0 , 2 ^ { k } - 1 ]$ , resulting in $\textbf { \em q } \in \ \{ 0 , 1 , \ldots , \bar { 2 } ^ { k } - 1 \}$ and $\pmb { x } ^ { q } \in \{ 0 , \alpha , \dots , ( 2 ^ { k - 1 } - 1 ) \alpha \}$ , respectively.
+
+As defined above, we use $\alpha$ as our quantization parameter. And we optimize it with:
+
+$$
+\alpha ^ { * } = \arg \operatorname* { m i n } _ { \alpha } \int p ( \pmb { x } ^ { r } ) | \pmb { x } ^ { q } - \pmb { x } ^ { r } | ^ { l }
+$$
+
+Table 2: Comparison of SG and EMA.   
+
+<table><tr><td>Method</td><td>VGGSmallw4a4</td><td>VGGSmallw2a2</td><td>ResNet18w4a4</td><td>ResNet18w3a3</td></tr><tr><td>EMA</td><td>93.95</td><td>92.78</td><td>69.48</td><td>66.80</td></tr><tr><td>SG</td><td>94.34</td><td>93.31</td><td>69.84</td><td>68.08</td></tr></table>
+
+VGGSmall is trained on Cifar10 and ResNet18 is on ImageNet.
+
+where $\mathbf { \boldsymbol { x } } ^ { r } , \mathbf { \boldsymbol { x } } ^ { q }$ are the same factors defined in Equation 1, $p ( \pmb { x } ^ { r } )$ is the probability density distribution of $\pmb { x } ^ { r }$ , and $l \in \{ 1 , 2 \}$ is an optional constraint (We use 2 in our experiments). In Figure 2, we present the relationship between quantization error and $\alpha$ . When fixing weights $\pmb { x } ^ { r }$ , we can find the optimal $\alpha ^ { * }$ by using the brute-force search approach, which induces high computation cost. Besides, the weights are updated during the re-training phase and the optimal value $\alpha ^ { * }$ changes accordingly. In other words, the optimal value for the factors is not fixed and it takes considerable computational overhead to find the dynamic optimal value.
+
+Inspired by Zhang et al. (2018a), we find through experiments that there is no need to find the optimal value $\alpha ^ { * }$ , and it works well enough to find a near-optimal value $\tilde { \alpha } ^ { * }$ . Generally, quantization can be considered as a regularization of the networks, and the quantization parameter $\alpha$ needs only to be adjusted to a near-optimal value to preserve the network capacity.
+
+Then the problem becomes how to find a $\tilde { \alpha } ^ { * }$ in the forward pass of the network training phase. At the beginning of training, we assign $\alpha$ an initial value. Then in every training iteration, we explore between $2 \alpha$ and $\alpha / 2$ to find a better search direction $d _ { b e t t e r } \in \{ - 1 , 0 , 1 \}$ , and use $- \alpha ^ { 2 } d _ { b e t t e r }$ as the simulated gradient (SG) of $\alpha$ which is detailed in Equation 9. The gradients of other parameters are still obtained by backpropagation algorithm. After that, we update all parameters with the gradients or simulated gradients. Another method is updating $\alpha$ by the exponential moving average (EMA). We experiment both of the methods, and the results show that SG is generally better than EMA on various networks (See Table 2). If not specifically stated, we use the SG method in experiments. The re-training process with weights and activations quantized is summarized in Step 1 of Algorithm 1.
+
+![](images/e686ab91bd9834460084d50fa5a530a5652517ef6dd02d9f3b062233971cf826.jpg)  
+Figure 2: L2 distance of quantization. The data is from weights of the first FC layer in AlexNet. As shown in the figure, the optimal $\alpha ^ { * }$ changes with the updating of weights.
+
+# 3.4 BN LAYER FUSION OF QUANTIZED NETWORKS
+
+As described in Section 1, merging the $B N$ layers into convolutional layers can reduce the latency of network inference by removing additional computation overhead. The operator of quantized $C o n \nu ^ { 1 }$ and $F C$ layers can be expressed as
+
+$$
+\pmb { o } = \alpha _ { a } \pmb { q } _ { a } \alpha _ { w } \pmb { q } _ { w } + b
+$$
+
+where $\alpha , \mathbf { \pmb q }$ are the same as Equation 1, $\alpha _ { a } \mathbf { q } _ { a }$ , $\alpha _ { w } \pmb { q } _ { w }$ donate the quantized activations and weights, while $b$ is the bias and $^ o$ is the output feature vector. Note that $\alpha _ { a } , \alpha _ { w }$ and $b$ are full precision values. And the $B N$ layer can be formulated as follows:
+
+$$
+{ \pmb y } = \frac { { \pmb o } - { \pmb \mu } } { \sqrt { \sigma ^ { 2 } + \epsilon } } \gamma + \beta
+$$
+
+where $\mu$ and $\sigma ^ { 2 }$ are EMA statistics, $\gamma$ and $\beta$ are learned parameters in $B N$ layers.
+
+Obviously, we can merge $B N$ layers and figure out the corrected parameters:
+
+$$
+\hat { \alpha } _ { w } = \zeta \alpha _ { w } ; ~ \hat { b } = ( b - \mu ) \zeta + \beta ; ~ \zeta = \frac { \gamma } { \sqrt { \sigma ^ { 2 } + \epsilon } }
+$$
+
+# 3.5 BIAS AND SCALING FACTOR QUANTIZATION FOR LOW-BIT INTEGER ONLY ARITHMETIC
+
+Further, the outputs of layers are quantized according to Equation 1. For integer-only-arithmetic, the bias use $\alpha _ { a } \alpha _ { w }$ as its scaling factor. And for the multiplier $\frac { \alpha _ { a } \alpha _ { w } } { \alpha _ { o } }$ , we use bit-shift quantization (See Equation 10) so that no multiplication but bit-shift operation is needed in hardware.
+
+$$
+\begin{array} { c } { { \alpha _ { o } { \bf q } _ { o } = \alpha _ { a } { \bf q } _ { a } \hat { \alpha } _ { w } { \bf q } _ { w } + \hat { b } } } \\ { { { \displaystyle \bf q } _ { o } = \frac { \alpha _ { a } \hat { \alpha } _ { w } } { \alpha _ { o } } ( { \bf q } _ { a } { \bf q } _ { w } + { \bf q } _ { b } ) } } \\ { { { \displaystyle \alpha _ { b h e r e } q _ { b } = c l a m p ( \bigl \lfloor \frac \hat { b } { \alpha _ { a } \hat { \alpha } _ { w } } \bigr \rceil , - 2 ^ { k _ { b } - 1 } , 2 ^ { k _ { b } - 1 } - 1 ) } } } \end{array}
+$$
+
+Note that $\alpha _ { a } \alpha _ { w }$ is a very small number, resulting in large quantization noise when adopting the clamp operation. In addition, the quantization of the scaling factors $\alpha$ can also raise the quantization noise of weights and activations. Parameter re-training summarized in Step 2 of Algorithm 1 is required.
+
+In the re-training phase, we adopt STE (Bengio et al., 2013) to realize the non-differentiable quantization function.
+
+For weights and bias, we have
+
+$$
+\frac { \partial y } { \partial w ^ { q } } \simeq \frac { \partial y } { \partial w ^ { r } } ; \frac { \partial y } { \partial b ^ { q } } \simeq \frac { \partial y } { \partial b ^ { r } }
+$$
+
+For activations, we have
+
+$$
+\frac { \partial y } { \partial a ^ { q } } \simeq \left\{ \begin{array} { l l } { \frac { \partial y } { \partial a ^ { r } } } & { i f 0 \leq a ^ { r } \leq ( 2 ^ { k } - 1 ) \alpha } \\ { 0 } & { o t h e r w i s e } \end{array} \right.
+$$
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, three sets of experiments on Cifar10, ImageNet and Pascal VOC datasets are presented. First, we conduct our proposed learned linear symmetric quantization (LLSQ) on weights and activations, leaving the first and last layers in full precision for a fair comparison with Zhang et al. (2018a). Second, we quantize the whole networks including the first and last layers, which is referred as LLSQF (LLSQ for Full network). Finally, we quantize the remaining bias and scaling factors. LLSQ is implemented in PyTorch (Paszke et al., 2017), and most of the baselines it uses in evaluation are from PyTorch Model $Z _ { 0 0 } { } ^ { 2 }$ .
+
+# 4.1 QUANTIZATION OF WEIGHTS AND ACTIVATIONS
+
+We firstly employ the VGG-Small network on Cifar10 to verify the LLSQ method. After that, we use AlexNet (Krizhevsky et al., 2012), ResNet18, ResNet34 (He et al., 2016), particularly the lightweight and hard-to-compress network architecture of MobileNet (Howard et al., 2017; Sandler et al., 2018) etc. to conduct more detailed experiments on the ImageNet dataset. Finally, we also quantize YOLOv2 (Redmon & Farhadi, 2017) to demonstrate that LLSQ also works well for complicated applications and especially the task adopting the activation functions like Leaky ReLU other than ReLU used in previous work.
+
+# VGG-SMALL ON CIFAR10
+
+The VGG-Small architecture is the same with Louizos et al. (2019); Zhang et al. (2018a), consisting of six Conv layers, three MaxPool layers, and one $F C$ layer. We adopt a cosine learn rate scheduler to train the VGG-Small reference and the quantized models. Specifically, we train the reference network for 400 epochs using an initial learning rate of 2e-2. And for the training of the quantized network, we use a warmup learning rate scheduler in the first ten epochs with an initial learning rate of 2e-3. In all quantization experiments, the total training epochs are 100. The VGG-Small quantization results are provided in Table 3. With 3-bit weights and 3-bit activations, the accuracy using our method is better than state-of-the-art method, LQ-Nets. And even when the first and last layers are all quantized in the same way, the loss of accuracy is minimal.
+
+Table 3: Comparison with the state-of-the-art low-bit quantization methods on CIFAR-10. The bitwidth for weights $( { \pmb w } )$ , activations $\mathbf { \Pi } ( \mathbf { a } )$ , bias $( b )$ and scaling factor $( \alpha )$ are given.   
+
+<table><tr><td>Method</td><td>#Bits wla/b/α</td><td>Acc(%)</td><td>Degradation(%)</td></tr><tr><td rowspan="3">LQ-Nets* (Zhang et al., 2018a)</td><td>Reference</td><td>93.8</td><td></td></tr><tr><td>3/3</td><td>93.8</td><td>0.0</td></tr><tr><td>2/2</td><td>93.5</td><td>0.3</td></tr><tr><td rowspan="3">RQ (Louizos et al., 2019)</td><td>Reference</td><td>93.05</td><td></td></tr><tr><td>8/8</td><td>93.30</td><td>-0.25</td></tr><tr><td>4/4 2/2</td><td>91.57</td><td>1.48 2.31</td></tr><tr><td rowspan="4">LLSQ*(ours)</td><td>Reference</td><td>90.92 93.34</td><td></td></tr><tr><td>4/4</td><td>94.34</td><td>-1.00</td></tr><tr><td>3/3</td><td>94.02</td><td>-0.68</td></tr><tr><td>2/2</td><td>93.31</td><td>0.03</td></tr><tr><td rowspan="4">LLSQF(ours)</td><td>4/4</td><td>94.30</td><td>-0.96</td></tr><tr><td>3/3</td><td>94.07</td><td>-0.73</td></tr><tr><td>2/2</td><td>93.12</td><td>0.22</td></tr><tr><td>4/4/8/8</td><td>93.84</td><td>-0.50</td></tr></table>
+
+\* first and last layer in full precision
+
+# IMAGENET DATASET
+
+We then quantize AlexNet, ResNet18, ResNet34 and MobileNetv2 on ILSVRC2012 dataset with different bitwidth configuration to demonstrate the effectiveness of the method. All of the pretrained float-point weights except MobileNetv $2 ^ { 3 }$ are downloaded from the PyTorch Model Zoo, and they are trained for 90 epochs with a step learning rate scheduler. After loading the pre-trained weights, we employ a warmup learning scheduler in the first three epochs and the cosine scheduler in the remained 57 epochs with an initial learning rate of 2e-2.
+
+As shown in Figure 3a, when quantizing both weights and activations, our degradation of accuracy is significantly smaller than LQ-Nets, PACT, and RQ. Especially, LLSQ outperforms the baselines when quantizing weights and activations into 4-bit. And it also outperforms other non-linear quantization methods with different bitwidth. Figure 3b shows that even with the first and last layers quantized, it can still achieve near baseline performance. In overall, the accuracy drop is less than $0 . 4 \%$ in ResNet18, ResNet34, and AlexNet when quantizing the whole network. We also quantize MobileNetv2, a more compact network, and obtain results that are significantly better than RQ. Please check Table 7 for detailed results.
+
+# OBJECT DETECTION ON PASCAL VOC
+
+We also apply the proposed LLSQ to YOLOv2. The backbone of YOLOv2 is Darknet-19, and its activation function is Leaky ReLU, so that the activations contain negative values. For YOLOv2 on Pascal VOC, we adopt the same quantization configuration (See Section 3.3) of the weights to the activations. Results are listed in Table 4. As shown in the table, LLSQ induces minor losses of $m A P$ in different bitwidth presentation. For comparison, we also quantize the activations into signed 5-bit integers using PACT, and consequently face considerable mAP losses $( 5 4 . 8 m A P )$ . Please note that we use the open-source PyTorch implementation of YOLOv2 4 as the baseline. We train the quantized model for 170 epochs (2/3 of baseline) with an initial learning rate of 1e-4 (1/10 of baseline).
+
+(a) Comparison with other state-of-art methods.   
+Lower is better.
+
+![](images/c77a4b3d1753549da14a63fb83a3565751683e1648bc66bbb2df309b4224a3cf.jpg)  
+Figure 3: Quantization results on different networks.
+
+![](images/a97efd5b33c1146e456371e6660c6f9710b806e60f52af3c35caf2df57ec3546.jpg)
+
+Table 4: LLSQ on YOLOv2 detector.   
+
+<table><tr><td>bitwidth</td><td>mAP</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>mbike</td><td>person plant</td><td></td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td></tr><tr><td>FP32</td><td>73.2</td><td>78.9</td><td>80.0</td><td>72.8</td><td>62.3</td><td>47.1</td><td>79.2</td><td>79.6</td><td>85.7</td><td>54.4</td><td>79.7</td><td>72.2</td><td>83.3</td><td>81.1</td><td>79.2</td><td>74.8</td><td>48.4</td><td>75.7</td><td>72.3</td><td>83.4</td><td>73.0</td></tr><tr><td>w4a5</td><td>70.3</td><td>73.9</td><td>76.1</td><td>67.8</td><td>57.3</td><td>39.9</td><td>81.2</td><td>79.1</td><td>82.6</td><td>51.8</td><td>75.7</td><td>68.3</td><td>80.3</td><td>83.9</td><td>78.7</td><td>70.6</td><td>42.6</td><td>72.2</td><td>71.6</td><td>83.5</td><td>69.5</td></tr><tr><td>w32a5</td><td>71.2</td><td>75.5</td><td>75.9</td><td>71.4</td><td>60.4</td><td>42.4</td><td>80.6</td><td>80.0</td><td>83.3</td><td>53.5</td><td>75.8</td><td>68.1</td><td>70.8</td><td>82.6</td><td>79.5</td><td>71.6</td><td>45.5</td><td>69.9</td><td>72.1</td><td>84.6</td><td>70.4</td></tr><tr><td>w4a8</td><td>73.4</td><td>74.5</td><td>79.1</td><td>75.5</td><td>60.6</td><td>43.8</td><td>80.9</td><td>80.7</td><td>85.8</td><td>56.6</td><td>80.0</td><td>70.9</td><td>83.5</td><td>84.5</td><td>81.0</td><td>74.5</td><td>47.5</td><td>74.8</td><td>75.2</td><td>84.1</td><td>73.6</td></tr><tr><td>w4a32</td><td>74.2</td><td>74.6</td><td>78.6</td><td>75.5</td><td>66.0</td><td>47.4</td><td>80.8</td><td>83.2</td><td>87.4</td><td>57.3</td><td>80.3</td><td>70.8</td><td>83.7</td><td>84.3</td><td>83.0</td><td>74.8</td><td>49.4</td><td>74.2</td><td>73.8</td><td>85.2</td><td>73.5</td></tr></table>
+
+# 4.2 BN FUSION AND QUANTIZATION OF BIAS AND THE SCALING FACTOR
+
+We adopt BN fusion in the PostAct $C o n \nu {  } B N {  } R e L U )$ networks according to the formula in Section 3.4. And the scaling factor of bias is the product of the corresponding scaling factors belonging to the activations and the weights, respectively. After that, we visualize the bias value distribution of VGG-Small. Figure 4 shows $b / \alpha _ { b }$ is distributed between a vast range (-1000, 1000), resulting in overflows of low bitwidth accumulators. And the overflow phenomena have a significantly harmful impact on the network performance. To deal with this issue, we quantize the bias and the scaling factors to 8-bit words, and then fine-tune the networks to restore the original performance. Generally, we need fine-tuning for one epoch only. After the quantization of bias and scaling factor, we have a fully quantized model and have it deployed onto our integer-only accelerator with 16 bitwidth accumulators. Table 3 and 7 show that the accuracy loss is negligible with $\pmb { w } 4 \pmb { a } 4 b 8 \pmb { \alpha } 8$ quantization on both VGG-Small and AlexNet.
+
+![](images/9e1b4c6d38ebf4bf95490ae5f2d98a8de7f18a1e2a205101e4a89dd2655e5f46.jpg)  
+Figure 4: Distribution of the bias/scaling factor. The data is from VGG-Small with w4a4 quantization.
+
+# 4.3 DEPLOYMENT ONTO REALISTIC HARDWARE
+
+The introduced linear symmetric quantization is intended to deploy the quantized networks to specialized integer-only-arithmetic CNN accelerators or other integer-only hardware. Our accelerators adopt the typical 2D systolic array architecture (Chen et al., 2018), but they are featured with 4-bit or 2-bit low-precision operation. As shown in Figure 1, the 8/4/2-bit accelerator has a $3 2 \mathrm { x } 7 $ array of processing elements (PE). And the MAC unit in each PE consists of a 4-bit multiplier and a 16-bit accumulator. For the 4-bit accelerator, we use INT4 representation for weights, UINT4 for activations, INT8 for the bias and scaling factors, respectively. For the 2-bit accelerator, we use INT2 for weights, UINT2 for activations, INT8 for the bias and scaling factors, respectively. Through the quantization process described in the paper, we can have a fully quantized network that works directly on the CNN accelerator. In addition, as we use linear symmetric quantization, we can use a straight-forward way to conduct multiply-accumulate operations without introducing shifters or lookup tables, which means the quantized models can run on state-of-the-art integer accelerators and ensures that their output accuracy degradation is minimal as presented in the above sections. Finally, we implement the 8/4/2-bit integer neural network processors with Synopsys Design Compiler (DC) under the $4 0 \mathrm { n m }$ technology, clocked at 800MHz. Table 5 shows that the $4 / 2$ -bit implementation achieves up to $2 . 5 6 \mathrm { x }$ lower silicon area and 5.56x lower power compared to that of the 8-bit baseline.
+
+Table 5: Comparison of our low-precision integer Neural Network Processors.   
+
+<table><tr><td>Bitwidth</td><td>#MAC Unit</td><td>Throughput (GOps/sec)</td><td>Silicon Area (mm²)</td><td>Power (mW)</td></tr><tr><td>8-bit</td><td>224</td><td>179.2</td><td>4.71</td><td>228</td></tr><tr><td>4-bit</td><td>224</td><td>179.2</td><td>2.80</td><td>93</td></tr><tr><td>2-bit</td><td>224</td><td>179.2</td><td>1.84</td><td>41</td></tr></table>
+
+Implemented and synthesized with Synopsys Design Compiler (DC) under the $4 0 \mathrm { n m }$ technology.
+
+# 5 CONCLUSIONS
+
+In this paper, we introduced a learned linear symmetric quantization (LLSQ) to quantize the whole network including the bias and scaling factors. We also use BN fusion and low bitwidth accumulators to reduce the network inference overhead and the hardware resources in integer neural accelerators. We show that our proposed method performs well for various networks on Cifar10, ImageNet, and Pascal VOC datasets. We also show that even the linear symmetric quantizer can obtain better results than asymmetric or non-linear quantization in the case of 4-bit networks. Finally, we deploy the quantized network onto our specialized integer-only neural network accelerator. Currently, the bitwidth of every layer in a network is all the same. Prior researches empirically find that different layers have different sensitivity to bitwidth of quantization. Hence in the future, we will explore a framework to support more flexible bitwidth for different layers or finer-grained quantization.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by the National Natural Science Foundation of China under Grant 61874124 and Grant 61902375.
+
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+
+# APPENDIX
+
+IMAGENET DETAILED
+
+We conduct experiments on AlexNet, ResNet18, ResNer34, and MobileNetv2. For all of the experiments, we adopt channel-wise quantization for Conv layers and layer-wise quantization for $F C$ layers as well as the activations. The AlexNet architecture is the same as the PyTorch Model Zoo, and it consists of five Conv layers, three $F C$ layers, three MaxPool layers, and two Dropout layers. To prevent over-fitting, we keep the Dropout layers when quantizing AlexNet. As shown in Figure 5d, we use the same learning rate scheduler for all experiments on ImageNet. The test curves are also shown in Figure 5. As we begin with the pre-trained full-precision weights, the test accuracy is already acceptable after one-epoch training. The final results are listed in Table 7.
+
+Table 6: Train Time of ResNet18   
+
+<table><tr><td rowspan=1 colspan=1>#Training Process</td><td rowspan=1 colspan=1>training time</td></tr><tr><td rowspan=1 colspan=1>Train the fp32 network from scratch</td><td rowspan=1 colspan=1>1.0x</td></tr><tr><td rowspan=1 colspan=1>Quantize w/a to 4/4 according to Step1 of Alg. 1</td><td rowspan=1 colspan=1>0.69x</td></tr><tr><td rowspan=1 colspan=1>Quantize w/a to 3/3 according to Step1 of Alg. 1</td><td rowspan=1 colspan=1>0.69x</td></tr><tr><td rowspan=1 colspan=1>Quantize b/α to 8/8 according to Step2 of Alg. 1</td><td rowspan=1 colspan=1>0.01x</td></tr></table>
+
+Training time. The proposed LLSQ requires about $2 / 3$ training epochs than that of floating-point network training. In each training iteration, LLSQ needs extra computation cost to optimize the quantizers. Specifically, the simulated gradients generation of the scaling factors is the major cost. Table 6 shows the total training time comparison of ResNet18 network. The quantization training time is $70 \%$ of baseline only.
+
+![](images/f6612721c7554999187d6d1ad01ee13872e2853d1471e362664f2ff4ea2e0ffc.jpg)  
+Figure 5: Test curves for AlexNet, ResNet18, and ResNet34 on ImageNet.
+
+# THE LLSQ ALGORITHM
+
+Generate simulate gradients for $\alpha$ :
+
+$$
+\begin{array} { c } { { E _ { m } = \displaystyle \sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \mid \alpha ) ) ^ { 2 } } } \\ { { E _ { l } = \displaystyle \sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \mid \frac \alpha 2 ) ) ^ { 2 } } } \\ { { E _ { r } = \displaystyle \sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \mid 2 \alpha ) ) ^ { 2 } } } \\ { { d _ { b e t t e r } = \displaystyle \operatorname * { a r g m i n } ( [ E _ { l } , E _ { m } , E _ { r } ] ) - 1 } } \\ { { \Delta G _ { \alpha } = - \alpha ^ { 2 } d _ { b e t t e r } } } \end{array}
+$$
+
+where $\pmb { x } ^ { r }$ is one kernel of weights or one layer of activations. $\arg \operatorname* { m i n } ( [ E _ { l } , E _ { m } , E _ { r } ] ) \in \{ 0 , 1 , 2 \}$   
+selects the index of the smallest number in the array $[ E _ { l } , E _ { m } , E _ { r } ]$ .
+
+The bit-shift quantization can be formulated as:
+
+$$
+\begin{array} { l } { { \pmb { \alpha } ^ { q } = S Q _ { k } ( \pmb { \alpha } ) } } \\ { { \nonumber } } \\ { { \vphantom { \int } } = \frac { \mathrm { c l a m p } ( \mathrm { r o u n d } ( 2 ^ { q c o d e } \cdot { \pmb { \alpha } } ) , - 2 ^ { k - 1 } , 2 ^ { k - 1 } - 1 ) } { 2 ^ { q c o d e } } } \end{array}
+$$
+
+where $\pmb { \alpha } \in \mathbb { R } ^ { + l e n ( \pmb { \alpha } ) }$ is the scaling factors to be quantized, $k \in \mathbb { Z }$ is the bitwidth, and $q c o d e \in \mathbb { Z }$ is the parameter of the bit-shift quantizer simply obtained by:
+
+$$
+q c o d e = k - \mathrm { c e i l } ( \log _ { 2 } ( \operatorname* { m a x } ( \pmb { \alpha } ) ) + 1 - 1 0 ^ { - 5 } )
+$$
+
+# Algorithm 1 LLSQ
+
+Input: Dataset $( { \pmb x } , { \pmb y } )$ , where $_ { \textbf { \em x } }$ is input and $\textbf {  { y } }$ is label; Pre-trained full-precision parameters $( w , b )$ , where $\pmb { w }$ is weights and $^ { b }$ is bias; Suppose the network consists of $L$ layers, ${ \pmb w } _ { l } ^ { ( i ) }$ represents the $i _ { t h }$ kernel of weights of the $l _ { t h }$ layer while $\mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \mathbf { } \mathbf { } \mathbf \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf { } \mathbf \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \Psi \Psi \mathbf \Psi \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \Psi \Psi \mathbf \Psi \mathbf \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \mathbf \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi \Psi $ is the output of the $l _ { t h }$ layer.
+
+Output: The quantized scaling factors: $\hat { \pmb { \alpha } } _ { \pmb { w } } ^ { q } = [ [ \hat { \alpha } _ { \pmb { w } _ { 0 } ^ { ( 0 ) } } ^ { q } , \dots ] , \dots , [ \hat { \alpha } _ { \pmb { w } _ { L - 1 } ^ { q } } ^ { q } , \dots ] ]$ $\pmb { \alpha _ { a } ^ { q } } = [ \alpha _ { { \pmb { a } } _ { 0 } } ^ { q } , \dots , \alpha _ { { \pmb { a } } _ { L - 1 } } ^ { q } ]$ ; The quantized weights $\pmb { w } ^ { q }$ and bias $\hat { b } ^ { q }$ .
+
+# Step 1: Quantize weights and activations and $\pmb { R e }$ -training
+
+// Re-training of quantized networks can converge faster and end with a higher accuracy due to the mechanism of $B N$ layers.
+
+repeat Forward: $\mathbf { } a _ { 0 } \gets$ input for $l = 1 , \cdots , L$ do for w(i)l in wl do // This is accelerated in parallel when implemented. $\pmb { w } _ { l } ^ { ( i ) q }  \mathrm { Q u a n t i z e } ( \pmb { w } _ { l } ^ { ( i ) } \mid \alpha _ { \pmb { w } _ { l } ^ { ( i ) } } )$ per Eq. (1) Generate simulated gradients for $\alpha _ { w _ { l } ^ { ( i ) } }$ per Eq. (9) end for $\begin{array} { r l } & { \mathbf { \Lambda } _ { { \pmb { w } } _ { l } ^ { q }  \mathrm { C o n c a t } } ^ { \pmb { \mathrm { c m u } } \bullet \mathbf { \Lambda } _ { { \pmb { w } } _ { l } ^ { q } } } } \\ & { \mathbf { \Lambda } _ { { \pmb { a } } _ { l } }  R e L U ( B N ( C o n v ( \mathbf { a } _ { l - 1 } ^ { q } , \mathbf { w } _ { l } ^ { q } , \mathbf { b } _ { l } ) ) ) } \end{array}$ $\mathbf { \pmb { a } } _ { l } ^ { q } \gets \mathrm { Q u a n t i z e } ( a _ { l } \mid \alpha _ { \pmb { a } _ { l } } )$ per Eq. (1) Generate simulated gradients for $\alpha _ { { \pmb a } _ { l } }$ per Eq. (9)
+
+# end for
+
+# Backward:
+
+Generate $\Delta G$ for weights and bias and $\Delta E$ for activation per Eq. (7), (8) and backpropagation algorithm. Update $w , b , \alpha _ { w } , \alpha _ { a }$ $i t e r \gets i t e r + 1$ ntil iter $\geq i t e r _ { m a x } / /$ need about 60 epochs, e.g. $i t e r _ { m a x } = 6 0 \frac { \mathrm { l e n } ( d a t a s e t ) } { b a t c h s i z e }$
+
+Step 2: Quantize bias and scaling factor after BN fusion and $\pmb { R e }$ -training   
+iter $\gets 0$   
+$\hat { \alpha _ { w } } , \hat { b } \gets \mathbf { B } \mathbf { N }$ fusion per Eq. (5)   
+repeat Forward: $\mathbf { } \mathbf { } \mathbf { 0 } \gets$ input for $l = 1 , \cdots , L \mathbf { d }$ o $\hat { \alpha } _ { { w _ { l } } } ^ { q } \gets \mathrm { S Q } ( \hat { \alpha } _ { { w _ { l } } } )$ per Eq. (10) for ${ \pmb w } _ { l } ^ { ( i ) }$ in ${ \pmb w } _ { l }$ do // This is accelerated in parallel when implemented. $\mathbf { \dot { \mathbf { w } } } _ { l } ^ { ( i ) q }  \mathrm { Q u a n t i z e } ( \mathbf { w } _ { l } ^ { ( i ) } \mid \hat { \alpha } _ { \mathbf { \pmb { w } } _ { l } ^ { ( i ) } } ^ { q } )$ αˆqw(i) ) per Eq. (1)
+
+$\hat { b } _ { l } ^ { ( i ) q } \gets \mathrm { Q u a n t i z e } ( \hat { b } _ { l } ^ { ( i ) } \mid \alpha _ { a _ { l - 1 } } ^ { q } \hat { \alpha } _ { { \pmb w } _ { l } ^ { ( i ) } } ^ { q } )$ (i) ) per Eq. (1) Generate simulated gradients for $\hat { \alpha } _ { { \pmb w } _ { l } ^ { ( i ) } }$ per Eq. (9) end for $\begin{array} { r l } & { \mathbf { \Lambda } _ { u } ^ { \mathrm { c u n t \ : 1 0 1 } } } \\ & { \mathbf { \Lambda } _ { u } ^ { w _ { l } ^ { q } }  \mathrm { C o n c a t } \ w _ { l } ^ { ( i ) q } } \\ & { \hat { b } _ { l } ^ { q }  \mathrm { C o n c a t } \ \hat { b } _ { l } ^ { ( i ) q } } \\ & { a _ { l }  R e L U ( C o n v ( a _ { l - 1 } ^ { q } , { \pmb { w } } _ { l } ^ { q } , \hat { b } _ { l } ^ { q } ) ) } \end{array}$ $\alpha _ { { \pmb a } _ { l } } ^ { q }  \mathrm { S Q } ( \alpha _ { { \pmb a } _ { l } } )$ −  per Eq. (10) $\mathbf { \em a } _ { l } ^ { q } \gets \mathrm { Q u a n t i z e } ( \mathbf { \em a } _ { i } \mid \alpha _ { \mathbf { \em a } _ { l } } ^ { q } )$ per Eq. (1) Generate simulated gradients for $\alpha _ { { \pmb a } _ { l } }$ per Eq. (9) end for Backward: Generate $\Delta G$ for weights and bias and $\Delta E$ for activation per Eq. (7), (8) and backpropagation algorithm. Update $\bar { w } , \bar { b } , \hat { \alpha } _ { w } , \bar { \alpha } _ { a }$ $i t e r \gets i t e r + 1$ until iter $\geq i t e r _ { m a x } / /$ only need one epoch, e.g. $i t e r _ { m a x } = \frac { \mathbf { l e n } ( d a t a s e t ) } { b a t c h s i z e }$
+
+Table 7: Comparison with state-of-the-art quantization methods on ImageNet. Top1, Top5 accuracy $\% )$ and degradation of Top1 are given.   
+
+<table><tr><td>Method</td><td>Model</td><td>#Bits w/a/b/α</td><td></td><td></td><td>Top1(%) Top5(%) Degradation</td></tr><tr><td rowspan="3">LQ-Nets* (Zhang et al., 2018a)</td><td>ResNet18</td><td>Reference</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>70.3 69.3</td><td>89.5</td><td></td></tr><tr><td></td><td>4/4 3/3</td><td>68.2</td><td>88.8</td><td>1.0</td></tr><tr><td rowspan="3">RQ (Louizos et al., 2019)</td><td>ResNet18</td><td>Reference</td><td>69.54</td><td>87.9</td><td>2.1</td></tr><tr><td></td><td></td><td>69.97</td><td>89.19</td><td></td></tr><tr><td></td><td>8/8 4/4</td><td>61.52</td><td>89.44 83.99</td><td>-0.43</td></tr><tr><td>RQ+ST</td><td></td><td>8/8</td><td>69.63</td><td>89.33</td><td>8.02 -0.09</td></tr><tr><td>(Louizos et al., 2019)</td><td></td><td>4/4</td><td>62.46</td><td>84.78</td><td>7.08</td></tr><tr><td rowspan="4">LLSQ(ours)*</td><td rowspan="4">ResNet18</td><td>Reference</td><td>69.76</td><td>89.08</td><td></td></tr><tr><td>4/4</td><td>69.84</td><td>89.14</td><td>-0.08</td></tr><tr><td>3/3</td><td>68.08</td><td>88.20</td><td>1.68</td></tr><tr><td>4/4</td><td>69.40</td><td>88.72</td><td>0.36</td></tr><tr><td rowspan="2">LLSQF(ours) LQ-Nets*</td><td>ResNet18</td><td>3/3</td><td>66.67</td><td>87.42</td><td>3.09</td></tr><tr><td>ResNet34</td><td>Reference</td><td>73.80</td><td>91.40</td><td></td></tr><tr><td>(Zhang et al., 2018a) LLSQ(ours)*</td><td>ResNet34</td><td>3/3</td><td>71.9</td><td>90.2</td><td>1.9</td></tr><tr><td rowspan="3"></td><td></td><td>Reference</td><td>73.30</td><td>91.42</td><td></td></tr><tr><td></td><td>4/4</td><td>73.60</td><td>91.28</td><td>-0.30</td></tr><tr><td></td><td>3/3</td><td>72.02</td><td>90.66</td><td>1.28</td></tr><tr><td>LLSQF(ours)</td><td>ResNet34</td><td>4/4 3/3</td><td>72.94 70.97</td><td>91.20</td><td>0.36</td></tr><tr><td rowspan="4">LLSQ(ours)*</td><td>AlexNet</td><td>Reference</td><td>56.55</td><td>89.95</td><td>2.33</td></tr><tr><td></td><td>4/4</td><td>56.57</td><td>79.09</td><td></td></tr><tr><td></td><td></td><td></td><td>79.02</td><td>-0.02</td></tr><tr><td></td><td>4/4/8/8</td><td>56.45</td><td>80.15</td><td>0.10</td></tr><tr><td rowspan="4">LLSQF(ours)</td><td>AlexNet</td><td>3/3</td><td>55.36</td><td>78.20</td><td>0.19</td></tr><tr><td></td><td>4/4</td><td>56.40</td><td>78.85</td><td>0.15</td></tr><tr><td></td><td>4/4/8/8</td><td>55.58</td><td>77.47</td><td>0.97</td></tr><tr><td></td><td>3/3</td><td>54.28</td><td>77.65</td><td>2.27</td></tr><tr><td rowspan="4">RQ (Louizos et al., 2019)</td><td>Mobilenet(v1)</td><td>Reference</td><td>70.61</td><td>89.47</td><td></td></tr><tr><td></td><td>8/8</td><td>70.43</td><td>89.42</td><td>0.18</td></tr><tr><td></td><td>6/6</td><td>68.02</td><td>88.00</td><td>2.59</td></tr><tr><td></td><td>5/5</td><td>61.38</td><td>83.73</td><td>9.23</td></tr><tr><td rowspan="3">RQ+ST (Louizos et al., 2019)</td><td>Mobilenet(v1)</td><td>8/8</td><td>70.06</td><td>89.52</td><td>0.55</td></tr><tr><td></td><td>6/6</td><td>67.62</td><td>87.78</td><td>2.99</td></tr><tr><td></td><td>5/5</td><td>56.85</td><td>80.35</td><td>13.76</td></tr><tr><td rowspan="4">LLSQ(ours)*</td><td>MobileNet(v2)</td><td>Reference</td><td>71.80</td><td>90.37</td><td></td></tr><tr><td></td><td>6/6</td><td>71.20</td><td>89.99</td><td>0.60</td></tr><tr><td></td><td>5/5</td><td>70.45</td><td>89.69</td><td>1.35</td></tr><tr><td></td><td>4/4</td><td>67.37</td><td>87.99</td><td>4.43</td></tr></table>
+
+\* First and last layers in full precision.

parse/train/H1lBj2VFPS/H1lBj2VFPS_content_list.json

@@ -0,0 +1,1635 @@
+[
+    {
+        "type": "text",
+        "text": "LINEAR SYMMETRIC QUANTIZATION OF NEURAL NETWORKS FOR LOW-PRECISION INTEGER HARDWARE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Xiandong $\\mathbf { Z } \\mathbf { h } \\mathbf { a } \\mathbf { 0 } ^ { 1 , 2 }$ , Ying Wang1,3∗ , Xuyi Cai1,2, Cheng Liu1, Lei Zhang1 ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Institute of Computing Technology, Chinese Academy of Sciences1   \nUniversity of Chinese Academy of Sciences2   \nState Key Laboratory of Computer Architecture3   \n{zhaoxiandong,wangying2009,caixuyi18s,liucheng,zlei}@ict.ac.cn ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            319
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With the proliferation of specialized neural network processors that operate on low-precision integers, the performance of Deep Neural Network inference becomes increasingly dependent on the result of quantization. Despite plenty of prior work on the quantization of weights or activations for neural networks, there is still a wide gap between the software quantizers and the low-precision accelerator implementation, which degrades either the efficiency of networks or that of the hardware for the lack of software and hardware coordination at designphase. In this paper, we propose a learned linear symmetric quantizer for integer neural network processors, which not only quantizes neural parameters and activations to low-bit integer but also accelerates hardware inference by using batch normalization fusion and low-precision accumulators (e.g., 16-bit) and multipliers (e.g., 4-bit). We use a unified way to quantize weights and activations, and the results outperform many previous approaches for various networks such as AlexNet, ResNet, and lightweight models like MobileNet while keeping friendly to the accelerator architecture. Additional, we also apply the method to object detection models and witness high performance and accuracy in YOLO-v2. Finally, we deploy the quantized models on our specialized integer-arithmetic-only DNN accelerator to show the effectiveness of the proposed quantizer. We show that even with linear symmetric quantization, the results can be better than asymmetric or non-linear methods in 4-bit networks. In evaluation, the proposed quantizer induces less than $0 . 4 \\%$ accuracy drop in ResNet18, ResNet34, and AlexNet when quantizing the whole network as required by the integer processors. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks have shown excellent performance on various computer vision and natural language processing tasks, such as classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), object detection (Girshick, 2015; Redmon et al., 2016; He et al., 2017), segmentation (Long et al., 2015; Noh et al., 2015), machine translation (Zhang et al., 2018b), speech recognition (Nassif et al., 2019), etc. While the past few years witnessed the success of DNNs on cloud and server-end computers, neural networks have been recently pushed to embedded and mobile areas to enable edge intelligence. For these scenarios, the power provision and computational strength on the edge computing devices are limited. As a result, it is essential to have more efficient network architectures and less expensive inference overhead. Therefore, there is increasing attention from the research community to study the compression of modern deep neural networks that are typically over-parameterized and computationally costly. ",
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+    },
+    {
+        "type": "text",
+        "text": "Several categories of approaches are proposed to decrease the computational overhead of neural networks, such as lightweight neural network architectures (Howard et al., 2017), neural architecture search (NAS) (Elsken et al., 2018), and network pruning (Han et al., 2015; 2016; Wen et al., 2016; ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/61ab447deea38b9cb961435457a765789a69c77b3b9a0830b5b5c660bbd7a7fc.jpg",
+        "table_caption": [
+            "Table 1: Comparison between different quantizers: All-Layer $( A L )$ denotes quantizing all the parameters of all the operators in networks, including weights, bias, activations, and the scaling factor for low-precision networks; $\\pmb { B N }$ donates that the BN operation is only invoked in training but merged into weights and induces no overhead in integer inference; Linear-Symmetric $( L S )$ denotes linear symmetric quantization; Activation Functions $( A F )$ donates the support of Leaky $R e L U$ and activation functions besides ReLU. Structure-Intact $( S I )$ indicates the network structure is unmodified. "
+        ],
+        "table_footnote": [
+            "a Naive linear quantization, which finds min-max value at runtime. b Clustering-based approaches to quantize weights. c DoReFa falls into linear asymmetric quantizer due to the need for offset. d In Choi (2018), they use PACT to quantize activations, and DoReFa to quantize weights. e DoReFa, RQ, WRPN, and PACT are designed for $R e L U$ , but they can be extended to support other activation functions in theory. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>AL</td><td>BN</td><td>LS</td><td>AFe</td><td>SI</td></tr><tr><td>Deep Compression (Han et al.,2016)b</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>WQ (Park et al., 2017)b</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>LQ-Nets (Zhang et al., 2018a)</td><td></td><td></td><td></td><td></td><td>√</td></tr><tr><td>Min-Max Linear Quantizationa</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoReFa (Zhou et al., 2016)c</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RQ (Louizos et al., 2019)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>WRPN (Mishra et al., 2018)</td><td></td><td></td><td>√</td><td></td><td></td></tr><tr><td>PACT (Choi, 2018)d</td><td></td><td></td><td></td><td>√</td><td></td></tr><tr><td>LLSQ(ours)</td><td>厂</td><td></td><td>√</td><td>1</td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Molchanov et al., 2017). Besides these techniques, quantizing high-precision floating-point networks to lower bitwidth representation can also drastically decrease both the static parameters and the intermediate data generated during the network inference, resulting in reduced memory footprint and also computational intensity. And this paper focuses on the quantization of neural networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Quantization technique is also closely related to the implementation of specialized hardware that maps the procedure of network inference onto the energy-efficient low-precision integer or fixedpoint arithmetic circuits. In the hardware perspective, low-precision integer accelerators or processors are dominating the solutions targeted on neural network inference, especially for mobile and embedded scenarios. Google’s Tensor Processing Unit 1.0 (TPU) (Jouppi et al., 2017), Unified Deep Neural Network Accelerator (UNPU) (Lee et al., 2018), Eyeriss (Chen et al., 2018), Stripes (Judd et al., 2016), Pragmatic(Albericio et al., 2017) and many other newly proposed hardware implementations are generally reliant on the effectiveness of the underlying quantization techniques, which are especially crucial for the low-precision integer hardware designed to process binary, ternary, 4-bit or 8-bit networks. In other words, quantization is not only a method to reduce the memory footprint as in traditional work, but also a mandatory step to make the network deployable on integer hardware. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Though there is a lot of prior work that investigates low-precision quantization, they mainly target on reducing the memory overhead caused by floating or high precision data representation in the networks, but not focus on specialized integer hardware for network inference. To enable the neural network processors to work with low-precision integer operands and minimize the accuracy losses, a good network quantizer must satisfy the constraints as enlisted in Table 1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "First, all the parameters, including weights, bias, activations, partial results that eventually accumulate to an activation, and even the scaling factors, which are indispensable for low-precision networks like binary and ternary representation, must be quantized into low bitwidth integers as required by the underlying specialized hardware. In some prior work (Zhou et al., 2016; Zhu et al., 2017; Zhang et al., 2018a; Mishra et al., 2018; Choi, 2018), they either leave bias and scaling factors unquantized or keep the first and last layer in full or high precision. Besides, some designs rely on high-precision internal register or ALUs to support high-precision partial results that are generated during computation before the final output of activations or features. For example, Krishnamoorthi (2018), which quantizes the weights and activations to 8-bit, directly use 32-bit accumulators to cache the intermediate values or partial results to avoid overflows. However, for 4-bit and lower bitwidth, the integer accelerators cannot afford high bitwidth accumulators, which indicates higher silicon area and power cost. For integer-only-arithmetic, we quantize the bias to fixed-point numbers by using a straight-forward method. The value range of these numbers is wide, resulting in overflows of the low bitwidth accumulators. To overcome this problem, we quantize the bias to 8-bit and finetune the bias of the model. As shown in Figure 1, the bitwidth of accumulators can be reduced to 16-bit. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Second, the BatchNorm (BN) layer does not necessarily need to be processed during inference for the reduction of computation and memory cost. For most of the convolutional neural networks, BN layers are often after the Conv or $F C$ layers. In these situations, $B N$ can be merged into the weights and biases of the corresponding Conv or $F C$ layers. However, in Zhou et al. (2016); Zhang et al. (2018a), they use asymmetric or non-linear quantization, causing barriers to $B N$ fusion. There are two ways to overcome this obstacle. One is $\\mathbf { \\bar { \\Sigma } } ^ { 6 6 }$ folded training”(Krishnamoorthi, 2018), which adopts BN fusion before weights quantization in every training step; the other is to use symmetric linear quantization. However, the first method doubles the training time, while the second one has no additional computational overhead, which will be introduced in Section 3.4. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Third, linear quantization is necessary for state-of-the-art accelerators. There are many non-linear quantization methods which achieve excellent bitwidth reduction efficacy and accuracy tradeoffs. In these cases, it requires additional transformation to have correct arithmetic results after quantizing the value into non-linear distribution. For example, as in Han et al. (2016); Park et al. (2017), it necessitates the operation of table lookup to have correct multiplication between quantized values. However, the linear quantization can make full use of the low-precision arithmetic components in off-the-shelf accelerators. Further, linear quantization can be divided into symmetric mode and asymmetric mode. Asymmetric quantization has one more parameter (e.g., zero-point (Krishnamoorthi, 2018)) than symmetric quantization, and it requires additional subtraction or linearoperation before multiplication. As a result, the symmetrical mode is compatible with the mainstream integer accelerator chip design and do not require the redesign of datapath in these hardware. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fourth, different CNNs or applications usually use a variety of activation functions. For instance, the object detection model Redmon et al. (2016) typically uses Leaky ReLU. And the bottleneck of ResNet block does not use any activation function. The quantization methods are expected to be adapted to these situations. However, Zhang et al. (2018a); Park et al. (2017) only focus on the quantization of activations after ReLU. In this paper, we demonstrate our method is friendly to different activation methods such as Leaky ReLU. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Some of the previous researches change the network structure for better quantization performance, e.g., Mishra et al. (2018) double or even triple the convolutional filters to reduce accuracy degradation. For the energy-efficient integer neural network chips, it needs to remap the changed network architecture to hardware and adds to computational and memory access overhead due to the increased filters and parameters. As a result, keeping the network structure intact is important. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Concerning all the factors above, in this paper, we present a learned linear symmetric quantization (LLSQ) method and also evaluate it on a low-precision neural network accelerator through hardware-software co-design. Specifically, our mainly contributions are: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Unlike most of other quantization methods, we quantize the whole network including the first and last layers. We also quantize bias and scaling factors, in support of the low bitwidth integer arithmetic units and accumulators on the accelerator.   \nWe adopt learned linear symmetric quantization schemes which are hardware friendly (such as the convenience of $B N$ fusion implementation) while achieving state-of-the-art prediction accuracy. We design a specialized low-precision CNN inference accelerator to validate the methodology, which supports 2/4/8 integer operating and work with high efficiency. We then deploy our quantization model on the accelerator to illustrate the efficacy of the workflow. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 MOTIVATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Edge or embedded neural network accelerators generally have three primary design goals— smallfootprint, high-throughput/low-latency, and low-power. For different applications and scenarios, the prior researches on specialized deep learning processors are often falling into different categories: cloud-oriented hardware for warehouse machines, low power mobile processors and ultra-low power accelerators for IoT or cyber-physical devices. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For mobile and embedded usage, specialized neural network processors are becoming increasingly popular as an efficient hardware solution of inference. DianNao (Chen et al., 2014) is proposed for fast inference of DNNs and it uses 16-bit fixed-point multipliers for small silicon area and lowenergy. Later, ShiDianNao (Du et al., 2015) is introduced and it burns extremely low energy consumption by putting all weights onto the SRAM to eliminate considerable DRAM accesses. Besides, DeepBurning (Wang et al., 2016) simplifies the design flow of accelerator for different NN models. Eyeriss (Chen et al., 2018) is also another representative of low-power accelerators. And it presents a row-stationary (RS) dataflow to minimize data movement energy consumption on a spatial architecture. To further reduce computation overhead, EIE (Han et al., 2016) exploits the sparsity and low-bit compression of the NNs and achieves better throughput, energy and area efficiency. These typical edge neural network processors are accepting fixed-point data input and using fixed-point processing elements to reduce the power and chip area overhead caused by floating-point arithmetic components and memory. For the cloud scenarios, specialized architectures like TPU (Jouppi et al., 2017) and FPGA-based accelerator cards are also replacing conventional GPGPU and CPU for highthroughput inference tasks. Even for cloud-oriented inference architectures, fixed-point processing architectures like TPU are favored because they are able to deliver much higher throughput for the given power budget and silicon area overhead. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, for the fixed-point or integer hardware targeted on neural network acceleration, quantization is prerequisite to convert the floating-point network model into the fixed-point format compatible with the specialized hardware, and it is also a critical step to ensure the accuracy of the network after conversion. Many prior quantization methods are intended to reduce the running overhead of networks but ignore the architecture and working mechanism of integer neural network processors, as illustrated in Table 1, and they sometimes face considerable accuracy losses, or performance penalty or even fail to be supported on the realistic integer datapath due to the unconsciousness of the underlying hardware. This problem becomes particularly important for the hardware that is designed to run low bitwidth networks such as binary, ternary, and 2/4-bit models. For instance, Deep compression and WQ are clustering-based quantization methods, and they still need high-precision values to represent the weights, bias, and activations. As a result, they are not compatible with the hardware that only supports low-precision computing. LQ-Nets uses non-linear quantization based on the binary code and basis vector, and it can theoretically calculate the inner products between quantized weights and activations by bitwise operations only. However, it requires intensive modifications to the design of current processors by adding a lot of look-up tables in the datapath. Further, bias and scaling factors are not quantized in PACT and WRPN, resulting in performance penalty when employing additional high-precision or float-point ALUs to deal with them. In contrast, our LLSQ is designed to ease the model quantization flow for the specialized integer neural network processors by conforming to the constraints specified in Table 1. To validate the importance of hardware-aware quantizer and software/hardware co-design, we also design a specialized CNN accelerator for wearable applications. And the specialized accelerator supports 2/4/8 integer operation and adopts the dataflow of low latency and energy design. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 NETWORKS WITH LEARNED LINEAR SYMMETRIC QUANTIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we firstly give the overview of the proposed quantization scheme. Then we detail the scheme including low-precision representation, quantized network training, and the deployment of quantization model on our specialized integer-only CNN accelerator for fast inference. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 OVERVIEW OF LLSQ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Many of the previous researches focus on the quantization-aware training in GPU, showing the potential of low-bit quantization on CNNs. Han et al. (2016); Park et al. (2017); Zhang et al. (2018a) propose non-linear quantization methods but lacks of a detailed description of the hardware feasibility. Krishnamoorthi (2018) provides a quantization scheme that quantizes weights and activations into 8-bit integers and integer-arithmetic-only implementation on ARM CPUs. The method achieves evident hardware acceleration effects, but does not fully exploit lower-precision quantization. Based on the researches, we propose a quantization scheme for state-of-the-art specialized accelerators operating on low-precision integers only. Figure 1 shows an overview of the proposed scheme. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c9cfdd978cc908a0f869aa6ced258efd9a3f69064b2bf31b5fca1215f8c520d7.jpg",
+        "image_caption": [
+            "Figure 1: An overview of LLSQ: using pre-trained weights for fast convergence; Retraining of the network with quantized weights and activations; BN fusion for efficient inference; Quantization of bias and scaling factors; Deployment of the quantized model to our accelerator. As shown in this figure, weights, activations, bias, and scaling factors are quantized to low-bit integers. And the bandwidth of accumulator can be set to lower (e.g., 16-bit in our experiments). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Compared with prior work, our proposed quantization scheme pays more attention to the constraints imposed by real hardware. We use a unified learned linear symmetric quantizer to quantize weights and activations. And the quantizer has only one parameter, known as the scaling factor. Linear symmetric quantization consumes little additional resources based on the mainstream integer accelerator designs while achieving state-of-the-art accuracy in various networks. After that, we adopt BN fusion for fast inference on hardware. As for bias and scaling factors, we also quantize them to low-bitwidth integers. The integer accelerator illustrated in Figure 1 is an illustrative case of 4-bit quantization and hardware acceleration. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 MAKING FULL USE OF THE PRE-TRAINED PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In experiments, we find that it is more efficient to start with the pre-trained full-precision parameters before quantization. Louizos et al. (2019); Zhou et al. (2017) use pre-trained weights for fast convergence and deployment, while Zhang et al. (2018a); Choi (2018); Cai et al. (2017) train quantized network from scratch to show the robustness of the algorithm. However, for some object detection models, the backbone models and pre-trained weights are essential to the detection performance. Redmon et al. (2016) shows that the pre-trained high-resolution classification network gives an increase of almost $4 \\% m A P$ . To have better performance in classification, object detection, and other CNN based tasks, in this paper, we use pre-trained parameters to initialize the networks. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 LOW-PRECISION REPRESENTATION AND QUANTIZATION ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use channel-wise quantization for $C o n \\nu$ layers and layer-wise quantization for $F C$ layers and activations. And we adopt the symmetric linear quantization to quantize weights or activations into $k$ bits words(e.g., 4-bit), which can be defined as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1c3213c0d833aede11b67e5f1b8a6938c5b80887a6ac52e4e5a89db4739ad6b.jpg",
+        "text": "$$\n\\begin{array} { c } { { { \\pmb x } ^ { q } = Q u a n t i z e _ { k } ( { \\pmb x } ^ { r } \\mid \\alpha ) } } \\\\ { { { \\pmb q } = \\displaystyle \\frac { { \\pmb x } ^ { q } } { \\alpha } = c l a m p ( \\lfloor \\displaystyle \\frac { { \\pmb x } ^ { r } } { \\alpha } \\rceil , - 2 ^ { k - 1 } , 2 ^ { k - 1 } - 1 ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            727,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { x } ^ { r } \\in \\mathbb { R }$ is one kernel of weights or one layer of activations, the variable $\\alpha \\in \\mathbb { R } ^ { + }$ is the quantization parameter, known as the scaling factor, while $\\pmb { q } \\in \\{ - 2 ^ { k - 1 } , \\dotsc , 0 , 1 , \\dotsc , 2 ^ { k - 1 } - 1 \\}$ is the integer values flowing in the integer accelerator and $\\pmb { x } ^ { q } \\in \\{ - 2 ^ { k - 1 } \\alpha , \\dotsc , 0 , \\alpha , \\dotsc , ( 2 ^ { k - 1 } - 1 ) \\alpha \\}$ is the quantized weights or activations. Note that for activations, which are non-negative values if the ActFun is $R e L U$ , we clamp them to $[ 0 , 2 ^ { k } - 1 ]$ , resulting in $\\textbf { \\em q } \\in \\ \\{ 0 , 1 , \\ldots , \\bar { 2 } ^ { k } - 1 \\}$ and $\\pmb { x } ^ { q } \\in \\{ 0 , \\alpha , \\dots , ( 2 ^ { k - 1 } - 1 ) \\alpha \\}$ , respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As defined above, we use $\\alpha$ as our quantization parameter. And we optimize it with: ",
+        "bbox": [
+            174,
+            875,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/090766b9ed013bd75be34fd134ca3ff39fcee1231ca8214811936c0eee07d084.jpg",
+        "text": "$$\n\\alpha ^ { * } = \\arg \\operatorname* { m i n } _ { \\alpha } \\int p ( \\pmb { x } ^ { r } ) | \\pmb { x } ^ { q } - \\pmb { x } ^ { r } | ^ { l }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/493298ce7ededc24a2f0ac4c094582e2a6fa96a27598987b661e7aac6574c352.jpg",
+        "table_caption": [
+            "Table 2: Comparison of SG and EMA. "
+        ],
+        "table_footnote": [
+            "VGGSmall is trained on Cifar10 and ResNet18 is on ImageNet. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>VGGSmallw4a4</td><td>VGGSmallw2a2</td><td>ResNet18w4a4</td><td>ResNet18w3a3</td></tr><tr><td>EMA</td><td>93.95</td><td>92.78</td><td>69.48</td><td>66.80</td></tr><tr><td>SG</td><td>94.34</td><td>93.31</td><td>69.84</td><td>68.08</td></tr></table>",
+        "bbox": [
+            192,
+            132,
+            805,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { \\boldsymbol { x } } ^ { r } , \\mathbf { \\boldsymbol { x } } ^ { q }$ are the same factors defined in Equation 1, $p ( \\pmb { x } ^ { r } )$ is the probability density distribution of $\\pmb { x } ^ { r }$ , and $l \\in \\{ 1 , 2 \\}$ is an optional constraint (We use 2 in our experiments). In Figure 2, we present the relationship between quantization error and $\\alpha$ . When fixing weights $\\pmb { x } ^ { r }$ , we can find the optimal $\\alpha ^ { * }$ by using the brute-force search approach, which induces high computation cost. Besides, the weights are updated during the re-training phase and the optimal value $\\alpha ^ { * }$ changes accordingly. In other words, the optimal value for the factors is not fixed and it takes considerable computational overhead to find the dynamic optimal value. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Inspired by Zhang et al. (2018a), we find through experiments that there is no need to find the optimal value $\\alpha ^ { * }$ , and it works well enough to find a near-optimal value $\\tilde { \\alpha } ^ { * }$ . Generally, quantization can be considered as a regularization of the networks, and the quantization parameter $\\alpha$ needs only to be adjusted to a near-optimal value to preserve the network capacity. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then the problem becomes how to find a $\\tilde { \\alpha } ^ { * }$ in the forward pass of the network training phase. At the beginning of training, we assign $\\alpha$ an initial value. Then in every training iteration, we explore between $2 \\alpha$ and $\\alpha / 2$ to find a better search direction $d _ { b e t t e r } \\in \\{ - 1 , 0 , 1 \\}$ , and use $- \\alpha ^ { 2 } d _ { b e t t e r }$ as the simulated gradient (SG) of $\\alpha$ which is detailed in Equation 9. The gradients of other parameters are still obtained by backpropagation algorithm. After that, we update all parameters with the gradients or simulated gradients. Another method is updating $\\alpha$ by the exponential moving average (EMA). We experiment both of the methods, and the results show that SG is generally better than EMA on various networks (See Table 2). If not specifically stated, we use the SG method in experiments. The re-training process with weights and activations quantized is summarized in Step 1 of Algorithm 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/e686ab91bd9834460084d50fa5a530a5652517ef6dd02d9f3b062233971cf826.jpg",
+        "image_caption": [
+            "Figure 2: L2 distance of quantization. The data is from weights of the first FC layer in AlexNet. As shown in the figure, the optimal $\\alpha ^ { * }$ changes with the updating of weights. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 BN LAYER FUSION OF QUANTIZED NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            631,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As described in Section 1, merging the $B N$ layers into convolutional layers can reduce the latency of network inference by removing additional computation overhead. The operator of quantized $C o n \\nu ^ { 1 }$ and $F C$ layers can be expressed as ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ebba015f6a840f471a73c626075dfeb380ce381b545fd6f39b65616c5dd9a1a.jpg",
+        "text": "$$\n\\pmb { o } = \\alpha _ { a } \\pmb { q } _ { a } \\alpha _ { w } \\pmb { q } _ { w } + b\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha , \\mathbf { \\pmb q }$ are the same as Equation 1, $\\alpha _ { a } \\mathbf { q } _ { a }$ , $\\alpha _ { w } \\pmb { q } _ { w }$ donate the quantized activations and weights, while $b$ is the bias and $^ o$ is the output feature vector. Note that $\\alpha _ { a } , \\alpha _ { w }$ and $b$ are full precision values. And the $B N$ layer can be formulated as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14348f271a7b8be6846e12a58b4115632984efca337f87a51c5b24e0a7f85848.jpg",
+        "text": "$$\n{ \\pmb y } = \\frac { { \\pmb o } - { \\pmb \\mu } } { \\sqrt { \\sigma ^ { 2 } + \\epsilon } } \\gamma + \\beta\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            772,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu$ and $\\sigma ^ { 2 }$ are EMA statistics, $\\gamma$ and $\\beta$ are learned parameters in $B N$ layers. ",
+        "bbox": [
+            173,
+            811,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Obviously, we can merge $B N$ layers and figure out the corrected parameters: ",
+        "bbox": [
+            173,
+            833,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddf4d3ddf78ad9272685f5372976df34eadf889b5fd026e52ee40cc3961d135b.jpg",
+        "text": "$$\n\\hat { \\alpha } _ { w } = \\zeta \\alpha _ { w } ; ~ \\hat { b } = ( b - \\mu ) \\zeta + \\beta ; ~ \\zeta = \\frac { \\gamma } { \\sqrt { \\sigma ^ { 2 } + \\epsilon } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 BIAS AND SCALING FACTOR QUANTIZATION FOR LOW-BIT INTEGER ONLY ARITHMETIC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Further, the outputs of layers are quantized according to Equation 1. For integer-only-arithmetic, the bias use $\\alpha _ { a } \\alpha _ { w }$ as its scaling factor. And for the multiplier $\\frac { \\alpha _ { a } \\alpha _ { w } } { \\alpha _ { o } }$ , we use bit-shift quantization (See Equation 10) so that no multiplication but bit-shift operation is needed in hardware. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ffe3df26f52e6a90607ba9fd9c1e3ee27fbc61174fa3ca2cc1013733b6e59c1.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\alpha _ { o } { \\bf q } _ { o } = \\alpha _ { a } { \\bf q } _ { a } \\hat { \\alpha } _ { w } { \\bf q } _ { w } + \\hat { b } } } \\\\ { { { \\displaystyle \\bf q } _ { o } = \\frac { \\alpha _ { a } \\hat { \\alpha } _ { w } } { \\alpha _ { o } } ( { \\bf q } _ { a } { \\bf q } _ { w } + { \\bf q } _ { b } ) } } \\\\ { { { \\displaystyle \\alpha _ { b h e r e } q _ { b } = c l a m p ( \\bigl \\lfloor \\frac \\hat { b } { \\alpha _ { a } \\hat { \\alpha } _ { w } } \\bigr \\rceil , - 2 ^ { k _ { b } - 1 } , 2 ^ { k _ { b } - 1 } - 1 ) } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            191,
+            666,
+            284
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\alpha _ { a } \\alpha _ { w }$ is a very small number, resulting in large quantization noise when adopting the clamp operation. In addition, the quantization of the scaling factors $\\alpha$ can also raise the quantization noise of weights and activations. Parameter re-training summarized in Step 2 of Algorithm 1 is required. ",
+        "bbox": [
+            176,
+            287,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the re-training phase, we adopt STE (Bengio et al., 2013) to realize the non-differentiable quantization function. ",
+        "bbox": [
+            174,
+            349,
+            823,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For weights and bias, we have ",
+        "bbox": [
+            174,
+            385,
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+            400
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eca94318137ae2933f22ffb1f9586eec080ee57f557a5e7ffbd77f05017cf217.jpg",
+        "text": "$$\n\\frac { \\partial y } { \\partial w ^ { q } } \\simeq \\frac { \\partial y } { \\partial w ^ { r } } ; \\frac { \\partial y } { \\partial b ^ { q } } \\simeq \\frac { \\partial y } { \\partial b ^ { r } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            397,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For activations, we have ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62919ade818968744d30209cba88706bfa6ca7aaff3357bc3a8198e8f133d070.jpg",
+        "text": "$$\n\\frac { \\partial y } { \\partial a ^ { q } } \\simeq \\left\\{ \\begin{array} { l l } { \\frac { \\partial y } { \\partial a ^ { r } } } & { i f 0 \\leq a ^ { r } \\leq ( 2 ^ { k } - 1 ) \\alpha } \\\\ { 0 } & { o t h e r w i s e } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, three sets of experiments on Cifar10, ImageNet and Pascal VOC datasets are presented. First, we conduct our proposed learned linear symmetric quantization (LLSQ) on weights and activations, leaving the first and last layers in full precision for a fair comparison with Zhang et al. (2018a). Second, we quantize the whole networks including the first and last layers, which is referred as LLSQF (LLSQ for Full network). Finally, we quantize the remaining bias and scaling factors. LLSQ is implemented in PyTorch (Paszke et al., 2017), and most of the baselines it uses in evaluation are from PyTorch Model $Z _ { 0 0 } { } ^ { 2 }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 QUANTIZATION OF WEIGHTS AND ACTIVATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We firstly employ the VGG-Small network on Cifar10 to verify the LLSQ method. After that, we use AlexNet (Krizhevsky et al., 2012), ResNet18, ResNet34 (He et al., 2016), particularly the lightweight and hard-to-compress network architecture of MobileNet (Howard et al., 2017; Sandler et al., 2018) etc. to conduct more detailed experiments on the ImageNet dataset. Finally, we also quantize YOLOv2 (Redmon & Farhadi, 2017) to demonstrate that LLSQ also works well for complicated applications and especially the task adopting the activation functions like Leaky ReLU other than ReLU used in previous work. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "VGG-SMALL ON CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The VGG-Small architecture is the same with Louizos et al. (2019); Zhang et al. (2018a), consisting of six Conv layers, three MaxPool layers, and one $F C$ layer. We adopt a cosine learn rate scheduler to train the VGG-Small reference and the quantized models. Specifically, we train the reference network for 400 epochs using an initial learning rate of 2e-2. And for the training of the quantized network, we use a warmup learning rate scheduler in the first ten epochs with an initial learning rate of 2e-3. In all quantization experiments, the total training epochs are 100. The VGG-Small quantization results are provided in Table 3. With 3-bit weights and 3-bit activations, the accuracy using our method is better than state-of-the-art method, LQ-Nets. And even when the first and last layers are all quantized in the same way, the loss of accuracy is minimal. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/574a3ab8e7c43029b38313d8c303c558a48cd147b68f29b9f5dd5a8dae9df196.jpg",
+        "table_caption": [
+            "Table 3: Comparison with the state-of-the-art low-bit quantization methods on CIFAR-10. The bitwidth for weights $( { \\pmb w } )$ , activations $\\mathbf { \\Pi } ( \\mathbf { a } )$ , bias $( b )$ and scaling factor $( \\alpha )$ are given. "
+        ],
+        "table_footnote": [
+            "\\* first and last layer in full precision "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>#Bits wla/b/α</td><td>Acc(%)</td><td>Degradation(%)</td></tr><tr><td rowspan=\"3\">LQ-Nets* (Zhang et al., 2018a)</td><td>Reference</td><td>93.8</td><td></td></tr><tr><td>3/3</td><td>93.8</td><td>0.0</td></tr><tr><td>2/2</td><td>93.5</td><td>0.3</td></tr><tr><td rowspan=\"3\">RQ (Louizos et al., 2019)</td><td>Reference</td><td>93.05</td><td></td></tr><tr><td>8/8</td><td>93.30</td><td>-0.25</td></tr><tr><td>4/4 2/2</td><td>91.57</td><td>1.48 2.31</td></tr><tr><td rowspan=\"4\">LLSQ*(ours)</td><td>Reference</td><td>90.92 93.34</td><td></td></tr><tr><td>4/4</td><td>94.34</td><td>-1.00</td></tr><tr><td>3/3</td><td>94.02</td><td>-0.68</td></tr><tr><td>2/2</td><td>93.31</td><td>0.03</td></tr><tr><td rowspan=\"4\">LLSQF(ours)</td><td>4/4</td><td>94.30</td><td>-0.96</td></tr><tr><td>3/3</td><td>94.07</td><td>-0.73</td></tr><tr><td>2/2</td><td>93.12</td><td>0.22</td></tr><tr><td>4/4/8/8</td><td>93.84</td><td>-0.50</td></tr></table>",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "IMAGENET DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We then quantize AlexNet, ResNet18, ResNet34 and MobileNetv2 on ILSVRC2012 dataset with different bitwidth configuration to demonstrate the effectiveness of the method. All of the pretrained float-point weights except MobileNetv $2 ^ { 3 }$ are downloaded from the PyTorch Model Zoo, and they are trained for 90 epochs with a step learning rate scheduler. After loading the pre-trained weights, we employ a warmup learning scheduler in the first three epochs and the cosine scheduler in the remained 57 epochs with an initial learning rate of 2e-2. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 3a, when quantizing both weights and activations, our degradation of accuracy is significantly smaller than LQ-Nets, PACT, and RQ. Especially, LLSQ outperforms the baselines when quantizing weights and activations into 4-bit. And it also outperforms other non-linear quantization methods with different bitwidth. Figure 3b shows that even with the first and last layers quantized, it can still achieve near baseline performance. In overall, the accuracy drop is less than $0 . 4 \\%$ in ResNet18, ResNet34, and AlexNet when quantizing the whole network. We also quantize MobileNetv2, a more compact network, and obtain results that are significantly better than RQ. Please check Table 7 for detailed results. ",
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+    },
+    {
+        "type": "text",
+        "text": "OBJECT DETECTION ON PASCAL VOC ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We also apply the proposed LLSQ to YOLOv2. The backbone of YOLOv2 is Darknet-19, and its activation function is Leaky ReLU, so that the activations contain negative values. For YOLOv2 on Pascal VOC, we adopt the same quantization configuration (See Section 3.3) of the weights to the activations. Results are listed in Table 4. As shown in the table, LLSQ induces minor losses of $m A P$ in different bitwidth presentation. For comparison, we also quantize the activations into signed 5-bit integers using PACT, and consequently face considerable mAP losses $( 5 4 . 8 m A P )$ . Please note that we use the open-source PyTorch implementation of YOLOv2 4 as the baseline. We train the quantized model for 170 epochs (2/3 of baseline) with an initial learning rate of 1e-4 (1/10 of baseline). ",
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+    {
+        "type": "text",
+        "text": "(a) Comparison with other state-of-art methods.   \nLower is better. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c77a4b3d1753549da14a63fb83a3565751683e1648bc66bbb2df309b4224a3cf.jpg",
+        "image_caption": [
+            "Figure 3: Quantization results on different networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/a97efd5b33c1146e456371e6660c6f9710b806e60f52af3c35caf2df57ec3546.jpg",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4a514cff63218f620ab0a792e76fa2af826a26be03b8a441959593422657046f.jpg",
+        "table_caption": [
+            "Table 4: LLSQ on YOLOv2 detector. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>bitwidth</td><td>mAP</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>mbike</td><td>person plant</td><td></td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td></tr><tr><td>FP32</td><td>73.2</td><td>78.9</td><td>80.0</td><td>72.8</td><td>62.3</td><td>47.1</td><td>79.2</td><td>79.6</td><td>85.7</td><td>54.4</td><td>79.7</td><td>72.2</td><td>83.3</td><td>81.1</td><td>79.2</td><td>74.8</td><td>48.4</td><td>75.7</td><td>72.3</td><td>83.4</td><td>73.0</td></tr><tr><td>w4a5</td><td>70.3</td><td>73.9</td><td>76.1</td><td>67.8</td><td>57.3</td><td>39.9</td><td>81.2</td><td>79.1</td><td>82.6</td><td>51.8</td><td>75.7</td><td>68.3</td><td>80.3</td><td>83.9</td><td>78.7</td><td>70.6</td><td>42.6</td><td>72.2</td><td>71.6</td><td>83.5</td><td>69.5</td></tr><tr><td>w32a5</td><td>71.2</td><td>75.5</td><td>75.9</td><td>71.4</td><td>60.4</td><td>42.4</td><td>80.6</td><td>80.0</td><td>83.3</td><td>53.5</td><td>75.8</td><td>68.1</td><td>70.8</td><td>82.6</td><td>79.5</td><td>71.6</td><td>45.5</td><td>69.9</td><td>72.1</td><td>84.6</td><td>70.4</td></tr><tr><td>w4a8</td><td>73.4</td><td>74.5</td><td>79.1</td><td>75.5</td><td>60.6</td><td>43.8</td><td>80.9</td><td>80.7</td><td>85.8</td><td>56.6</td><td>80.0</td><td>70.9</td><td>83.5</td><td>84.5</td><td>81.0</td><td>74.5</td><td>47.5</td><td>74.8</td><td>75.2</td><td>84.1</td><td>73.6</td></tr><tr><td>w4a32</td><td>74.2</td><td>74.6</td><td>78.6</td><td>75.5</td><td>66.0</td><td>47.4</td><td>80.8</td><td>83.2</td><td>87.4</td><td>57.3</td><td>80.3</td><td>70.8</td><td>83.7</td><td>84.3</td><td>83.0</td><td>74.8</td><td>49.4</td><td>74.2</td><td>73.8</td><td>85.2</td><td>73.5</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.2 BN FUSION AND QUANTIZATION OF BIAS AND THE SCALING FACTOR ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We adopt BN fusion in the PostAct $C o n \\nu {  } B N {  } R e L U )$ networks according to the formula in Section 3.4. And the scaling factor of bias is the product of the corresponding scaling factors belonging to the activations and the weights, respectively. After that, we visualize the bias value distribution of VGG-Small. Figure 4 shows $b / \\alpha _ { b }$ is distributed between a vast range (-1000, 1000), resulting in overflows of low bitwidth accumulators. And the overflow phenomena have a significantly harmful impact on the network performance. To deal with this issue, we quantize the bias and the scaling factors to 8-bit words, and then fine-tune the networks to restore the original performance. Generally, we need fine-tuning for one epoch only. After the quantization of bias and scaling factor, we have a fully quantized model and have it deployed onto our integer-only accelerator with 16 bitwidth accumulators. Table 3 and 7 show that the accuracy loss is negligible with $\\pmb { w } 4 \\pmb { a } 4 b 8 \\pmb { \\alpha } 8$ quantization on both VGG-Small and AlexNet. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/9e1b4c6d38ebf4bf95490ae5f2d98a8de7f18a1e2a205101e4a89dd2655e5f46.jpg",
+        "image_caption": [
+            "Figure 4: Distribution of the bias/scaling factor. The data is from VGG-Small with w4a4 quantization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            640,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 DEPLOYMENT ONTO REALISTIC HARDWARE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The introduced linear symmetric quantization is intended to deploy the quantized networks to specialized integer-only-arithmetic CNN accelerators or other integer-only hardware. Our accelerators adopt the typical 2D systolic array architecture (Chen et al., 2018), but they are featured with 4-bit or 2-bit low-precision operation. As shown in Figure 1, the 8/4/2-bit accelerator has a $3 2 \\mathrm { x } 7 $ array of processing elements (PE). And the MAC unit in each PE consists of a 4-bit multiplier and a 16-bit accumulator. For the 4-bit accelerator, we use INT4 representation for weights, UINT4 for activations, INT8 for the bias and scaling factors, respectively. For the 2-bit accelerator, we use INT2 for weights, UINT2 for activations, INT8 for the bias and scaling factors, respectively. Through the quantization process described in the paper, we can have a fully quantized network that works directly on the CNN accelerator. In addition, as we use linear symmetric quantization, we can use a straight-forward way to conduct multiply-accumulate operations without introducing shifters or lookup tables, which means the quantized models can run on state-of-the-art integer accelerators and ensures that their output accuracy degradation is minimal as presented in the above sections. Finally, we implement the 8/4/2-bit integer neural network processors with Synopsys Design Compiler (DC) under the $4 0 \\mathrm { n m }$ technology, clocked at 800MHz. Table 5 shows that the $4 / 2$ -bit implementation achieves up to $2 . 5 6 \\mathrm { x }$ lower silicon area and 5.56x lower power compared to that of the 8-bit baseline. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8854ba753884fcd4329127c6483156dde9496e12274da6d1e276654249566f05.jpg",
+        "table_caption": [
+            "Table 5: Comparison of our low-precision integer Neural Network Processors. "
+        ],
+        "table_footnote": [
+            "Implemented and synthesized with Synopsys Design Compiler (DC) under the $4 0 \\mathrm { n m }$ technology. "
+        ],
+        "table_body": "<table><tr><td>Bitwidth</td><td>#MAC Unit</td><td>Throughput (GOps/sec)</td><td>Silicon Area (mm²)</td><td>Power (mW)</td></tr><tr><td>8-bit</td><td>224</td><td>179.2</td><td>4.71</td><td>228</td></tr><tr><td>4-bit</td><td>224</td><td>179.2</td><td>2.80</td><td>93</td></tr><tr><td>2-bit</td><td>224</td><td>179.2</td><td>1.84</td><td>41</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduced a learned linear symmetric quantization (LLSQ) to quantize the whole network including the bias and scaling factors. We also use BN fusion and low bitwidth accumulators to reduce the network inference overhead and the hardware resources in integer neural accelerators. We show that our proposed method performs well for various networks on Cifar10, ImageNet, and Pascal VOC datasets. We also show that even the linear symmetric quantizer can obtain better results than asymmetric or non-linear quantization in the case of 4-bit networks. Finally, we deploy the quantized network onto our specialized integer-only neural network accelerator. Currently, the bitwidth of every layer in a network is all the same. Prior researches empirically find that different layers have different sensitivity to bitwidth of quantization. Hence in the future, we will explore a framework to support more flexible bitwidth for different layers or finer-grained quantization. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the National Natural Science Foundation of China under Grant 61874124 and Grant 61902375. ",
+        "bbox": [
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+    },
+    {
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+    {
+        "type": "text",
+        "text": "Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. CoRR, abs/1702.03044, 2017. URL http://arxiv.org/abs/1702.03044. ",
+        "bbox": [
+            176,
+            665,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Shuchang Zhou, Zekun Ni, Xinyu Zhou, He Wen, Yuxin Wu, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. arXiv: Neural and Evolutionary Computing, 2016. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. international conference on learning representations, 2017. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "IMAGENET DETAILED ",
+        "bbox": [
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+            869
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on AlexNet, ResNet18, ResNer34, and MobileNetv2. For all of the experiments, we adopt channel-wise quantization for Conv layers and layer-wise quantization for $F C$ layers as well as the activations. The AlexNet architecture is the same as the PyTorch Model Zoo, and it consists of five Conv layers, three $F C$ layers, three MaxPool layers, and two Dropout layers. To prevent over-fitting, we keep the Dropout layers when quantizing AlexNet. As shown in Figure 5d, we use the same learning rate scheduler for all experiments on ImageNet. The test curves are also shown in Figure 5. As we begin with the pre-trained full-precision weights, the test accuracy is already acceptable after one-epoch training. The final results are listed in Table 7. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f357404ca1cbe59eee6c8e4c7c8e7d2397fdb8082b4242c6fd4bb591fa6853ae.jpg",
+        "table_caption": [
+            "Table 6: Train Time of ResNet18 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>#Training Process</td><td rowspan=1 colspan=1>training time</td></tr><tr><td rowspan=1 colspan=1>Train the fp32 network from scratch</td><td rowspan=1 colspan=1>1.0x</td></tr><tr><td rowspan=1 colspan=1>Quantize w/a to 4/4 according to Step1 of Alg. 1</td><td rowspan=1 colspan=1>0.69x</td></tr><tr><td rowspan=1 colspan=1>Quantize w/a to 3/3 according to Step1 of Alg. 1</td><td rowspan=1 colspan=1>0.69x</td></tr><tr><td rowspan=1 colspan=1>Quantize b/α to 8/8 according to Step2 of Alg. 1</td><td rowspan=1 colspan=1>0.01x</td></tr></table>",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training time. The proposed LLSQ requires about $2 / 3$ training epochs than that of floating-point network training. In each training iteration, LLSQ needs extra computation cost to optimize the quantizers. Specifically, the simulated gradients generation of the scaling factors is the major cost. Table 6 shows the total training time comparison of ResNet18 network. The quantization training time is $70 \\%$ of baseline only. ",
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f6612721c7554999187d6d1ad01ee13872e2853d1471e362664f2ff4ea2e0ffc.jpg",
+        "image_caption": [
+            "Figure 5: Test curves for AlexNet, ResNet18, and ResNet34 on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "THE LLSQ ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Generate simulate gradients for $\\alpha$ : ",
+        "bbox": [
+            174,
+            762,
+            400,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f3ef4719326b420c9f663769e0706a0ff5897dc17bb10c706a7a55c676e39d5.jpg",
+        "text": "$$\n\\begin{array} { c } { { E _ { m } = \\displaystyle \\sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \\mid \\alpha ) ) ^ { 2 } } } \\\\ { { E _ { l } = \\displaystyle \\sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \\mid \\frac \\alpha 2 ) ) ^ { 2 } } } \\\\ { { E _ { r } = \\displaystyle \\sum _ { i } ( x _ { i } ^ { r } - q u a n t i z e _ { k } ( x _ { i } ^ { r } \\mid 2 \\alpha ) ) ^ { 2 } } } \\\\ { { d _ { b e t t e r } = \\displaystyle \\operatorname * { a r g m i n } ( [ E _ { l } , E _ { m } , E _ { r } ] ) - 1 } } \\\\ { { \\Delta G _ { \\alpha } = - \\alpha ^ { 2 } d _ { b e t t e r } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { x } ^ { r }$ is one kernel of weights or one layer of activations. $\\arg \\operatorname* { m i n } ( [ E _ { l } , E _ { m } , E _ { r } ] ) \\in \\{ 0 , 1 , 2 \\}$   \nselects the index of the smallest number in the array $[ E _ { l } , E _ { m } , E _ { r } ]$ . ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The bit-shift quantization can be formulated as: ",
+        "bbox": [
+            173,
+            138,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1aea6ad223594b09c2d4bf08914ae6c905d6b33dbf0d31f98c0ad0006ec61c2.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\pmb { \\alpha } ^ { q } = S Q _ { k } ( \\pmb { \\alpha } ) } } \\\\ { { \\nonumber } } \\\\ { { \\vphantom { \\int } } = \\frac { \\mathrm { c l a m p } ( \\mathrm { r o u n d } ( 2 ^ { q c o d e } \\cdot { \\pmb { \\alpha } } ) , - 2 ^ { k - 1 } , 2 ^ { k - 1 } - 1 ) } { 2 ^ { q c o d e } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { \\alpha } \\in \\mathbb { R } ^ { + l e n ( \\pmb { \\alpha } ) }$ is the scaling factors to be quantized, $k \\in \\mathbb { Z }$ is the bitwidth, and $q c o d e \\in \\mathbb { Z }$ is the parameter of the bit-shift quantizer simply obtained by: ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95a360f5c500680a9607b461dd31368121f2a6f4f37e173df5ede871818a2088.jpg",
+        "text": "$$\nq c o d e = k - \\mathrm { c e i l } ( \\log _ { 2 } ( \\operatorname* { m a x } ( \\pmb { \\alpha } ) ) + 1 - 1 0 ^ { - 5 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 LLSQ ",
+        "text_level": 1,
+        "bbox": [
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+            280,
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Input: Dataset $( { \\pmb x } , { \\pmb y } )$ , where $_ { \\textbf { \\em x } }$ is input and $\\textbf {  { y } }$ is label; Pre-trained full-precision parameters $( w , b )$ , where $\\pmb { w }$ is weights and $^ { b }$ is bias; Suppose the network consists of $L$ layers, ${ \\pmb w } _ { l } ^ { ( i ) }$ represents the $i _ { t h }$ kernel of weights of the $l _ { t h }$ layer while $\\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\mathbf { } \\mathbf { } \\mathbf \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf { } \\mathbf \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\Psi \\Psi \\mathbf \\Psi \\mathbf \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\mathbf \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi \\Psi $ is the output of the $l _ { t h }$ layer. ",
+        "bbox": [
+            174,
+            299,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Output: The quantized scaling factors: $\\hat { \\pmb { \\alpha } } _ { \\pmb { w } } ^ { q } = [ [ \\hat { \\alpha } _ { \\pmb { w } _ { 0 } ^ { ( 0 ) } } ^ { q } , \\dots ] , \\dots , [ \\hat { \\alpha } _ { \\pmb { w } _ { L - 1 } ^ { q } } ^ { q } , \\dots ] ]$ $\\pmb { \\alpha _ { a } ^ { q } } = [ \\alpha _ { { \\pmb { a } } _ { 0 } } ^ { q } , \\dots , \\alpha _ { { \\pmb { a } } _ { L - 1 } } ^ { q } ]$ ; The quantized weights $\\pmb { w } ^ { q }$ and bias $\\hat { b } ^ { q }$ . ",
+        "bbox": [
+            176,
+            345,
+            478,
+            414
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Step 1: Quantize weights and activations and $\\pmb { R e }$ -training ",
+        "text_level": 1,
+        "bbox": [
+            189,
+            426,
+            576,
+            440
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "// Re-training of quantized networks can converge faster and end with a higher accuracy due to the mechanism of $B N$ layers. ",
+        "bbox": [
+            189,
+            441,
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+            468
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "repeat Forward: $\\mathbf { } a _ { 0 } \\gets$ input for $l = 1 , \\cdots , L$ do for w(i)l in wl do // This is accelerated in parallel when implemented. $\\pmb { w } _ { l } ^ { ( i ) q }  \\mathrm { Q u a n t i z e } ( \\pmb { w } _ { l } ^ { ( i ) } \\mid \\alpha _ { \\pmb { w } _ { l } ^ { ( i ) } } )$ per Eq. (1) Generate simulated gradients for $\\alpha _ { w _ { l } ^ { ( i ) } }$ per Eq. (9) end for $\\begin{array} { r l } & { \\mathbf { \\Lambda } _ { { \\pmb { w } } _ { l } ^ { q }  \\mathrm { C o n c a t } } ^ { \\pmb { \\mathrm { c m u } } \\bullet \\mathbf { \\Lambda } _ { { \\pmb { w } } _ { l } ^ { q } } } } \\\\ & { \\mathbf { \\Lambda } _ { { \\pmb { a } } _ { l } }  R e L U ( B N ( C o n v ( \\mathbf { a } _ { l - 1 } ^ { q } , \\mathbf { w } _ { l } ^ { q } , \\mathbf { b } _ { l } ) ) ) } \\end{array}$ $\\mathbf { \\pmb { a } } _ { l } ^ { q } \\gets \\mathrm { Q u a n t i z e } ( a _ { l } \\mid \\alpha _ { \\pmb { a } _ { l } } )$ per Eq. (1) Generate simulated gradients for $\\alpha _ { { \\pmb a } _ { l } }$ per Eq. (9) ",
+        "bbox": [
+            199,
+            477,
+            697,
+            660
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "end for ",
+        "text_level": 1,
+        "bbox": [
+            214,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Backward: ",
+        "text_level": 1,
+        "bbox": [
+            214,
+            672,
+            294,
+            685
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Generate $\\Delta G$ for weights and bias and $\\Delta E$ for activation per Eq. (7), (8) and backpropagation algorithm. Update $w , b , \\alpha _ { w } , \\alpha _ { a }$ $i t e r \\gets i t e r + 1$ ntil iter $\\geq i t e r _ { m a x } / /$ need about 60 epochs, e.g. $i t e r _ { m a x } = 6 0 \\frac { \\mathrm { l e n } ( d a t a s e t ) } { b a t c h s i z e }$ ",
+        "bbox": [
+            207,
+            686,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Step 2: Quantize bias and scaling factor after BN fusion and $\\pmb { R e }$ -training   \niter $\\gets 0$   \n$\\hat { \\alpha _ { w } } , \\hat { b } \\gets \\mathbf { B } \\mathbf { N }$ fusion per Eq. (5)   \nrepeat Forward: $\\mathbf { } \\mathbf { } \\mathbf { 0 } \\gets$ input for $l = 1 , \\cdots , L \\mathbf { d }$ o $\\hat { \\alpha } _ { { w _ { l } } } ^ { q } \\gets \\mathrm { S Q } ( \\hat { \\alpha } _ { { w _ { l } } } )$ per Eq. (10) for ${ \\pmb w } _ { l } ^ { ( i ) }$ in ${ \\pmb w } _ { l }$ do // This is accelerated in parallel when implemented. $\\mathbf { \\dot { \\mathbf { w } } } _ { l } ^ { ( i ) q }  \\mathrm { Q u a n t i z e } ( \\mathbf { w } _ { l } ^ { ( i ) } \\mid \\hat { \\alpha } _ { \\mathbf { \\pmb { w } } _ { l } ^ { ( i ) } } ^ { q } )$ αˆqw(i) ) per Eq. (1) ",
+        "bbox": [
+            189,
+            771,
+            697,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\hat { b } _ { l } ^ { ( i ) q } \\gets \\mathrm { Q u a n t i z e } ( \\hat { b } _ { l } ^ { ( i ) } \\mid \\alpha _ { a _ { l - 1 } } ^ { q } \\hat { \\alpha } _ { { \\pmb w } _ { l } ^ { ( i ) } } ^ { q } )$ (i) ) per Eq. (1) Generate simulated gradients for $\\hat { \\alpha } _ { { \\pmb w } _ { l } ^ { ( i ) } }$ per Eq. (9) end for $\\begin{array} { r l } & { \\mathbf { \\Lambda } _ { u } ^ { \\mathrm { c u n t \\ : 1 0 1 } } } \\\\ & { \\mathbf { \\Lambda } _ { u } ^ { w _ { l } ^ { q } }  \\mathrm { C o n c a t } \\ w _ { l } ^ { ( i ) q } } \\\\ & { \\hat { b } _ { l } ^ { q }  \\mathrm { C o n c a t } \\ \\hat { b } _ { l } ^ { ( i ) q } } \\\\ & { a _ { l }  R e L U ( C o n v ( a _ { l - 1 } ^ { q } , { \\pmb { w } } _ { l } ^ { q } , \\hat { b } _ { l } ^ { q } ) ) } \\end{array}$ $\\alpha _ { { \\pmb a } _ { l } } ^ { q }  \\mathrm { S Q } ( \\alpha _ { { \\pmb a } _ { l } } )$ −  per Eq. (10) $\\mathbf { \\em a } _ { l } ^ { q } \\gets \\mathrm { Q u a n t i z e } ( \\mathbf { \\em a } _ { i } \\mid \\alpha _ { \\mathbf { \\em a } _ { l } } ^ { q } )$ per Eq. (1) Generate simulated gradients for $\\alpha _ { { \\pmb a } _ { l } }$ per Eq. (9) end for Backward: Generate $\\Delta G$ for weights and bias and $\\Delta E$ for activation per Eq. (7), (8) and backpropagation algorithm. Update $\\bar { w } , \\bar { b } , \\hat { \\alpha } _ { w } , \\bar { \\alpha } _ { a }$ $i t e r \\gets i t e r + 1$ until iter $\\geq i t e r _ { m a x } / /$ only need one epoch, e.g. $i t e r _ { m a x } = \\frac { \\mathbf { l e n } ( d a t a s e t ) } { b a t c h s i z e }$ ",
+        "bbox": [
+            187,
+            99,
+            727,
+            367
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/bdb876056b25e72196785c29bed1b5821e6cd345dd9fd94c365a4f2c0f0e9d82.jpg",
+        "table_caption": [
+            "Table 7: Comparison with state-of-the-art quantization methods on ImageNet. Top1, Top5 accuracy $\\% )$ and degradation of Top1 are given. "
+        ],
+        "table_footnote": [
+            "\\* First and last layers in full precision. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Model</td><td>#Bits w/a/b/α</td><td></td><td></td><td>Top1(%) Top5(%) Degradation</td></tr><tr><td rowspan=\"3\">LQ-Nets* (Zhang et al., 2018a)</td><td>ResNet18</td><td>Reference</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>70.3 69.3</td><td>89.5</td><td></td></tr><tr><td></td><td>4/4 3/3</td><td>68.2</td><td>88.8</td><td>1.0</td></tr><tr><td rowspan=\"3\">RQ (Louizos et al., 2019)</td><td>ResNet18</td><td>Reference</td><td>69.54</td><td>87.9</td><td>2.1</td></tr><tr><td></td><td></td><td>69.97</td><td>89.19</td><td></td></tr><tr><td></td><td>8/8 4/4</td><td>61.52</td><td>89.44 83.99</td><td>-0.43</td></tr><tr><td>RQ+ST</td><td></td><td>8/8</td><td>69.63</td><td>89.33</td><td>8.02 -0.09</td></tr><tr><td>(Louizos et al., 2019)</td><td></td><td>4/4</td><td>62.46</td><td>84.78</td><td>7.08</td></tr><tr><td rowspan=\"4\">LLSQ(ours)*</td><td rowspan=\"4\">ResNet18</td><td>Reference</td><td>69.76</td><td>89.08</td><td></td></tr><tr><td>4/4</td><td>69.84</td><td>89.14</td><td>-0.08</td></tr><tr><td>3/3</td><td>68.08</td><td>88.20</td><td>1.68</td></tr><tr><td>4/4</td><td>69.40</td><td>88.72</td><td>0.36</td></tr><tr><td rowspan=\"2\">LLSQF(ours) LQ-Nets*</td><td>ResNet18</td><td>3/3</td><td>66.67</td><td>87.42</td><td>3.09</td></tr><tr><td>ResNet34</td><td>Reference</td><td>73.80</td><td>91.40</td><td></td></tr><tr><td>(Zhang et al., 2018a) LLSQ(ours)*</td><td>ResNet34</td><td>3/3</td><td>71.9</td><td>90.2</td><td>1.9</td></tr><tr><td rowspan=\"3\"></td><td></td><td>Reference</td><td>73.30</td><td>91.42</td><td></td></tr><tr><td></td><td>4/4</td><td>73.60</td><td>91.28</td><td>-0.30</td></tr><tr><td></td><td>3/3</td><td>72.02</td><td>90.66</td><td>1.28</td></tr><tr><td>LLSQF(ours)</td><td>ResNet34</td><td>4/4 3/3</td><td>72.94 70.97</td><td>91.20</td><td>0.36</td></tr><tr><td rowspan=\"4\">LLSQ(ours)*</td><td>AlexNet</td><td>Reference</td><td>56.55</td><td>89.95</td><td>2.33</td></tr><tr><td></td><td>4/4</td><td>56.57</td><td>79.09</td><td></td></tr><tr><td></td><td></td><td></td><td>79.02</td><td>-0.02</td></tr><tr><td></td><td>4/4/8/8</td><td>56.45</td><td>80.15</td><td>0.10</td></tr><tr><td rowspan=\"4\">LLSQF(ours)</td><td>AlexNet</td><td>3/3</td><td>55.36</td><td>78.20</td><td>0.19</td></tr><tr><td></td><td>4/4</td><td>56.40</td><td>78.85</td><td>0.15</td></tr><tr><td></td><td>4/4/8/8</td><td>55.58</td><td>77.47</td><td>0.97</td></tr><tr><td></td><td>3/3</td><td>54.28</td><td>77.65</td><td>2.27</td></tr><tr><td rowspan=\"4\">RQ (Louizos et al., 2019)</td><td>Mobilenet(v1)</td><td>Reference</td><td>70.61</td><td>89.47</td><td></td></tr><tr><td></td><td>8/8</td><td>70.43</td><td>89.42</td><td>0.18</td></tr><tr><td></td><td>6/6</td><td>68.02</td><td>88.00</td><td>2.59</td></tr><tr><td></td><td>5/5</td><td>61.38</td><td>83.73</td><td>9.23</td></tr><tr><td rowspan=\"3\">RQ+ST (Louizos et al., 2019)</td><td>Mobilenet(v1)</td><td>8/8</td><td>70.06</td><td>89.52</td><td>0.55</td></tr><tr><td></td><td>6/6</td><td>67.62</td><td>87.78</td><td>2.99</td></tr><tr><td></td><td>5/5</td><td>56.85</td><td>80.35</td><td>13.76</td></tr><tr><td rowspan=\"4\">LLSQ(ours)*</td><td>MobileNet(v2)</td><td>Reference</td><td>71.80</td><td>90.37</td><td></td></tr><tr><td></td><td>6/6</td><td>71.20</td><td>89.99</td><td>0.60</td></tr><tr><td></td><td>5/5</td><td>70.45</td><td>89.69</td><td>1.35</td></tr><tr><td></td><td>4/4</td><td>67.37</td><td>87.99</td><td>4.43</td></tr></table>",
+        "bbox": [
+            200,
+            226,
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+            818
+        ],
+        "page_idx": 15
+    }
+]

parse/train/HJKkY35le/HJKkY35le.md

@@ -0,0 +1,321 @@
+# MODE REGULARIZED GENERATIVE ADVERSARIAL NETWORKS
+
+†Tong Che∗, ‡Yanran Li∗, †,§Athul Paul Jacob, †Yoshua Bengio, ‡Wenjie Li   
+†Montreal Institute for Learning Algorithms, Universite de Montr ´ eal, Montr ´ eal, QC H3T 1J4, Canada ´   
+‡Department of Computing, The Hong Kong Polytechnic University, Hong Kong   
+§David R. Cheriton School of Computer Science, University Of Waterloo, Waterloo, ON N2L 3G1, Canada   
+{tong.che,ap.jacob,yoshua.bengio}@umontreal.ca   
+{csyli,cswjli}@comp.polyu.edu.hk
+
+# ABSTRACT
+
+Although Generative Adversarial Networks achieve state-of-the-art results on a variety of generative tasks, they are regarded as highly unstable and prone to miss modes. We argue that these bad behaviors of GANs are due to the very particular functional shape of the trained discriminators in high dimensional spaces, which can easily make training stuck or push probability mass in the wrong direction, towards that of higher concentration than that of the data generating distribution. We introduce several ways of regularizing the objective, which can dramatically stabilize the training of GAN models. We also show that our regularizers can help the fair distribution of probability mass across the modes of the data generating distribution, during the early phases of training and thus providing a unified solution to the missing modes problem.
+
+# 1 INTRODUCTION
+
+Generative adversarial networks (GAN) (Goodfellow et al., 2014) have demonstrated their potential on various tasks, such as image generation, image super-resolution, 3D object generation, and video prediction (Radford et al., 2015; Ledig et al., 2016; Sønderby et al., 2016; Nguyen et al., 2016; Wu et al., 2016; Mathieu et al., 2015). The objective is to train a parametrized function (the generator) which maps noise samples (e.g., uniform or Gaussian) to samples whose distribution is close to that of the data generating distribution. The basic scheme of the GAN training procedure is to train a discriminator which assigns higher probabilities to real data samples and lower probabilities to generated data samples, while simultaneously trying to move the generated samples towards the real data manifold using the gradient information provided by the discriminator. In a typical setting, the generator and the discriminator are represented by deep neural networks.
+
+Despite their success, GANs are generally considered as very hard to train due to training instability and sensitivity to hyper-parameters. On the other hand, a common failure pattern observed while training GANs is the collapsing of large volumes of probability mass onto a few modes. Namely, although the generators produce meaningful samples, these samples are often from just a few modes (small regions of high probability under the data distribution). Behind this phenomenon is the missing modes problem, which is widely conceived as a major problem for training GANs: many modes of the data generating distribution are not at all represented in the generated samples, yielding a much lower entropy distribution, with less variety than the data generating distribution.
+
+This issue has been the subject of several recent papers proposing several tricks and new architectures to stabilize GAN’s training and encourage its samples’ diversity. However, we argue that a general cause behind these problems is the lack of control on the discriminator during GAN training. We would like to encourage the manifold of the samples produced by the generator to move towards that of real data, using the discriminator as a metric. However, even if we train the discriminator to distinguish between these two manifolds, we have no control over the shape of the discriminator function in between these manifolds. In fact, the shape of the discriminator function in the data space can be very non-linear with bad plateaus and wrong maxima and this can therefore hurt the training of GANs (Figure 1).
+
+![](images/369f34d05cb3594aa3e375375091c128e3752c2e993fd9075254cddcb9f00c0b.jpg)  
+Figure 1: Samples with very high discrimination values $\mathrm { ( D = } 1 . 0$ ) in DCGAN model trained on CelebA dataset.
+
+To remedy this problem, we propose a novel regularizer for the GAN training target. The basic idea is simple yet powerful: in addition to the gradient information provided by the discriminator, we want the generator to take advantage of other similarity metrics with much more predictable behavior, such as the $L _ { 2 }$ norm. Differentiating these similarity metrics will provide us with more stable gradients to train our generator. Combining this idea with an approach meant to penalize the missing modes, we propose a family of additional regularizers for the GAN objective. We then design a set of metrics to evaluate the generated samples in terms of both the diversity of modes and the distribution fairness of the probability mass. These metrics are shown to be more robust in judging complex generative models, including those which are well-trained and collapsed ones.
+
+Regularizers usually bring a trade-off between model variance and bias. Our results have shown that, when correctly applied, our regularizers can dramatically reduce model variance, stabilize the training, and fix the missing mode problem all at once, with positive or at the least no negative effects on the generated samples. We also discuss a variant of the regularized GAN algorithm, which can even improve sample quality as compared to the DCGAN baseline.
+
+# 2 RELATED WORK
+
+The GAN approach was initially proposed by Goodfellow et al. (2014) where both the generator and the discriminator are defined by deep neural networks.
+
+In Goodfellow et al. (2014), the GAN is able to generate interesting local structure but globally incoherent images on various datasets. Mirza & Osindero (2014) enlarges GAN’s representation capacity by introducing an extra vector to allow the generator to produce samples conditioned on other beneficial information. Motivated from this, several conditional variants of GAN has been applied to a wide range of tasks, including image prediction from a normal map Wang & Gupta (2016), image synthesis from text Reed et al. (2016) and edge map Isola et al. (2016), real-time image manipulation Zhu et al. (2016), temporal image generation Zhou & Berg (2016); Saito & Matsumoto (2016); Vondrick et al. (2016), texture synthesis, style transfer, and video stylization Li & Wand (2016).
+
+Researchers also aim at stretching GAN’s limit to generate higher-resolution, photo-realistic images. Denton et al. (2015) initially apply a Laplacian pyramid framework on GAN to generate images of high resolution. At each level of their LAPGAN, both the generator and the discriminator are convolutional networks. As an alternative to LAPGAN, Radford et al. (2015) successfully designs a class of deep convolutional generative adversarial networks which has led to significant improvements on unsupervised image representation learning. Another line of work aimed at improving GANs are through feature learning, including features from the latent space and image space. The motivation is that features from different spaces are complementary for generating perceptual and natural-looking images. With this perspective, some researchers use distances between learned features as losses for training objectives for generative models. Larsen et al. (2015) combine a variational autoencoder objective with a GAN and utilize the learned features from the discriminator in the GANs for better image similarity metrics. It is shown that the learned distance from the discriminator is of great help for the sample visual fidelity. Recent literature have also shown impressive results on image super-resolution to infer photo-realistic natural images for $4 \mathbf { x }$ upscaling factors Ledig et al. (2016); Sønderby et al. (2016); Nguyen et al. (2016).
+
+Despite these promising successes, GANs are notably hard to train. Although Radford et al. (2015) provide a class of empirical architectural choices that are critical to stabilize GAN’s training, it would be even better to train GANs more robustly and systematically. Salimans et al. (2016) propose feature matching technique to stabilize GAN’s training. The generator is required to match the statistics of intermediate features of the discriminator. Similar idea is adopted by Zhao et al. (2016).
+
+In addition to feature distances, Dosovitskiy & Brox (2016) found that the counterpart loss in image space further improves GAN’s training stability. Furthermore, some researchers make use of information in both spaces in a unified learning procedure (Dumoulin et al., 2016; Donahue et al., 2016). In Dumoulin et al. (2016), one trains not just a generator but also an encoder, and the discriminator is trained to distinguish between two joint distributions over image and latent spaces produced either by the application of the encoder on the training data or by the application of the generator (decoder) to the latent prior. This is in contrast with the regular GAN training, in which the discriminator only attempts to separate the distributions in the image space. Parallelly, Metz et al. (2016) stabilize GANs by unrolling the optimization of discriminator, which can be considered as an orthogonal work with ours.
+
+Our work is related to VAEGAN (Larsen et al., 2015) in terms of training an autoencoder or VAE jointly with the GAN model. However, the variational autoencoder (VAE) in VAEGAN is used to generate samples whereas our autoencoder based losses serves as a regularizer to penalize missing modes and thus improving GAN’s training stability and sample qualities. We demonstrate detailed differences from various aspects in Appendix D.
+
+# 3 MODE REGULARIZERS FOR GANS
+
+The GAN training procedure can be viewed as a non-cooperative two player game, in which the discriminator $D$ tries to distinguish real and generated examples, while the generator $G$ tries to fool the discriminator by pushing the generated samples towards the direction of higher discrimination values. Training the discriminator $D$ can be viewed as training an evaluation metric on the sample space. Then the generator $G$ has to take advantage of the local gradient $\nabla \log D ( G )$ provided by the discriminator to improve itself, namely to move towards the data manifold.
+
+We now take a closer look at the root cause of the instabilities while training GANs. The discriminator is trained on both generated and real examples. As pointed out by Goodfellow et al. (2014); Denton et al. (2015); Radford et al. (2015), when the data manifold and the generation manifold are disjoint (which is true in almost all practical situations), it is equivalent to training a characteristic function to be very close to 1 on the data manifold, and 0 on the generation manifold. In order to pass good gradient information to the generator, it is important that the trained discriminator produces stable and smooth gradients. However, since the discriminator objective does not directly depend on the behavior of the discriminator in other parts of the space, training can easily fail if the shape of the discriminator function is not as expected. As an example,Denton et al. (2015) noted a common failure pattern for training GANs which is the vanishing gradient problem, in which the discriminator $D$ perfectly classifies real and fake examples, such that around the fake examples, $D$ is nearly zero. In such cases, the generator will receive no gradient to improve itself.1
+
+Another important problem while training GANs is mode missing. In theory, if the generated data and the real data come from the same low dimensional manifold, the discriminator can help the generator distribute its probability mass, because the missing modes will not have near-0 probability under the generator and so the samples in these areas can be appropriately concentrated towards regions where $D$ is closer to 1. However, in practice since the two manifolds are disjoint, $D$ tends to be near 1 on all the real data samples, so large modes usually have a much higher chance of attracting the gradient of discriminator. For a typical GAN model, since all modes have similar $D$ values, there is no reason why the generator cannot collapse to just a few major modes. In other words, since the discriminator’s output is nearly 0 and 1 on fake and real data respectively, the generator is not penalized for missing modes.
+
+# 3.1 GEOMETRIC METRICS REGULARIZER
+
+Compared with the objective for the GAN generator, the optimization targets for supervised learning are more stable from an optimization point of view. The difference is clear: the optimization target for the GAN generator is a learned discriminator. While in supervised models, the optimization targets are distance functions with nice geometric properties. The latter usually provides much easier training gradients than the former, especially at the early stages of training.
+
+Inspired by this observation, we propose to incorporate a supervised training signal as a regularizer on top of the discriminator target. Assume the generator $G ( z ) : Z \to X$ generates samples by sampling first from a fixed prior distribution in space $Z$ followed by a deterministic trainable transformation $G$ into the sample space $X$ . Together with $G$ , we also jointly train an encoder $E ( x ) : X  Z$ . Assume $d$ is some similarity metric in the data space, we add $\mathbb { E } _ { { x } \sim { p } _ { d } } [ d ( { x } , G \circ E ( { x } ) ) ]$ as a regularizer, where $p _ { d }$ is the data generating distribution. The encoder itself is trained by minimizing the same reconstruction error.
+
+In practice, there are many options for the distance measure $d$ . For instance, the pixel-wise $L ^ { 2 }$ distance, or the distance of learned features by the discriminator (Dumoulin et al., 2016) or by other networks, such as a VGG classifier. (Ledig et al., 2016)
+
+The geometric intuition for this regularizer is straight-forward. We are trying to move the generated manifold to the real data manifold using gradient descent. In addition to the gradient provided by the discriminator, we can also try to match the two manifolds by other geometric distances, say, $L ^ { s }$ metric. The idea of adding an encoder is equivalent to first training a point to point mapping $G ( E ( x ) )$ between the two manifolds and then trying to minimize the expected distance between the points on these two manifolds.
+
+# 3.2 MODE REGULARIZER
+
+In addition to the metric regularizer, we propose a mode regularizer to further penalize missing modes. In traditional GANs, the optimization target for the generator is the empirical sum $\begin{array} { r } { \sum _ { i } { \nabla _ { \theta } \log { D ( G _ { \theta } ( z _ { i } ) ) } } } \end{array}$ . The missing mode problem is caused by the conjunction of two facts: (1) the areas near missing modes are rarely visited by the generator, by definition, thus providing very few examples to improve the generator around those areas, and (2) both missing modes and nonmissing modes tend to correspond to a high value of $D$ , because the generator is not perfect so that the discriminator can take strong decisions locally and obtain a high value of $D$ even near non-missing modes.
+
+As an example, consider the situation in Figure 2. For most $z$ , the gradient of the generator $\nabla _ { \theta } \log D ( G _ { \theta } ( z ) )$ pushes the generator towards the major mode $M _ { 1 }$ . Only when $G ( z )$ is very close to the mode $M _ { 2 }$ can the generator get gradients to push itself towards the minor mode $M _ { 2 }$ . However, it is possible that such $z$ is of low or zero probability in the prior distribution $p _ { 0 }$ .
+
+![](images/7385e91fa5a2d1dc34cba4cf1f79b9eaf272d966fa73d5daac595e2758dd374d.jpg)  
+Figure 2: Illustration of missing modes problem.
+
+Given this observation, consider a regularized GAN model with the metric regularizer. Assume $M _ { 0 }$ is a minor mode of the data generating distribution. For $x \ \in \ M _ { 0 }$ , we know that if $G \circ E$ is a good autoencoder, $G ( E ( x ) )$ will be located very close to mode $M _ { 0 }$ . Since there are sufficient training examples of mode $M _ { 0 }$ in the training data, we add the mode regularizer $\mathbb { E } _ { { x } \sim p _ { d } } [ \log { \bar { D } } ( G \circ E ( x ) ) ]$ to our optimization target for the generator, to encourage $G ( E ( x ) )$
+
+to move towards a nearby mode of the data generating distribution. In this way, we can achieve fair probability mass distribution across different modes.
+
+In short, our regularized optimization target for the generator and the encoder becomes:
+
+$$
+\begin{array} { r } { T _ { G } = - \mathbb { E } _ { z } [ \log D ( G ( z ) ) ] + \mathbb { E } _ { x \sim p _ { d } } [ \lambda _ { 1 } d ( x , G \circ E ( x ) ) + \lambda _ { 2 } \log D ( G \circ E ( x ) ) ] } \\ { T _ { E } = \mathbb { E } _ { x \sim p _ { d } } [ \lambda _ { 1 } d ( x , G \circ E ( x ) ) + \lambda _ { 2 } \log D ( G \circ E ( x ) ) ] } \end{array}
+$$
+
+# 3.3 MANIFOLD-DIFFUSION TRAINING FOR REGULARIZED GANS
+
+On some large scale datasets, CelebA for example, the regularizers we have discussed do improve the diversity of generated samples, but the quality of samples may not be as good without carefully tuning the hyperparameters. Here we propose a new algorithm for training metric-regularized GANs, which is very stable and much easier to tune for producing good samples.
+
+The proposed algorithm divides the training procedure of GANs into two steps: a manifold step and a diffusion step. In the manifold step, we try to match the generation manifold and the real data manifold with the help of an encoder and the geometric metric loss. In the diffusion step, we try to distribute the probability mass on the generation manifold fairly according to the real data distribution.
+
+An example of manifold-diffusion training of GAN (MDGAN for short) is as follows: we train a discriminator $D _ { 1 }$ which separates between the samples $x$ and $G \circ E ( x )$ , for $x$ from the data, and we optimize $G$ with respect to the regularized GAN loss $\mathbb { E } [ \log D _ { 1 } ( G \circ E ( x ) ) + \lambda d ( x , G \circ E ( x ) ) ]$ in order to match the two manifolds. In the diffusion step we train a discriminator $D _ { 2 }$ between distributions $G ( z )$ and $G \circ E ( x )$ , and we train $G$ to maximize $\log { D _ { 2 } ( G ( z ) ) }$ . Since these two distributions are now nearly on the same low dimensional manifold, the discriminator $D _ { 2 }$ provides much smoother and more stable gradients. The detailed training procedure is given in Appendix A. See Figure 6 for the quality of generated samples.
+
+# 3.4 EVALUATION METRICS FOR MODE MISSING
+
+In order to estimate both the missing modes and the sample qualities in our experiments, we used several different metrics for different experiments instead of human annotators.
+
+The inception score (Salimans et al., 2016) was considered as a good assessment for sample quality from a labelled dataset:
+
+$$
+\exp { ( \mathbb { E } _ { x } K L ( p ( y | \mathbf { x } ) | | p ^ { * } ( y ) ) ) }
+$$
+
+Where $\mathbf { x }$ denotes one sample, $p ( y | x )$ is the softmax output of a trained classifier of the labels, and $p ^ { * } ( y )$ is the overall label distribution of generated samples. The intuition behind this score is that a strong classifier usually has a high confidence for good samples. However, the inception score is sometimes not a good metric for our purpose. Assume a generative model that collapse to a very bad image. Although the model is very bad, it can have a perfect inception score, because $p ( y | x )$ can have a high entropy and $p ^ { * } ( y )$ can have a low entropy. So instead, for labelled datasets, we propose another assessment for both visual quality and variety of samples, the MODE score:
+
+$$
+\exp \left( \mathbb { E } _ { x } K L ( p ( y | \mathbf { x } ) | | p ( y ) ) - K L ( p ^ { * } ( y ) | | p ( y ) ) \right)
+$$
+
+where $p ( y )$ is the distribution of labels in the training data. According to our human evaluation experiences, the MODE score successfully measures two important aspects of generative models, i.e., variety and visual quality, in one metric.
+
+However, in datasets without labels (LSUN) or where the labels are not sufficient to characterize every data mode (CelebA), the above metric does not work well. We instead train a third party discriminator between the real data and the generated data from the model. It is similar to the GAN discriminator but is not used to train the generator. We can view the output of the discriminator as an estimator for the quantity (See (Goodfellow et al., 2014) for proof):
+
+$$
+D ^ { * } ( s ) \approx \frac { p _ { g } ( s ) } { p _ { g } ( s ) + p _ { d } ( s ) }
+$$
+
+Where $p _ { g }$ is the probability density of the generator and $p _ { d }$ is the density of the data distribution. To prevent $D ^ { * }$ from learning a perfect 0-1 separation of $p _ { g }$ and $p _ { d }$ , we inject a zero-mean Gaussian noise to the inputs when training $D ^ { * }$ . After training, we test $D ^ { * }$ on the test set $T$ of the real dataset. If for any test sample $t \in T$ , the discrimination value $D ( t )$ is close to 1, we can conclude that the mode corresponding to $t$ is missing. In this way, although we cannot measure exactly the number of modes that are missing, we have a good estimator of the total probability mass of all the missing modes.
+
+# 4 EXPERIMENTS
+
+# 4.1 MNIST
+
+We perform two classes of experiments on MNIST. For the MNIST dataset, we can assume that the data generating distribution can be approximated with ten dominant modes, if we define the term “mode” here as a connected component of the data manifold.
+
+# 4.1.1 GRID SEARCH FOR MNIST GAN MODELS
+
+Table 1: Grid Search for Hyperparameters.   
+
+<table><tr><td>nLayerG</td><td>[2,3,4]</td></tr><tr><td>nLayerD</td><td>[2,3,4]</td></tr><tr><td>sizeG sizeD</td><td>[400,800,1600,3200]</td></tr><tr><td></td><td>[256, 512, 1024]</td></tr><tr><td>dropoutD</td><td>[True,False]</td></tr><tr><td>optimG</td><td>[SGD,Adam]</td></tr><tr><td>optimD</td><td>[SGD,Adam]</td></tr><tr><td>lr</td><td>[1e-2,1e-3,1e-4]</td></tr></table>
+
+In order to systemically explore the effect of our proposed regularizers on GAN models in terms of improving stability and sample quality, we use a large scale grid search of different GAN hyper-parameters on the MNIST dataset. The grid search is based on a pair of randomly selected loss weights: $\lambda _ { 1 } = 0 . 2$ and $\lambda _ { 2 } ~ = ~ 0 . 4$ . We use the same hyper-parameter settings for both GAN and Regularized GAN, and list the search ranges in Table 1. Our grid search is similar to those proposed in Zhao et al. (2016). Please refer to it for detailed explanations regarding these hyper-parameters.
+
+For evaluation, we first train a 4-layer CNN classifier on the MNIST digits, and then apply it to compute the MODE scores for the generated samples from all these models. The resulting distribution of MODE score is shown in Figure 3. Clearly, our proposed regularizer significantly improves the MODE scores and thus demonstrates its benefits on stabilizing GANs and improving sample qualities.
+
+![](images/23691178a9332efa7d0b890615c2cf021c092595979c93013beab8b2b47d1ea1.jpg)  
+Figure 3: The distributions of MODE scores for GAN and regularized GAN.
+
+To illustrate the effect of regularizers with different coefficients, we randomly pick an architecture and train it with different $\lambda _ { 1 } = \lambda _ { 2 }$ . The results are shown in Figure 4.
+
+![](images/ba0462a6a90a59f8d012d3b775739a6a034addf655922b4e85364f2d3f1a8aa3.jpg)  
+Figure 4: (Left 1-5) Different hyperparameters for MNIST generation. The values of the $\lambda _ { 1 }$ and $\lambda _ { 2 }$ in our Regularized GAN are listed below the corresponding samples. (Right 6-7) Best samples through grid search for GAN and Regularized GAN.
+
+# 4.1.2 COMPOSITIONAL MNIST DATA WITH 1000 MODES
+
+In order to quantitatively study the effect of our regularizers on the missing modes, we concatenate three MNIST digits to a number in [0,999] in a single 64x64 image, and then train DCGAN as a baseline model on the 1000 modes dataset. The digits on the image are sampled with different probabilities, in order to test the model’s capability to preserve small modes in generation. We again use a pre-trained classifier for MNIST instead of a human to evaluate the models.
+
+Table 2: Results for Compositional MNIST with 1000 modes. The proposed regularization (RegDCGAN) allows to substantially reduce the number of missed modes as well as the KL divergence that measures the plausibility of the generated samples (like in the Inception score).   
+
+<table><tr><td rowspan="2"></td><td colspan="2">Set 1</td><td colspan="2">Set 2</td><td colspan="2">Set3</td><td colspan="2">Set4</td></tr><tr><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td></tr><tr><td>DCGAN</td><td>204.7</td><td>77.9</td><td>204.3</td><td>60.2</td><td>103.4</td><td>75.9</td><td>89.3</td><td>77.8</td></tr><tr><td>Reg-DCGAN</td><td>32.1</td><td>62.3</td><td>71.5</td><td>58.9</td><td>42.7</td><td>68.4</td><td>31.6</td><td>67.8</td></tr></table>
+
+The performances on the compositional experiment are measured by two metrics. #Miss represents the classifier-reported number of missing modes, which is the size of the set of numbers that the model never generates. KL stands for the KL divergence between the classifier-reported distribution of generated numbers and the distribution of numbers in the training data (as for the Inception score). The results are shown in Table 2. With the help of our proposed regularizer, both the number of missing modes and KL divergence drop dramatically among all the sets of the compositional MNIST dataset, which again proves the effectiveness of our regularizer for preventing the missing modes problem.
+
+# 4.2 CELEBA
+
+To test the effectiveness of our proposal on harder problems, we implement an encoder for the DCGAN algorithm and train our model with different hyper-parameters together with the DCGAN baseline on the CelebA dataset. We provide the detailed architecture of our regularized DCGAN in Appendix B.
+
+# 4.2.1 MISSING MODES ESTIMATION ON CELEBA
+
+We also employ a third party discriminator trained with injected noise as a metric for missing mode estimation. To implement this, we add noise in the input layer in the discriminator network. For each GAN model to be estimated, we independently train this noisy discriminator, as mode estimator, with the same architecture and hyper-parameters on the generated data and the training data. We then apply the mode estimator to the test data. The images which have high mode estimator outputs can be viewed as on the missing modes.
+
+Table 3: Number of images on the missing modes on CelebA estimated by a third-party discriminator. The numbers in the brackets indicate the dimension of prior $z$ . $\sigma$ denotes the standard deviation of the added Gaussian noise applied at the input of the discriminator to regularize it. MDGAN achieves a very high reduction in the number of missing modes, in comparison to other methods .   
+
+<table><tr><td>0</td><td>DCGAN (100)</td><td>DCGAN (200)</td><td>Reg-GAN (100)</td><td>Reg-GAN (200)</td><td>MDGAN (200)</td></tr><tr><td>3.5</td><td>5463</td><td>17089</td><td>754</td><td>3644</td><td>74</td></tr><tr><td>4.0</td><td>590</td><td>15832</td><td>42</td><td>391</td><td>13</td></tr></table>
+
+The comparison result is shown in Table 3. Both our proposed Regularized-GAN and MDGAN outperform baseline DCGAN models on all settings. Especially, MDGAN suppresses other models, showing its superiority on modes preserving. We also find that, although sharing the same architecture, the DCGAN with 200-dimensional noise performs quite worse than that with 100-dimensional noise as input. On the contrary, our regularized GAN performs more consistently.
+
+To get a better understanding of the models’ performance, we want to figure out when and where these models miss the modes. Visualizing the test images associated with missed modes is instructive. In Figure 5, the left three images are missed by all models. It is rare to see in the training data the cap in the second image and the type of background in the third, which thus can be viewed as small modes under this situation. These three images should be considered as the hardest test data for GAN to learn. Nonetheless, our best model, MDGAN still capture certain small modes. The seven images on the right in Figure 5 are only missed by DCGAN. The sideface, paleface, black, and the berets are special attributes among these images, but our proposed MDGAN performs well on all of them.
+
+![](images/dd60ba26724f057d75e53e8f9051cfa7ab0d2f40c846afeeccfbb3ecd6ec1b15.jpg)  
+Figure 5: Test set images that are on missing mode. Left: Both MDGAN and DCGAN missing. Right: Only DCGAN missing.
+
+# 4.2.2 QUALITATIVE EVALUATION OF GENERATED SAMPLES
+
+After quantitative evaluation, we manually examine the generated samples by our regularized GAN to see whether the proposed regularizer has side-effects on sample quality. We compare our model with ALI (Dumoulin et al., 2016), VAEGAN (Larsen et al., 2015), and DCGAN (Radford et al., 2015) in terms of sample visual quality and mode diversity. Samples generated from these models are shown in Figure $6 ^ { 2 ^ { \circ } }$ .
+
+![](images/601224511ba5ab1ca605c826bf8588ec4a9779fd983509775ed6e2be4b574de3.jpg)  
+Figure 6: Samples generated from different generative models. For each compared model, we directly take ten decent samples reported in their corresponding papers and code repositories. Note how MDGAN samples are both globally more coherent and locally have sharp textures.
+
+Both MDGAN and Regularized-GAN generate clear and natural-looking face images. Although ALI’s samples are plausible, they are sightly deformed in comparison with those from MDGAN. The samples from VAEGAN and DCGAN seem globally less coherent and locally less sharp.
+
+As to sample quality, it is worth noting that the samples from MDGAN enjoy fewer distortions. With all four other models, the majority of generated samples suffer from some sort of distortion. However, for the samples generated by MDGAN, the level of distortion is lower compared with the other four compared models. We attribute it to the help of the autoencoder as the regularizer to alter the generation manifolds. In this way, the generator is able to learn fine-grained details such as face edges. As a result, MDGAN is able to reduce distortions.
+
+![](images/7ccfe9752c1edde7bf368397e5fc0a44880f5ada2037abd62a3f910d669af1b5.jpg)  
+Figure 7: Sideface samples generated by Regularized-GAN and MDGAN.
+
+In terms of missing modes problem, we instructed five individuals to conduct human evaluation on the generated samples. They achieve consensus that MDGAN wins in terms of mode diversities. Two people pointed out that MDGAN generates a larger amount of samples with side faces than other models. We select several of these side face samples in Figure 7. Clearly, our samples maintain acceptable visual fidelity meanwhile share diverse modes. Combined with the above quantitative results, it is convincing that our regularizers bring benefits for both training stability and mode variety without the loss of sample quality.
+
+# 5 CONCLUSIONS
+
+Although GANs achieve state-of-the-art results on a large variety of unsupervised learning tasks, training them is considered highly unstable, very difficult and sensitive to hyper-parameters, all the while, missing modes from the data distribution or even collapsing large amounts of probability mass on some modes. Successful GAN training usually requires large amounts of human and computing efforts to fine tune the hyper-parameters, in order to stabilize training and avoid collapsing. Researchers usually rely on their own experience and published tricks and hyper-parameters instead of systematic methods for training GANs.
+
+We provide systematic ways to measure and avoid the missing modes problem and stabilize training with the proposed autoencoder-based regularizers. The key idea is that some geometric metrics can provide more stable gradients than trained discriminators, and when combined with the encoder, they can be used as regularizers for training. These regularizers can also penalize missing modes and encourage a fair distribution of probability mass on the generation manifold.
+
+# ACKNOWLEDGEMENTS
+
+We thank Naiyan Wang, Jianbo Ye, Yuchen Ding, Saboya Yang for their GPU support. We also want to thank Huiling Zhen for helpful discussions, Junbo Zhao for providing the details of grid search experiments on the EBGAN model, as well as Anders Boesen Lindbo Larsen for kindly helping us on running VAEGAN experiments. We appreciate for the valuable suggestions and comments from the anonymous reviewers. The work described in this paper was partially supported by NSERC, Calcul Quebec, Compute Canada, the Canada Research Chairs, CIFAR, National Natural Science Foundation of China (61672445 and 61272291), Research Grants Council of Hong Kong (PolyU 152094/14E), and The Hong Kong Polytechnic University (G-YBP6).
+
+# REFERENCES
+
+Emily L Denton, Soumith Chintala, Rob Fergus, et al. Deep generative image models using a laplacian pyramid of adversarial networks. In Advances in neural information processing systems, pp. 1486–1494, 2015.   
+Jeff Donahue, Philipp Krahenb ¨ uhl, and Trevor Darrell. Adversarial feature learning. ¨ arXiv preprint arXiv:1605.09782, 2016.   
+Alexey Dosovitskiy and Thomas Brox. Generating images with perceptual similarity metrics based on deep networks. arXiv preprint arXiv:1602.02644, 2016.   
+Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially learned inference. arXiv preprint arXiv:1606.00704, 2016.
+
+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.
+
+Phillip Isola, Jun-Yan Zhu, Tinghui Zhou, and Alexei A Efros. Image-to-image translation with conditional adversarial networks. arxiv, 2016.
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+Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, Hugo Larochelle, and Ole Winther. Autoencoding beyond pixels using a learned similarity metric. arXiv preprint arXiv:1512.09300, 2015.
+
+Christian Ledig, Lucas Theis, Ferenc Huszar, Jose Caballero, Andrew Aitken, Alykhan Tejani, Jo- ´ hannes Totz, Zehan Wang, and Wenzhe Shi. Photo-realistic single image super-resolution using a generative adversarial network. arXiv preprint arXiv:1609.04802, 2016.
+
+Chuan Li and Michael Wand. Precomputed real-time texture synthesis with markovian generative adversarial networks. arXiv preprint arXiv:1604.04382, 2016.
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+Michael Mathieu, Camille Couprie, and Yann LeCun. Deep multi-scale video prediction beyond mean square error. arXiv preprint arXiv:1511.05440, 2015.
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+Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163, 2016.
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+Anh Nguyen, Jason Yosinski, Yoshua Bengio, Alexey Dosovitskiy, and Jeff Clune. Plug & play generative networks: Conditional iterative generation of images in latent space. arXiv preprint arXiv:1612.00005, 2016.
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+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.
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+Scott Reed, Zeynep Akata, Xinchen Yan, Lajanugen Logeswaran, Bernt Schiele, and Honglak Lee. Generative adversarial text to image synthesis. arXiv preprint arXiv:1605.05396, 2016.
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+Masaki Saito and Eiichi Matsumoto. Temporal generative adversarial nets. arXiv preprint arXiv:1611.06624, 2016.
+
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. arXiv preprint arXiv:1606.03498, 2016.
+
+Casper Kaae Sønderby, Jose Caballero, Lucas Theis, Wenzhe Shi, and Ferenc Huszar. Amortised ´ map inference for image super-resolution. arXiv preprint arXiv:1610.04490, 2016.
+
+Carl Vondrick, Hamed Pirsiavash, and Antonio Torralba. Generating videos with scene dynamics. In Advances In Neural Information Processing Systems, pp. 613–621, 2016.
+
+Xiaolong Wang and Abhinav Gupta. Generative image modeling using style and structure adversarial networks. In ECCV, 2016.
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+Jiajun Wu, Chengkai Zhang, Tianfan Xue, William T Freeman, and Joshua B Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In Neural Information Processing Systems (NIPS), 2016.
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+Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016.
+
+Yipin Zhou and Tamara L Berg. Learning temporal transformations from time-lapse videos. In European Conference on Computer Vision, pp. 262–277. Springer, 2016.
+
+Jun-Yan Zhu, Philipp Krahenb ¨ uhl, Eli Shechtman, and Alexei A. Efros. Generative visual manipula- ¨ tion on the natural image manifold. In Proceedings of European Conference on Computer Vision (ECCV), 2016.
+
+# A APPENDIX: PSEUDO CODE FOR MDGAN
+
+In this Appendix, we give the detailed training procedure of an MDGAN example we discuss in Section 3.3.
+
+# Manifold Step:
+
+1. Sample $\left\{ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \cdots \mathbf { x } _ { m } \right\}$ from data generating distribution $p _ { d a t a } ( x )$
+
+2. Update discriminator $D _ { 1 }$ using SGD with gradient ascent:
+
+$$
+\nabla _ { \theta _ { d } ^ { 1 } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ \log D _ { 1 } ( \mathbf { x } _ { i } ) + \log ( 1 - D _ { 1 } ( G ( E ( \mathbf { x } _ { i } ) ) ) ) ]
+$$
+
+3. Update generator $G$ using SGD with gradient ascent:
+
+$$
+\nabla _ { \theta _ { g } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ \lambda \log D _ { 1 } ( G ( E ( \mathbf { x } _ { i } ) ) ) - | | \mathbf { x } _ { i } - G ( E ( \mathbf { x } _ { i } ) ) | | ^ { 2 } ]
+$$
+
+# Diffusion Step:
+
+4. Sample $\left\{ \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \cdots \mathbf { x } _ { m } \right\}$ from data generating distribution $p _ { d a t a } ( x )$ .   
+5. Sample $\{ \mathbf { z } _ { 1 } , \mathbf { z } _ { 2 } , \cdots \mathbf { z } _ { m } \}$ from prior distribution $p _ { \sigma } ( z )$ .   
+6. Update discriminator $D _ { 2 }$ using SGD with gradient ascent:
+
+$$
+\nabla _ { \theta _ { d } ^ { 2 } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ \log D _ { 2 } ( G ( E ( \mathbf { x } _ { i } ) ) ) + \log ( 1 - D _ { 2 } ( \mathbf { z } _ { i } ) ) ]
+$$
+
+7. Update generator $G$ using SGD with gradient ascent:
+
+$$
+\nabla _ { \boldsymbol { \theta } _ { g } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } [ \log D _ { 2 } ( G ( \mathbf { z } _ { i } ) ) ]
+$$
+
+# B APPENDIX: ARCHITECTURE FOR EXPERIMENTS
+
+We use similar architectures for Compositional MNIST and CelebA experiments. The architecture is based on that found in DCGAN Radford et al. (2015). Apart from the discriminator and generator which are the same as DCGAN, we add an encoder which is the ”inverse” of the generator, by reversing the order of layers and replacing the de-convolutional layers with convolutional layers.
+
+One has to pay particular attention to batch normalization layers. In DCGAN, there are batch normalization layers both in the generator and the discriminator. However, two classes of data go through the batch normalization layers in the generator. One come from sampled noise $z$ , the other one come from the encoder. In our implementation, we separate the batch statistics for these two classes of data in the generator, while keeping the parameters of BN layer to be shared. In this way, the batch statistics of these two kinds of batches cannot interfere with each other.
+
+# C APPENDIX: ADDITIONAL SYNTHESIZED EXPERIMENTS
+
+To demonstrate the effectiveness of mode-regularized GANs proposed in this paper, we train a very simple GAN architecture on synthesized 2D dataset, following Metz et al. (2016).
+
+The data is sampled from a mixture of 6 Gaussians, with standard derivation of 0.1. The means of the Gaussians are placed around a circle with radius 5. The generator network has two ReLU hidden layers with 128 neurons. It generates 2D output samples from 3D uniform noise from [0,1]. The discriminator consists of only one fully connected layer of ReLU neurons, mapping the 2D input to a real 1D number. Both networks are optimized with the Adam optimizer with the learning rate of 1e-4.
+
+In the regularized version, we choose $\lambda _ { 1 } = \lambda _ { 2 } = 0 . 0 0 5$ . The comparison between the generator distribution from standard GAN and our proposed regularized GAN are shown in Figure 9.
+
+![](images/2acc16f21eb9e8b9d0333914b869f7d2c9aea0b818b3a87da7a5c415e117f9f3.jpg)  
+Figure 9: Comparison results on a toy 2D mixture of Gaussians dataset. The columns on the left shows heatmaps of the generator distributions as the number of training epochs increases, whereas the rightmost column presents the target, the original data distribution. The top row shows standard GAN result. The generator has a hard time oscillating among the modes of the data distribution, and is only able to “recover” a single data mode at once. In contrast, the bottom row shows results of our regularized GAN. Its generator quickly captures the underlying multiple modes and fits the target distribution.
+
+# D APPENDIX: COMPARISON WITH VAEGAN
+
+In this appendix section, we demonstrate the effectiveness and uniqueness of mode-regularized GANs proposed in this paper as compared to Larsen et al. (2015) in terms of its theoretical difference, sample quality and number of missing modes.
+
+With regard to the theoretical difference, the optimization of VAEGAN relies on the probabilistic variational bound, namely $p ( x ) \geq \mathbb { E } _ { q ( z | x ) } [ \log p ( x | z ) ] - { \mathrm { K L } } ( q ( z | x ) | | p ( z ) )$ . This variational bound together with a GAN loss is optimized with several assumptions imposed in VAEGAN:
+
+1. In general, VAE is based on the assumption that the true posterior $p ( z | x )$ can be well approximated by factorized Gaussian distribution $q$ .   
+2. As to VAEGAN, It is also assumed that the maximum likelihood objectives does not conflict with GAN objective in terms of probabilistic framework.
+
+The first assumption does not necessarily hold for GANs. We have found that in some trained models of DCGANs, the real posterior $p ( z | x )$ is even not guaranteed to have only one mode, not to mention it is anything close to factorized Gaussian. We believe that this difference in probabilistic framework is an essential obstacle when one tries to use the objective of VAEGAN as a regularizer. However, in our algorithm, where we use a plain auto-encoder instead of VAE as the objective. Plain auto-encooders works better than VAE for our purposes because as long as the model $G ( z )$ is able to generate training samples, there always exists a function $E ^ { \ast } ( x )$ such that $G ( E ( x ) ) = x$ . Our encoder can therefore be viewed as being trained to approximate this real encoder $E ^ { * }$ . There are no conflicts between a good GAN generator and our regularization objective. Hence, our objectives can be used as regularizers for encoding the prior knowledge that good models should be able to generate the training samples. This is why our work is essentially different from VAEGAN. In our experiments, we also believe that this is the reason why VAEGAN generates worse samples than a carefully tuned regularized GANs.
+
+In terms of sample quality and missing modes, we run the official code of VAEGAN 3 with their default setting. We train VAEGAN for 30 epochs 4 and our models for only 20 epochs. For fairness, their model was run 3 times and the trained model with the best sample visual quality was taken for the comparison.
+
+The generated samples are shown in Figure 10. The most obvious difference between our samples and VAEGAN’s samples is the face distortion, which is consistent with our experimental results in Section 4.2.2. We conjecture that the distortions of VAEGAN’s samples are due to the conflicts between the two objectives, as we present above. In other words, the way we introduce auto-encoders as regularizers for GAN models is different from VAEGAN’s. The difference is that the second assumption mentioned above is not required in our approaches. In our framework, the auto-encoders helps alter the generation manifolds, leading to fewer distortions in fine-grained details in our generated samples.
+
+![](images/e0082f3127f2aab9defa52d7d88f547f2ca4aa33600ecc6c7008814eebed363b.jpg)  
+Figure 10: Samples generated by our models and VAEGAN. The third line are samples generated by our self-trained VAEGAN model, with default settings. The last line are generated samples reported in the original VAEGAN paper. We depict both of them here for a fair comparison.
+
+In terms of the missing modes problem, we use the same method described in Section 4.2.1 for computing the number of images with missing modes. The results are shown below.
+
+Table 4: Number of images on the missing modes on CelebA estimated by a third-party discriminator. The numbers in the brackets indicate the dimension of prior $z$ . $\sigma$ denotes the standard deviation of the added Gaussian noise applied at the input of the discriminator to regularize it. MDGAN achieves a very high reduction in the number of missing modes, in comparison to VAEGAN.   
+
+<table><tr><td>0</td><td>VAEGAN (100)</td><td>Reg-GAN (100)</td><td>Reg-GAN (200)</td><td>MDGAN (200)</td></tr><tr><td>3.5</td><td>9720</td><td>754</td><td>3644</td><td>74</td></tr><tr><td>4.0</td><td>5862</td><td>42</td><td>391</td><td>13</td></tr></table>
+
+We see that using our proposed regularizers results in a huge drop in the number of missing modes. We conjecture that the reason why VAEGAN performs very bad in our metric for missing modes is because the samples generated are of low quality, so the discriminator classifies the samples as “not on mode”. Namely, the data generated is too far away from many real data modes. Essentially if a model generates very bad samples, we can say that the model misses all or most modes.
+
+To conduct more fair evaluation between VAEGAN and our methods, we also perform a blind human evaluation. Again we instructed five individuals to conduct this evaluation of sample variability. Without telling them which is generated by VAEGAN and which is generated by our methods, four people agree that our method wins in terms of sample diversity. One person thinks the samples are equally diverse.
+
+In conclusion, we demonstrate that our proposed mode-regularized GANs, i.e., Reg-GAN and MDGAN, are different from VAEGAN theoretically as discussed above. Such differences empirically result in better sample quality and mode preserving ability, which are our main contributions.

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+        "text": "MODE REGULARIZED GENERATIVE ADVERSARIAL NETWORKS ",
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+        "text": "†Tong Che∗, ‡Yanran Li∗, †,§Athul Paul Jacob, †Yoshua Bengio, ‡Wenjie Li   \n†Montreal Institute for Learning Algorithms, Universite de Montr ´ eal, Montr ´ eal, QC H3T 1J4, Canada ´   \n‡Department of Computing, The Hong Kong Polytechnic University, Hong Kong   \n§David R. Cheriton School of Computer Science, University Of Waterloo, Waterloo, ON N2L 3G1, Canada   \n{tong.che,ap.jacob,yoshua.bengio}@umontreal.ca   \n{csyli,cswjli}@comp.polyu.edu.hk ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
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+        "text": "Although Generative Adversarial Networks achieve state-of-the-art results on a variety of generative tasks, they are regarded as highly unstable and prone to miss modes. We argue that these bad behaviors of GANs are due to the very particular functional shape of the trained discriminators in high dimensional spaces, which can easily make training stuck or push probability mass in the wrong direction, towards that of higher concentration than that of the data generating distribution. We introduce several ways of regularizing the objective, which can dramatically stabilize the training of GAN models. We also show that our regularizers can help the fair distribution of probability mass across the modes of the data generating distribution, during the early phases of training and thus providing a unified solution to the missing modes problem. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Generative adversarial networks (GAN) (Goodfellow et al., 2014) have demonstrated their potential on various tasks, such as image generation, image super-resolution, 3D object generation, and video prediction (Radford et al., 2015; Ledig et al., 2016; Sønderby et al., 2016; Nguyen et al., 2016; Wu et al., 2016; Mathieu et al., 2015). The objective is to train a parametrized function (the generator) which maps noise samples (e.g., uniform or Gaussian) to samples whose distribution is close to that of the data generating distribution. The basic scheme of the GAN training procedure is to train a discriminator which assigns higher probabilities to real data samples and lower probabilities to generated data samples, while simultaneously trying to move the generated samples towards the real data manifold using the gradient information provided by the discriminator. In a typical setting, the generator and the discriminator are represented by deep neural networks. ",
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+        "type": "text",
+        "text": "Despite their success, GANs are generally considered as very hard to train due to training instability and sensitivity to hyper-parameters. On the other hand, a common failure pattern observed while training GANs is the collapsing of large volumes of probability mass onto a few modes. Namely, although the generators produce meaningful samples, these samples are often from just a few modes (small regions of high probability under the data distribution). Behind this phenomenon is the missing modes problem, which is widely conceived as a major problem for training GANs: many modes of the data generating distribution are not at all represented in the generated samples, yielding a much lower entropy distribution, with less variety than the data generating distribution. ",
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+        "text": "This issue has been the subject of several recent papers proposing several tricks and new architectures to stabilize GAN’s training and encourage its samples’ diversity. However, we argue that a general cause behind these problems is the lack of control on the discriminator during GAN training. We would like to encourage the manifold of the samples produced by the generator to move towards that of real data, using the discriminator as a metric. However, even if we train the discriminator to distinguish between these two manifolds, we have no control over the shape of the discriminator function in between these manifolds. In fact, the shape of the discriminator function in the data space can be very non-linear with bad plateaus and wrong maxima and this can therefore hurt the training of GANs (Figure 1). ",
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+        "image_caption": [
+            "Figure 1: Samples with very high discrimination values $\\mathrm { ( D = } 1 . 0$ ) in DCGAN model trained on CelebA dataset. "
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+        "text": "To remedy this problem, we propose a novel regularizer for the GAN training target. The basic idea is simple yet powerful: in addition to the gradient information provided by the discriminator, we want the generator to take advantage of other similarity metrics with much more predictable behavior, such as the $L _ { 2 }$ norm. Differentiating these similarity metrics will provide us with more stable gradients to train our generator. Combining this idea with an approach meant to penalize the missing modes, we propose a family of additional regularizers for the GAN objective. We then design a set of metrics to evaluate the generated samples in terms of both the diversity of modes and the distribution fairness of the probability mass. These metrics are shown to be more robust in judging complex generative models, including those which are well-trained and collapsed ones. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Regularizers usually bring a trade-off between model variance and bias. Our results have shown that, when correctly applied, our regularizers can dramatically reduce model variance, stabilize the training, and fix the missing mode problem all at once, with positive or at the least no negative effects on the generated samples. We also discuss a variant of the regularized GAN algorithm, which can even improve sample quality as compared to the DCGAN baseline. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The GAN approach was initially proposed by Goodfellow et al. (2014) where both the generator and the discriminator are defined by deep neural networks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In Goodfellow et al. (2014), the GAN is able to generate interesting local structure but globally incoherent images on various datasets. Mirza & Osindero (2014) enlarges GAN’s representation capacity by introducing an extra vector to allow the generator to produce samples conditioned on other beneficial information. Motivated from this, several conditional variants of GAN has been applied to a wide range of tasks, including image prediction from a normal map Wang & Gupta (2016), image synthesis from text Reed et al. (2016) and edge map Isola et al. (2016), real-time image manipulation Zhu et al. (2016), temporal image generation Zhou & Berg (2016); Saito & Matsumoto (2016); Vondrick et al. (2016), texture synthesis, style transfer, and video stylization Li & Wand (2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Researchers also aim at stretching GAN’s limit to generate higher-resolution, photo-realistic images. Denton et al. (2015) initially apply a Laplacian pyramid framework on GAN to generate images of high resolution. At each level of their LAPGAN, both the generator and the discriminator are convolutional networks. As an alternative to LAPGAN, Radford et al. (2015) successfully designs a class of deep convolutional generative adversarial networks which has led to significant improvements on unsupervised image representation learning. Another line of work aimed at improving GANs are through feature learning, including features from the latent space and image space. The motivation is that features from different spaces are complementary for generating perceptual and natural-looking images. With this perspective, some researchers use distances between learned features as losses for training objectives for generative models. Larsen et al. (2015) combine a variational autoencoder objective with a GAN and utilize the learned features from the discriminator in the GANs for better image similarity metrics. It is shown that the learned distance from the discriminator is of great help for the sample visual fidelity. Recent literature have also shown impressive results on image super-resolution to infer photo-realistic natural images for $4 \\mathbf { x }$ upscaling factors Ledig et al. (2016); Sønderby et al. (2016); Nguyen et al. (2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite these promising successes, GANs are notably hard to train. Although Radford et al. (2015) provide a class of empirical architectural choices that are critical to stabilize GAN’s training, it would be even better to train GANs more robustly and systematically. Salimans et al. (2016) propose feature matching technique to stabilize GAN’s training. The generator is required to match the statistics of intermediate features of the discriminator. Similar idea is adopted by Zhao et al. (2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In addition to feature distances, Dosovitskiy & Brox (2016) found that the counterpart loss in image space further improves GAN’s training stability. Furthermore, some researchers make use of information in both spaces in a unified learning procedure (Dumoulin et al., 2016; Donahue et al., 2016). In Dumoulin et al. (2016), one trains not just a generator but also an encoder, and the discriminator is trained to distinguish between two joint distributions over image and latent spaces produced either by the application of the encoder on the training data or by the application of the generator (decoder) to the latent prior. This is in contrast with the regular GAN training, in which the discriminator only attempts to separate the distributions in the image space. Parallelly, Metz et al. (2016) stabilize GANs by unrolling the optimization of discriminator, which can be considered as an orthogonal work with ours. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our work is related to VAEGAN (Larsen et al., 2015) in terms of training an autoencoder or VAE jointly with the GAN model. However, the variational autoencoder (VAE) in VAEGAN is used to generate samples whereas our autoencoder based losses serves as a regularizer to penalize missing modes and thus improving GAN’s training stability and sample qualities. We demonstrate detailed differences from various aspects in Appendix D. ",
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+    },
+    {
+        "type": "text",
+        "text": "3 MODE REGULARIZERS FOR GANS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The GAN training procedure can be viewed as a non-cooperative two player game, in which the discriminator $D$ tries to distinguish real and generated examples, while the generator $G$ tries to fool the discriminator by pushing the generated samples towards the direction of higher discrimination values. Training the discriminator $D$ can be viewed as training an evaluation metric on the sample space. Then the generator $G$ has to take advantage of the local gradient $\\nabla \\log D ( G )$ provided by the discriminator to improve itself, namely to move towards the data manifold. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now take a closer look at the root cause of the instabilities while training GANs. The discriminator is trained on both generated and real examples. As pointed out by Goodfellow et al. (2014); Denton et al. (2015); Radford et al. (2015), when the data manifold and the generation manifold are disjoint (which is true in almost all practical situations), it is equivalent to training a characteristic function to be very close to 1 on the data manifold, and 0 on the generation manifold. In order to pass good gradient information to the generator, it is important that the trained discriminator produces stable and smooth gradients. However, since the discriminator objective does not directly depend on the behavior of the discriminator in other parts of the space, training can easily fail if the shape of the discriminator function is not as expected. As an example,Denton et al. (2015) noted a common failure pattern for training GANs which is the vanishing gradient problem, in which the discriminator $D$ perfectly classifies real and fake examples, such that around the fake examples, $D$ is nearly zero. In such cases, the generator will receive no gradient to improve itself.1 ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Another important problem while training GANs is mode missing. In theory, if the generated data and the real data come from the same low dimensional manifold, the discriminator can help the generator distribute its probability mass, because the missing modes will not have near-0 probability under the generator and so the samples in these areas can be appropriately concentrated towards regions where $D$ is closer to 1. However, in practice since the two manifolds are disjoint, $D$ tends to be near 1 on all the real data samples, so large modes usually have a much higher chance of attracting the gradient of discriminator. For a typical GAN model, since all modes have similar $D$ values, there is no reason why the generator cannot collapse to just a few major modes. In other words, since the discriminator’s output is nearly 0 and 1 on fake and real data respectively, the generator is not penalized for missing modes. ",
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+    {
+        "type": "text",
+        "text": "3.1 GEOMETRIC METRICS REGULARIZER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Compared with the objective for the GAN generator, the optimization targets for supervised learning are more stable from an optimization point of view. The difference is clear: the optimization target for the GAN generator is a learned discriminator. While in supervised models, the optimization targets are distance functions with nice geometric properties. The latter usually provides much easier training gradients than the former, especially at the early stages of training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Inspired by this observation, we propose to incorporate a supervised training signal as a regularizer on top of the discriminator target. Assume the generator $G ( z ) : Z \\to X$ generates samples by sampling first from a fixed prior distribution in space $Z$ followed by a deterministic trainable transformation $G$ into the sample space $X$ . Together with $G$ , we also jointly train an encoder $E ( x ) : X  Z$ . Assume $d$ is some similarity metric in the data space, we add $\\mathbb { E } _ { { x } \\sim { p } _ { d } } [ d ( { x } , G \\circ E ( { x } ) ) ]$ as a regularizer, where $p _ { d }$ is the data generating distribution. The encoder itself is trained by minimizing the same reconstruction error. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In practice, there are many options for the distance measure $d$ . For instance, the pixel-wise $L ^ { 2 }$ distance, or the distance of learned features by the discriminator (Dumoulin et al., 2016) or by other networks, such as a VGG classifier. (Ledig et al., 2016) ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The geometric intuition for this regularizer is straight-forward. We are trying to move the generated manifold to the real data manifold using gradient descent. In addition to the gradient provided by the discriminator, we can also try to match the two manifolds by other geometric distances, say, $L ^ { s }$ metric. The idea of adding an encoder is equivalent to first training a point to point mapping $G ( E ( x ) )$ between the two manifolds and then trying to minimize the expected distance between the points on these two manifolds. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 MODE REGULARIZER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In addition to the metric regularizer, we propose a mode regularizer to further penalize missing modes. In traditional GANs, the optimization target for the generator is the empirical sum $\\begin{array} { r } { \\sum _ { i } { \\nabla _ { \\theta } \\log { D ( G _ { \\theta } ( z _ { i } ) ) } } } \\end{array}$ . The missing mode problem is caused by the conjunction of two facts: (1) the areas near missing modes are rarely visited by the generator, by definition, thus providing very few examples to improve the generator around those areas, and (2) both missing modes and nonmissing modes tend to correspond to a high value of $D$ , because the generator is not perfect so that the discriminator can take strong decisions locally and obtain a high value of $D$ even near non-missing modes. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As an example, consider the situation in Figure 2. For most $z$ , the gradient of the generator $\\nabla _ { \\theta } \\log D ( G _ { \\theta } ( z ) )$ pushes the generator towards the major mode $M _ { 1 }$ . Only when $G ( z )$ is very close to the mode $M _ { 2 }$ can the generator get gradients to push itself towards the minor mode $M _ { 2 }$ . However, it is possible that such $z$ is of low or zero probability in the prior distribution $p _ { 0 }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/7385e91fa5a2d1dc34cba4cf1f79b9eaf272d966fa73d5daac595e2758dd374d.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of missing modes problem. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Given this observation, consider a regularized GAN model with the metric regularizer. Assume $M _ { 0 }$ is a minor mode of the data generating distribution. For $x \\ \\in \\ M _ { 0 }$ , we know that if $G \\circ E$ is a good autoencoder, $G ( E ( x ) )$ will be located very close to mode $M _ { 0 }$ . Since there are sufficient training examples of mode $M _ { 0 }$ in the training data, we add the mode regularizer $\\mathbb { E } _ { { x } \\sim p _ { d } } [ \\log { \\bar { D } } ( G \\circ E ( x ) ) ]$ to our optimization target for the generator, to encourage $G ( E ( x ) )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "to move towards a nearby mode of the data generating distribution. In this way, we can achieve fair probability mass distribution across different modes. ",
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+    {
+        "type": "text",
+        "text": "In short, our regularized optimization target for the generator and the encoder becomes: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/465989454cd64079a3138096b97174902b00b2a7418ff0302361cf98fad5e65d.jpg",
+        "text": "$$\n\\begin{array} { r } { T _ { G } = - \\mathbb { E } _ { z } [ \\log D ( G ( z ) ) ] + \\mathbb { E } _ { x \\sim p _ { d } } [ \\lambda _ { 1 } d ( x , G \\circ E ( x ) ) + \\lambda _ { 2 } \\log D ( G \\circ E ( x ) ) ] } \\\\ { T _ { E } = \\mathbb { E } _ { x \\sim p _ { d } } [ \\lambda _ { 1 } d ( x , G \\circ E ( x ) ) + \\lambda _ { 2 } \\log D ( G \\circ E ( x ) ) ] } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "3.3 MANIFOLD-DIFFUSION TRAINING FOR REGULARIZED GANS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On some large scale datasets, CelebA for example, the regularizers we have discussed do improve the diversity of generated samples, but the quality of samples may not be as good without carefully tuning the hyperparameters. Here we propose a new algorithm for training metric-regularized GANs, which is very stable and much easier to tune for producing good samples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The proposed algorithm divides the training procedure of GANs into two steps: a manifold step and a diffusion step. In the manifold step, we try to match the generation manifold and the real data manifold with the help of an encoder and the geometric metric loss. In the diffusion step, we try to distribute the probability mass on the generation manifold fairly according to the real data distribution. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "An example of manifold-diffusion training of GAN (MDGAN for short) is as follows: we train a discriminator $D _ { 1 }$ which separates between the samples $x$ and $G \\circ E ( x )$ , for $x$ from the data, and we optimize $G$ with respect to the regularized GAN loss $\\mathbb { E } [ \\log D _ { 1 } ( G \\circ E ( x ) ) + \\lambda d ( x , G \\circ E ( x ) ) ]$ in order to match the two manifolds. In the diffusion step we train a discriminator $D _ { 2 }$ between distributions $G ( z )$ and $G \\circ E ( x )$ , and we train $G$ to maximize $\\log { D _ { 2 } ( G ( z ) ) }$ . Since these two distributions are now nearly on the same low dimensional manifold, the discriminator $D _ { 2 }$ provides much smoother and more stable gradients. The detailed training procedure is given in Appendix A. See Figure 6 for the quality of generated samples. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 EVALUATION METRICS FOR MODE MISSING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to estimate both the missing modes and the sample qualities in our experiments, we used several different metrics for different experiments instead of human annotators. ",
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+    },
+    {
+        "type": "text",
+        "text": "The inception score (Salimans et al., 2016) was considered as a good assessment for sample quality from a labelled dataset: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aafb5f122f72c0dacec80549a0858d3a76e7f09d5bb4288fc0bd84578458da4e.jpg",
+        "text": "$$\n\\exp { ( \\mathbb { E } _ { x } K L ( p ( y | \\mathbf { x } ) | | p ^ { * } ( y ) ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            502,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where $\\mathbf { x }$ denotes one sample, $p ( y | x )$ is the softmax output of a trained classifier of the labels, and $p ^ { * } ( y )$ is the overall label distribution of generated samples. The intuition behind this score is that a strong classifier usually has a high confidence for good samples. However, the inception score is sometimes not a good metric for our purpose. Assume a generative model that collapse to a very bad image. Although the model is very bad, it can have a perfect inception score, because $p ( y | x )$ can have a high entropy and $p ^ { * } ( y )$ can have a low entropy. So instead, for labelled datasets, we propose another assessment for both visual quality and variety of samples, the MODE score: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea827145ec213a569ba151ea7be7a1aa8937e0834acccd1b4e6bdb379c69f849.jpg",
+        "text": "$$\n\\exp \\left( \\mathbb { E } _ { x } K L ( p ( y | \\mathbf { x } ) | | p ( y ) ) - K L ( p ^ { * } ( y ) | | p ( y ) ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $p ( y )$ is the distribution of labels in the training data. According to our human evaluation experiences, the MODE score successfully measures two important aspects of generative models, i.e., variety and visual quality, in one metric. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, in datasets without labels (LSUN) or where the labels are not sufficient to characterize every data mode (CelebA), the above metric does not work well. We instead train a third party discriminator between the real data and the generated data from the model. It is similar to the GAN discriminator but is not used to train the generator. We can view the output of the discriminator as an estimator for the quantity (See (Goodfellow et al., 2014) for proof): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14b1003ef2971d36815e1ae844f95a6c5c592bfed0e2c3a4fbecf876b0206658.jpg",
+        "text": "$$\nD ^ { * } ( s ) \\approx \\frac { p _ { g } ( s ) } { p _ { g } ( s ) + p _ { d } ( s ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where $p _ { g }$ is the probability density of the generator and $p _ { d }$ is the density of the data distribution. To prevent $D ^ { * }$ from learning a perfect 0-1 separation of $p _ { g }$ and $p _ { d }$ , we inject a zero-mean Gaussian noise to the inputs when training $D ^ { * }$ . After training, we test $D ^ { * }$ on the test set $T$ of the real dataset. If for any test sample $t \\in T$ , the discrimination value $D ( t )$ is close to 1, we can conclude that the mode corresponding to $t$ is missing. In this way, although we cannot measure exactly the number of modes that are missing, we have a good estimator of the total probability mass of all the missing modes. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 MNIST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We perform two classes of experiments on MNIST. For the MNIST dataset, we can assume that the data generating distribution can be approximated with ten dominant modes, if we define the term “mode” here as a connected component of the data manifold. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 GRID SEARCH FOR MNIST GAN MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/d70c7cc33fae7d44469f8d2702b9394c1ad7c0ee2d873ba68c10369f49c34ba8.jpg",
+        "table_caption": [
+            "Table 1: Grid Search for Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>nLayerG</td><td>[2,3,4]</td></tr><tr><td>nLayerD</td><td>[2,3,4]</td></tr><tr><td>sizeG sizeD</td><td>[400,800,1600,3200]</td></tr><tr><td></td><td>[256, 512, 1024]</td></tr><tr><td>dropoutD</td><td>[True,False]</td></tr><tr><td>optimG</td><td>[SGD,Adam]</td></tr><tr><td>optimD</td><td>[SGD,Adam]</td></tr><tr><td>lr</td><td>[1e-2,1e-3,1e-4]</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "In order to systemically explore the effect of our proposed regularizers on GAN models in terms of improving stability and sample quality, we use a large scale grid search of different GAN hyper-parameters on the MNIST dataset. The grid search is based on a pair of randomly selected loss weights: $\\lambda _ { 1 } = 0 . 2$ and $\\lambda _ { 2 } ~ = ~ 0 . 4$ . We use the same hyper-parameter settings for both GAN and Regularized GAN, and list the search ranges in Table 1. Our grid search is similar to those proposed in Zhao et al. (2016). Please refer to it for detailed explanations regarding these hyper-parameters. ",
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+    {
+        "type": "text",
+        "text": "For evaluation, we first train a 4-layer CNN classifier on the MNIST digits, and then apply it to compute the MODE scores for the generated samples from all these models. The resulting distribution of MODE score is shown in Figure 3. Clearly, our proposed regularizer significantly improves the MODE scores and thus demonstrates its benefits on stabilizing GANs and improving sample qualities. ",
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+    {
+        "type": "image",
+        "img_path": "images/23691178a9332efa7d0b890615c2cf021c092595979c93013beab8b2b47d1ea1.jpg",
+        "image_caption": [
+            "Figure 3: The distributions of MODE scores for GAN and regularized GAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To illustrate the effect of regularizers with different coefficients, we randomly pick an architecture and train it with different $\\lambda _ { 1 } = \\lambda _ { 2 }$ . The results are shown in Figure 4. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/ba0462a6a90a59f8d012d3b775739a6a034addf655922b4e85364f2d3f1a8aa3.jpg",
+        "image_caption": [
+            "Figure 4: (Left 1-5) Different hyperparameters for MNIST generation. The values of the $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ in our Regularized GAN are listed below the corresponding samples. (Right 6-7) Best samples through grid search for GAN and Regularized GAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1.2 COMPOSITIONAL MNIST DATA WITH 1000 MODES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to quantitatively study the effect of our regularizers on the missing modes, we concatenate three MNIST digits to a number in [0,999] in a single 64x64 image, and then train DCGAN as a baseline model on the 1000 modes dataset. The digits on the image are sampled with different probabilities, in order to test the model’s capability to preserve small modes in generation. We again use a pre-trained classifier for MNIST instead of a human to evaluate the models. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/71666736033d81d0147714d74aafcb68fb1cd93f319564f27940fd29a5bfa2c2.jpg",
+        "table_caption": [
+            "Table 2: Results for Compositional MNIST with 1000 modes. The proposed regularization (RegDCGAN) allows to substantially reduce the number of missed modes as well as the KL divergence that measures the plausibility of the generated samples (like in the Inception score). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Set 1</td><td colspan=\"2\">Set 2</td><td colspan=\"2\">Set3</td><td colspan=\"2\">Set4</td></tr><tr><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td><td>#Miss</td><td>KL</td></tr><tr><td>DCGAN</td><td>204.7</td><td>77.9</td><td>204.3</td><td>60.2</td><td>103.4</td><td>75.9</td><td>89.3</td><td>77.8</td></tr><tr><td>Reg-DCGAN</td><td>32.1</td><td>62.3</td><td>71.5</td><td>58.9</td><td>42.7</td><td>68.4</td><td>31.6</td><td>67.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The performances on the compositional experiment are measured by two metrics. #Miss represents the classifier-reported number of missing modes, which is the size of the set of numbers that the model never generates. KL stands for the KL divergence between the classifier-reported distribution of generated numbers and the distribution of numbers in the training data (as for the Inception score). The results are shown in Table 2. With the help of our proposed regularizer, both the number of missing modes and KL divergence drop dramatically among all the sets of the compositional MNIST dataset, which again proves the effectiveness of our regularizer for preventing the missing modes problem. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 CELEBA ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To test the effectiveness of our proposal on harder problems, we implement an encoder for the DCGAN algorithm and train our model with different hyper-parameters together with the DCGAN baseline on the CelebA dataset. We provide the detailed architecture of our regularized DCGAN in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.2.1 MISSING MODES ESTIMATION ON CELEBA ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We also employ a third party discriminator trained with injected noise as a metric for missing mode estimation. To implement this, we add noise in the input layer in the discriminator network. For each GAN model to be estimated, we independently train this noisy discriminator, as mode estimator, with the same architecture and hyper-parameters on the generated data and the training data. We then apply the mode estimator to the test data. The images which have high mode estimator outputs can be viewed as on the missing modes. ",
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+    {
+        "type": "table",
+        "img_path": "images/2eeaa39d40381cd97d1072398ead88c0c090eb7bf31f51b9d762d27e14c5e133.jpg",
+        "table_caption": [
+            "Table 3: Number of images on the missing modes on CelebA estimated by a third-party discriminator. The numbers in the brackets indicate the dimension of prior $z$ . $\\sigma$ denotes the standard deviation of the added Gaussian noise applied at the input of the discriminator to regularize it. MDGAN achieves a very high reduction in the number of missing modes, in comparison to other methods . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>0</td><td>DCGAN (100)</td><td>DCGAN (200)</td><td>Reg-GAN (100)</td><td>Reg-GAN (200)</td><td>MDGAN (200)</td></tr><tr><td>3.5</td><td>5463</td><td>17089</td><td>754</td><td>3644</td><td>74</td></tr><tr><td>4.0</td><td>590</td><td>15832</td><td>42</td><td>391</td><td>13</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The comparison result is shown in Table 3. Both our proposed Regularized-GAN and MDGAN outperform baseline DCGAN models on all settings. Especially, MDGAN suppresses other models, showing its superiority on modes preserving. We also find that, although sharing the same architecture, the DCGAN with 200-dimensional noise performs quite worse than that with 100-dimensional noise as input. On the contrary, our regularized GAN performs more consistently. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To get a better understanding of the models’ performance, we want to figure out when and where these models miss the modes. Visualizing the test images associated with missed modes is instructive. In Figure 5, the left three images are missed by all models. It is rare to see in the training data the cap in the second image and the type of background in the third, which thus can be viewed as small modes under this situation. These three images should be considered as the hardest test data for GAN to learn. Nonetheless, our best model, MDGAN still capture certain small modes. The seven images on the right in Figure 5 are only missed by DCGAN. The sideface, paleface, black, and the berets are special attributes among these images, but our proposed MDGAN performs well on all of them. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/dd60ba26724f057d75e53e8f9051cfa7ab0d2f40c846afeeccfbb3ecd6ec1b15.jpg",
+        "image_caption": [
+            "Figure 5: Test set images that are on missing mode. Left: Both MDGAN and DCGAN missing. Right: Only DCGAN missing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 QUALITATIVE EVALUATION OF GENERATED SAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "After quantitative evaluation, we manually examine the generated samples by our regularized GAN to see whether the proposed regularizer has side-effects on sample quality. We compare our model with ALI (Dumoulin et al., 2016), VAEGAN (Larsen et al., 2015), and DCGAN (Radford et al., 2015) in terms of sample visual quality and mode diversity. Samples generated from these models are shown in Figure $6 ^ { 2 ^ { \\circ } }$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/601224511ba5ab1ca605c826bf8588ec4a9779fd983509775ed6e2be4b574de3.jpg",
+        "image_caption": [
+            "Figure 6: Samples generated from different generative models. For each compared model, we directly take ten decent samples reported in their corresponding papers and code repositories. Note how MDGAN samples are both globally more coherent and locally have sharp textures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Both MDGAN and Regularized-GAN generate clear and natural-looking face images. Although ALI’s samples are plausible, they are sightly deformed in comparison with those from MDGAN. The samples from VAEGAN and DCGAN seem globally less coherent and locally less sharp. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As to sample quality, it is worth noting that the samples from MDGAN enjoy fewer distortions. With all four other models, the majority of generated samples suffer from some sort of distortion. However, for the samples generated by MDGAN, the level of distortion is lower compared with the other four compared models. We attribute it to the help of the autoencoder as the regularizer to alter the generation manifolds. In this way, the generator is able to learn fine-grained details such as face edges. As a result, MDGAN is able to reduce distortions. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ccfe9752c1edde7bf368397e5fc0a44880f5ada2037abd62a3f910d669af1b5.jpg",
+        "image_caption": [
+            "Figure 7: Sideface samples generated by Regularized-GAN and MDGAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In terms of missing modes problem, we instructed five individuals to conduct human evaluation on the generated samples. They achieve consensus that MDGAN wins in terms of mode diversities. Two people pointed out that MDGAN generates a larger amount of samples with side faces than other models. We select several of these side face samples in Figure 7. Clearly, our samples maintain acceptable visual fidelity meanwhile share diverse modes. Combined with the above quantitative results, it is convincing that our regularizers bring benefits for both training stability and mode variety without the loss of sample quality. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Although GANs achieve state-of-the-art results on a large variety of unsupervised learning tasks, training them is considered highly unstable, very difficult and sensitive to hyper-parameters, all the while, missing modes from the data distribution or even collapsing large amounts of probability mass on some modes. Successful GAN training usually requires large amounts of human and computing efforts to fine tune the hyper-parameters, in order to stabilize training and avoid collapsing. Researchers usually rely on their own experience and published tricks and hyper-parameters instead of systematic methods for training GANs. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We provide systematic ways to measure and avoid the missing modes problem and stabilize training with the proposed autoencoder-based regularizers. The key idea is that some geometric metrics can provide more stable gradients than trained discriminators, and when combined with the encoder, they can be used as regularizers for training. These regularizers can also penalize missing modes and encourage a fair distribution of probability mass on the generation manifold. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Naiyan Wang, Jianbo Ye, Yuchen Ding, Saboya Yang for their GPU support. We also want to thank Huiling Zhen for helpful discussions, Junbo Zhao for providing the details of grid search experiments on the EBGAN model, as well as Anders Boesen Lindbo Larsen for kindly helping us on running VAEGAN experiments. We appreciate for the valuable suggestions and comments from the anonymous reviewers. The work described in this paper was partially supported by NSERC, Calcul Quebec, Compute Canada, the Canada Research Chairs, CIFAR, National Natural Science Foundation of China (61672445 and 61272291), Research Grants Council of Hong Kong (PolyU 152094/14E), and The Hong Kong Polytechnic University (G-YBP6). ",
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+    {
+        "type": "text",
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+        "text": "Jiajun Wu, Chengkai Zhang, Tianfan Xue, William T Freeman, and Joshua B Tenenbaum. Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In Neural Information Processing Systems (NIPS), 2016. ",
+        "bbox": [
+            173,
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+            823,
+            791
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Junbo Zhao, Michael Mathieu, and Yann LeCun. Energy-based generative adversarial network. arXiv preprint arXiv:1609.03126, 2016. ",
+        "bbox": [
+            173,
+            800,
+            821,
+            829
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yipin Zhou and Tamara L Berg. Learning temporal transformations from time-lapse videos. In European Conference on Computer Vision, pp. 262–277. Springer, 2016. ",
+        "bbox": [
+            173,
+            838,
+            823,
+            867
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Philipp Krahenb ¨ uhl, Eli Shechtman, and Alexei A. Efros. Generative visual manipula- ¨ tion on the natural image manifold. In Proceedings of European Conference on Computer Vision (ECCV), 2016. ",
+        "bbox": [
+            173,
+            876,
+            821,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX: PSEUDO CODE FOR MDGAN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            553,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this Appendix, we give the detailed training procedure of an MDGAN example we discuss in Section 3.3. ",
+        "bbox": [
+            176,
+            136,
+            826,
+            164
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Manifold Step: ",
+        "text_level": 1,
+        "bbox": [
+            183,
+            190,
+            289,
+            205
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. Sample $\\left\\{ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\cdots \\mathbf { x } _ { m } \\right\\}$ from data generating distribution $p _ { d a t a } ( x )$ ",
+        "bbox": [
+            187,
+            205,
+            645,
+            219
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "2. Update discriminator $D _ { 1 }$ using SGD with gradient ascent: ",
+        "bbox": [
+            184,
+            220,
+            588,
+            233
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/445b11f8bd62bdf4ac6522c54f1a637fdb3ae774a443c250db6dabcfce3adc0d.jpg",
+        "text": "$$\n\\nabla _ { \\theta _ { d } ^ { 1 } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ \\log D _ { 1 } ( \\mathbf { x } _ { i } ) + \\log ( 1 - D _ { 1 } ( G ( E ( \\mathbf { x } _ { i } ) ) ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            237,
+            679,
+            279
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "3. Update generator $G$ using SGD with gradient ascent: ",
+        "bbox": [
+            183,
+            286,
+            549,
+            303
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6092f2109d49563770fd7b7db563677e0b9796a1abf84c389f0a914d2d026e2.jpg",
+        "text": "$$\n\\nabla _ { \\theta _ { g } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ \\lambda \\log D _ { 1 } ( G ( E ( \\mathbf { x } _ { i } ) ) ) - | | \\mathbf { x } _ { i } - G ( E ( \\mathbf { x } _ { i } ) ) | | ^ { 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            306,
+            694,
+            347
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Diffusion Step: ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            351,
+            290,
+            364
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4. Sample $\\left\\{ \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\cdots \\mathbf { x } _ { m } \\right\\}$ from data generating distribution $p _ { d a t a } ( x )$ .   \n5. Sample $\\{ \\mathbf { z } _ { 1 } , \\mathbf { z } _ { 2 } , \\cdots \\mathbf { z } _ { m } \\}$ from prior distribution $p _ { \\sigma } ( z )$ .   \n6. Update discriminator $D _ { 2 }$ using SGD with gradient ascent: ",
+        "bbox": [
+            183,
+            366,
+            648,
+            407
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4796e4aa9fbaee3194a2fe04b9a6bda852d047a18bc5f8177f403fa8b8d58875.jpg",
+        "text": "$$\n\\nabla _ { \\theta _ { d } ^ { 2 } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ \\log D _ { 2 } ( G ( E ( \\mathbf { x } _ { i } ) ) ) + \\log ( 1 - D _ { 2 } ( \\mathbf { z } _ { i } ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            410,
+            678,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7. Update generator $G$ using SGD with gradient ascent: ",
+        "bbox": [
+            184,
+            460,
+            549,
+            476
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d9aa6c6f9f119feb023def7c6ce57ec7aa7749d5aba893e73fdc40b897543f73.jpg",
+        "text": "$$\n\\nabla _ { \\boldsymbol { \\theta } _ { g } } \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } [ \\log D _ { 2 } ( G ( \\mathbf { z } _ { i } ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            479,
+            599,
+            520
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B APPENDIX: ARCHITECTURE FOR EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            592,
+            609,
+            608
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use similar architectures for Compositional MNIST and CelebA experiments. The architecture is based on that found in DCGAN Radford et al. (2015). Apart from the discriminator and generator which are the same as DCGAN, we add an encoder which is the ”inverse” of the generator, by reversing the order of layers and replacing the de-convolutional layers with convolutional layers. ",
+        "bbox": [
+            174,
+            626,
+            825,
+            683
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "One has to pay particular attention to batch normalization layers. In DCGAN, there are batch normalization layers both in the generator and the discriminator. However, two classes of data go through the batch normalization layers in the generator. One come from sampled noise $z$ , the other one come from the encoder. In our implementation, we separate the batch statistics for these two classes of data in the generator, while keeping the parameters of BN layer to be shared. In this way, the batch statistics of these two kinds of batches cannot interfere with each other. ",
+        "bbox": [
+            173,
+            689,
+            825,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C APPENDIX: ADDITIONAL SYNTHESIZED EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            799,
+            663,
+            814
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of mode-regularized GANs proposed in this paper, we train a very simple GAN architecture on synthesized 2D dataset, following Metz et al. (2016). ",
+        "bbox": [
+            174,
+            832,
+            820,
+            861
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The data is sampled from a mixture of 6 Gaussians, with standard derivation of 0.1. The means of the Gaussians are placed around a circle with radius 5. The generator network has two ReLU hidden layers with 128 neurons. It generates 2D output samples from 3D uniform noise from [0,1]. The discriminator consists of only one fully connected layer of ReLU neurons, mapping the 2D input to a real 1D number. Both networks are optimized with the Adam optimizer with the learning rate of 1e-4. ",
+        "bbox": [
+            176,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the regularized version, we choose $\\lambda _ { 1 } = \\lambda _ { 2 } = 0 . 0 0 5$ . The comparison between the generator distribution from standard GAN and our proposed regularized GAN are shown in Figure 9. ",
+        "bbox": [
+            171,
+            138,
+            823,
+            167
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/2acc16f21eb9e8b9d0333914b869f7d2c9aea0b818b3a87da7a5c415e117f9f3.jpg",
+        "image_caption": [
+            "Figure 9: Comparison results on a toy 2D mixture of Gaussians dataset. The columns on the left shows heatmaps of the generator distributions as the number of training epochs increases, whereas the rightmost column presents the target, the original data distribution. The top row shows standard GAN result. The generator has a hard time oscillating among the modes of the data distribution, and is only able to “recover” a single data mode at once. In contrast, the bottom row shows results of our regularized GAN. Its generator quickly captures the underlying multiple modes and fits the target distribution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            178,
+            825,
+            306
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D APPENDIX: COMPARISON WITH VAEGAN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            462,
+            563,
+            478
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this appendix section, we demonstrate the effectiveness and uniqueness of mode-regularized GANs proposed in this paper as compared to Larsen et al. (2015) in terms of its theoretical difference, sample quality and number of missing modes. ",
+        "bbox": [
+            174,
+            493,
+            825,
+            536
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "With regard to the theoretical difference, the optimization of VAEGAN relies on the probabilistic variational bound, namely $p ( x ) \\geq \\mathbb { E } _ { q ( z | x ) } [ \\log p ( x | z ) ] - { \\mathrm { K L } } ( q ( z | x ) | | p ( z ) )$ . This variational bound together with a GAN loss is optimized with several assumptions imposed in VAEGAN: ",
+        "bbox": [
+            176,
+            542,
+            825,
+            585
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. In general, VAE is based on the assumption that the true posterior $p ( z | x )$ can be well approximated by factorized Gaussian distribution $q$ .   \n2. As to VAEGAN, It is also assumed that the maximum likelihood objectives does not conflict with GAN objective in terms of probabilistic framework. ",
+        "bbox": [
+            212,
+            597,
+            825,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The first assumption does not necessarily hold for GANs. We have found that in some trained models of DCGANs, the real posterior $p ( z | x )$ is even not guaranteed to have only one mode, not to mention it is anything close to factorized Gaussian. We believe that this difference in probabilistic framework is an essential obstacle when one tries to use the objective of VAEGAN as a regularizer. However, in our algorithm, where we use a plain auto-encoder instead of VAE as the objective. Plain auto-encooders works better than VAE for our purposes because as long as the model $G ( z )$ is able to generate training samples, there always exists a function $E ^ { \\ast } ( x )$ such that $G ( E ( x ) ) = x$ . Our encoder can therefore be viewed as being trained to approximate this real encoder $E ^ { * }$ . There are no conflicts between a good GAN generator and our regularization objective. Hence, our objectives can be used as regularizers for encoding the prior knowledge that good models should be able to generate the training samples. This is why our work is essentially different from VAEGAN. In our experiments, we also believe that this is the reason why VAEGAN generates worse samples than a carefully tuned regularized GANs. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            849
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In terms of sample quality and missing modes, we run the official code of VAEGAN 3 with their default setting. We train VAEGAN for 30 epochs 4 and our models for only 20 epochs. For fairness, their model was run 3 times and the trained model with the best sample visual quality was taken for the comparison. ",
+        "bbox": [
+            176,
+            857,
+            823,
+            886
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The generated samples are shown in Figure 10. The most obvious difference between our samples and VAEGAN’s samples is the face distortion, which is consistent with our experimental results in Section 4.2.2. We conjecture that the distortions of VAEGAN’s samples are due to the conflicts between the two objectives, as we present above. In other words, the way we introduce auto-encoders as regularizers for GAN models is different from VAEGAN’s. The difference is that the second assumption mentioned above is not required in our approaches. In our framework, the auto-encoders helps alter the generation manifolds, leading to fewer distortions in fine-grained details in our generated samples. ",
+        "bbox": [
+            173,
+            138,
+            825,
+            251
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e0082f3127f2aab9defa52d7d88f547f2ca4aa33600ecc6c7008814eebed363b.jpg",
+        "image_caption": [
+            "Figure 10: Samples generated by our models and VAEGAN. The third line are samples generated by our self-trained VAEGAN model, with default settings. The last line are generated samples reported in the original VAEGAN paper. We depict both of them here for a fair comparison. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            261,
+            825,
+            443
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In terms of the missing modes problem, we use the same method described in Section 4.2.1 for computing the number of images with missing modes. The results are shown below. ",
+        "bbox": [
+            174,
+            534,
+            820,
+            563
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/638af6856108c1e83a2058aaa4598e9b7d6e823382760a255985202d54539fc0.jpg",
+        "table_caption": [
+            "Table 4: Number of images on the missing modes on CelebA estimated by a third-party discriminator. The numbers in the brackets indicate the dimension of prior $z$ . $\\sigma$ denotes the standard deviation of the added Gaussian noise applied at the input of the discriminator to regularize it. MDGAN achieves a very high reduction in the number of missing modes, in comparison to VAEGAN. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>0</td><td>VAEGAN (100)</td><td>Reg-GAN (100)</td><td>Reg-GAN (200)</td><td>MDGAN (200)</td></tr><tr><td>3.5</td><td>9720</td><td>754</td><td>3644</td><td>74</td></tr><tr><td>4.0</td><td>5862</td><td>42</td><td>391</td><td>13</td></tr></table>",
+        "bbox": [
+            228,
+            640,
+            769,
+            704
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We see that using our proposed regularizers results in a huge drop in the number of missing modes. We conjecture that the reason why VAEGAN performs very bad in our metric for missing modes is because the samples generated are of low quality, so the discriminator classifies the samples as “not on mode”. Namely, the data generated is too far away from many real data modes. Essentially if a model generates very bad samples, we can say that the model misses all or most modes. ",
+        "bbox": [
+            174,
+            720,
+            825,
+            791
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To conduct more fair evaluation between VAEGAN and our methods, we also perform a blind human evaluation. Again we instructed five individuals to conduct this evaluation of sample variability. Without telling them which is generated by VAEGAN and which is generated by our methods, four people agree that our method wins in terms of sample diversity. One person thinks the samples are equally diverse. ",
+        "bbox": [
+            174,
+            797,
+            825,
+            867
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In conclusion, we demonstrate that our proposed mode-regularized GANs, i.e., Reg-GAN and MDGAN, are different from VAEGAN theoretically as discussed above. Such differences empirically result in better sample quality and mode preserving ability, which are our main contributions. ",
+        "bbox": [
+            174,
+            875,
+            825,
+            917
+        ],
+        "page_idx": 12
+    }
+]

parse/train/Hkg-xgrYvH/Hkg-xgrYvH.md

@@ -0,0 +1,497 @@
+# EMPIRICAL BAYES TRANSDUCTIVE META-LEARNING WITH SYNTHETIC GRADIENTS
+
+Shell Xu $\mathbf { H } \mathbf { u } ^ { 1 }$ Pablo G. Moreno2 Yang Xiao1 Xi Shen1   
+Guillaume Obozinski3 Neil D. Lawrence4 Andreas Damianou2
+
+1École des Ponts ParisTech   
+Champs-sur-Marne, France   
+{xu.hu, yang.xiao, xi.shen}@enpc.fr   
+2Amazon   
+Cambridge, United Kingdom   
+{morepabl, damianou}@amazon.com
+
+3Swiss Data Science Center Lausanne, Switzerland guillaume.obozinski@epfl.ch
+
+4University of Cambridge Cambridge, United Kingdom ndl21@cam.ac.uk
+
+# ABSTRACT
+
+We propose a meta-learning approach that learns from multiple tasks in a transductive setting, by leveraging the unlabeled query set in addition to the support set to generate a more powerful model for each task. To develop our framework, we revisit the empirical Bayes formulation for multi-task learning. The evidence lower bound of the marginal log-likelihood of empirical Bayes decomposes as a sum of local KL divergences between the variational posterior and the true posterior on the query set of each task. We derive a novel amortized variational inference that couples all the variational posteriors via a meta-model, which consists of a synthetic gradient network and an initialization network. Each variational posterior is derived from synthetic gradient descent to approximate the true posterior on the query set, although where we do not have access to the true gradient. Our results on the Mini-ImageNet and CIFAR-FS benchmarks for episodic few-shot classification outperform previous state-of-the-art methods. Besides, we conduct two zero-shot learning experiments to further explore the potential of the synthetic gradient.
+
+# 1 INTRODUCTION
+
+While supervised learning of deep neural networks can achieve or even surpass human-level performance (He et al., 2015; Devlin et al., 2018), they can hardly extrapolate the learned knowledge beyond the domain where the supervision is provided. The problem of solving rapidly a new task after learning several other similar tasks is called meta-learning (Schmidhuber, 1987; Bengio et al., 1991; Thrun & Pratt, 1998); typically, the data is presented in a two-level hierarchy such that each data point at the higher level is itself a dataset associated with a task, and the goal is to learn a meta-model that generalizes across tasks. In this paper, we mainly focus on few-shot learning (Vinyals et al., 2016), an instance of meta-learning problems, where a task $t$ consists of a query set $d _ { t } : = \{ ( x _ { t , i } , y _ { t , i } ) \} _ { i = 1 } ^ { n }$ serving as the test-set of the task and a support set $d _ { t } ^ { l } { : = } \{ ( x _ { t , i } ^ { l } , y _ { t , i } ^ { l } ) \} _ { i = 1 } ^ { n ^ { l } }$ serving as the train-set. In meta-testing1, one is given the support set and the inputs of the query set $x _ { t } : = \{ x _ { t , i } \} _ { i = 1 } ^ { n }$ , and asked to predict the labels $y _ { t } : = \{ y _ { t , i } \} _ { i = 1 } ^ { n }$ . In meta-training, $y _ { t }$ is provided as the ground truth. The setup of few-shot learning is summarized in Table 1.
+
+A important distinction to make is whether a task is solved in a transductive or inductive manner, that is, whether $x _ { t }$ is used. The inductive setting is what was originally proposed by Vinyals et al. (2016), in which only $d _ { t } ^ { l }$ is used to generate a model. The transductive setting, as an alternative, has the advantage of being able to see partial or all points in $x _ { t }$ before making predictions. In fact,
+
+<table><tr><td rowspan="3">d={wtoy1</td><td rowspan="3">Support set</td><td colspan="2">Query set</td></tr><tr><td>xt :={xt,i=1</td><td>yt={yt,ii=1</td></tr><tr><td>Meta-training</td><td>√</td><td>√</td><td>×</td></tr><tr><td>Meta-testing</td><td></td><td></td><td></td></tr></table>
+
+Table 1: The setup of few-shot learning. If task $t$ is used for meta-testing, $y _ { t }$ is not given to the model.
+
+Nichol et al. (2018) notice that most of the existing meta-learning methods involve transduction unintentionally since they use $x _ { t }$ implicitly via the batch normalization (Ioffe & Szegedy, 2015). Explicit transduction is less explored in meta-learning, the exception is Liu et al. (2018), who adapted the idea of label propagation (Zhu et al., 2003) from graph-based semi-supervised learning methods. We take a totally different path that meta-learn the “gradient” descent on $x _ { t }$ without using $y _ { t }$ .
+
+Due to the hierarchical structure of the data, it is natural to formulate meta-learning by a hierarchical Bayes (HB) model (Good, 1980; Berger, 1985), or alternatively, an empirical Bayes (EB) model (Robbins, 1985; Kucukelbir & Blei, 2014). The difference is that the latter restricts the learning of meta-parameters to point estimates. In this paper, we focus on the EB model, as it largely simplifies the training and testing without losing the strength of the HB formulation.
+
+The idea of using HB or EB for meta-learning is not new: Amit & Meir (2018) derive an objective similar to that of HB using PAC-Bayesian analysis; Grant et al. (2018) show that MAML (Finn et al., 2017) can be understood as a EB method; Ravi & Beatson (2018) consider a HB extension to MAML and compute posteriors via amortized variational inference. However, unlike our proposal, these methods do not make full use of the unlabeled data in query set. Roughly speaking, they construct the variational posterior as a function of the labeled set $d _ { t } ^ { l }$ without taking advantage of the unlabeled set $x _ { t }$ . The situation is similar in gradient based meta-learning methods (Finn et al., 2017; Ravi & Larochelle, 2016; Li et al., 2017b; Nichol et al., 2018; Flennerhag et al., 2019) and many other meta-learning methods (Vinyals et al., 2016; Snell et al., 2017; Gidaris & Komodakis, 2018), where the mechanisms used to generate the task-specific parameters rely on groundtruth labels, thus, there is no place for the unlabeled set to contribute. We argue that this is a suboptimal choice, which may lead to overfitting when the labeled set is small and hinder the possibility of zero-shot learning (when the labeled set is not provided).
+
+In this paper, we propose to use synthetic gradient (Jaderberg et al., 2017) to enable transduction, such that the variational posterior is implemented as a function of the labeled set $d _ { t } ^ { l }$ and the unlabeled set $x _ { t }$ . The synthetic gradient is produced by chaining the output of a gradient network into autodifferentiation, which yields a surrogate of the inaccessible true gradient. The optimization process is similar to the inner gradient descent in MAML, but it iterates on the unlabeled $x _ { t }$ rather than on the labeled $d _ { t } ^ { l }$ , since it does not rely on $y _ { t }$ to compute the true gradient. The labeled set for generating the model for an unseen task is now optional, which is only used to compute the initialization of model weights in our case. In summary, our main contributions are the following:
+
+1. In section 2 and section 3, we develop a novel empirical Bayes formulation with transduction for meta-learning. To perform amortized variational inference, we propose a parameterization for the variational posterior based on synthetic gradient descent, which incoporates the contextual information from all the inputs of the query set.
+
+2. In section 4, we show in theory that a transductive variational posterior yields better generalization performance. The generalization analysis is done through the connection between empirical Bayes formulation and a multitask extension of the information bottleneck principle. In light of this, we name our method synthetic information bottleneck (SIB).
+
+3. In section 5, we verify our proposal empirically. Our experimental results demonstrate that our method significantly outperforms the state-of-the-art meta-learning methods on few-shot classification benchmarks under the one-shot setting.
+
+![](images/a8c0136a8d5013b7eb6ec73954c0e41cc23cacc636d828c6e3ff83801f2330e5.jpg)  
+Figure 1: (a) The generative and inference processes of the empirical Bayes model are depicted in solid and dashed arrows respectively, where the meta-parameters are denoted by dashed circles due to the point estimates. A comparison between MAML (6) and our method (SIB) (10) is shown in $\mathbf { ( b ) }$ and (c). MAML is an inductive method since, for a task $t$ , it first constructs the variational posterior (with parameter $\theta ^ { K }$ ) as a function of the support set $d _ { t } ^ { l }$ , and then test on the unlabeled $x _ { t }$ ; while SIB uses a better variational posterior as a function of both $\dot { d } _ { t } ^ { l }$ and $x _ { t }$ : it starts from an initialization $\theta _ { t } ^ { 0 } ( d _ { t } ^ { l } )$ generated using $d _ { t } ^ { l }$ , and then yields $\theta _ { t } ^ { K }$ by running $K$ synthetic gradient steps on $x _ { t }$ .
+
+# 2 META-LEARNING WITH TRANSDUCTIVE INFERENCE
+
+The goal of meta-learning is to train a meta-model on a collection of tasks, such that it works well on another disjoint collection of tasks. Suppose that we are given a collection of $N$ tasks for training. The associated data is denoted by $\mathcal { D } : = \bar { \{ d _ { t } : = ( x _ { t } , y _ { t } ) \} } _ { t = 1 } ^ { N ^ { - } }$ . In the case of few-shot learning, we are given in addition a support set $d _ { t } ^ { \tilde { l } }$ in each task. In this section, we revisit the classical empirical Bayes model for meta-learning. Then, we propose to use a transductive scheme in the variational inference by implementing the variational posterior as a function of $x _ { t }$ .
+
+# 2.1 EMPIRICAL BAYES MODEL
+
+Due to the hierarchical structure among data, it is natural to consider a hierarchical Bayes model with the marginal likelihood
+
+$$
+p _ { f } ( \mathcal D ) = \int _ { \psi } p _ { f } ( \mathcal D | \psi ) p ( \psi ) = \int _ { \psi } \Big [ \prod _ { t = 1 } ^ { N } \int _ { w _ { t } } p _ { f } ( d _ { t } | w _ { t } ) p ( w _ { t } | \psi ) \Big ] p ( \psi ) .
+$$
+
+The generative process is illustrated in Figure 1 (a, in red arrows): first, a meta-parameter $\psi$ (i.e., hyper-parameter) is sampled from the hyper-prior $p ( \psi )$ ; then, for each task, a task-specific parameter $w _ { t }$ is sampled from the prior $p ( w _ { t } | \psi )$ ; finally, the dataset is drawn from the likelihood $p _ { f } ( d _ { t } | w _ { t } )$ . Without loss of generality, we assume the log-likelihood model factorizes as
+
+$$
+\begin{array} { l } { { \displaystyle \log p _ { f } ( d _ { t } | w _ { t } ) = \sum _ { i = 1 } ^ { n } \log p _ { f } ( y _ { t , i } | x _ { t , i } , w _ { t } ) + \log p ( x _ { t , i } | w _ { t } ) } , \ ~ } \\ { { \displaystyle = \sum _ { i = 1 } ^ { n } - \frac { 1 } { n } \ell _ { t } \big ( \hat { y } _ { t , i } ( f ( x _ { t , i } ) , w _ { t } ) , y _ { t , i } \big ) + \mathrm { c o n s t a n t } } . \ } \end{array}
+$$
+
+It is the setting advocated by Minka (2005), in which the generative model $p ( x _ { t , i } | w _ { t } )$ can be safely ignored since it is irrelevant to the prediction of $y _ { t }$ . To simplify the presentation, we still keep the notation $p _ { f } ( d _ { t } | w _ { t } )$ for the likelihood of the task $t$ and use $\ell _ { t }$ to specify the discriminative model, which is also referred to as the task-specific loss, e.g., the cross entropy loss. The first argument in $\ell _ { t }$ is the prediction, denoted by $\hat { y } _ { t , i } = \hat { y } _ { t , i } ( f ( x _ { t , i } ) , w _ { t } )$ , which depends on the feature representation $f ( x _ { t , i } )$ and the task-specific weight $w _ { t }$ .
+
+Note that rather than following a fully Bayesian approach, we leave some random variables to be estimated in a frequentist way, e.g., $f$ is a meta-parameter of the likelihood model shared by all tasks, for which we use a point estimate. As such, the posterior inference about these variables will be largely simplified. For the same reason, we derive the empirical Bayes (Robbins, 1985; Kucukelbir & Blei, 2014) by taking a point estimate on $\psi$ . The marginal likelihood
+
+now reads as
+
+$$
+p _ { \psi , f } ( \mathcal D ) = \prod _ { t = 1 } ^ { N } \int _ { w _ { t } } p _ { f } ( d _ { t } | w _ { t } ) p _ { \psi } ( w _ { t } ) .
+$$
+
+We highlight the meta-parameters as subscripts of the corresponding distributions to distinguish from random variables. Indeed, we are not the first to formulate meta-learning as empirical Bayes. The overall model formulation is essentially the same as the ones considered by Amit $\&$ Meir (2018); Grant et al. (2018); Ravi & Beatson (2018). Our contribution lies in the variational inference for empirical Bayes.
+
+# 2.2 AMORTIZED INFERENCE WITH TRANSDUCTION
+
+As in standard probabilistic modeling, we derive an evidence lower bound (ELBO) on the log version of (3) by introducing a variational distribution $q _ { \theta _ { t } } ( w _ { t } )$ for each task with parameter $\theta _ { t }$ :
+
+$$
+\log p _ { \psi , f } ( \mathcal { D } ) \geq \sum _ { t = 1 } ^ { N } \left[ \mathbb { E } _ { w _ { t } \sim q _ { \theta _ { t } } } \left[ \log p _ { f } ( d _ { t } | w _ { t } ) \right] - D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } } ( w _ { t } ) \| p _ { \psi } ( w _ { t } ) \big ) \right] .
+$$
+
+The variational inference amounts to maximizing the ELBO with respect to $\theta _ { 1 } , \ldots , \theta _ { N }$ , which together with the maximum likelihood estimation of the meta-parameters, we have the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { \psi , f } \operatorname* { m i n } _ { \theta _ { 1 } , \ldots , \theta _ { N } } \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \left[ \mathbb { E } _ { w _ { t } \sim q _ { \theta _ { t } } } \big [ - \log p _ { f } ( d _ { t } | w _ { t } ) \big ] + D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } } ( w _ { t } ) | | p _ { \psi } ( w _ { t } ) \big ) \right] .
+$$
+
+However, the optimization in (5), as $N$ increases, becomes more and more expensive in terms of the memory footprint and the computational cost. We therefore wish to bypass this heavy optimization and to take advantage of the fact that individual KL terms indeed share the same structure. To this end, instead of introducing $N$ different variational distributions, we consider a parameterized family of distributions in the form of $q _ { \phi ( \cdot ) }$ , which is defined implicitly by a deep neural network $\phi$ taking as input either $d _ { t } ^ { l }$ or $d _ { t } ^ { l }$ plus $x _ { t }$ , that is, $q _ { \phi ( d _ { t } ^ { l } ) }$ or $q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) }$ . Note that we cannot use entire $d _ { t }$ , since we do not have access to $y _ { t }$ during meta-testing. This amortization technique was first introduced in the case of variational autoencoders (Kingma & Welling, 2013; Rezende et al., 2014), and has been extended to Bayesian inference in the case of neural processes (Garnelo et al., 2018).
+
+Since $d _ { t } ^ { l }$ and $x _ { t }$ are disjoint, the inference scheme is inductive for a variational posterior $q _ { \phi ( d _ { t } ^ { l } ) }$ . As an example, MAML (Finn et al., 2017) takes $q _ { \phi ( d _ { t } ^ { l } ) }$ as the Dirac delta distribution, where $\phi ( d _ { t } ^ { l } ) = \theta _ { t } ^ { K }$ , is the $K$ -th iterate of the stochastic gradient descent
+
+$$
+\begin{array} { r } { \theta _ { t } ^ { k + 1 } = \theta _ { t } ^ { k } + \eta \nabla _ { \theta } \mathbb { E } _ { w _ { t } \sim q _ { \theta _ { t } ^ { k } } } \Big [ \log p ( d _ { t } ^ { l } | w _ { t } ) \Big ] \ \mathrm { w i t h } \ \theta _ { t } ^ { 0 } = \phi , \ \mathrm { a l e a r n a b l e ~ i n i t i a l i z a t i o n } . } \end{array}
+$$
+
+In this work, we consider a transductive inference scheme with variational posterior $q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) }$ . The inference process is shown in Figure 1(a, in green arrows). Replacing each $q _ { \theta _ { t } }$ in (5) by $q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) }$ , the optimization problem becomes
+
+$$
+\operatorname* { m i n } _ { \psi , f } \operatorname* { m i n } _ { \phi } \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \left[ \mathbb { E } _ { w _ { t } \sim q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) } } \left[ - \log p _ { f } ( d _ { t } | w _ { t } ) \right] + D _ { \mathrm { K L } } \big ( q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) } ( w _ { t } ) \| p _ { \psi } ( w _ { t } ) \big ) \right] .
+$$
+
+In a nutshell, the meta-model to be optimized includes the feature network $f$ , the hyper-parameter $\psi$ from the empirical Bayes formulation and the amortization network $\phi$ from the variational inference.
+
+# 3 UNROLLING EXACT INFERENCE WITH SYNTHETIC GRADIENTS
+
+It is however non-trivial to design a proper network architecture for $\phi ( d _ { t } ^ { l } , x _ { t } )$ , since $d _ { t } ^ { l }$ and $x _ { t }$ are both sets. The strategy adopted by neural processes (Garnelo et al., 2018) is to aggregate the information from all individual examples via an averaging function. However, as pointed out by Kim et al.
+
+(2019), such a strategy tends to underfit $x _ { t }$ because the aggregation does not necessarily attain the most relevant information for identifying the task-specific parameter. Extensions, such as attentive neural process (Kim et al., 2019) and set transformer (Lee et al., 2019a), may alleviate this issue but come at a price of $O ( n ^ { 2 } )$ time complexity. We instead design $\phi ( d _ { t } ^ { l } , x _ { t } )$ to mimic the exact inference a $\begin{array} { r } { \mathrm { { \cdot g } m i n } _ { \theta _ { t } } D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } } ( w _ { t } ) \| p _ { \psi , f } ( w _ { t } | d _ { t } ) \big ) } \end{array}$ by parameterizing the optimization process with respect to $\theta _ { t }$ . More specifically, consider the gradient descent on $\theta _ { t }$ with step size $\eta$ :
+
+$$
+\begin{array} { r } { \theta _ { t } ^ { k + 1 } = \theta _ { t } ^ { k } - \eta \nabla _ { \theta _ { t } } D _ { \mathrm { K L } } \Big ( q _ { \theta _ { t } ^ { k } } ( w ) \| p _ { \psi , f } ( w | d _ { t } ) \Big ) . } \end{array}
+$$
+
+We would like to unroll this optimization dynamics up to the $K$ -th step such that ${ \theta } _ { t } ^ { K } = \phi ( d _ { t } ^ { l } , x _ { t } )$ while make sure that $\theta _ { t } ^ { K }$ is a good approximation to the optimum $\theta _ { t } ^ { \star }$ , which consists of parameterizing
+
+By doing so, $\theta _ { t } ^ { K }$ becomes a function of $\phi , \psi$ and ${ x _ { t } } ^ { 2 }$ , we therefore realize $q _ { \phi ( d _ { t } ^ { l } , x _ { t } ) }$ as $q _ { \theta _ { t } ^ { K } }$
+
+For (a), we opt to either let $\theta _ { t } ^ { 0 } = \lambda$ to be a global data-independent initialization as in MAML (Finn et al., 2017) or let $\theta _ { t } ^ { 0 } = \mathsf { \bar { \lambda } } ( d _ { t } ^ { l } )$ with a few supervisions from the support set, where $\lambda$ can be implemented by a permutation invariant network described in Gidaris & Komodakis (2018). In the second case, the features of the support set will be first averaged in terms of their labels and then scaled by a learnable vector of the same size.
+
+For (b), the fundamental reason that we parameterize the gradient is because we do not have access to $y _ { t }$ during the meta-testing phase, although we are able to follow (8) in meta-training to obtain ${ q _ { \theta _ { t } ^ { \star } } ( w _ { t } ) \propto p _ { f } ( d _ { t } | w _ { t } ) p _ { \psi } ( w _ { t } ) }$ . To make a consistent parameterization in both meta-training and meta-testing, we thus do not touch $y _ { t }$ when constructing the variational posterior. Recall that the true gradient decomposes as
+
+$$
+\nabla _ { \theta _ { t } } D _ { \mathrm { K L } } \Big ( q _ { \theta _ { t } } \| p _ { \psi , f } \Big ) = \mathbb { E } _ { \epsilon } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \partial \ell _ { t } ( \hat { y } _ { t , i } , y _ { t , i } ) } { \partial \hat { y } _ { t , i } } \frac { \partial \hat { y } _ { t , i } } { \partial w _ { t } } \frac { \partial w _ { t } ( \theta _ { t } , \epsilon ) } { \partial \theta _ { t } } \Big ] + \nabla _ { \theta _ { t } } D _ { \mathrm { K L } } \Big ( q _ { \theta _ { t } } \| p _ { \psi } \Big )
+$$
+
+under a reparameterization $w _ { t } = w _ { t } ( \theta _ { t } , \epsilon )$ with $\epsilon \sim p ( \epsilon )$ , where all the terms can be computed without $y _ { t }$ except for ∂\`t∂yˆ . Thus, we introduce a deep neural network ξ(ˆyt,i) to synthesize it. The idea of synthetic gradients was originally proposed by Jaderberg et al. (2017) to parallelize the back-propagation. Here, the purpose of $\xi ( \hat { y } _ { t , i } )$ is to update $\theta _ { t }$ regardless of the groundtruth labels, which is slightly different from its original purpose. Besides, we do not introduce an additional loss between $\xi ( \hat { y } _ { t , i } )$ and $\frac { \partial \ell _ { t } } { \partial \hat { y } _ { t , i } }$ since $\xi ( \hat { y } _ { t , i } )$ will be driven by the objective in (7). As an intermediate computation, the synthetic gradient is not necessarily a good approximation to the true gradient.
+
+To sum up, we have derived a particular implementation of $\phi ( d _ { t } ^ { l } , x _ { t } )$ by parameterizing the exact inference update, namely (8), without using the labels of the query set, where the meta-model $\phi$ includes an initialization network $\lambda$ and a synthetic gradient network $\xi$ , such that $\phi ( x _ { t } ) = \theta _ { t } ^ { K }$ , the $K$ -th iterate of the following update:
+
+$$
+\theta _ { t } ^ { k + 1 } = \theta _ { t } ^ { k } - \eta \left[ \mathbb { E } _ { \epsilon } \Big [ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi ( \hat { y } _ { t , i } ) \frac { \partial \hat { y } _ { t , i } } { \partial w _ { t } } \frac { \partial w _ { t } ( \theta _ { t } ^ { k } , \epsilon ) } { \partial \theta _ { t } } \Big ] + \nabla _ { \theta _ { t } } D _ { \mathrm { K L } } \Big ( q _ { \theta _ { t } ^ { k } } \| p _ { \psi } \Big ) \right] .
+$$
+
+The overall algorithm is depicted in Algorithm 1. We also make a side-by-side comparison with MAML shown in Figure 1. Rather than viewing (10) as an optimization process, it may be more precise to think of it as a part of the computation graph created in the forward-propagation. The computation graph of the amortized inference is shown in Figure 2,
+
+As an extension, if we were deciding to estimate the feature network $f$ in a Bayesian manner, we would have to compute higher-order gradients as in the case of MAML. This is inpractical from a computational point of view and needs technical simplifications (Nichol et al., 2018). By introducing a series of synthetic gradient networks in a way similar to Jaderberg et al. (2017), the computation will be decoupled into computations within each layer, and thus becomes more feasible. We see this as a potential advantage of our method and leave this to our future work3.
+
+![](images/0217cb2fb1d7515b7ec763b17a8d50f1068c924386793ffb7d007f26c9e4c62d.jpg)  
+Figure 2: The computation graph to compute the negative ELBO, where the input and output of the synthetic gradient module are highlighted in red. The detach() is used to stop the back-propagation down to the feature network. Note that we do not include every computation for simplicity.
+
+Algorithm 1 Variational inference with synthetic gradients for empirical Bayes
+
+1: Input: the dataset $\mathcal { D }$ ; the step size $\eta$ ; the number of inner iterations $K$ ; pretrained $f$ .   
+2: Initialize the meta-models $\psi$ , and $\phi = ( \lambda , \xi )$ .   
+3: while not converged do   
+4: Sample a task $t$ and the associated query set $d _ { t }$ (plus optionally the support set $d _ { t } ^ { l } ,$ ).   
+5: Compute the initialization $\theta _ { t } ^ { 0 } = \lambda$ or $\theta _ { t } ^ { 0 } = \lambda ( d _ { t } ^ { l } )$ .   
+6: for $k = 1 , \ldots , K$ do   
+7: Compute $\theta _ { t } ^ { k }$ via (10).   
+8: end for   
+9: Compute $w _ { t } = w _ { t } ( \theta _ { t } ^ { K } , \epsilon )$ with $\epsilon \sim p ( \epsilon )$ .   
+10: Update $\psi  \psi - \eta \nabla _ { \psi } D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } ^ { K } ( \psi ) } \| p _ { \psi } \big )$ .   
+11: Update $\phi  \phi - \eta \nabla _ { \phi } D _ { \mathrm { K L } } ( q _ { \phi ( x _ { t } ) } | | p _ { f } \cdot p _ { \psi } )$ .   
+12: Optionally, update $f \gets f + \eta \nabla _ { f } \log p _ { f } ( d _ { t } | w _ { t } )$ .   
+13: end while
+
+# 4 GENERALIZATION ANALYSIS OF EMPIRICAL BAYES VIA THE CONNECTION TO INFORMATION BOTTLENECK
+
+The learning of empirical Bayes (EB) models follows the frequentist’s approach, therefore, we can use frequentist’s tool to analyze the model. In this section, we study the generalization ability of the empirical Bayes model through its connection to a variant of the information bottleneck principle Achille & Soatto (2017); Tishby et al. (2000).
+
+Abstract form of empirical Bayes From (3), we see that the empirical Bayes model implies a simpler joint distribution since
+
+$$
+\log p _ { \psi , f } ( w _ { 1 } , \dots , w _ { N } , \mathcal { D } ) = \sum _ { t = 1 } ^ { N } \log p _ { f } ( d _ { t } | w _ { t } ) + \log p _ { \psi } ( w _ { t } ) ,
+$$
+
+which is equal to the log-density of $N$ iid samples drawn from the joint distribution
+
+$$
+p ( w , d , t ) \equiv p _ { \psi , f } ( w , d , t ) = p _ { f } ( d | w , t ) p _ { \psi , f } ( w ) p ( t ) ^ { 4 }
+$$
+
+up to a constant if we introduce a random variable to represent the task and assume $p ( t )$ is an uniform distribution. We thus see that this joint distribution embodies the generative process of empirical Bayes. Correspondingly, there is another graphical model of the joint distribution characterizes the
+
+inference process of the empirical Bayes:
+
+$$
+q ( w , d , t ) \equiv q _ { \phi } ( w , d , t ) = q _ { \phi } ( w | d , t ) q ( d | t ) q ( t ) ,
+$$
+
+where $q _ { \phi } ( w | d , t )$ is the abstract form of the variational posterior with amortization, includes both the inductive form and the transductive form. The coupling between $p ( w , d , t )$ and $\scriptstyle q ( w , d , t )$ is due to $p ( t ) \equiv q ( t )$ as we only have access to tasks through sampling.
+
+We are interested in the case that the number of tasks $N \to \infty$ , such as the few-shot learning paradigm proposed by Vinyals et al. (2016), in which the objective of (7) can be rewritten in an abstract form of
+
+$$
+\begin{array} { r } { \mathbb E _ { q ( t ) } \mathbb E _ { q ( d | t ) } \Big [ \mathbb E _ { q ( w | d , t ) } \big [ - \log p ( d | w , t ) \big ] + D _ { \mathrm { K L } } \big ( q ( w | d , t ) \| p ( w ) \big ) \Big ] . } \end{array}
+$$
+
+In fact, optimizing this objective is the same as optimizing (7) from a stochastic gradient descent point of view.
+
+The learning of empirical Bayes with amortized variational inference can be understood as a variational EM in the sense that the $\mathrm { E }$ -step amounts to aligning $q ( w | d , t )$ with $p ( w | d , t )$ while the M-step amounts to adjusting the likelihood $\bar { p } ( d | w , t )$ and the prior $p ( w )$ .
+
+Connection to information bottleneck The following theorem shows the connection between (14) and the information bottleneck principle.
+
+Theorem 1. Given distributions $q ( w | d , t ) , q ( d | t ) , q ( t ) , p ( w )$ and $p ( d | w , t )$ , we have
+
+$$
+\begin{array} { r } { ( 1 4 ) \geq I _ { q } ( w ; d | t ) + H _ { q } ( d | w , t ) , } \end{array}
+$$
+
+where $I _ { q } ( w ; d | t ) : = D _ { K L } \big ( q ( w , d | t ) | | q ( w | t ) q ( d | t ) \big )$ is the conditional mutual information and $H _ { q } ( w | d , t ) : = \mathbb { E } _ { q ( w , d , t ) } [ - \log q ( w | d , t ) ]$ is the conditional entropy. The equality holds when
+
+$$
+\forall t \colon D _ { K L } ( q ( w | t ) \| p ( w ) ) = 0 a n d D _ { K L } ( q ( d | w , t ) \| p ( d | w , t ) ) = 0 .
+$$
+
+In fact, the lower bound on (14) is an extention of the information bottleneck principle (Achille & Soatto, 2017) under the multi-task setting, which, together with the synthetic gradient based variational posterior, inspire the name synthetic information bottleneck of our method. The tightness of the lower bound depends on both the parameterizations of $p _ { f } ( d | w , t )$ and $p _ { \psi } ( w )$ as well as the optimization of (14). It thus can be understood as how well we can align the inference process with the generative process. From an inference process point of view, for a given $q ( w | d , t )$ , the optimal likelihood and prior have been determined. In theory, we only need to find the optimal $q ( w | d , t )$ using the information bottleneck in (19). However, in practice, minimizing (14) is more straightforward.
+
+Generalization of empirical Bayes meta-learning The connection to information bottleneck suggests that we can eliminate $p ( d | w , t )$ and $p ( w )$ from the generalization analysis of empirical Bayes meta-learning and define the generalization error by $\scriptstyle q ( w , d , t )$ only. To this end, we first identify the empirical risk for a single task $t$ with respect to particular weights $w$ and dataset $d$ as
+
+$$
+L _ { t } ( w , d ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell _ { t } ( \hat { y } _ { i } ( f ( x _ { i } ) , w ) , y _ { i } ) .
+$$
+
+The true risk for task $t$ with respect to $w$ is then the expected empirical risk $\mathbb { E } _ { d \sim q ( d | t ) } L _ { t } ( w , d )$ . Now, we define the generalization error with respect to $\scriptstyle q ( w , d , t )$ as the average of the difference between the true risk and the empirical risk over all possible $t , d , w$ :
+
+$$
+\begin{array} { r l } & { \mathrm { g e n } ( q ) : = {  { \mathbb E } } _ { q ( t ) q ( d | t ) q ( w | d , t ) } \biggl [ {  { \mathbb E } } _ { d \sim q ( d | t ) } L _ { t } ( w , d ) - L _ { t } ( w , d ) \biggr ] } \\ & { \qquad = {  { \mathbb E } } _ { q ( t ) q ( d | t ) q ( w | t ) } L _ { t } ( w , d ) - {  { \mathbb E } } _ { q ( t ) q ( d | t ) q ( w | d , t ) } L _ { t } ( w , d ) , } \end{array}
+$$
+
+where $q ( w | t )$ is the aggregated posterior of task $t$ .
+
+Next, we extend the result from $\mathrm { X u }$ & Raginsky (2017) and derive a data-dependent upper bound for $\mathrm { g e n } ( q )$ using mutual information.
+
+Theorem 2. Denote by $z = ( x , y )$ . If $\ell _ { t } ( \hat { y } _ { i } ( f ( x _ { i } ) , w ) , y _ { i } ) \equiv \ell _ { t } ( w , z _ { i } )$ is $\sigma$ -subgaussian under $q ( w | t ) q ( z | t )$ , then $L _ { t } ( w , d )$ is $\sigma / { \sqrt { n } }$ -subgaussian under $q ( w | t ) q ( d | t )$ due to the iid assumption, and
+
+$$
+| g e n ( q ) \bigr | \leq \sqrt { \frac { 2 \sigma ^ { 2 } } { n } I _ { q } ( w ; d | t ) } .
+$$
+
+Plugging this back to Theorem 1, we obtain a different interpretation for the empirical Bayes ELBO.
+
+Corollary 1. If $\ell _ { t }$ is chosen to be the negative log-likelihood, minimizing the population objective of empirical Bayes meta-learning amounts to minimizing both the expected generalization error and the expected empirical risk:
+
+$$
+( 1 4 ) \geq \frac { n } { 2 \sigma ^ { 2 } } g e n ( q ) ^ { 2 } + \mathbb { E } _ { q ( t ) q ( d | t ) q ( w | d , t ) } L _ { t } ( w , d ) .
+$$
+
+The Corollary 1 suggests that (14) amounts to minimizing a regularized empirical risk minimization. In general, there is a tradeoff between the generalization error and the empirical risk controlled by the coefficient $\textstyle { \frac { n } { 2 \sigma ^ { 2 } } }$ , where $n = | d |$ is the cardinality of $d$ . If $n$ is small, then we are in the overfitting regime. This is the case of the inductive inference with variational posterior $q ( w | d ^ { l } , t )$ , where the support set $d ^ { l }$ is fairly small by the definition of few-shot learning. On the other hand, if we were following the transductive setting, we expect to achieve a small generalization error since the implemented variational posterior is a better approximation to $q ( w | d , t )$ . However, keeping increasing $n$ will potentially over-regularize the model and thus yield negative effect. An empirical study on varying $n$ can be found in Table 5 in Appendix D.
+
+# 5 EXPERIMENTS
+
+In this section, we first validate our method on few-shot learning, and then on zero-shot learning (no support set and no class description are available). Note that many meta-learning methods cannot do zero-shot learning since they rely on the support set.
+
+# 5.1 FEW-SHOT CLASSIFICATION
+
+We compare SIB with state-of-the-art methods on few-shot classification problems. Our code is available at https://github.com/amzn/xfer.
+
+# 5.1.1 SETUP
+
+Datasets We choose standard benchmarks of few-shot classification for this experiment. Each benchmark is composed of disjoint training, validation and testing classes. MiniImageNet is proposed by Vinyals et al. (2016), which contains 100 classes, split into 64 training classes, 16 validation classes and 20 testing classes; each image is of size $8 4 \times 8 4$ . CIFAR-FS is proposed by Bertinetto et al. (2018), which is created by dividing the original CIFAR-100 into 64 training classes, 16 validation classes and 20 testing classes; each image is of size $3 2 \times 3 2$ .
+
+Evaluation metrics In few-shot classification, a task (aka episode) $t$ consists of a query set $d _ { t }$ and a support set $d _ { t } ^ { l }$ . When we say the task $t$ is $k$ -way- $. n ^ { l }$ -shot we mean that $d _ { t } ^ { l }$ is formed by first sampling $k$ classes from a pool of classes; then, for each sampled class, $n ^ { l }$ examples are drawn and a new label taken from $\{ 0 , \ldots , k - 1 \}$ is assigned to these examples. By default, each query set contains $1 5 k$ examples. The goal of this problem is to predict the labels of the query set, which are provided as ground truth during training. The evaluation is the average classification accuracy over tasks.
+
+Network architectures Following Gidaris & Komodakis (2018); Qiao et al. (2018); Gidaris et al. (2019), we implement $f$ by a 4-layer convolutional network (Conv-4-64 or ${ \bf C o n v - } 4 { \bf - } 1 2 8 ^ { 5 } ,$ or a WideResNet (WRN-28-10) (Zagoruyko & Komodakis, 2016). We pretrain the feature network $f ( \cdot )$ on the 64 training classes for a stardard 64-way classification. We reuse the feature averaging network proposed by Gidaris & Komodakis (2018) as our initialization network $\lambda ( \cdot )$ , which basically averages the feature vectors of all data points from the same class and then scales each feature dimension differently by a learned coefficient. For the synthetic gradient network $\xi ( \cdot )$ , we implement a three-layer MLP with hidden-layer size $8 k$ . Finally, for the predictor $\hat { y } _ { i j } ( \cdot , w _ { i } )$ , we adopt the cosine-similarity based classifier advocated by Chen et al. (2019) and Gidaris $\&$ Komodakis (2018).
+
+Table 2: Average classification accuracies (with $9 5 \%$ confidence intervals) on the test-set of MiniImageNet and CIFAR-FS. For evaluation, we sample 2000 and 5000 episodes respectively for MiniImageNet and CIFAR-FS and test three different architectures as the feature extractor: Conv-4- 64, Conv-4-128 and WRN-28-10. We train SIB with learning rate 0.001 and try different numbers of synthetic gradient steps $K$ .   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="2">MiniImageNet, 5-way</td><td colspan="2">CIFAR-FS,5-way</td></tr><tr><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>Matching Net (Vinyals et al.,2016)</td><td>Conv-4-64</td><td>44.2%</td><td>57%</td><td>1</td><td>一</td></tr><tr><td>MAML (Finn et ai., 2017)</td><td>Conv-4-64</td><td>48.7±1.8%</td><td>63.1±0.9%</td><td>58.9±1.9%</td><td>71.5±1.0%</td></tr><tr><td>Prototypical Net (Snell et al., 2017)</td><td>Conv-4-64</td><td>49.4±0.8%</td><td>68.2±0.7%</td><td>55.5±0.7%</td><td>72.0±0.6%</td></tr><tr><td>Relation Net (Sung et al.,2018)</td><td>Conv-4-64</td><td>50.4±0.8%</td><td>65.3±0.7%</td><td>55.0±1.0%</td><td>69.3±0.8%</td></tr><tr><td>GNN (Satorras &amp; Bruna,2017)</td><td>Conv-4-64</td><td>50.3%</td><td>66.4%</td><td>61.9%</td><td>75.3%</td></tr><tr><td>R2-D2 (Bertinetto et al.,2018)</td><td>Conv-4-64</td><td>49.5±0.2%</td><td>65.4±0.2%</td><td>62.3±0.2%</td><td>77.4±0.2%</td></tr><tr><td>TPN (Liu et al., 2018)</td><td>Conv-4-64</td><td>55.5%</td><td>69.9%</td><td></td><td></td></tr><tr><td>Gidaris et al. (2019)</td><td>Conv-4-64</td><td>54.8±0.4%</td><td>71.9±0.3%</td><td>63.5±0.3%</td><td>79.8±0.2%</td></tr><tr><td>SIBK=O (Pre-trained feature)</td><td>Conv-4-64</td><td>50.0±0.4%</td><td>67.0±0.4%</td><td>59.2±0.5%</td><td>75.4±0.4%</td></tr><tr><td>SIB n=le-3,K=3</td><td>Conv-4-64</td><td>58.0±0.6%</td><td>70.7±0.4%</td><td>68.7±0.6%</td><td>77.1±0.4%</td></tr><tr><td>SIB n=1e-3,K=0</td><td>Conv-4-128</td><td>53.62 ± 0.79%</td><td>71.48 ± 0.64%</td><td>一</td><td>一</td></tr><tr><td>SIB n=le-3, K=1</td><td>Conv-4-128</td><td>58.74 ± 0.89%</td><td>74.12 ± 0.63%</td><td></td><td></td></tr><tr><td>SIB n=le-3, K=3</td><td>Conv-4-128</td><td>62.59 ± 1.02%</td><td>75.43 ± 0.67%</td><td>一</td><td></td></tr><tr><td> SIB n=le-3, K=5</td><td>Conv-4-128</td><td>63.26 ± 1.07 %</td><td>75.73 ± 0.71%</td><td>一</td><td></td></tr><tr><td>TADAM (Oreshkin et al.,2018)</td><td>ResNet-12</td><td>58.5±0.3%</td><td>76.7±0.3%</td><td></td><td>一</td></tr><tr><td>SNAIL (Santoro et al., 2017)</td><td>ResNet-12</td><td>55.7±1.0%</td><td>68.9±0.9%</td><td></td><td></td></tr><tr><td>MetaOptNet-RR( (Lee etal., 2019b)</td><td>ResNet-12</td><td>61.4±0.6%</td><td>77.9±0.5%</td><td>72.6±0.7%</td><td>84.3±0.5%</td></tr><tr><td>MetaOptNet-SVM</td><td>ResNet-12</td><td>62.6±0.6%</td><td>78.6±0.5%</td><td>72.0±0.7%</td><td>84.2±0.5%</td></tr><tr><td>CTM (Li et al., 2019)</td><td>ResNet-18</td><td>64.1±0.8%</td><td>80.5±0.1%</td><td>二</td><td>1</td></tr><tr><td>Qiao et al. (2018)</td><td>WRN-28-10</td><td>59.6±0.4%</td><td>73.7±0.2%</td><td></td><td>二</td></tr><tr><td>LEO (Rusu et al., 2019)</td><td>WRN-28-10</td><td>61.8±0.1%</td><td>77.6±0.1%</td><td></td><td></td></tr><tr><td>Gidaris et al. (2019)</td><td>WRN-28-10</td><td>62.9±0.5%</td><td>79.9±0.3%</td><td>73.6±0.3%</td><td>86.1±0.2%</td></tr><tr><td>SIBK=O (Pre-trained feature)</td><td>WRN-28-10</td><td>60.6±0.4%</td><td>77.5±0.3%</td><td>70.0±0.5%</td><td>83.5±0.4%</td></tr><tr><td>SIB n=le-3,K=1</td><td>WRN-28-10</td><td>67.3±0.5%</td><td>78.8±0.4%</td><td>76.8±0.5%</td><td>84.9±0.4%</td></tr><tr><td>SIB n=le-3, K=3</td><td>WRN-28-10</td><td>69.6±0.6 %</td><td>78.9±0.4%</td><td>78.4±0.6%</td><td>85.3±0.4%</td></tr><tr><td> SIB n=le-3, K=5</td><td>WRN-28-10</td><td>70.0±0.6%</td><td>79.2±0.4%</td><td>80.0±0.6%</td><td>85.3±0.4%</td></tr></table>
+
+Training details We run SGD with batch size 8 for 40000 steps, where the learning rate is fixed to $1 0 ^ { - 3 }$ . During training, we freeze the feature network. To select the best hyper-parameters, we sample 1000 tasks from the validation classes and reuse them at each training epoch.
+
+# 5.1.2 COMPARISON TO STATE-OF-THE-ART META-LEARNING METHODS
+
+In Table 2, we show a comparison between the state-of-the-art approaches and several variants of our method (varying $K$ or $f ( \cdot ) )$ . Apart from SIB, TPN (Liu et al., 2018) and CTM (Li et al., 2019) are also transductive methods.
+
+First of all, comparing SIB ( $K = 3$ ) to SIB $K = 0$ ), we observe a clear improvement, which suggests that, by taking a few synthetic gradient steps, we do obtain a better variational posterior as promised. For 1-shot learning, SIB (when $K = 3$ or $K = 5$ ) significantly outperforms previous methods on both MiniImageNet and CIFAR-FS. For 5-shot learning, the results are also comparable. It should be noted that the performance boost is consistently observed with different feature networks, which suggests that SIB is a general method for few-shot learning.
+
+However, we also observe a potential limitation of SIB: when the support set is relatively large, e.g., 5-shot, with a good feature network like WRN-28-10, the initialization $\theta _ { t } ^ { 0 }$ may already be close to some local minimum, making the updates later less important.
+
+For 5-shot learning, SIB is sligtly worse than CTM (Li et al., 2019) and/or Gidaris et al. (2019). CMT (Li et al., 2019) can be seen as an alternative way to incorporate transduction – it measures the similarity between a query example and the support set while making use of intra- and inter-class relationships. Gidaris et al. (2019) uses in addition the self-supervision as an auxilary loss to learn a richer and more transferable feature model. Both ideas are complementary to SIB. We leave these extensions to our future work.
+
+![](images/6bdd85c7eb813de96c66a4f1984fc780a0c74ca167b0d20c0776930236c724bb.jpg)  
+Figure 3: Left: the mean-square errors on $D _ { \mathrm { t e s t } }$ , $\mathbb { E } _ { t } D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } ^ { K } } ( w _ { t } ) \| p ( w _ { t } | d _ { t } ) \big )$ , $D _ { \mathrm { K L } } ( p _ { \psi } ( w ) \Vert p ( w ) )$ and the estimate of $I ( w ; d ) \approx \mathbb { E } _ { t } D _ { \mathrm { K L } } \big ( q _ { \theta _ { t } ^ { K } } ( w _ { t } ) \| p _ { \psi } ( w _ { t } ) \big )$ . Middle: the predicted $y$ ’s by $y = \theta _ { t } ^ { k } x$ for $k = 0 , \ldots , 4$ . Right: the predictions of SIB.
+
+Since our variational posterior relies only on $x _ { t }$ , SIB is also applicable to zero-shot problems (i.e., no support set available). We first look at a toy multi-task problem, where $I ( w _ { t } ; d _ { t } )$ is tractable.
+
+Denote by $D _ { \mathrm { t r a i n } } : = \{ d _ { t } \} _ { t = 1 } ^ { N }$ the train set, which consists of datasets of size $n$ : $d = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } ,$ . We construct a dataset $d$ by firstly sampling iid Gaussian random variables as inputs: $x _ { i } \sim \mathcal N ( \mu , \sigma ^ { 2 } )$ . Then, we generate the weight for each dataset by calculating the mean of the inputs and shifting with a Gaussian random variable $\epsilon _ { w }$ : $\begin{array} { r } { w = \frac { 1 } { n } \sum _ { i } ^ { } x _ { i } + \epsilon _ { w } } \end{array}$ , $\epsilon _ { w } \sim \mathcal { N } ( \mu _ { w } , \sigma _ { w } ^ { 2 } )$ . The output for $x _ { i }$ is $y _ { i } = w \cdot x _ { i }$ . We decide ahead of time the hyperparameters $\mu , \sigma , \mu _ { w } , \sigma _ { w }$ for generating $x _ { i }$ and $y _ { i }$ . Recall that a weighted sum of iid Gaussian random variables is still a Gaussian random variable. Specifically, if $\begin{array} { r } { w = \sum _ { i } { c _ { i } x _ { i } } } \end{array}$ and $x _ { i } \sim \mathcal N ( \mu _ { i } , \sigma _ { i } ^ { 2 } )$ , then $\begin{array} { r } { w \sim \mathcal { N } ( \sum _ { i } c _ { i } \mu _ { i } , \sum _ { i } c _ { i } ^ { 2 } \sigma _ { i } ^ { 2 } ) } \end{array}$ . Therefore, we have $\begin{array} { r } { p ( w ) = \mathcal { N } ( \mu + \mu _ { w } , \frac { 1 } { n } \sigma ^ { 2 } + \sigma _ { w } ^ { 2 } ) } \end{array}$ i  i. On the other hand, if we are given a dataset $d$ of size $n$ , the only uncertainty about $w$ comes from $\epsilon _ { w }$ , that is, we should consider $x _ { i }$ as a constant given $d$ . Therefore, the posterior $\begin{array} { r } { p ( w | d ) = \mathcal { N } ( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i } + \mu _ { w } , \sigma _ { w } ^ { 2 } ) } \end{array}$ . We use a simple implementation for SIB: The variational posterior is realized by
+
+$$
+q _ { \theta _ { t } ^ { K } } ( w ) = \mathcal { N } ( \theta _ { t } ^ { K } , \sigma _ { w } ) , \ \theta _ { t } ^ { k + 1 } = \theta _ { t } ^ { k } - 1 0 ^ { - 3 } \sum _ { i = 1 } ^ { n } x _ { i } \xi ( \theta _ { t } ^ { k } x _ { i } ) , \ \mathrm { a n d } \ \theta _ { t } ^ { 0 } = \lambda \in \mathbb { R } ;
+$$
+
+$\ell _ { t }$ is a mean squared error, implies that $p ( y | x , w ) = \mathcal { N } ( w x , 1 )$ ; $p _ { \psi } ( w )$ is a Gaussian distribution with parameters $\psi \in \mathbb { R } ^ { 2 }$ ; The synthetic gradient network $\xi$ is a three-layer MLP with hidden size 8.
+
+In the experiment, we sample 240 tasks respectively for both $D _ { \mathrm { t r a i n } }$ and $D _ { \mathrm { t e s t } }$ . We learn SIB and BNN on $D _ { \mathrm { t r a i n } }$ for 150 epochs using the ADAM optimizer (Kingma & Ba, 2014), with learning rate $1 0 ^ { - 3 }$ and batch size 8. Other hyperparameters are specified as follows: $n = 3 2 , K = 3 , \mu = 0 , \sigma =$ $1 , \mu _ { w } = 1 , \sigma _ { w } = 0 . 1$ . The results are shown in Figure 3. On the left, both $D _ { \mathrm { K L } } ( q _ { \theta _ { t } ^ { K } } ( w _ { t } ) \lVert p ( w _ { t } | d _ { t } ) )$ and $D _ { \mathrm { K L } } ( p _ { \psi } ( w ) \Vert p ( w ) )$ are close to zero indicating the success of the learning. More interestingly, in the middle, we see that $\theta _ { t } ^ { 0 } , \theta _ { t } ^ { 1 } , \ldots , \theta _ { t } ^ { 4 }$ evolves gradually towards the ground truth, which suggests that the synthetic gradient network is able to identify the descent direction after meta-learning.
+
+# 6 CONCLUSION
+
+We have presented an empirical Bayesian framework for meta-learning. To enable an efficient variational inference, we followed the amortized inference paradigm, and proposed to use a transductive scheme for constructing the variational posterior. To implement the transductive inference, we make use of two neural networks: a synthetic gradient network and an initialization network, which together enables a synthetic gradient descent on the unlabeled data to generate the parameters of the amortized variational posterior dynamically. We have studied the theoretical properties of the proposed framework and shown that it yields performance boost on MiniImageNet and CIFAR-FS for few-shot classification.
+
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+Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000.
+
+Oriol Vinyals, Charles Blundell, Timothy Lillicrap, koray kavukcuoglu, and Daan Wierstra. Matching networks for one shot learning. In Advances in Neural Information Processing Systems 29, pp. 3630–3638. Curran Associates, Inc., 2016.
+
+Aolin Xu. Information-theoretic limitations of distributed information processing. PhD thesis, University of Illinois at Urbana-Champaign, 2016.
+
+Aolin Xu and Maxim Raginsky. Information-theoretic analysis of generalization capability of learning algorithms. In Advances in Neural Information Processing Systems, pp. 2524–2533, 2017.
+
+Jiaolong Xu, Liang Xiao, and Antonio M López. Self-supervised domain adaptation for computer vision tasks. IEEE Access, 7:156694–156706, 2019.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
+
+Xiaojin Zhu, Zoubin Ghahramani, and John D Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In Proceedings of the 20th International conference on Machine learning (ICML-03), pp. 912–919, 2003.
+
+# APPENDIX
+
+A PROOFS
+
+Theorem 1. Given distributions $q ( w | d , t ) , q ( d | t ) , q ( t ) , p ( w )$ and $p ( d | w , t )$ , we have
+
+$$
+\begin{array} { r } { ( 1 4 ) \geq I _ { q } ( w ; d | t ) + H _ { q } ( d | w , t ) , } \end{array}
+$$
+
+where $I _ { q } ( w ; d | t ) : = D _ { K L } \big ( q ( w , d | t ) | | q ( w | t ) q ( d | t ) \big )$ is the conditional mutual information and $H _ { q } ( w | d , t ) : = \mathbb { E } _ { q ( w , d , t ) } [ - \log q ( w | d , t ) ]$ is the conditional entropy. The equality holds when
+
+$$
+\forall t \colon D _ { K L } ( q ( w | t ) \| p ( w ) ) = 0 a n d D _ { K L } ( q ( d | w , t ) \| p ( d | w , t ) ) = 0 .
+$$
+
+Proof. Denote by $q ( w | t ) : = \mathbb { E } _ { q ( d | t ) } q ( w | d , t ) q ( d | t )$ the aggregated posterior of task $t$ . (14) can be decomposed as
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { q ( t ) } \mathbb { E } _ { q ( d | t ) } \Big [ \mathbb { E } _ { q ( w | d , t ) } \big [ - \log p ( d | w , t ) \big ] + D _ { \mathrm { K L } } \big ( q ( w | d , t ) | | p ( w ) \big ) \Big ] } \\ & { = \mathbb { E } _ { q ( t ) } \mathbb { E } _ { q ( d | t ) } \mathbb { E } _ { q ( w | d , t ) } \Big [ \log \frac { q ( w | d , t ) q ( w | t ) } { p ( d | w , t ) p ( w ) q ( w | t ) } \Big ] } \\ & { = \mathbb { E } _ { q ( t ) } \mathbb { E } _ { q ( d | t ) } \mathbb { E } _ { q ( w | d , t ) } \Big [ \log \frac { q ( w | d , t ) } { q ( w | t ) } \Big ] + \mathbb { E } _ { q ( t ) } \mathbb { E } _ { q ( d | t ) } \mathbb { E } _ { q ( w | d , t ) } \Big [ - \log p ( d | w , t ) \Big ] } \\ & { \quad + \mathbb { E } _ { q ( t ) } \mathbb { E } _ { q ( d | t ) } \mathbb { E } _ { q ( w | d , t ) } \Big [ \log \frac { q ( w | t ) } { p ( w ) } \Big ] } \\ & { = I _ { q } ( w ; d | t ) + H _ { q , p } ( d | w , t ) + \mathbb { E } _ { q ( t ) } D _ { \mathrm { K L } } ( q ( w | t ) | p ( w ) ) } \\ & { > I _ { c } ( w ; d | t ) + H _ { c , n } ( d | w , t ) . } \end{array}
+$$
+
+The inequality is because $D _ { \mathrm { K L } } ( q ( w | t ) | | p ( w ) ) \geq 0$ for all $t$ ’s. Besides, we used the notation $H _ { q , p }$ , which is the conditional cross entropy. Recall that $D _ { \mathrm { K L } } \big ( q ( d | w , t ) | | p ( d | w , t ) \big ) = - H _ { q } ( d | w , t ) ^ { - } +$ $H _ { q , p } ( d | w , t ) \geq 0$ . We attain the lower bound as desired if this inequality is applied to replace $H _ { q , p } ( d | w , t )$ by $H _ { q } ( d | w , t )$ . □
+
+The following lemma and theorem show the connection between $I _ { q } ( w ; d | t )$ and the generalization error. We first extend $\mathrm { X u }$ (2016, Lemma 4.2).
+
+Lemma 1. If, for all $t$ , $f _ { t } ( X , Y )$ is $\sigma$ -subgaussain under $P _ { X } \otimes P _ { Y }$ , then
+
+$$
+\begin{array} { r } { \left| \mathbb { E } _ { P ( T ) } \Big [ \mathbb { E } _ { P ( X , Y \mid T ) } f _ { T } ( X , Y ) - \mathbb { E } _ { P ( X \mid T ) P ( Y \mid T ) } f _ { T } ( X , Y ) \Big ] \right| \leq \sqrt { 2 \sigma ^ { 2 } I ( X ; Y \mid T ) } . } \end{array}
+$$
+
+Proof. The proof is adapted from the proof of $\mathrm { X u }$ (2016, Lemma 4.2).
+
+$$
+\begin{array} { r l } & { L H S \leq \mathbb { E } _ { P ( T ) } \Big | \mathbb { E } _ { P ( X , Y \mid T ) } f _ { T } ( X , Y ) - \mathbb { E } _ { P ( X \mid T ) P ( Y \mid T ) } f _ { T } ( X , Y ) \Big | } \\ & { \qquad \leq \mathbb { E } _ { P ( T ) } \sqrt { 2 \sigma ^ { 2 } D _ { \mathrm { K L } } ( P ( X , Y \mid T ) \| P ( X \mid T ) P ( Y \mid T ) ) } } \\ & { \qquad \leq \sqrt { 2 \sigma ^ { 2 } \mathbb { E } _ { P ( T ) } D _ { \mathrm { K L } } ( P ( X , Y \mid T ) \| P ( X \mid T ) P ( Y \mid T ) ) } } \\ & { \qquad = \sqrt { 2 \sigma ^ { 2 } I ( X ; Y \mid T ) } . } \end{array}
+$$
+
+The second inequality was due to the Donsker-Varadhan variational representation of KL divergence and the definition of subgaussain random variable. □
+
+Theorem 2. Denote by $z = ( x , y )$ . If $\ell _ { t } ( \hat { y } _ { i } ( f ( x _ { i } ) , w ) , y _ { i } ) \equiv \ell _ { t } ( w , z _ { i } )$ is $\sigma$ -subgaussian under $q ( w | t ) q ( z | t )$ , then $L _ { t } ( w , d )$ is $\sigma / { \sqrt { n } }$ -subgaussian under $q ( w | t ) q ( d | t )$ due to the iid assumption, and
+
+$$
+\left| g e n ( q ) \right| \leq { \sqrt { { \frac { 2 \sigma ^ { 2 } } { n } } I _ { q } ( w ; d | t ) } } .
+$$
+
+Proof. First, if $\ell _ { t } ( \hat { y } ( f ( x ) , w ) , y )$ is $\sigma$ -subgaussian under $q ( w | t ) q ( z | t )$ , by definition,
+
+$$
+\mathbb { E } _ { q ( w | t ) q ( z | t ) } \exp ( \lambda \ell _ { t } ( w , z ) ) \le \exp ( \lambda \mathbb { E } _ { q ( w | t ) q ( z | t ) } \ell _ { t } ( w , z ) ) \exp ( \lambda ^ { 2 } \sigma ^ { 2 } / 2 )
+$$
+
+It is straightforward to show $L _ { t } ( w , d )$ is $\sigma / { \sqrt { n } }$ -subgaussian since
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { q ( w | t ) q ( d | t ) } \exp ( \lambda L _ { t } ( w , d ) ) = \displaystyle \prod _ { i = 1 } ^ { n } \mathbb { E } _ { w , z _ { i } } \exp ( \frac { \lambda } { n } \ell _ { t } ( w , z _ { i } ) ) } \\ & { \qquad \le \displaystyle \prod _ { i = 1 } ^ { n } \exp \Big ( \frac { \lambda } { n } \mathbb { E } _ { w , z _ { i } } \ell _ { t } ( w , z _ { i } ) + \frac { \lambda ^ { 2 } \sigma ^ { 2 } } { 2 n ^ { 2 } } \Big ) } \\ & { \qquad = \exp \Big ( \lambda \mathbb { E } _ { w , z } \ell _ { t } ( w , z ) \Big ) \exp ( \frac { \lambda ^ { 2 } \sigma ^ { 2 } } { 2 n } ) } \\ & { \qquad = \exp \Big ( \lambda \mathbb { E } _ { q ( w | t ) q ( d | t ) } L _ { t } ( w , d ) \Big ) \exp ( \frac { \lambda ^ { 2 } ( \sigma / \sqrt { n } ) ^ { 2 } } { 2 } ) . } \end{array}
+$$
+
+<table><tr><td>Method</td><td>Art</td><td>Cartoon</td><td>Sketch</td><td>Photo</td><td>Average</td></tr><tr><td>JiGen (Carlucci et al., 2019)</td><td>84.9%</td><td>81.1%</td><td>79.1%</td><td>98.0%</td><td>85.7%</td></tr><tr><td>Rot (Xu et al.,2019)</td><td>88.7%</td><td>86.4%</td><td>74.9%</td><td>98.0%</td><td>87.0%</td></tr><tr><td>SIB-Rot K = 0</td><td>85.7%</td><td>86.6%</td><td>80.3%</td><td>98.3%</td><td>87.7%</td></tr><tr><td> SIB-Rot K = 3</td><td>88.9%</td><td>89.0%</td><td>82.2%</td><td>98.3%</td><td>89.6%</td></tr></table>
+
+Table 3: Multi-source domain adaptation results on PACS with ResNet-18 features. Three domains are used as the source domains keeping the fourth one as target.
+
+By Lemma 1, we have
+
+$$
+\begin{array} { r l } & { \left| \mathrm { g e n } ( q ) \right| = \left| \mathbb { E } _ { q ( t ) } \left[ \mathbb { E } _ { q ( w | d , t ) q ( d | t ) } L _ { t } ( w , d ) - \mathbb { E } _ { q ( w | t ) q ( d | t ) } L _ { t } ( w , d ) \right] \right| } \\ & { \qquad \leq \sqrt { \frac { 2 \sigma ^ { 2 } } { n } I ( w ; d | t ) } } \end{array}
+$$
+
+as desired.
+
+# B ZERO-SHOT CLASSIFICATION: UNSUPERVISED MULTI-SOURCE DOMAIN ADAPTATION
+
+A more interesting zero-shot multi-task problem is unsupervised domain adaptation. We consider the case where there exists multiple source domains and a unlabeled target domain. In this case, we treat each minibatch as a task. This makes sense because the difference in statistics between two minibatches are much larger than in the traditional supervised learning. The experimental setup is similar to few-shot classification described in Section 5.1, except that we do not have a support set and the class labels between two tasks are the same. Hence, it is possible to explore the relationship between class labels and self-supervised labels to implement the initialization network $\lambda$ without resorting to support set. We reuse the same model implementation for SIB as described in Section 5.1. The only difference is the initialization network. Denote by $z _ { t } : = \{ z _ { t , i } \} _ { i = 1 } ^ { n }$ the set of self-supervised labels of task $t$ , the initialization network $\lambda$ is implemented as follows:
+
+$$
+\begin{array} { r } { \theta _ { t } ^ { 0 } = \lambda - \eta \nabla _ { \theta } L _ { t } \Big ( \hat { z } _ { t } \big ( \hat { y } _ { t } ( f ( x _ { t } ) , w _ { t } ( \theta , \epsilon ) ) , f ( x _ { t } ) \big ) , z _ { t } \Big ) , } \end{array}
+$$
+
+where $\lambda ^ { 6 }$ is a global initialization similar to the one used by MAML; $L _ { t }$ is the self-supervised loss, $\hat { z } _ { t }$ is the set of predictions of the self-supervised labels. One may argue that $\theta _ { t } ^ { 0 } = \lambda$ would be a simpler solution. However, it is insufficient since the gap between two domains can be very large. The initial solution yielded by (38) is more dynamic in the sense that $\theta _ { t } ^ { 0 }$ is adapted taking into account the information from $x _ { t }$ .
+
+In terms of experiments, we test SIB on the PACS dataset (Li et al., 2017a), which has 7 object categories and 4 domains (Photo, Art Paintings, Cartoon and Sketches), and compare with stateof-the-art algorithms for unsupervised domain adaptation. We follow the standard experimental setting (Carlucci et al., 2019), where the feature network is implemented by ResNet-18. We assign a self-supervised label $z _ { t , i }$ to image $i$ by rotating the image by a predicted degree. This idea was originally proposed by Gidaris et al. (2018) for representation learning and adopted by Xu et al. (2019) for domain adaptation. The training is done by running ADAM for 60 epochs with learning rate $1 0 ^ { - 4 }$ . We take each domain in turns as the target domain. The results are shown in Table 3. It can be seen that SIB-Rot $K = 3$ ) improves upon the baseline SIB-Rot $K = 0$ ) for zero-shot classification, which also outperforms state-of-the-art methods when the baseline is comparable.
+
+# C IMPORTANCE OF SYNTHETIC GRADIENTS
+
+To further verify the effectiveness of the synthetic gradient descent, we implement an inductive version of SIB inspired by MAML, where the initialization $\theta _ { t } ^ { 0 }$ is generated exactly the same way as SIB using $\lambda ( d _ { t } ^ { l } )$ , but it then follows the iterations in (6) as in MAML rather than follows the iterations in (10) as in standard SIB.
+
+We conduct an experiment on CIFAR-FS using Conv-4-64 feature network. The results are shown in Table 4. It can be seen that there is no improvement over SIB $K = 0$ ) suggesting that the inductive approach is insufficient.   
+
+<table><tr><td colspan="7">inductive SIB Training on 1-shot</td></tr><tr><td colspan="3"></td><td colspan="2">Training on 1-shot Testing on</td><td colspan="2">Training on 5-shot Testing on</td></tr><tr><td>K</td><td>n</td><td>Testing on 1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td><td>1-shot</td><td>5-shot</td></tr><tr><td>0</td><td>=</td><td>59.7±0.5%</td><td>75.5±0.4%</td><td>59.2±0.5%</td><td>75.4±0.4%</td><td>59.2±0.5%</td><td>75.4±0.4%</td></tr><tr><td>1</td><td>1e-1</td><td>59.8±0.5%</td><td>71.2±0.4%</td><td>65.3±0.6%</td><td>75.8±0.4%</td><td>64.5±0.6%</td><td>77.3±0.4%</td></tr><tr><td>3</td><td>1e-1</td><td>59.6±0.5%</td><td>75.9±0.4%</td><td>65.0±0.6%</td><td>75.0±0.4%</td><td>64.0±0.6%</td><td>77.0±0.4%</td></tr><tr><td>5</td><td>1e-1</td><td>59.9±0.5%</td><td>74.9±0.4%</td><td>66.0±0.6%</td><td>76.3±0.4%</td><td>64.0±0.5%</td><td>76.8±0.4%</td></tr><tr><td></td><td>1e-2</td><td>59.7±0.5%</td><td>75.5±0.4%</td><td>67.8±0.6%</td><td>74.3±0.4%</td><td>63.6±0.6%</td><td>77.3±0.4%</td></tr><tr><td>3</td><td>1e-2</td><td>59.5±0.5%</td><td>75.8±0.4%</td><td>68.6±0.6%</td><td>77.4±0.4%</td><td>67.8±0.6%</td><td>78.5±0.4%</td></tr><tr><td>5</td><td>1e-2</td><td>59.8±0.5%</td><td>75.7±0.4%</td><td>67.4±0.6%</td><td>72.6±0.6%</td><td>67.7±0.7%</td><td>77.7±0.4%</td></tr><tr><td></td><td>1e-3</td><td>59.5±0.5%</td><td>75.6±0.4%</td><td>66.2±0.6%</td><td>75.7±0.4%</td><td>64.6±0.6%</td><td>78.1±0.4%</td></tr><tr><td>3</td><td>1e-3</td><td>59.9±0.5%</td><td>75.9±0.4%</td><td>68.7±0.6%</td><td>77.1±0.4%</td><td>66.8±0.6%</td><td>78.4±0.4%</td></tr><tr><td>5</td><td>1e-3</td><td>59.4±0.5%</td><td>75.7±0.4%</td><td>69.1±0.6%</td><td>76.7±0.4%</td><td>66.7±0.6%</td><td>78.5±0.4%</td></tr><tr><td>1</td><td>1e-4</td><td>58.8±0.5%</td><td>75.5±0.4%</td><td>59.0±0.5%</td><td>75.7±0.4%</td><td>59.3±0.5%</td><td>75.7±0.4%</td></tr><tr><td>3</td><td>1e-4</td><td>59.4±0.5%</td><td>75.9±0.4%</td><td>58.9±0.5%</td><td>75.6±0.4%</td><td>59.3±0.5%</td><td>75.9±0.4%</td></tr><tr><td>5</td><td>1e-4</td><td>59.3±0.5%</td><td>75.3±0.4%</td><td>60.1±0.5%</td><td>76.0±0.4%</td><td>60.5±0.5%</td><td>76.4±0.4%</td></tr></table>
+
+Table 4: Average 5-way classification accuracies (with $9 5 \%$ confidence intervals) with Conv-4-64 on the test set of CIFAR-FS. For each test, we sample 5000 episodes containing 5 categories (5-way) and 15 queries in each category. We report the results with using different learning rate $\eta$ as well as different number of updates $K$ . Note that $K = 0$ is the performance only using the pre-trained feature.
+
+# D VARYING THE SIZE OF THE QUERY SET
+
+We notice that changing the size of $d _ { t }$ (i.e., $n$ ) during training does make a difference on testing. The results are shown in Table 5.   
+Table 5: Average classification accuracies on the validation set and the test set of Mini-ImageNet with backbone Conv-4-128. We modify the number of query images, i.e., $n$ , for each episode to study the effect on generalization.   
+
+<table><tr><td rowspan="2">n</td><td colspan="2">5-way,5-shot</td><td colspan="2">5-way,1-shot</td><td rowspan="2">Test</td></tr><tr><td>Validation</td><td>Test</td><td>Validation</td><td></td></tr><tr><td>3</td><td>77.97 ± 0.34%</td><td>75.91 ± 0.66%</td><td>63.60 ± 0.52%</td><td></td><td>61.32 ± 1.02%</td></tr><tr><td>5</td><td>78.14 ± 0.35%</td><td>76.01 ± 0.66%</td><td>64.67 ± 0.55%</td><td></td><td>62.50 ± 1.02%</td></tr><tr><td>10</td><td>78.30 ± 0.35%</td><td>76.22 ± 0.66%</td><td>65.34 ± 0.56%</td><td></td><td>63.22 ± 1.04%</td></tr><tr><td>15</td><td>77.53 ± 0.35%</td><td>75.43 ± 0.67%</td><td>65.14 ± 0.55%</td><td></td><td>62.59 ± 1.02%</td></tr><tr><td>30</td><td>76.21 ± 0.35%</td><td>74.04 ± 0.67%</td><td>63.37 ± 0.53%</td><td></td><td>60.96 ± 0.98%</td></tr><tr><td>45</td><td>75.65 ± 0.36%</td><td>73.27 ± 0.66%</td><td>62.08 ± 0.51%</td><td></td><td>59.59 ± 0.93%</td></tr></table>

parse/train/HkwoSDPgg/HkwoSDPgg.md

@@ -0,0 +1,398 @@
+# SEMI-SUPERVISED KNOWLEDGE TRANSFERFOR DEEP LEARNING FROM PRIVATE TRAINING DATA
+
+Nicolas Papernot∗ Pennsylvania State University ngp5056@cse.psu.edu
+
+Mart´ın Abadi   
+Google Brain   
+abadi@google.com
+
+Ulfar Erlingsson ´ Google ulfar@google.com
+
+Ian Goodfellow   
+Google Brain†   
+goodfellow@google.com   
+Kunal Talwar   
+Google Brain   
+kunal@google.com
+
+# ABSTRACT
+
+Some machine learning applications involve training data that is sensitive, such as the medical histories of patients in a clinical trial. A model may inadvertently and implicitly store some of its training data; careful analysis of the model may therefore reveal sensitive information.
+
+To address this problem, we demonstrate a generally applicable approach to providing strong privacy guarantees for training data: Private Aggregation of Teacher Ensembles (PATE). The approach combines, in a black-box fashion, multiple models trained with disjoint datasets, such as records from different subsets of users. Because they rely directly on sensitive data, these models are not published, but instead used as “teachers” for a “student” model. The student learns to predict an output chosen by noisy voting among all of the teachers, and cannot directly access an individual teacher or the underlying data or parameters. The student’s privacy properties can be understood both intuitively (since no single teacher and thus no single dataset dictates the student’s training) and formally, in terms of differential privacy. These properties hold even if an adversary can not only query the student but also inspect its internal workings.
+
+Compared with previous work, the approach imposes only weak assumptions on how teachers are trained: it applies to any model, including non-convex models like DNNs. We achieve state-of-the-art privacy/utility trade-offs on MNIST and SVHN thanks to an improved privacy analysis and semi-supervised learning.
+
+# 1 INTRODUCTION
+
+Some machine learning applications with great benefits are enabled only through the analysis of sensitive data, such as users’ personal contacts, private photographs or correspondence, or even medical records or genetic sequences (Alipanahi et al., 2015; Kannan et al., 2016; Kononenko, 2001; Sweeney, 1997). Ideally, in those cases, the learning algorithms would protect the privacy of users’ training data, e.g., by guaranteeing that the output model generalizes away from the specifics of any individual user. Unfortunately, established machine learning algorithms make no such guarantee; indeed, though state-of-the-art algorithms generalize well to the test set, they continue to overfit on specific training examples in the sense that some of these examples are implicitly memorized.
+
+Recent attacks exploiting this implicit memorization in machine learning have demonstrated that private, sensitive training data can be recovered from models. Such attacks can proceed directly, by analyzing internal model parameters, but also indirectly, by repeatedly querying opaque models to gather data for the attack’s analysis. For example, Fredrikson et al. (2015) used hill-climbing on the output probabilities of a computer-vision classifier to reveal individual faces from the training data.
+
+Because of those demonstrations—and because privacy guarantees must apply to worst-case outliers, not only the average—any strategy for protecting the privacy of training data should prudently assume that attackers have unfettered access to internal model parameters.
+
+To protect the privacy of training data, this paper improves upon a specific, structured application of the techniques of knowledge aggregation and transfer (Breiman, 1994), previously explored by Nissim et al. (2007), Pathak et al. (2010), and particularly Hamm et al. (2016). In this strategy, first, an ensemble (Dietterich, 2000) of teacher models is trained on disjoint subsets of the sensitive data. Then, using auxiliary, unlabeled non-sensitive data, a student model is trained on the aggregate output of the ensemble, such that the student learns to accurately mimic the ensemble. Intuitively, this strategy ensures that the student does not depend on the details of any single sensitive training data point (e.g., of any single user), and, thereby, the privacy of the training data is protected even if attackers can observe the student’s internal model parameters.
+
+This paper shows how this strategy’s privacy guarantees can be strengthened by restricting student training to a limited number of teacher votes, and by revealing only the topmost vote after carefully adding random noise. We call this strengthened strategy PATE, for Private Aggregation of Teacher Ensembles. Furthermore, we introduce an improved privacy analysis that makes the strategy generally applicable to machine learning algorithms with high utility and meaningful privacy guarantees—in particular, when combined with semi-supervised learning.
+
+To establish strong privacy guarantees, it is important to limit the student’s access to its teachers, so that the student’s exposure to teachers’ knowledge can be meaningfully quantified and bounded. Fortunately, there are many techniques for speeding up knowledge transfer that can reduce the rate of student/teacher consultation during learning. We describe several techniques in this paper, the most effective of which makes use of generative adversarial networks (GANs) (Goodfellow et al., 2014) applied to semi-supervised learning, using the implementation proposed by Salimans et al. (2016). For clarity, we use the term PATE-G when our approach is combined with generative, semisupervised methods. Like all semi-supervised learning methods, PATE-G assumes the student has access to additional, unlabeled data, which, in this context, must be public or non-sensitive. This assumption should not greatly restrict our method’s applicability: even when learning on sensitive data, a non-overlapping, unlabeled set of data often exists, from which semi-supervised methods can extract distribution priors. For instance, public datasets exist for text and images, and for medical data.
+
+It seems intuitive, or even obvious, that a student machine learning model will provide good privacy when trained without access to sensitive training data, apart from a few, noisy votes from a teacher quorum. However, intuition is not sufficient because privacy properties can be surprisingly hard to reason about; for example, even a single data item can greatly impact machine learning models trained on a large corpus (Chaudhuri et al., 2011). Therefore, to limit the effect of any single sensitive data item on the student’s learning, precisely and formally, we apply the well-established, rigorous standard of differential privacy (Dwork & Roth, 2014). Like all differentially private algorithms, our learning strategy carefully adds noise, so that the privacy impact of each data item can be analyzed and bounded. In particular, we dynamically analyze the sensitivity of the teachers’ noisy votes; for this purpose, we use the state-of-the-art moments accountant technique from Abadi et al. (2016), which tightens the privacy bound when the topmost vote has a large quorum. As a result, for MNIST and similar benchmark learning tasks, our methods allow students to provide excellent utility, while our analysis provides meaningful worst-case guarantees. In particular, we can bound the metric for privacy loss (the differential-privacy $\varepsilon$ ) to a range similar to that of existing, real-world privacyprotection mechanisms, such as Google’s RAPPOR (Erlingsson et al., 2014).
+
+Finally, it is an important advantage that our learning strategy and our privacy analysis do not depend on the details of the machine learning techniques used to train either the teachers or their student. Therefore, the techniques in this paper apply equally well for deep learning methods, or any such learning methods with large numbers of parameters, as they do for shallow, simple techniques. In comparison, Hamm et al. (2016) guarantee privacy only conditionally, for a restricted class of student classifiers—in effect, limiting applicability to logistic regression with convex loss. Also, unlike the methods of Abadi et al. (2016), which represent the state-of-the-art in differentiallyprivate deep learning, our techniques make no assumptions about details such as batch selection, the loss function, or the choice of the optimization algorithm. Even so, as we show in experiments on
+
+![](images/d7717c03d9df57663d58cc01cb8ea2daa4d640b2149e5038c16b69b12d684e45.jpg)  
+Figure 1: Overview of the approach: (1) an ensemble of teachers is trained on disjoint subsets of the sensitive data, (2) a student model is trained on public data labeled using the ensemble.
+
+MNIST and SVHN, our techniques provide a privacy/utility tradeoff that equals or improves upon bespoke learning methods such as those of Abadi et al. (2016).
+
+Section 5 further discusses the related work. Building on this related work, our contributions are as follows:
+
+• We demonstrate a general machine learning strategy, the PATE approach, that provides differential privacy for training data in a “black-box” manner, i.e., independent of the learning algorithm, as demonstrated by Section 4 and Appendix C.   
+We improve upon the strategy outlined in Hamm et al. (2016) for learning machine models that protect training data privacy. In particular, our student only accesses the teachers’ top vote and the model does not need to be trained with a restricted class of convex losses.   
+• We explore four different approaches for reducing the student’s dependence on its teachers, and show how the application of GANs to semi-supervised learning of Salimans et al. (2016) can greatly reduce the privacy loss by radically reducing the need for supervision.   
+• We present a new application of the moments accountant technique from Abadi et al. (2016) for improving the differential-privacy analysis of knowledge transfer, which allows the training of students with meaningful privacy bounds. We evaluate our framework on MNIST and SVHN, allowing for a comparison of our results with previous differentially private machine learning methods. Our classifiers achieve an $( \varepsilon , \delta )$ differential-privacy bound of $( 2 . 0 4 , 1 0 ^ { - 5 } )$ for MNIST and $( 8 . 1 9 , 1 0 ^ { - 6 } ) $ for SVHN, respectively with accuracy of $9 8 . 0 0 \%$ and $9 0 . 6 6 \%$ . In comparison, for MNIST, Abadi et al. (2016) obtain a looser $( 8 , 1 0 ^ { - 5 } )$ privacy bound and $9 7 \%$ accuracy. For SVHN, Shokri & Shmatikov (2015) report approx. $9 2 \%$ accuracy with $\varepsilon > 2$ per each of 300,000 model parameters, naively making the total $\varepsilon > 6 0 0 , 0 0 0$ , which guarantees no meaningful privacy. Finally, we show that the PATE approach can be successfully applied to other model structures and to datasets with different characteristics. In particular, in Appendix C PATE protects the privacy of medical data used to train a model based on random forests.
+
+Our results are encouraging, and highlight the benefits of combining a learning strategy based on semi-supervised knowledge transfer with a precise, data-dependent privacy analysis. However, the most appealing aspect of this work is probably that its guarantees can be compelling to both an expert and a non-expert audience. In combination, our techniques simultaneously provide both an intuitive and a rigorous guarantee of training data privacy, without sacrificing the utility of the targeted model. This gives hope that users will increasingly be able to confidently and safely benefit from machine learning models built from their sensitive data.
+
+# 2 PRIVATE LEARNING WITH ENSEMBLES OF TEACHERS
+
+In this section, we introduce the specifics of the PATE approach, which is illustrated in Figure 1. We describe how the data is partitioned to train an ensemble of teachers, and how the predictions made by this ensemble are noisily aggregated. In addition, we discuss how GANs can be used in training the student, and distinguish PATE-G variants that improve our approach using generative, semi-supervised methods.
+
+# 2.1 TRAINING THE ENSEMBLE OF TEACHERS
+
+Data partitioning and teachers: Instead of training a single model to solve the task associated with dataset $( X , Y )$ , where $X$ denotes the set of inputs, and $Y$ the set of labels, we partition the data in $n$ disjoint sets $( X _ { n } , Y _ { n } )$ and train a model separately on each set. As evaluated in Section 4.1, assuming that $n$ is not too large with respect to the dataset size and task complexity, we obtain $n$ classifiers $f _ { i }$ called teachers. We then deploy them as an ensemble making predictions on unseen inputs $x$ by querying each teacher for a prediction $f _ { i } ( x )$ and aggregating these into a single prediction.
+
+Aggregation: The privacy guarantees of this teacher ensemble stems from its aggregation. Let $m$ be the number of classes in our task. The label count for a given class $j \in [ m ]$ and an input $\vec { x }$ is the number of teachers that assigned class $j$ to input $\vec { x }$ : $n _ { j } ( \vec { x } ) = | \{ i : i \in [ \bar { n } ] , \bar { f } _ { i } ( \vec { x } ) = j \} |$ . If we simply apply plurality—use the label with the largest count—the ensemble’s decision may depend on a single teacher’s vote. Indeed, when two labels have a vote count differing by at most one, there is a tie: the aggregated output changes if one teacher makes a different prediction. We add random noise to the vote counts $n _ { j }$ to introduce ambiguity:
+
+$$
+f ( x ) = \arg \operatorname* { m a x } _ { j } \left\{ n _ { j } ( \vec { x } ) + L a p \left( \frac { 1 } { \gamma } \right) \right\}
+$$
+
+In this equation, $\gamma$ is a privacy parameter and $L a p ( b )$ the Laplacian distribution with location 0 and scale $b$ . The parameter $\gamma$ influences the privacy guarantee we can prove. Intuitively, a large $\gamma$ leads to a strong privacy guarantee, but can degrade the accuracy of the labels, as the noisy maximum $f$ above can differ from the true plurality.
+
+While we could use an $f$ such as above to make predictions, the noise required would increase as we make more predictions, making the model useless after a bounded number of queries. Furthermore, privacy guarantees do not hold when an adversary has access to the model parameters. Indeed, as each teacher $f _ { i }$ was trained without taking into account privacy, it is conceivable that they have sufficient capacity to retain details of the training data. To address these limitations, we train another model, the student, using a fixed number of labels predicted by the teacher ensemble.
+
+# 2.2 SEMI-SUPERVISED TRANSFER OF THE KNOWLEDGE FROM AN ENSEMBLE TO A STUDENT
+
+We train a student on nonsensitive and unlabeled data, some of which we label using the aggregation mechanism. This student model is the one deployed, in lieu of the teacher ensemble, so as to fix the privacy loss to a value that does not grow with the number of user queries made to the student model. Indeed, the privacy loss is now determined by the number of queries made to the teacher ensemble during student training and does not increase as end-users query the deployed student model. Thus, the privacy of users who contributed to the original training dataset is preserved even if the student’s architecture and parameters are public or reverse-engineered by an adversary.
+
+We considered several techniques to trade-off the student model’s quality with the number of labels it needs to access: distillation, active learning, semi-supervised learning (see Appendix B). Here, we only describe the most successful one, used in PATE-G: semi-supervised learning with GANs.
+
+Training the student with GANs: The GAN framework involves two machine learning models, a generator and a discriminator. They are trained in a competing fashion, in what can be viewed as a two-player game (Goodfellow et al., 2014). The generator produces samples from the data distribution by transforming vectors sampled from a Gaussian distribution. The discriminator is trained to distinguish samples artificially produced by the generator from samples part of the real data distribution. Models are trained via simultaneous gradient descent steps on both players’ costs. In practice, these dynamics are often difficult to control when the strategy set is non-convex (e.g., a DNN). In their application of GANs to semi-supervised learning, Salimans et al. (2016) made the following modifications. The discriminator is extended from a binary classifier (data vs. generator sample) to a multi-class classifier (one of $k$ classes of data samples, plus a class for generated samples). This classifier is then trained to classify labeled real samples in the correct class, unlabeled real samples in any of the $k$ classes, and the generated samples in the additional class.
+
+Although no formal results currently explain why yet, the technique was empirically demonstrated to greatly improve semi-supervised learning of classifiers on several datasets, especially when the classifier is trained with feature matching loss (Salimans et al., 2016).
+
+Training the student in a semi-supervised fashion makes better use of the entire data available to the student, while still only labeling a subset of it. Unlabeled inputs are used in unsupervised learning to estimate a good prior for the distribution. Labeled inputs are then used for supervised learning.
+
+# 3 PRIVACY ANALYSIS OF THE APPROACH
+
+We now analyze the differential privacy guarantees of our PATE approach. Namely, we keep track of the privacy budget throughout the student’s training using the moments accountant (Abadi et al., 2016). When teachers reach a strong quorum, this allows us to bound privacy costs more strictly.
+
+# 3.1 DIFFERENTIAL PRIVACY PRELIMINARIES AND A SIMPLE ANALYSIS OF PATE
+
+Differential privacy (Dwork et al., 2006b; Dwork, 2011) has established itself as a strong standard. It provides privacy guarantees for algorithms analyzing databases, which in our case is a machine learning training algorithm processing a training dataset. Differential privacy is defined using pairs of adjacent databases: in the present work, these are datasets that only differ by one training example. Recall the following variant of differential privacy introduced in Dwork et al. (2006a).
+
+Definition 1. A randomized mechanism $\mathcal { M }$ with domain $\mathcal { D }$ and range $\mathcal { R }$ satisfies $( \varepsilon , \delta )$ -differential privacy if for any two adjacent inputs $d , d ^ { \prime } \in \mathcal { D }$ and for any subset of outputs $S \subseteq \mathcal { R }$ it holds that:
+
+$$
+\operatorname* { P r } [ \mathcal { M } ( d ) \in S ] \leq e ^ { \varepsilon } \operatorname* { P r } [ \mathcal { M } ( d ^ { \prime } ) \in S ] + \delta .
+$$
+
+It will be useful to define the privacy loss and the privacy loss random variable. They capture the differences in the probability distribution resulting from running $\mathcal { M }$ on $d$ and $d ^ { \prime }$ .
+
+Definition 2. Let $\mathcal { M } \colon \mathcal { D }  \mathcal { R }$ be a randomized mechanism and $d , d ^ { \prime }$ a pair of adjacent databases. Let aux denote an auxiliary input. For an outcome $o \in \mathcal { R }$ , the privacy loss at o is defined as:
+
+$$
+c ( o ; \mathcal { M } , a u x , d , d ^ { \prime } ) \overset { \Delta } { = } \log \frac { \operatorname* { P r } [ \mathcal { M } ( a u x , d ) = o ] } { \operatorname* { P r } [ \mathcal { M } ( a u x , d ^ { \prime } ) = o ] } .
+$$
+
+The privacy loss random variable $C ( \mathcal { M } , a u x , d , d ^ { \prime } )$ is defined as $c ( \mathcal { M } ( d ) ; \mathcal { M } , a u x , d , d ^ { \prime } )$ , i.e. the random variable defined by evaluating the privacy loss at an outcome sampled from $\mathcal M ( d )$ .
+
+A natural way to bound our approach’s privacy loss is to first bound the privacy cost of each label queried by the student, and then use the strong composition theorem (Dwork et al., 2010) to derive the total cost of training the student. For neighboring databases $d , d ^ { \prime }$ , each teacher gets the same training data partition (that is, the same for the teacher with $d$ and with $d ^ { \prime }$ , not the same across teachers), with the exception of one teacher whose corresponding training data partition differs. Therefore, the label counts $n _ { j } ( \vec { x } )$ for any example $\vec { x }$ , on $d$ and $d ^ { \prime }$ differ by at most 1 in at most two locations. In the next subsection, we show that this yields loose guarantees.
+
+# 3.2 THE MOMENTS ACCOUNTANT: A BUILDING BLOCK FOR BETTER ANALYSIS
+
+To better keep track of the privacy cost, we use recent advances in privacy cost accounting. The moments accountant was introduced by Abadi et al. (2016), building on previous work (Bun & Steinke, 2016; Dwork & Rothblum, 2016; Mironov, 2016).
+
+Definition 3. Let $\mathcal { M } \colon \mathcal { D }  \mathcal { R }$ be a randomized mechanism and $d , d ^ { \prime }$ a pair of adjacent databases. Let aux denote an auxiliary input. The moments accountant is defined as:
+
+$$
+\alpha _ { \mathcal { M } } ( \lambda ) \overset { \Delta } { = } \operatorname* { m a x } _ { a u x , d , d ^ { \prime } } \alpha _ { \mathcal { M } } ( \lambda ; a u x , d , d ^ { \prime } )
+$$
+
+where ${ \alpha } _ { \mathcal { M } } ( \lambda ; a u x , d , d ^ { \prime } ) \overset { \Delta } { = } \log \mathbb { E } [ \exp ( \lambda C ( \mathcal { M } , a u x , d , d ^ { \prime } ) ) ]$ is the moment generating function of the privacy loss random variable.
+
+The following properties of the moments accountant are proved in Abadi et al. (2016).
+
+Theorem 1. 1. [Composability] Suppose that a mechanism $\mathcal { M }$ consists of a sequence of adaptive mechanisms $\mathcal { M } _ { 1 } , \ldots , \mathcal { M } _ { k }$ where $\begin{array} { r } { \mathcal { M } _ { i } \colon \prod _ { j = 1 } ^ { i - 1 } \mathcal { R } _ { j } \times \mathcal { D } \to \mathcal { R } _ { i } } \end{array}$ . Then, for any output sequence $o _ { 1 } , \ldots , o _ { k - 1 }$ and any $\lambda$
+
+$$
+\alpha _ { \mathcal { M } } ( \lambda ; d , d ^ { \prime } ) = \sum _ { i = 1 } ^ { k } \alpha _ { \mathcal { M } _ { i } } ( \lambda ; o _ { 1 } , \ldots , o _ { i - 1 } , d , d ^ { \prime } ) ,
+$$
+
+where $\alpha _ { \mathcal { M } }$ is conditioned on $\mathcal { M } _ { i }$ ’s output being $o _ { i }$ for $i < k$ .
+
+2. [Tail bound] For any $\varepsilon > 0$ , the mechanism $\mathcal { M }$ is $( \varepsilon , \delta )$ -differentially private for
+
+$$
+\delta = \operatorname* { m i n } _ { \lambda } \exp ( \alpha _ { \mathcal { M } } ( \lambda ) - \lambda \varepsilon ) .
+$$
+
+We write down two important properties of the aggregation mechanism from Section 2. The first property is proved in Dwork & Roth (2014), and the second follows from Bun & Steinke (2016).
+
+Theorem 2. Suppose that on neighboring databases $d , d ^ { \prime }$ , the label counts $n _ { j }$ differ by at most $^ { l }$ in each coordinate. Let $\mathcal { M }$ be the mechanism that reports $\arg \operatorname* { m a x } _ { j } \left\{ n _ { j } + L a p ( \textstyle { \frac { 1 } { \gamma } } ) \right\}$ . Then $\mathcal { M }$ satisfies $( 2 \gamma , 0 )$ -differential privacy. Moreover, for any $l$ , aux, $d$ and $d ^ { \prime }$ ,
+
+$$
+\alpha ( l ; a u x , d , d ^ { \prime } ) \le 2 \gamma ^ { 2 } l ( l + 1 )
+$$
+
+At each step, we use the aggregation mechanism with noise $L a p ( \textstyle { \frac { 1 } { \gamma } } )$ which is $( 2 \gamma , 0 )$ -DP. Thus over $T$ steps, we get $( 4 T \gamma ^ { 2 } + 2 \gamma \sqrt { 2 T \ln \frac { 1 } { \delta } } , \delta )$ -differential privacy. This can be rather large: plugging in values that correspond to our SVHN result, $\gamma = 0 . 0 5 , T = 1 0 0 0 , \delta = 1 \mathrm { e } { - 6 }$ gives us $\varepsilon \approx 2 6$ or alternatively plugging in values that correspond to our MNIST result, $\gamma = 0 . 0 5 , T = 1 0 0 , \delta = 1 \mathrm { e } { - 5 }$ gives us $\varepsilon \approx 5 . 8 0$ .
+
+# 3.3 A PRECISE, DATA-DEPENDENT PRIVACY ANALYSIS OF PATE
+
+Our data-dependent privacy analysis takes advantage of the fact that when the quorum among the teachers is very strong, the majority outcome has overwhelming likelihood, in which case the privacy cost is small whenever this outcome occurs. The moments accountant allows us analyze the composition of such mechanisms in a unified framework.
+
+The following theorem, proved in Appendix A, provides a data-dependent bound on the moments of any differentially private mechanism where some specific outcome is very likely.
+
+ThLet . Letand $\mathcal { M }$ $( 2 \gamma , 0 )$ -differentially private and hen for any aux and any n $q \geq \operatorname* { P r } [ \mathcal { M } ( d ) \neq o ^ { * } ]$ for some outcome atisfies $o ^ { * }$ $l , \gamma \geq 0$ $\begin{array} { r } { q < \frac { e ^ { 2 \gamma } - 1 } { e ^ { 4 \gamma } - 1 } } \end{array}$ $d ^ { \prime }$ $\mathcal { M }$
+
+$$
+\alpha ( l ; a u x , d , d ^ { \prime } ) \le \log ( ( 1 - q ) \Big ( \frac { 1 - q } { 1 - e ^ { 2 \gamma } q } \Big ) ^ { l } + q \exp ( 2 \gamma l ) ) .
+$$
+
+To upper bound $q$ for our aggregation mechanism, we use the following simple lemma, also proved in Appendix A.
+
+Lemma 4. Let n be the label score vector for a database $d$ with $n _ { j ^ { * } } \geq n _ { j }$ for all $j$ . Then
+
+$$
+\operatorname* { P r } [ \mathcal { M } ( d ) \neq j ^ { * } ] \le \sum _ { j \neq j ^ { * } } \frac { 2 + \gamma ( n _ { j ^ { * } } - n _ { j } ) } { 4 \exp ( \gamma ( n _ { j ^ { * } } - n _ { j } ) ) }
+$$
+
+This allows us to upper bound $q$ for a specific score vector $\mathbf { n }$ , and hence bound specific moments. We take the smaller of the bounds we get from Theorems 2 and 3. We compute these moments for a few values of $\lambda$ (integers up to 8). Theorem 1 allows us to add these bounds over successive steps, and derive an $( \varepsilon , \delta )$ guarantee from the final $\alpha$ . Interested readers are referred to the script that we used to empirically compute these bounds, which is released along with our code: https://github. com/tensorflow/models/tree/master/differential_privacy/multiple_teachers
+
+Since the privacy moments are themselves now data dependent, the final $\varepsilon$ is itself data-dependent and should not be revealed. To get around this, we bound the smooth sensitivity (Nissim et al., 2007) of the moments and add noise proportional to it to the moments themselves. This gives us a differentially private estimate of the privacy cost. Our evaluation in Section 4 ignores this overhead and reports the un-noised values of $\varepsilon$ . Indeed, in our experiments on MNIST and SVHN, the scale of the noise one needs to add to the released $\varepsilon$ is smaller than 0.5 and 1.0 respectively.
+
+How does the number of teachers affect the privacy cost? Recall that the student uses a noisy label computed in (1) which has a parameter $\gamma$ . To ensure that the noisy label is likely to be the correct one, the noise scale $\frac { 1 } { \gamma }$ should be small compared to the the additive gap between the two largest vales of $n _ { j }$ . While the exact dependence of $\gamma$ on the privacy cost in Theorem 3 is subtle, as a general principle, a smaller $\gamma$ leads to a smaller privacy cost. Thus, a larger gap translates to a smaller privacy cost. Since the gap itself increases with the number of teachers, having more teachers would lower the privacy cost. This is true up to a pofraction of the training data. For large enough t. With , each te $n$ teachers, each teacher only trains on a hers will have too little training data to $\textstyle { \frac { 1 } { n } }$ $n$   
+accurate.
+
+To conclude, we note that our analysis is rather conservative in that it pessimistically assumes that, even if just one example in the training set for one teacher changes, the classifier produced by that teacher may change arbitrarily. One advantage of our approach, which enables its wide applicability, is that our analysis does not require any assumptions about the workings of the teachers. Nevertheless, we expect that stronger privacy guarantees may perhaps be established in specific settings—when assumptions can be made on the learning algorithm used to train the teachers.
+
+# 4 EVALUATION
+
+In our evaluation of PATE and its generative variant PATE-G, we first train a teacher ensemble for each dataset. The trade-off between the accuracy and privacy of labels predicted by the ensemble is greatly dependent on the number of teachers in the ensemble: being able to train a large set of teachers is essential to support the injection of noise yielding strong privacy guarantees while having a limited impact on accuracy. Second, we minimize the privacy budget spent on learning the student by training it with as few queries to the ensemble as possible.
+
+Our experiments use MNIST and the extended SVHN datasets. Our MNIST model stacks two convolutional layers with max-pooling and one fully connected layer with ReLUs. When trained on the entire dataset, the non-private model has a $9 9 . 1 8 \%$ test accuracy. For SVHN, we add two hidden layers.1 The non-private model achieves a $9 2 . 8 \%$ test accuracy, which is shy of the state-of-the-art. However, we are primarily interested in comparing the private student’s accuracy with the one of a non-private model trained on the entire dataset, for different privacy guarantees. The source code for reproducing the results in this section is available on GitHub.2
+
+# 4.1 TRAINING AN ENSEMBLE OF TEACHERS PRODUCING PRIVATE LABELS
+
+As mentioned above, compensating the noise introduced by the Laplacian mechanism presented in Equation 1 requires large ensembles. We evaluate the extent to which the two datasets considered can be partitioned with a reasonable impact on the performance of individual teachers. Specifically, we show that for MNIST and SVHN, we are able to train ensembles of 250 teachers. Their aggregated predictions are accurate despite the injection of large amounts of random noise to ensure privacy. The aggregation mechanism output has an accuracy of $9 3 . 1 8 \%$ for MNIST and $8 7 . 7 9 \%$ for SVHN, when evaluated on their respective test sets, while each query has a low privacy budget of $\varepsilon = 0 . 0 5$ .
+
+Prediction accuracy: All other things being equal, the number $n$ of teachers is limited by a tradeoff between the classification task’s complexity and the available data. We train $n$ teachers by partitioning the training data $n$ -way. Larger values of $n$ lead to larger absolute gaps, hence potentially allowing for a larger noise level and stronger privacy guarantees. At the same time, a larger $n$ implies a smaller training dataset for each teacher, potentially reducing the teacher accuracy. We empirically find appropriate values of $n$ for the MNIST and SVHN datasets by measuring the test set accuracy of each teacher trained on one of the $n$ partitions of the training data. We find that even for $n = 2 5 0$ , the average test accuracy of individual teachers is $8 3 . 8 6 \%$ for MNIST and $8 3 . 1 8 \%$ for SVHN. The larger size of SVHN compensates its increased task complexity.
+
+![](images/3d0f2414761710708a40157a3a73fa2cbadf0de052c528d51a20b9d0479cf0ce.jpg)  
+Figure 2: How much noise can be injected to a query? Accuracy of the noisy aggregation for three MNIST and SVHN teacher ensembles and varying $\gamma$ value per query. The noise introduced to achieve a given $\gamma$ scales inversely proportionally to the value of $\gamma$ : small values of $\gamma$ on the left of the axis correspond to large noise amplitudes and large $\gamma$ values on the right to small noise.
+
+![](images/a66ae0d26ae3c231407e574c5df899a89e0dda53ec25eee418372deea7ce68b9.jpg)  
+Figure 3: How certain is the aggregation of teacher predictions? Gap between the number of votes assigned to the most and second most frequent labels normalized by the number of teachers in an ensemble. Larger gaps indicate that the ensemble is confident in assigning the labels, and will be robust to more noise injection. Gaps were computed by averaging over the test data.
+
+Prediction confidence: As outlined in Section 3, the privacy of predictions made by an ensemble of teachers intuitively requires that a quorum of teachers generalizing well agree on identical labels. This observation is reflected by our data-dependent privacy analysis, which provides stricter privacy bounds when the quorum is strong. We study the disparity of labels assigned by teachers. In other words, we count the number of votes for each possible label, and measure the difference in votes between the most popular label and the second most popular label, i.e., the gap. If the gap is small, introducing noise during aggregation might change the label assigned from the first to the second. Figure 3 shows the gap normalized by the total number of teachers $n$ . As $n$ increases, the gap remains larger than $6 0 \%$ of the teachers, allowing for aggregation mechanisms to output the correct label in the presence of noise.
+
+Noisy aggregation: For MNIST and SVHN, we consider three ensembles of teachers with varying number of teachers $n \in \{ 1 0 , 1 0 0 , 2 5 0 \}$ . For each of them, we perturb the vote counts with Laplacian noise of inversed scale $\gamma$ ranging between 0.01 and 1. This choice is justified below in Section 4.2. We report in Figure 2 the accuracy of test set labels inferred by the noisy aggregation mechanism for these values of $\varepsilon$ . Notice that the number of teachers needs to be large to compensate for the impact of noise injection on the accuracy.
+
+# 4.2 SEMI-SUPERVISED TRAINING OF THE STUDENT WITH PRIVACY
+
+The noisy aggregation mechanism labels the student’s unlabeled training set in a privacy-preserving fashion. To reduce the privacy budget spent on student training, we are interested in making as few label queries to the teachers as possible. We therefore use the semi-supervised training approach described previously. Our MNIST and SVHN students with $( \varepsilon , \delta )$ differential privacy of $( 2 . 0 4 , 1 0 ^ { - 5 } )$ ) and (8.19, $1 0 ^ { - 6 } )$ achieve accuracies of $9 8 . 0 0 \%$ and $9 0 . 6 6 \%$ . These results improve the differential privacy state-of-the-art for these datasets. Abadi et al. (2016) previously obtained $9 7 \%$ accuracy with a $( 8 , 1 0 ^ { - 5 } )$ bound on MNIST, starting from an inferior baseline model without privacy. Shokri & Shmatikov (2015) reported about $9 2 \%$ accuracy on SVHN with $\varepsilon > 2$ per model parameter and a model with over 300,000 parameters. Naively, this corresponds to a total $\varepsilon > 6 0 0 , 0 0 0$ .
+
+Figure 4: Utility and privacy of the semi-supervised students: each row is a variant of the student model trained with generative adversarial networks in a semi-supervised way, with a different number of label queries made to the teachers through the noisy aggregation mechanism. The last column reports the accuracy of the student and the second and third column the bound $\varepsilon$ and failure probability $\delta$ of the $( \varepsilon , \delta )$ differential privacy guarantee.   
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>Queries</td><td rowspan=1 colspan=1>Non-Private Baseline</td><td rowspan=1 colspan=1> Student Accuracy</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>2.04</td><td rowspan=1 colspan=1>10-5</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>99.18%</td><td rowspan=1 colspan=1>98.00%</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>8.03</td><td rowspan=1 colspan=1>10-5</td><td rowspan=1 colspan=1>1000</td><td rowspan=1 colspan=1>99.18%</td><td rowspan=1 colspan=1>98.10%</td></tr><tr><td rowspan=1 colspan=1>SVHN</td><td rowspan=1 colspan=1>5.04</td><td rowspan=1 colspan=1>10-6</td><td rowspan=1 colspan=1>500</td><td rowspan=1 colspan=1>92.80%</td><td rowspan=1 colspan=1>82.72%</td></tr><tr><td rowspan=1 colspan=1>SVHN</td><td rowspan=1 colspan=1>8.19</td><td rowspan=1 colspan=1>10-6</td><td rowspan=1 colspan=1>1000</td><td rowspan=1 colspan=1>92.80%</td><td rowspan=1 colspan=1>90.66%</td></tr></table>
+
+We apply semi-supervised learning with GANs to our problem using the following setup for each dataset. In the case of MNIST, the student has access to 9,000 samples, among which a subset of either 100, 500, or 1,000 samples are labeled using the noisy aggregation mechanism discussed in Section 2.1. Its performance is evaluated on the 1,000 remaining samples of the test set. Note that this may increase the variance of our test set accuracy measurements, when compared to those computed over the entire test data. For the MNIST dataset, we randomly shuffle the test set to ensure that the different classes are balanced when selecting the (small) subset labeled to train the student. For SVHN, the student has access to 10,000 training inputs, among which it labels 500 or 1,000 samples using the noisy aggregation mechanism. Its performance is evaluated on the remaining 16,032 samples. For both datasets, the ensemble is made up of 250 teachers. We use Laplacian scale of 20 to guarantee an individual query privacy bound of $\varepsilon = 0 . 0 5$ . These parameter choices are motivated by the results from Section 4.1.
+
+In Figure 4, we report the values of the $( \varepsilon , \delta )$ differential privacy guarantees provided and the corresponding student accuracy, as well as the number of queries made by each student. The MNIST student is able to learn a $9 8 \%$ accurate model, which is shy of $1 \%$ when compared to the accuracy of a model learned with the entire training set, with only 100 label queries. This results in a strict differentially private bound of $\varepsilon = 2 . 0 4$ for a failure probability fixed at $1 0 ^ { - 5 }$ . The SVHN student achieves $9 0 . 6 6 \%$ accuracy, which is also comparable to the $9 2 . 8 0 \%$ accuracy of one teacher learned with the entire training set. The corresponding privacy bound is $\varepsilon = 8 . 1 9$ , which is higher than for the MNIST dataset, likely because of the larger number of queries made to the aggregation mechanism.
+
+We observe that our private student outperforms the aggregation’s output in terms of accuracy, with or without the injection of Laplacian noise. While this shows the power of semi-supervised learning, the student may not learn as well on different kinds of data (e.g., medical data), where categories are not explicitly designed by humans to be salient in the input space. Encouragingly, as Appendix C illustrates, the PATE approach can be successfully applied to at least some examples of such data.
+
+# 5 DISCUSSION AND RELATED WORK
+
+Several privacy definitions are found in the literature. For instance, $k$ -anonymity requires information about an individual to be indistinguishable from at least $k - 1$ other individuals in the dataset (L. Sweeney, 2002). However, its lack of randomization gives rise to caveats (Dwork & Roth, 2014), and attackers can infer properties of the dataset (Aggarwal, 2005). An alternative definition, differential privacy, established itself as a rigorous standard for providing privacy guarantees (Dwork et al., 2006b). In contrast to $k$ -anonymity, differential privacy is a property of the randomized algorithm and not the dataset itself.
+
+A variety of approaches and mechanisms can guarantee differential privacy. Erlingsson et al. (2014) showed that randomized response, introduced by Warner (1965), can protect crowd-sourced data collected from software users to compute statistics about user behaviors. Attempts to provide differential privacy for machine learning models led to a series of efforts on shallow machine learning models, including work by Bassily et al. (2014); Chaudhuri & Monteleoni (2009); Pathak et al. (2011); Song et al. (2013), and Wainwright et al. (2012).
+
+A privacy-preserving distributed SGD algorithm was introduced by Shokri & Shmatikov (2015). It applies to non-convex models. However, its privacy bounds are given per-parameter, and the large number of parameters prevents the technique from providing a meaningful privacy guarantee. Abadi et al. (2016) provided stricter bounds on the privacy loss induced by a noisy SGD by introducing the moments accountant. In comparison with these efforts, our work increases the accuracy of a private MNIST model from $9 7 \%$ to ${ \bar { 9 } } 8 \%$ while improving the privacy bound $\varepsilon$ from 8 to 1.9. Furthermore, the PATE approach is independent of the learning algorithm, unlike this previous work. Support for a wide range of architecture and training algorithms allows us to obtain good privacy bounds on an accurate and private SVHN model. However, this comes at the cost of assuming that nonprivate unlabeled data is available, an assumption that is not shared by (Abadi et al., 2016; Shokri & Shmatikov, 2015).
+
+Pathak et al. (2010) first discussed secure multi-party aggregation of locally trained classifiers for a global classifier hosted by a trusted third-party. Hamm et al. (2016) proposed the use of knowledge transfer between a collection of models trained on individual devices into a single model guaranteeing differential privacy. Their work studied linear student models with convex and continuously differentiable losses, bounded and $c$ -Lipschitz derivatives, and bounded features. The PATE approach of this paper is not constrained to such applications, but is more generally applicable.
+
+Previous work also studied semi-supervised knowledge transfer from private models. For instance, Jagannathan et al. (2013) learned privacy-preserving random forests. A key difference is that their approach is tailored to decision trees. PATE works well for the specific case of decision trees, as demonstrated in Appendix C, and is also applicable to other machine learning algorithms, including more complex ones. Another key difference is that Jagannathan et al. (2013) modified the classic model of a decision tree to include the Laplacian mechanism. Thus, the privacy guarantee does not come from the disjoint sets of training data analyzed by different decision trees in the random forest, but rather from the modified architecture. In contrast, partitioning is essential to the privacy guarantees of the PATE approach.
+
+# 6 CONCLUSIONS
+
+To protect the privacy of sensitive training data, this paper has advanced a learning strategy and a corresponding privacy analysis. The PATE approach is based on knowledge aggregation and transfer from “teacher” models, trained on disjoint data, to a “student” model whose attributes may be made public. In combination, the paper’s techniques demonstrably achieve excellent utility on the MNIST and SVHN benchmark tasks, while simultaneously providing a formal, state-of-the-art bound on users’ privacy loss. While our results are not without limits—e.g., they require disjoint training data for a large number of teachers (whose number is likely to increase for tasks with many output classes)—they are encouraging, and highlight the advantages of combining semi-supervised learning with precise, data-dependent privacy analysis, which will hopefully trigger further work. In particular, such future work may further investigate whether or not our semi-supervised approach will also reduce teacher queries for tasks other than MNIST and SVHN, for example when the discrete output categories are not as distinctly defined by the salient input space features.
+
+A key advantage is that this paper’s techniques establish a precise guarantee of training data privacy in a manner that is both intuitive and rigorous. Therefore, they can be appealing, and easily explained, to both an expert and non-expert audience. However, perhaps equally compelling are the techniques’ wide applicability. Both our learning approach and our analysis methods are “blackbox,” i.e., independent of the learning algorithm for either teachers or students, and therefore apply, in general, to non-convex, deep learning, and other learning methods. Also, because our techniques do not constrain the selection or partitioning of training data, they apply when training data is naturally and non-randomly partitioned—e.g., because of privacy, regulatory, or competitive concerns— or when each teacher is trained in isolation, with a different method. We look forward to such further applications, for example on RNNs and other sequence-based models.
+
+# ACKNOWLEDGMENTS
+
+Nicolas Papernot is supported by a Google PhD Fellowship in Security. The authors would like to thank Ilya Mironov and Li Zhang for insightful discussions about early drafts of this document.
+
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+
+# A MISSING DETAILS ON THE ANALYSIS
+
+We provide missing proofs from Section 3.
+
+ThLet . Letand $\mathcal { M }$ $( 2 \gamma , 0 )$ -differentially private and hen for any aux and any n $q \geq \operatorname* { P r } [ \mathcal { M } ( d ) \neq o ^ { * } ]$ for some outcome atisfies $o ^ { * }$ . $l , \gamma \geq 0$ $q < \frac { e ^ { 2 \gamma } - 1 } { e ^ { 4 \gamma } - 1 }$ $d ^ { \prime }$ $\mathcal { M }$
+
+$$
+\alpha ( l ; a u x , d , d ^ { \prime } ) \le \log ( ( 1 - q ) \Big ( \frac { 1 - q } { 1 - e ^ { 2 \gamma } q } \Big ) ^ { l } + q \exp ( 2 \gamma l ) ) .
+$$
+
+Proof. Since M is 2γ-differentially private, for every outcome o, P r[M(d)=o]P r[M(d0)=o] $\frac { P r [ M ( d ) = o ] } { P r [ M ( d ^ { \prime } ) = o ] } \ \leq \ \exp ( 2 \gamma )$ . Let $q ^ { \prime } = P r [ M ( d ) \neq o ^ { * } ]$ . Then $P r [ M ( d ^ { \prime } ) \neq o ^ { * } ] \leq \exp ( 2 \gamma ) q ^ { \prime }$ . Thus
+
+$$
+\begin{array} { r l } {  { \operatorname { N p } ( \alpha ( l ; \mathbf { a u x } , d , d ^ { \prime } ) ) = \sum _ { o } \operatorname* { P r } [ M ( d ) = o ] \Big ( \frac { \operatorname* { P r } [ M ( d ) = o ] } { \operatorname* { P r } [ M ( d ^ { \prime } ) = o ] } \Big ) ^ { l } } } \\ & { = \operatorname* { P r } [ M ( d ) = o ^ { * } ] \Big ( \frac { \operatorname* { P r } [ M ( d ) = o ^ { * } ] } { \operatorname* { P r } [ M ( d ^ { \prime } ) = o ^ { * } ] } \Big ) ^ { l } + \sum _ { o \neq o ^ { * } } \operatorname* { P r } [ M ( d ) = o ] \Big ( \frac { \operatorname* { P r } [ M ( d ) = o ] } { \operatorname* { P r } [ M ( d ^ { \prime } ) = o ] } \Big ) } \\ & { \leq ( 1 - q ^ { \prime } ) \Big ( \frac { 1 - q ^ { \prime } } { 1 - e ^ { 2 \gamma } q ^ { \prime } } \Big ) ^ { l } + \sum _ { o \neq o ^ { * } } \operatorname* { P r } [ M ( d ) = o ] ( e ^ { 2 \gamma } ) ^ { l } } \\ & { \leq ( 1 - q ^ { \prime } ) \big ( \frac { 1 - q ^ { \prime } } { 1 - e ^ { 2 \gamma } q ^ { \prime } } \big ) ^ { l } + q ^ { \prime } e ^ { 2 \gamma l } . } \end{array}
+$$
+
+Now consider the function
+
+$$
+f ( z ) = ( 1 - z ) \Bigl ( \frac { 1 - z } { 1 - e ^ { 2 \gamma } z } \Bigr ) ^ { l } + z e ^ { 2 \gamma l } .
+$$
+
+We next argue that this function is non-decreasing in $( 0 , { \frac { e ^ { 2 \gamma } - 1 } { e ^ { 4 \gamma } - 1 } } )$ under the conditions of the lemma. Towards this goal, define
+
+$$
+g ( z , w ) = ( 1 - z ) \Big ( \frac { 1 - w } { 1 - e ^ { 2 \gamma } w } \Big ) ^ { l } + z e ^ { 2 \gamma l } ,
+$$
+
+and observe that $f ( z ) = g ( z , z )$ . We can easily verify by differentiation that $g ( z , w )$ is increasing individually in $z$ and in $w$ in the range of interest. This implies that $f ( q ^ { \prime } ) \leq f ( q )$ completing the proof. □
+
+Lemma 4. Let n be the label score vector for a database $d$ with $n _ { j ^ { * } } \geq n _ { j }$ for all $j$ . Then
+
+$$
+\operatorname* { P r } [ \mathcal { M } ( d ) \neq j ^ { * } ] \leq \sum _ { j \neq j ^ { * } } \frac { 2 + \gamma ( n _ { j ^ { * } } - n _ { j } ) } { 4 \exp ( \gamma ( n _ { j ^ { * } } - n _ { j } ) ) }
+$$
+
+Proof. The probability that $\begin{array} { r } { n _ { j ^ { * } } + L a p ( \frac { 1 } { \gamma } ) < n _ { j } + L a p ( \frac { 1 } { \gamma } ) } \end{array}$ is equal to the probability that the sum of two independent $L a p ( 1 )$ random variables exceeds $\gamma ( n _ { j ^ { * } } - n _ { j } )$ . The sum of two independent $L a p ( 1 )$ variables has the same distribution as the difference of two $G a m m a ( 2 , 1 )$ random variables. Recalling that the $G a m m a ( 2 , 1 )$ distribution has pdf $x e ^ { - x }$ , we can compute the pdf of the difference via convolution as
+
+$$
+\int _ { y = 0 } ^ { \infty } ( y + | x | ) e ^ { - y - | x | } y e ^ { - y } d y = { \frac { 1 } { e ^ { | x | } } } \int _ { y = 0 } ^ { \infty } ( y ^ { 2 } + y | x | ) e ^ { - 2 y } d y = { \frac { 1 + | x | } { 4 e ^ { | x | } } } .
+$$
+
+The probability mass in the tail can then be computed by integration as $\frac { 2 + \gamma ( n _ { j ^ { * } } - n _ { j } ) } { 4 \exp ( \gamma ( n _ { j ^ { * } } - n _ { j } ) }$ . Taking a union bound over the various candidate $j$ ’s gives the claimed bound. □
+
+# B APPENDIX: TRAINING THE STUDENT WITH MINIMAL TEACHER QUERIES
+
+In this appendix, we describe approaches that were considered to reduce the number of queries made to the teacher ensemble by the student during its training. As pointed out in Sections 3 and 4, this effort is motivated by the direct impact of querying on the total privacy cost associated with student training. The first approach is based on distillation, a technique used for knowledge transfer and model compression (Hinton et al., 2015). The three other techniques considered were proposed in the context of active learning, with the intent of identifying training examples most useful for learning. In Sections 2 and 4, we described semi-supervised learning, which yielded the best results. The student models in this appendix differ from those in Sections 2 and 4, which were trained using GANs. In contrast, all students in this appendix were learned in a fully supervised fashion from a subset of public, labeled examples. Thus, the learning goal was to identify the subset of labels yielding the best learning performance.
+
+# B.1 TRAINING STUDENTS USING DISTILLATION
+
+Distillation is a knowledge transfer technique introduced as a means of compressing large models into smaller ones, while retaining their accuracy (Bucilua et al., 2006; Hinton et al., 2015). This is for instance useful to train models in data centers before deploying compressed variants in phones. The transfer is accomplished by training the smaller model on data that is labeled with probability vectors produced by the first model, which encode the knowledge extracted from training data. Distillation is parameterized by a temperature parameter $T$ , which controls the smoothness of probabilities output by the larger model: when produced at small temperatures, the vectors are discrete, whereas at high temperature, all classes are assigned non-negligible values. Distillation is a natural candidate to compress the knowledge acquired by the ensemble of teachers, acting as the large model, into a student, which is much smaller with $n$ times less trainable parameters compared to the $n$ teachers.
+
+To evaluate the applicability of distillation, we consider the ensemble of $n = 5 0$ teachers for SVHN. In this experiment, we do not add noise to the vote counts when aggregating the teacher predictions. We compare the accuracy of three student models: the first is a baseline trained with labels obtained by plurality, the second and third are trained with distillation at $T \in \{ 1 , 5 \}$ . We use the first 10,000 samples from the test set as unlabeled data. Figure 5 reports the accuracy of the student model on the last 16,032 samples from the test set, which were not accessible to the model during training. It is plotted with respect to the number of samples used to train the student (and hence the number of queries made to the teacher ensemble). Although applying distillation yields classifiers that perform more accurately, the increase in accuracy is too limited to justify the increased privacy cost of revealing the entire probability vector output by the ensemble instead of simply the class assigned the largest number of votes. Thus, we turn to an investigation of active learning.
+
+# B.2 ACTIVE LEARNING OF THE STUDENT
+
+Active learning is a class of techniques that aims to identify and prioritize points in the student’s training set that have a high potential to contribute to learning (Angluin, 1988; Baum, 1991). If the label of an input in the student’s training set can be predicted confidently from what we have learned so far by querying the teachers, it is intuitive that querying it is not worth the privacy budget spent. In our experiments, we made several attempts before converging to a simpler final formulation.
+
+Siamese networks: Our first attempt was to train a pair of siamese networks, introduced by Bromley et al. (1993) in the context of one-shot learning and later improved by Koch (2015). The siamese networks take two images as input and return 1 if the images are equal and 0 otherwise. They are two identical networks trained with shared parameters to force them to produce similar representations of the inputs, which are then compared using a distance metric to determine if the images are identical or not. Once the siamese models are trained, we feed them a pair of images where the first is unlabeled and the second labeled. If the unlabeled image is confidently matched with a known labeled image, we can infer the class of the unknown image from the labeled image. In our experiments, the siamese networks were able to say whether two images are identical or not, but did not generalize well: two images of the same class did not receive sufficiently confident matches. We also tried a variant of this approach where we trained the siamese networks to output 1 when the two images are of the same class and 0 otherwise, but the learning task proved too complicated to be an effective means for reducing the number of queries made to teachers.
+
+![](images/301e2803665289652cc9c4a671ef12c8391e1f4aac0f59022196ecd50bdf0862.jpg)  
+Figure 5: Influence of distillation on the accuracy of the SVHN student trained with respect to the initial number of training samples available to the student. The student is learning from $n = 5 0$ teachers, whose predictions are aggregated without noise: in case where only the label is returned, we use plurality, and in case a probability vector is returned, we sum the probability vectors output by each teacher before normalizing the resulting vector.
+
+Collection of binary experts: Our second attempt was to train a collection of binary experts, one per class. An expert for class $j$ is trained to output 1 if the sample is in class $j$ and 0 otherwise. We first trained the binary experts by making an initial batch of queries to the teachers. Using the experts, we then selected available unlabeled student training points that had a candidate label score below 0.9 and at least 4 other experts assigning a score above 0.1. This gave us about 500 unconfident points for 1700 initial label queries. After labeling these unconfident points using the ensemble of teachers, we trained the student. Using binary experts improved the student’s accuracy when compared to the student trained on arbitrary data with the same number of teacher queries. The absolute increases in accuracy were however too limited—between $1 . 5 \%$ and $2 . 5 \%$ .
+
+Identifying unconfident points using the student: This last attempt was the simplest yet the most effective. Instead of using binary experts to identify student training points that should be labeled by the teachers, we used the student itself. We asked the student to make predictions on each unlabeled training point available. We then sorted these samples by increasing values of the maximum probability assigned to a class for each sample. We queried the teachers to label these unconfident inputs first and trained the student again on this larger labeled training set. This improved the accuracy of the student when compared to the student trained on arbitrary data. For the same number of teacher queries, the absolute increases in accuracy of the student trained on unconfident inputs first when compared to the student trained on arbitrary data were in the order of $4 \% - 1 0 \%$ .
+
+# C APPENDIX: ADDITIONAL EXPERIMENTS ON THE UCI ADULT AND DIABETES DATASETS
+
+In order to further demonstrate the general applicability of our approach, we performed experiments on two additional datasets. While our experiments on MNIST and SVHN in Section 4 used convolutional neural networks and GANs, here we use random forests to train our teacher and student models for both of the datasets. Our new results on these datasets show that, despite the differing data types and architectures, we are able to provide meaningful privacy guarantees.
+
+UCI Adult dataset: The UCI Adult dataset is made up of census data, and the task is to predict when individuals make over $\$ 50\mathrm { k }$ per year. Each input consists of 13 features (which include the age, workplace, education, occupation—see the UCI website for a full list3). The only pre-processing we apply to these features is to map all categorical features to numerical values by assigning an integer value to each possible category. The model is a random forest provided by the scikit-learn Python package. When training both our teachers and student, we keep all the default parameter values, except for the number of estimators, which we set to 100. The data is split between a training set of 32,562 examples, and a test set of 16,282 inputs.
+
+UCI Diabetes dataset: The UCI Diabetes dataset includes de-identified records of diabetic patients and corresponding hospital outcomes, which we use to predict whether diabetic patients were readmitted less than 30 days after their hospital release. To the best of our knowledge, no particular classification task is considered to be a standard benchmark for this dataset. Even so, it is valuable to consider whether our approach is applicable to the likely classification tasks, such as readmission, since this dataset is collected in a medical environment—a setting where privacy concerns arise frequently. We select a subset of 18 input features from the 55 available in the dataset (to avoid features with missing values) and form a dataset balanced between the two output classes (see the UCI website for more details4). In class 0, we include all patients that were readmitted in a 30-day window, while class 1 includes all patients that were readmitted after 30 days or never readmitted at all. Our balanced dataset contains 34,104 training samples and 12,702 evaluation samples. We use a random forest model identical to the one described above in the presentation of the Adult dataset.
+
+Experimental results: We apply our approach described in Section 2. For both datasets, we train ensembles of $n = 2 5 0$ random forests on partitions of the training data. We then use the noisy aggregation mechanism, where vote counts are perturbed with Laplacian noise of scale 0.05 to privately label the first 500 test set inputs. We train the student random forest on these 500 test set inputs and evaluate it on the last 11,282 test set inputs for the Adult dataset, and 6,352 test set inputs for the Diabetes dataset. These numbers deliberately leave out some of the test set, which allowed us to observe how the student performance-privacy trade-off was impacted by varying the number of private labels, as well as the Laplacian scale used when computing these labels.
+
+For the Adult dataset, we find that our student model achieves an $8 3 \%$ accuracy for an $( \varepsilon , \delta ) =$ $( 2 . 6 6 , 1 0 ^ { - 5 } )$ differential privacy bound. Our non-private model on the dataset achieves $8 5 \%$ accuracy, which is comparable to the state-of-the-art accuracy of $8 6 \%$ on this dataset (Poulos & Valle, 2016). For the Diabetes dataset, we find that our privacy-preserving student model achieves a $9 3 . 9 4 \%$ accuracy for a $( \varepsilon , \delta ) = ( 1 . 4 4 , 1 0 ^ { - 5 } )$ differential privacy bound. Our non-private model on the dataset achieves $9 3 . 8 1 \%$ accuracy.

parse/train/HylUPnVKvH/HylUPnVKvH.md

@@ -0,0 +1,304 @@
+# MIX & MATCH: TRAINING CONVNETS WITH MIXED IMAGE SIZES FOR IMPROVED ACCURACY, SPEED AND SCALE RESILIENCY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Convolutional neural networks (CNNs) are commonly trained using a fixed spatial image size predetermined for a given model. Although trained on images of a specific size, it is well established that CNNs can be used to evaluate a wide range of image sizes at test time, by adjusting the size of intermediate feature maps.
+
+In this work, we describe and evaluate a novel mixed-size training regime that mixes several image sizes at training time. We demonstrate that models trained using our method are more resilient to image size changes and generalize well even on small images. This allows faster inference by using smaller images at test time. For instance, we receive a $7 6 . 4 3 \%$ top-1 accuracy using ResNet50 with an image size of 160, which matches the accuracy of the baseline model with $2 \times$ fewer computations. Furthermore, for a given image size used at test time, we show this method can be exploited either to accelerate training or the final test accuracy. For example, we are able to reach a $7 9 . 2 7 \%$ accuracy with a model evaluated at a 288 spatial size for a relative improvement of $1 4 \%$ over the baseline. Our PyTorch implementation and pre-trained models are publicly available1
+
+# 1 INTRODUCTION
+
+Convolutional neural networks are successfully used to solve various tasks across multiple domains such as visual (Krizhevsky et al., 2012; Ren et al., 2015), audio (van den Oord et al., 2016), language (Gehring et al., 2017) and speech (Abdel-Hamid et al., 2014). While scale-invariance is considered important for visual representations (Lowe, 1999), convolutional networks are not scale invariant with respect to the spatial resolution of the image input, as a change in image dimension may lead to a non-linear change of their output. Even though CNNs are able to achieve state-of-the
+
+![](images/1b579338f11246697fb783b19c8d0953fb640ad92da56184477f7a81c9df594d.jpg)  
+Figure 1: Test accuracy per image size, models trained on specific sizes (ResNet50, ImageNet).
+
+art results in many tasks and domains, their sensitivity to the image size is an inherent deficiency that limits practical use cases and requires evaluation inputs to match training image size. For example, Touvron et al. (2019) demonstrated that networks trained on specific image size, perform poorly on other image sizes at evaluation, as shown in Figure 1.
+
+Several works attempted to achieve scale invariance by modifying the network structure (Xu et al., 2014; Takahashi et al., 2017). However, the most common method is to artificially enlarge the dataset using a set of label-preserving transformations also known as ”data augmentation” (Krizhevsky et al., 2012; Howard, 2013). Several of these transformations scale and crop objects appearing within the data, thus increasing the network’s robustness to inputs of different scale.
+
+Although not explicitly trained to handle varying image sizes, CNNs are commonly evaluated on multiple scales post training, such as in the case of detection (Lin et al., 2017; Redmon & Farhadi,
+
+2018) and segmentation (He et al., 2017) tasks. In these tasks, a network that was pretrained with fixed image size for classification is used as the backbone of a larger model that is expected to adapt to a wide variety of image sizes.
+
+In this work, we will introduce a novel training regime, “MixSize” for convolutional networks that uses stochastic image and batch sizes. The main contributions of the MixSize regime are:
+
+• Reducing image size sensitivity. We show that the MixSize training regime can improve model performance on a wide range of sizes used at evaluation.   
+• Faster inference. As our mixed-size models can be evaluated at smaller image sizes, we show up to $2 \times$ reduction in computations required at inference to reach the same accuracy as the baseline model.   
+• Faster training vs. high accuracy. We show that reducing the average image size at training leads to a trade-off between the time required to train the model and its final accuracy.
+
+# 2 RELATED WORK
+
+# 2.1 USING MULTIPLE IMAGE SIZES
+
+Deep convolutional networks are traditionally trained using fixed-size inputs, with spatial dimensions $H \times W$ and a batch size $B$ . The network architecture is configured such that the spatial dimensions are reduced through strided pooling or convolutions, with the last classification layer applied on a $1 \times 1$ spatial dimension. Modern convolutional networks usually conclude with a final ”global” average pooling (Lin et al., 2013; Szegedy et al., 2015), that reduces any remaining spatial dimensions with a simple averaging operation. Modifying the spatial size of an input to a convolutional layer by a factor $\gamma$ , will yield an output with size scaled by the same factor $\gamma$ . This modification does not require any change to the number of parameters of the given convolutional layer, nor its underlying operation. Small changes in the expected size can occur, however, due to padding or strides performed by the layer. It was observed by practitioners and previous works that a network trained on a specific input dimension can still be used at inference using a modified image size to some extent (Simonyan & Zisserman, 2014). Moreover, evaluating with an image size that is larger than used for training can improve accuracy up to a threshold, after which it quickly deteriorates (Touvron et al., 2019).
+
+Recently, Tan & Le (2019) showed a computational-vs-accuracy trade-off in scaling image size used to train and evaluate with a convolutional network. This finding is consistent with past findings, which demonstrated that training with a larger image size can result in a larger classification error (Szegedy et al., 2016; Huang et al., 2018). In addition, previous works explored the notion of “progressive resizing” (Karras et al., 2017; Howard, 2018) — increasing image size as training progresses to improve model performance and time to convergence. More recently, Touvron et al. (2019) demonstrated that CNNs can be trained using a fixed small image size and fine-tuned posttraining to a larger size, with which evaluation will be performed. This procedure reduced the traintest discrepancy caused by the change in image size and allowed faster training time and improved accuracy — at the cost of additional fine-tuning procedure and additional computations at inference time.
+
+In this work we will further explore the notion of using multiple image sizes at training, so the CNN performance will be resilient to test time changes in the image size.
+
+# 2.2 LARGE BATCH TRAINING OF DEEP NETWORKS
+
+Deep neural network training can be distributed across many computational units and devices. The most common distribution method is by ”data-parallelism”—computing an average estimate of the gradients using multiple, separably computed data samples. As training NN models is done using batch-SGD method and its variants, scaling this process across more computational devices while maintaining similar utilization for each device inflates the global batch size.
+
+Large batch training is known to affect the generalization capabilities of the networks and to require modification of the regime used for its optimization. While several works claimed that large-batch training leads to an inherent ”generalization gap” (Keskar et al., 2016), more recent works demonstrated that this gap is largely caused from an insufficient number of optimization steps performed and can be partly mitigated by hyper-parameter tuning (Hoffer et al., 2017; Shallue et al., 2018). In order to cope with the changes in the training dynamics of the network, several modifications to the optimization procedure have been proposed such as a linear (Goyal et al., 2017) or a square-root (Hoffer et al., 2017) scaling of the learning rate with respect to the batch size growth. Other modifications include per-layer gradient scaling schemes (You et al., 2017) and optimizer modifications (Ginsburg et al., 2019). Several works also explored using incremented batch-sizes (Smith et al., 2018) in order to decrease the number of training iterations required to reach the desired accuracy.
+
+Recent work by Hoffer et al. (2019) introduced the notion of ”Batch Augmentation” (BA)— increasing the batch size by augmenting several instances of each sample within the same batch. BA aids generalization across a wide variety of models and tasks, with the expense of an increased computational effort per step. A similar method called “Repeated Augmentation” (RA) was proposed by Berman et al. (2019). It was also demonstrated that BA may allow to decrease the number of training steps needed to achieve a similar accuracy and also mitigate I/O throughput bottlenecks (Choi et al., 2019). As previous works investigated mostly homogeneous training settings (e.g., using a fixed batch size), an open question still exists on the utility of rapidly varying batch-sizes. We will explore this notion and suggest a new optimizer modification that enables training with multiple varying batch-sizes with limited hyper-parameter tuning.
+
+# 3 MIXSIZE: TRAINING WITH MULTIPLE IMAGE SCALES
+
+The traditional practice of training convolutional networks using fixed-size images holds several shortcomings. First, CNNs are commonly evaluated using a different size than that used for training (Lin et al., 2017; Redmon & Farhadi, 2018; He et al., 2017) and it was observed that classification accuracy may degrade above or below a certain size threshold (Touvron et al. (2019) and Figure 1). To remedy these issues, we suggest a stochastic training regime, where image sizes can change in each optimization step.
+
+Motivation. In order to motivate our method, we first evaluate the impact of the image size on the training progress of a CNN — by examining gradient statistics during training2. Specifically, in Table 1 we measured the correlation of the gradients across image sizes. We see that gradients computed across different scales of the same image have a strong correlation compared to those obtained across different images. This correlation is especially apparent during the first stages of training and decreases as the model converges. This suggests that the small image gradients can be used as an approximation of the full image gradients, with a smaller computational footprint. Therefore, using large images along the entire training process may be sub-optimal in terms of computational resource utilization. More specifically, as the gradients of images of different size are highly correlated at the initial steps of training, it may prove beneficial to sacrifice spatial size in favor of batch size that can be increased. To do so, we suggest the following.
+
+The MixSize training regime. We suggest ”MixSize”, a stochastic training regime, where input sizes can vary in each optimization step. In this regime, we modify the spatial dimensions $H , W$ (height and width) of the input image size3, as well as the batch size. The batch size is changed either by the number of samples used, denoted $B$ , or the number of batch-augmentations for each sample (Hoffer et al., 2019), denoted $D$ (”duplicates”). To simplify our notation and use-cases, we will follow the common practice of training on square images and use $S = H = W$ . Formally, in the MixSize regime, these sizes can be described as random variables sharing a single discrete distribution
+
+$$
+( \hat { S } , \hat { B } , \hat { D } ) = \left\{ ( S , B , D ) _ { i } ~ w . p . ~ p _ { i } \right\} ,
+$$
+
+where $\forall i : p _ { i } \geq 0$ and $\textstyle \sum _ { i } p _ { i } = 1$ .
+
+Table 1: ResNet-44 gradient correlation on CIFAR10. We measure the Spearman correlation coefficient $\rho$ between different spatial size of random images $\rho \left( x ^ { ( s _ { 1 } ) } , x ^ { ( s _ { 2 } ) } \right)$ , as well as non-identical random images of the same size $\rho \left( x ^ { ( s _ { 1 } ) } , y ^ { ( s _ { 1 } ) } \right)$ . We also compute the variance $V ( x )$ for the gradients of each spatial size.   
+
+<table><tr><td rowspan="2">Measure</td><td colspan="3">Network State</td></tr><tr><td>Initial</td><td>Partially Trained</td><td>Fully Trained</td></tr><tr><td>Epoch Test Accuracy</td><td>1 55.12%</td><td>50 87.56%</td><td>100 92.62%</td></tr><tr><td>p(x(32）,x(24))</td><td>0.2</td><td>0.08</td><td>0.03</td></tr><tr><td>(x(32),y(32)) p</td><td>0.086</td><td>0.02</td><td>-0.004</td></tr><tr><td>V (x(32)）</td><td>1.03e-6</td><td>1.44e-6</td><td>6.24e-7</td></tr><tr><td>V （(x(24））</td><td>1.95e-6</td><td>6.34e-6</td><td>2.26e-5</td></tr></table>
+
+As the computational cost of each training step is approximately proportional to $S ^ { 2 } { \cdot } B { \cdot } D$ , we choose these sizes to reflect an approximately fixed budget for any choice $i$ such that $S _ { i } ^ { 2 } B _ { i } D _ { i } \approx C o n s t$ Thus the computational and memory requirements for each step are constant.
+
+Benefits and Trade-offs. We will demonstrate that using such a MixSize regime can have a positive impact on the resiliency of trained networks to the image size used at evaluation. That is, mixed-size networks will be shown to have better accuracy across a wide range of sizes. This entails a considerable saving in computations needed for inference, especially when using smaller models. Furthermore, given a fixed budget of computational and time resources (per step), we can now modify our regime along spatial and batch axes. We will explore two trade-offs:
+
+• Decrease number of iterations per epoch – by enlarging $B$ at the expense of $S$ • Improve generalization per epoch – by enlarging $D$ at the expense of $S$ .
+
+# 4 IMPROVED TRAINING PRACTICES FOR MIXSIZE
+
+MixSize regimes continuously change the statistics of the model’s inputs, by modifying the image size as well as batch-size. This behavior may require hyper-parameter tuning and may also affect size-dependent layers such as batch normalization (Ioffe & Szegedy, 2015). To easily adapt training regimes to the use of MixSize as well as improve their final performance, we continue to describe two methods we found useful: Gradient Smoothing and Batch-norm calibration.
+
+# 4.1 GRADIENT SMOOTHING
+
+Training with varying batch and spatial sizes inadvertently leads to a change in the variance of the accumulated gradients. For example, in Table 1, the gradient variance is larger when computed over a small image size (unsurprisingly). This further suggests that the optimization regime should be adapted to smaller spatial sizes, in a manner similar to learning-rate adaptations that are used for large-batch training. This property was explored in previous works concerning large-batch regimes, in which a learning rate modification was suggested to compensate for the variance reduction for larger batch-sizes. Unfortunately, the nature of this modification can vary from task to task or across models (Shallue et al., 2018), with solutions such as a square-root scaling (Hoffer et al., 2017), linear scaling (Goyal et al., 2017) or a fixed norm ratio (You et al., 2017). Here we suggest changing both the spatial size as well as the batch size, which is also expected to modify the variance of gradients within each step and further complicates the choice of optimal scaling.
+
+Previous works suggested methods to control the gradient norm by gradient normalization (Hazan et al., 2015) and gradient clipping (Pascanu et al., 2013). These methods explicitly disable or limit the gradient’s norm used for each optimization step, but also limit naturally occurring variations in gradient statistics. We suggest an alternative solution to previous approaches, which we refer to as ”Gradient smoothing”. Gradient smoothing mitigates the variability of gradient statistics when image sizes are constantly changing across training.
+
+We introduce an exponentially moving weighted average of the gradients’ norm $\bar { g } _ { t }$ (scalar) which is updated according to
+
+$$
+\bar { g } _ { t } = \alpha \bar { g } _ { t - 1 } + ( 1 - \alpha ) g _ { t }
+$$
+
+where
+
+$$
+g _ { t } = \bigg | \bigg | \frac { \partial E } { \partial w _ { t } } \bigg | \bigg | _ { 2 } \mathrm { a n d } \bar { g } _ { 0 } = g _ { 0 } .
+$$
+
+We normalize the gradients used for each step by the smoothing coefficient, such that each consecutive step is performed with gradients of similar norm. For example, for the vanilla SGD step, we use a weight update rule of the form
+
+$$
+w _ { t + 1 } = w _ { t } - \eta \frac { \bar { g } _ { t } } { g _ { t } } \frac { \partial E } { \partial w _ { t } } .
+$$
+
+This running estimate of gradient norm is similar to the optimizer suggested by Ginsburg et al. (2019), which keeps a per-layer estimate of gradient moments. Gradient smoothing, however, is designed to adapt globally (across all layers) to the batch and spatial size modification and can be used regardless of the optimization method used.
+
+We found gradient smoothing to be mostly beneficial in regimes where multiple varying batch sizes are used. Figure 5a in the Appendix demonstrates how gradient smoothing reduces the gap between gradient norms of different sizes. Measuring test error on the same model shows a slight advantage for gradient-smoothing (Appendix Figure 5b).
+
+# 4.2 BATCH-NORM CALIBRATION FOR VARYING IMAGE SIZES
+
+As demonstrated by Touvron et al. (2019), using a different image size at evaluation may incur a discrepancy between training and evaluation protocols, caused by using different data pre-processing. Touvron et al. (2019) suggested a post-training procedure, where a network trained on a specific fixed-size is fine-tuned on another size, later used for evaluation. Their solution required 10s of training epochs, amounting to 1000s of full forward and back-propagation computations, along with parameter updates for batch-norm and classifier layers. In contrast, we surmise that for networks trained with mixed-regimes, discrepancy issues mainly arise from the use of the batch-norm layers (Ioffe & Szegedy, 2015) and can be solved by targeting them specifically.
+
+Batch-norm layers introduce a discrepancy between training and test evaluations (Ioffe, 2017), as at inference a running estimate of the mean and variance (of training data) are used instead of the actual mean and variance values. This difference is emphasized further in the use of varying image size, as changing the spatial size of an input map can significantly modify the measured variance of that map. While a fine-tuning process per image size can eliminate this discrepancy (Touvron et al., 2019), we offer a simpler alternative. For each evaluated size, we calibrate the mean and variance estimates used for that size by computing an average value over a small number of training examples. This calibration requires only a few (100s) feed-forward operations with no back-propagation or parameter update and takes only a few seconds on a single GPU.
+
+Interestingly, we highlight the fact that although this process has little or no effect on models trained using a fixed-size input, it does improve our mixed-size models considerably on a wide range of image sizes.
+
+# 5 EXPERIMENTS
+
+# 5.1 MIXSIZE WITH A FIXED IMAGE SIZE AT TEST-TIME: THE SPEED-ACCURACY TRADE-OFF
+
+CIFAR10/100. First, we examine our method using the common visual datasets CIFAR10/100 (Krizhevsky, 2009) that consist of $3 2 \times 3 2$ color images. We use the ResNet-44 model suggested by (He et al., 2016), Wide Resnet WRN-28-10 (Zagoruyko, 2016) and AmoebaNet (Real et al., 2019) with their original regime and batch size of 64. While for ResNet-44 we use the original augmentation protocol, we apply cutout (DeVries & Taylor, 2017) and auto-augment policies (Cubuk et al., 2018) on WRN-28-10 and AmoebaNet for both datasets (see Appendix A.1 for details).
+
+![](images/b136ba7cf484e737618f6844d373518d17c7d6b2033803cccf0222271dd940bc.jpg)  
+Figure 2: Training (dotted) and test accuracy vs optimization step (ResNet44, CIFAR10). We compare vanilla training with two computationally equivalent stochastic regimes: increased duplicates $\bar { ( } D ^ { + } )$ and increased batch $( B ^ { + } )$ . $\bar { B ^ { + } }$ regime achieves better test accuracy at a reduced number of iterations, while $D ^ { + }$ improves accuracy further at a similar computational cost.
+
+As CIFAR datasets are limited in size, we consider the following balanced stochastic regime chosen:
+
+$$
+S = \left\{ \begin{array} { l l l } { 4 0 , } & { \mathrm { w . p . } } & { p = 0 . 2 } \\ { 3 2 , } & { \mathrm { w . p . } } & { p = 0 . 3 } \\ { 2 4 , } & { \mathrm { w . p . } } & { p = 0 . 3 } \\ { 1 6 , } & { \mathrm { w . p . } } & { p = 0 . 2 } \end{array} \right.
+$$
+
+The regime was designed to be centered around the mean value of 28. As the original image size used for training is $3 2 \times 3 2$ , we are now able to increase either the batch size or number of duplicates for each training step by a factor of 322S2 such that $S ^ { 2 } \cdot B \cdot D$ is approximately constant. We denote our modified mixed-size regimes as $B ^ { + }$ for an increased effective batch-size and $D ^ { + }$ for an increased number of BA duplicates of the same ratio. We used our sampling strategy to train and compare our regime to the baseline results. We use the original hyper-parameters without modification. For the $B ^ { + }$ regime, use our gradient smoothing method, as described in Section 4.1. For each result, we measure our final test accuracy on the original $3 2 \times 3 2$ image size. We also perform batch-norm calibration as described in Section 4.2. From Table 2, we see that our MixSize regimes on CIFAR datasets yield two possible improvements:
+
+• Reduced number of training steps to achieve a similar test accuracy using $B ^ { + }$ regime.   
+• Better test accuracy when using $D ^ { + }$ regime.
+
+Training progress on the CIFAR10 using ResNet44 is depicted in Figure 2. Interestingly, although designed only to reduce training time, we can see that our $B ^ { + }$ regime also improves accuracy in some cases. This improvement can be attributed to a regularization effect induced by changing image sizes during training, also manifested by an increase in training error throughout its progress.
+
+ImageNet. We also perform large scale experiments using the ImageNet dataset (Deng et al., 2009) to confirm our findings. We used the ResNet-50 (He et al., 2016) model, with the training regime suggested by Goyal et al. (2017) that consists of base learning rate of 0.1, decreased by a factor of 10 on epochs 30, 60, 80, stopping at epoch 90. We used the base batch size of 256 over 4 devices and $L _ { 2 }$ regularization over weights of convolutional layers. We used the standard data augmentation and did not incorporate any additional regularization or augmentation techniques.
+
+Table 2: Test accuracy (Top-1) results for CIFAR and ImageNet. Each row represents models trained using the same computational and memory budget per step. Steps and accuracy are reported at the completion of a fixed epoch budget (e.g., 90 epochs for ResNet on ImageNet, 200 for ResNet on CIFAR). Accuracy is reported for model’s original size (32 for CIFAR, 224 for ImageNet).   
+
+<table><tr><td rowspan="2">Network</td><td rowspan="2">Dataset</td><td colspan="3">Steps</td><td colspan="3">Accuracy</td></tr><tr><td>Baseline</td><td>B+</td><td>D+</td><td>Baseline</td><td>B+</td><td>D+</td></tr><tr><td>ResNet-44</td><td>CIFAR10</td><td>156K</td><td>109K</td><td>156K</td><td>92.84%</td><td>94.30%</td><td>94.46%</td></tr><tr><td>WRN-28-10</td><td>CIFAR10</td><td>156K</td><td>109K</td><td>156K</td><td>96.60%</td><td>97.28%</td><td>97.68%</td></tr><tr><td>AmoebaNet</td><td>CIFAR10</td><td>469K</td><td>328K</td><td>469K</td><td>98.16%</td><td>98.14%</td><td>98.32%</td></tr><tr><td>ResNet-44</td><td>CIFAR100</td><td>156K</td><td>109K</td><td>156K</td><td>70.36%</td><td>72.19%</td><td>73.10%</td></tr><tr><td>WRN-28-10</td><td>CIFAR100</td><td>156K</td><td>109K</td><td>156K</td><td>79.85%</td><td>83.08%</td><td>83.52%</td></tr><tr><td>ResNet-50</td><td>ImageNet</td><td>450K</td><td>169K</td><td>450K</td><td>76.40%</td><td>76.61%</td><td>78.04%</td></tr><tr><td>EfficientNet-B0</td><td>ImageNet</td><td>1000K</td><td>376K</td><td>1000K</td><td>76.32%</td><td>76.29%</td><td>76.53%</td></tr></table>
+
+Additionally, we also used the EfficientNet-B0 model suggested by Tan & Le (2019). We used the same data augmentation and regularization as the original paper, but opted for a shorter training regime with a momentum-SGD optimizer that consisted of a cosine-annealed learning rate (Loshchilov & Hutter, 2016) over 200 epochs starting from an initial base 0.1 value.
+
+For the ImageNet dataset, we use the following stochastic regime found by cross-validation on several alternatives (see Appendix D):
+
+$$
+S ^ { ( 1 4 4 ) } : S = { \left\{ \begin{array} { l l } { 2 5 6 , } & { { \mathrm { w . p } } p = 0 . 1 } \\ { 2 2 4 , } & { { \mathrm { w . p } } p = 0 . 1 } \\ { 1 2 8 , } & { { \mathrm { w . p } } p = 0 . 6 } \\ { 9 6 , } & { { \mathrm { w . p } } p = 0 . 2 } \end{array} \right. }
+$$
+
+While the original training regime consisted of images of size $2 2 4 \times 2 2 4$ , our proposed regime makes for an average image size of $\bar { S } \times \bar { S } = 1 4 4 \times 1 4 \bar { 4 }$ . This regime was designed so that the reduced spatial size can be used to increase the corresponding batch size or the number of BA duplicates, as described in Section 3. We are first interested in accelerating the time needed for convergence of the tested models using our $B ^ { + }$ scheme. We enlarge the batch size used for each spatial size by a factor of 2242 such that $S ^ { 2 } \cdot B$ is kept approximately fixed. As the average batch size is larger than $B _ { o }$ , which was used with the original optimization hyper-parameters, we scale the learning rate linearly as suggested by Goyal et al. (2017) by a factor of $\bar { \frac { B } { B _ { o } } }$ . We note that for the proposed regimes we did not require any learning rate warm-up, due to the use of gradient smoothing.
+
+As can be seen in Figure 3, regime $B ^ { + }$ enables training with approximately $2 . 7 \times$ less training steps, while reaching a better-than-baseline accuracy of $7 6 . 6 1 \%$ . As sizes were chosen to reflect in approximately equal computational cost per iteration, $B ^ { + }$ regime offers a similar improvement in total wall-clock time.
+
+Next, we perform a similar experiment with a $D ^ { + }$ regime, where the number of BA duplicates is similarly increased with respect to $D _ { o }$ instead of the batch size. This scaling results with an average duplicates of $\bar { D } = 3$ .
+
+![](images/1a84b5090d53fd568c012d510a34e8d7fc621440369594acf75a993e3fa628b9.jpg)  
+Figure 3: Training (dotted) and test accuracy on ImageNet using the Baseline, $B ^ { + }$ and $D ^ { \bar { + } }$ regimes $2 2 4 \times 2 2 4$ evaluation size). All regimes required similar computational resources per step. $B ^ { + }$ regime required $\approx 2 . 7 \times$ less steps per epoch.
+
+As the computational cost for each step remains approximately constant, as well as the number of required steps per epochs, training a model under this regime requires an equal wall-clock
+
+time. However, the increased batch-augmentation improves the final test accuracy to $7 8 . 0 4 \%$ , approximately $7 \%$ relative improvement over the $7 6 . 4 \%$ baseline.
+
+# 5.2 INCREASING MODEL RESILIENCY TO TEST-TIME CHANGES IN IMAGE SIZE
+
+Next, we examine how MixSize affects the resulting model resiliency to changes in the image size during test-time. We evaluated the models by varying the test-time image sizes around the original 224 spatial size: $S = 2 2 4 + 3 2 \cdot m , m \in \{ - \bar { 6 } , . . . , 6 \}$ . The common evaluation procedure for ImageNet models first scales the image to a 256 smallest dimension and crops a center $2 2 4 \times 2 2 4$ image. We adapt this regime for other image sizes by scaling the smallest dimension to $\lfloor { \frac { 8 } { 7 } } S \rfloor$ (since $\frac { 8 } { 7 } \cdot 2 2 4 = 2 5 6 )$ and then cropping the center $S \times S$ patch. Models trained with a mixed regime were calibrated to a specific evaluation size by measuring batch-norm statistics for 200 batches of training samples. We note that for original fixed-size regimes this calibration procedure resulted with degraded results and so we report accuracy without calibration for these models. We did not use any fine-tuning procedure post training for any of the models.
+
+As can be seen in Figure 4a, the baseline model trained using a fixed size, reaches $7 6 . 4 \%$ top-1 accuracy at the same 224 spatial size it was trained on. As observed previously, the model continues to slightly improve beyond that size, to a maximum of $7 6 . 8 \%$ accuracy. However, it is apparent that the model’s performance quickly degrades when evaluating with sizes smaller than 224.
+
+We compare these results with a $D ^ { + }$ regime, trained with an average size of $\bar { S } = 1 4 4$ . As described earlier, this model requires the same time and computational resources as the baseline model. However, due to the decreased average size, we were able to leverage more than 1 duplicates per batch on average, which improved the model’s top-1 accuracy to $7 7 . 1 4 \%$ at size 224. Furthermore, we find that the model performs much more favorably at image sizes smaller than 224, scoring an improved (over baseline) accuracy of $7 6 . 4 3 \%$ at only $1 6 0 \times 1 6 0$ spatial size. We analyzed an alternative regime $S ^ { ( 2 0 8 ) }$ , where the average spatial size is larger at $2 0 8 \times 2 0 8$ (for more details see Appendix D). The model trained with the $S ^ { ( 2 0 8 ) }$ regime offers a similar improvement in accuracy, only across a larger spatial size, as it observed an average size of $2 0 8 \times 2 0 8$ during training. Figure 4a demonstrates that while all three models (Fixed with $S = 2 2 4$ , $S ^ { ( 1 4 4 ) }$ and $S ^ { ( 2 0 8 ) }$ ) were trained with the same compute and memory budget, mixed-size regimes offer superior accuracy over a wide range of evaluation sizes. Specifically, mixed-regime at $S = 2 0 8$ dominates the baseline fixed-size regime at all sizes, while our mixed regime at $S = 1 4 4$ achieves best results at sizes smaller than 224.
+
+![](images/db2b2b989b584c1ba941925f5f6a2fddc0143307bbbf6bef492dca374db7de46.jpg)  
+Figure 4: Left: Test accuracy on validation set per image size, all models trained using the same computational and memory resources (regime $D ^ { + }$ ). Right: Test accuracy per billion flop (at evaluation).
+
+We also compared the classification performance across evaluated image sizes, using networks trained on a variety of fixed sizes and our mixed regimes. As a baseline, we use results obtained by Touvron et al. (2019) (trained with repeated augmentations, without fine-tuning) and compare them with mixed-regime models trained with an equal computational budget, by setting the base number of BA duplicates to $D = 2$ . As can be seen in Figure 4b, mixed-regime trained models offer a wider range of resolutions with close-to-baseline accuracy (within a $2 \%$ change) and perform better than their fixed-size counterparts at all sizes. As the number of floating-point operations (flops) grows linearly with the number of pixels, using a mixed regime significantly improves accuracy per compute at evaluation. We further note that our $S ^ { ( 2 2 4 ) }$ model reaches a top accuracy of $7 9 . 2 7 \%$ at a $2 8 8 \times 2 8 8$ evaluation size.
+
+# 6 SUMMARY
+
+In this work, we introduced and examined a performance trade-off between computational load and classification accuracy governed by the input’s spatial size. We suggested stochastic image size regimes, which randomly change the spatial dimension as well as the batch size and the number of augmentation (duplicates) in the batch. Stochastic regime benefits are threefold: (1) reduced number of training iterations; or (2) improved model accuracy (generalization) and (3) improved model robustness to changing the image size. We believe this approach may have a profound impact on the practice of training convolutional networks. Given a computational and time budget, stochastic size regimes may enable to train networks faster, with better results, as well as to target specific image sizes that will be used at test time. As the average size chosen to train is reflected in the optimal operating point for evaluation resolution, mixed regimes can be used to create networks with better performance across multiple designated use cases.
+
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+# Appendix
+
+A EXPERIMENTAL SETTINGS
+
+A.1 CIFAR
+
+We used the common data augmentation technique as described by He et al. (2016). In this method, the input image is padded with 4 zero-valued pixels at each side, top and bottom. A random $3 2 \times 3 2$ part of the padded image is then cropped and with a 0.5 probability flipped horizontally. In order to adapt to varying input scales, we add an additional augmentation step, that resizes the images using bilinear interpolation to $S \times S$ , depending on a sampled size for each step. We note that this keeps the exact original augmentation procedure for $S = 3 2$ .
+
+# B IMPACT OF GRADIENT SMOOTHING
+
+(a) Gradient norm values with and without Gradsmoothing. Solid lines are gradients for $S ~ = ~ 3 2$ while dotted lines are for $S = 1 6$ .
+
+![](images/3b5a4bec38369a3a9f08bcde93793589da2f3699c013203d4024b4b0aa2b6387.jpg)  
+(b) Training and test error with and with out gradient smoothing. Solid lines are test errors while dotted lines are for training.
+
+![](images/b880a0f1614ff8f14f091c95c236b578af67e6f91fb93f032cd8ddadba161c4b.jpg)
+
+Figure 5: Impact of gradient smoothing on CIFAR10, ResNet-44. The training regime includes two image sizes: $3 2 \times 3 2$ and $1 6 \times 1 6$ (average size is $S = 2 4$ ). Using a $B ^ { + }$ regime creates two batch sizes: 256 and 2, 048 respectively. Gradient smoothing helps to reduce gap between gradient norms at difference batch sizes and improves final accuracy.
+
+# C VARYING IMAGE-SIZE TRAINING REGIMES
+
+We wish to consider training regimes with varying image sizes, such that the average image size is smaller than the desired evaluation size. For example, for the height dimension $H$ , we wish to obtain an average size of $\begin{array} { r } { \bar { H } = \sum _ { i } p _ { i } H _ { i } } \end{array}$ such that $\bar { H } < \mathbf { \bar { \Gamma } } H _ { o }$ . We consider three alternatives for image size variations:
+
+• Increase image size from small to large, where each image size is used for number of epochs $E _ { i } = p _ { i } E _ { \mathrm { t o t a l } }$ , where $E _ { \mathrm { t o t a l } }$ is the total number training epochs required. • Using a random image size for each epoch, keeping the epoch number for each size at $E _ { i }$ • Sampling image size per training step at probability $p _ { i }$
+
+As can be seen in Figure 6, we found that random sampling regimes performed better than scaling image size from small to large (Howard, 2018; Touvron et al., 2019). While sampling both at epoch and step time frames performed similarly, replacing sizes on each step seemed to converge faster and to have less noise in measured test accuracy. We note that these behaviours may partly stem from the use of batch-normalization (Ioffe & Szegedy, 2015) which is sensitive to the image size used at evaluation or insufficient hyper-parameter tuning for each specific size (e.g., spiking error at the end of the small-to-large regime). Considering these findings, we continue to perform our experiments using the third regime – sampling image size per training step.
+
+![](images/d9d9dea62f1564c1593f40644bbbaff8dfb4165f7fd1b7a845f4ec6fddccade1.jpg)  
+Figure 6: Test accuracy vs step for 3 size sampling regimes: (1) From small to large (2) Sample each Epoch (3) Sample each step. All methods reached a similar accuracy, but sampling each epoch was less noisy and did not require hyper-parameter tuning.
+
+# D ALTERNATIVE SIZE DISTRIBUTION REGIMES
+
+We used alternative size regimes balanced around 224, named $S ^ { ( 2 0 8 ) }$ and $S ^ { ( 2 2 4 ) }$ . They can be described by the following distributions:
+
+$$
+S ^ { ( 2 0 8 ) } : ~ S = \left\{ \begin{array} { l l } { 3 2 0 , } & { \mathrm { w . p } ~ p = 0 . 1 } \\ { 2 8 8 , } & { \mathrm { w . p } ~ p = 0 . 1 } \\ { 2 5 6 , } & { \mathrm { w . p } ~ p = 0 . 1 } \\ { 2 2 4 , } & { \mathrm { w . p } ~ p = 0 . 2 } \\ { 1 9 2 , } & { \mathrm { w . p } ~ p = 0 . 2 } \\ { 1 6 0 , } & { \mathrm { w . p } ~ p = 0 . 1 } \\ { 1 2 8 , } & { \mathrm { w . p } ~ p = 0 . 1 } \\ { 9 6 , } & { \mathrm { w . p } ~ p = 0 . 1 } \end{array} \right.
+$$

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+        "type": "text",
+        "text": "MIX & MATCH: TRAINING CONVNETS WITH MIXED IMAGE SIZES FOR IMPROVED ACCURACY, SPEED AND SCALE RESILIENCY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "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,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional neural networks (CNNs) are commonly trained using a fixed spatial image size predetermined for a given model. Although trained on images of a specific size, it is well established that CNNs can be used to evaluate a wide range of image sizes at test time, by adjusting the size of intermediate feature maps. ",
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+        "type": "text",
+        "text": "In this work, we describe and evaluate a novel mixed-size training regime that mixes several image sizes at training time. We demonstrate that models trained using our method are more resilient to image size changes and generalize well even on small images. This allows faster inference by using smaller images at test time. For instance, we receive a $7 6 . 4 3 \\%$ top-1 accuracy using ResNet50 with an image size of 160, which matches the accuracy of the baseline model with $2 \\times$ fewer computations. Furthermore, for a given image size used at test time, we show this method can be exploited either to accelerate training or the final test accuracy. For example, we are able to reach a $7 9 . 2 7 \\%$ accuracy with a model evaluated at a 288 spatial size for a relative improvement of $1 4 \\%$ over the baseline. Our PyTorch implementation and pre-trained models are publicly available1 ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional neural networks are successfully used to solve various tasks across multiple domains such as visual (Krizhevsky et al., 2012; Ren et al., 2015), audio (van den Oord et al., 2016), language (Gehring et al., 2017) and speech (Abdel-Hamid et al., 2014). While scale-invariance is considered important for visual representations (Lowe, 1999), convolutional networks are not scale invariant with respect to the spatial resolution of the image input, as a change in image dimension may lead to a non-linear change of their output. Even though CNNs are able to achieve state-of-the",
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+        "type": "image",
+        "img_path": "images/1b579338f11246697fb783b19c8d0953fb640ad92da56184477f7a81c9df594d.jpg",
+        "image_caption": [
+            "Figure 1: Test accuracy per image size, models trained on specific sizes (ResNet50, ImageNet). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "art results in many tasks and domains, their sensitivity to the image size is an inherent deficiency that limits practical use cases and requires evaluation inputs to match training image size. For example, Touvron et al. (2019) demonstrated that networks trained on specific image size, perform poorly on other image sizes at evaluation, as shown in Figure 1. ",
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+    {
+        "type": "text",
+        "text": "Several works attempted to achieve scale invariance by modifying the network structure (Xu et al., 2014; Takahashi et al., 2017). However, the most common method is to artificially enlarge the dataset using a set of label-preserving transformations also known as ”data augmentation” (Krizhevsky et al., 2012; Howard, 2013). Several of these transformations scale and crop objects appearing within the data, thus increasing the network’s robustness to inputs of different scale. ",
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+    {
+        "type": "text",
+        "text": "Although not explicitly trained to handle varying image sizes, CNNs are commonly evaluated on multiple scales post training, such as in the case of detection (Lin et al., 2017; Redmon & Farhadi, ",
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+    {
+        "type": "text",
+        "text": "2018) and segmentation (He et al., 2017) tasks. In these tasks, a network that was pretrained with fixed image size for classification is used as the backbone of a larger model that is expected to adapt to a wide variety of image sizes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we will introduce a novel training regime, “MixSize” for convolutional networks that uses stochastic image and batch sizes. The main contributions of the MixSize regime are: ",
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+    {
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+        "text": "• Reducing image size sensitivity. We show that the MixSize training regime can improve model performance on a wide range of sizes used at evaluation.   \n• Faster inference. As our mixed-size models can be evaluated at smaller image sizes, we show up to $2 \\times$ reduction in computations required at inference to reach the same accuracy as the baseline model.   \n• Faster training vs. high accuracy. We show that reducing the average image size at training leads to a trade-off between the time required to train the model and its final accuracy. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 USING MULTIPLE IMAGE SIZES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Deep convolutional networks are traditionally trained using fixed-size inputs, with spatial dimensions $H \\times W$ and a batch size $B$ . The network architecture is configured such that the spatial dimensions are reduced through strided pooling or convolutions, with the last classification layer applied on a $1 \\times 1$ spatial dimension. Modern convolutional networks usually conclude with a final ”global” average pooling (Lin et al., 2013; Szegedy et al., 2015), that reduces any remaining spatial dimensions with a simple averaging operation. Modifying the spatial size of an input to a convolutional layer by a factor $\\gamma$ , will yield an output with size scaled by the same factor $\\gamma$ . This modification does not require any change to the number of parameters of the given convolutional layer, nor its underlying operation. Small changes in the expected size can occur, however, due to padding or strides performed by the layer. It was observed by practitioners and previous works that a network trained on a specific input dimension can still be used at inference using a modified image size to some extent (Simonyan & Zisserman, 2014). Moreover, evaluating with an image size that is larger than used for training can improve accuracy up to a threshold, after which it quickly deteriorates (Touvron et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Recently, Tan & Le (2019) showed a computational-vs-accuracy trade-off in scaling image size used to train and evaluate with a convolutional network. This finding is consistent with past findings, which demonstrated that training with a larger image size can result in a larger classification error (Szegedy et al., 2016; Huang et al., 2018). In addition, previous works explored the notion of “progressive resizing” (Karras et al., 2017; Howard, 2018) — increasing image size as training progresses to improve model performance and time to convergence. More recently, Touvron et al. (2019) demonstrated that CNNs can be trained using a fixed small image size and fine-tuned posttraining to a larger size, with which evaluation will be performed. This procedure reduced the traintest discrepancy caused by the change in image size and allowed faster training time and improved accuracy — at the cost of additional fine-tuning procedure and additional computations at inference time. ",
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+    {
+        "type": "text",
+        "text": "In this work we will further explore the notion of using multiple image sizes at training, so the CNN performance will be resilient to test time changes in the image size. ",
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+    {
+        "type": "text",
+        "text": "2.2 LARGE BATCH TRAINING OF DEEP NETWORKS ",
+        "text_level": 1,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Deep neural network training can be distributed across many computational units and devices. The most common distribution method is by ”data-parallelism”—computing an average estimate of the gradients using multiple, separably computed data samples. As training NN models is done using batch-SGD method and its variants, scaling this process across more computational devices while maintaining similar utilization for each device inflates the global batch size. ",
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+        "text": "Large batch training is known to affect the generalization capabilities of the networks and to require modification of the regime used for its optimization. While several works claimed that large-batch training leads to an inherent ”generalization gap” (Keskar et al., 2016), more recent works demonstrated that this gap is largely caused from an insufficient number of optimization steps performed and can be partly mitigated by hyper-parameter tuning (Hoffer et al., 2017; Shallue et al., 2018). In order to cope with the changes in the training dynamics of the network, several modifications to the optimization procedure have been proposed such as a linear (Goyal et al., 2017) or a square-root (Hoffer et al., 2017) scaling of the learning rate with respect to the batch size growth. Other modifications include per-layer gradient scaling schemes (You et al., 2017) and optimizer modifications (Ginsburg et al., 2019). Several works also explored using incremented batch-sizes (Smith et al., 2018) in order to decrease the number of training iterations required to reach the desired accuracy. ",
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+    {
+        "type": "text",
+        "text": "Recent work by Hoffer et al. (2019) introduced the notion of ”Batch Augmentation” (BA)— increasing the batch size by augmenting several instances of each sample within the same batch. BA aids generalization across a wide variety of models and tasks, with the expense of an increased computational effort per step. A similar method called “Repeated Augmentation” (RA) was proposed by Berman et al. (2019). It was also demonstrated that BA may allow to decrease the number of training steps needed to achieve a similar accuracy and also mitigate I/O throughput bottlenecks (Choi et al., 2019). As previous works investigated mostly homogeneous training settings (e.g., using a fixed batch size), an open question still exists on the utility of rapidly varying batch-sizes. We will explore this notion and suggest a new optimizer modification that enables training with multiple varying batch-sizes with limited hyper-parameter tuning. ",
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+        "type": "text",
+        "text": "3 MIXSIZE: TRAINING WITH MULTIPLE IMAGE SCALES ",
+        "text_level": 1,
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+        "text": "The traditional practice of training convolutional networks using fixed-size images holds several shortcomings. First, CNNs are commonly evaluated using a different size than that used for training (Lin et al., 2017; Redmon & Farhadi, 2018; He et al., 2017) and it was observed that classification accuracy may degrade above or below a certain size threshold (Touvron et al. (2019) and Figure 1). To remedy these issues, we suggest a stochastic training regime, where image sizes can change in each optimization step. ",
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+        "text": "Motivation. In order to motivate our method, we first evaluate the impact of the image size on the training progress of a CNN — by examining gradient statistics during training2. Specifically, in Table 1 we measured the correlation of the gradients across image sizes. We see that gradients computed across different scales of the same image have a strong correlation compared to those obtained across different images. This correlation is especially apparent during the first stages of training and decreases as the model converges. This suggests that the small image gradients can be used as an approximation of the full image gradients, with a smaller computational footprint. Therefore, using large images along the entire training process may be sub-optimal in terms of computational resource utilization. More specifically, as the gradients of images of different size are highly correlated at the initial steps of training, it may prove beneficial to sacrifice spatial size in favor of batch size that can be increased. To do so, we suggest the following. ",
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+        "text": "The MixSize training regime. We suggest ”MixSize”, a stochastic training regime, where input sizes can vary in each optimization step. In this regime, we modify the spatial dimensions $H , W$ (height and width) of the input image size3, as well as the batch size. The batch size is changed either by the number of samples used, denoted $B$ , or the number of batch-augmentations for each sample (Hoffer et al., 2019), denoted $D$ (”duplicates”). To simplify our notation and use-cases, we will follow the common practice of training on square images and use $S = H = W$ . Formally, in the MixSize regime, these sizes can be described as random variables sharing a single discrete distribution ",
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+        "type": "equation",
+        "img_path": "images/cb2343be97af330271dba119bfddf253b2605ee09a10038a13682abec0cd62e0.jpg",
+        "text": "$$\n( \\hat { S } , \\hat { B } , \\hat { D } ) = \\left\\{ ( S , B , D ) _ { i } ~ w . p . ~ p _ { i } \\right\\} ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\forall i : p _ { i } \\geq 0$ and $\\textstyle \\sum _ { i } p _ { i } = 1$ . ",
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+    {
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+        "img_path": "images/d58e393545283a1324c2c02ce88c2cb44775308eee49f8ddf9f9b37e2370b081.jpg",
+        "table_caption": [
+            "Table 1: ResNet-44 gradient correlation on CIFAR10. We measure the Spearman correlation coefficient $\\rho$ between different spatial size of random images $\\rho \\left( x ^ { ( s _ { 1 } ) } , x ^ { ( s _ { 2 } ) } \\right)$ , as well as non-identical random images of the same size $\\rho \\left( x ^ { ( s _ { 1 } ) } , y ^ { ( s _ { 1 } ) } \\right)$ . We also compute the variance $V ( x )$ for the gradients of each spatial size. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Measure</td><td colspan=\"3\">Network State</td></tr><tr><td>Initial</td><td>Partially Trained</td><td>Fully Trained</td></tr><tr><td>Epoch Test Accuracy</td><td>1 55.12%</td><td>50 87.56%</td><td>100 92.62%</td></tr><tr><td>p(x(32）,x(24))</td><td>0.2</td><td>0.08</td><td>0.03</td></tr><tr><td>(x(32),y(32)) p</td><td>0.086</td><td>0.02</td><td>-0.004</td></tr><tr><td>V (x(32)）</td><td>1.03e-6</td><td>1.44e-6</td><td>6.24e-7</td></tr><tr><td>V （(x(24））</td><td>1.95e-6</td><td>6.34e-6</td><td>2.26e-5</td></tr></table>",
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+        "type": "text",
+        "text": "As the computational cost of each training step is approximately proportional to $S ^ { 2 } { \\cdot } B { \\cdot } D$ , we choose these sizes to reflect an approximately fixed budget for any choice $i$ such that $S _ { i } ^ { 2 } B _ { i } D _ { i } \\approx C o n s t$ Thus the computational and memory requirements for each step are constant. ",
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+        "type": "text",
+        "text": "Benefits and Trade-offs. We will demonstrate that using such a MixSize regime can have a positive impact on the resiliency of trained networks to the image size used at evaluation. That is, mixed-size networks will be shown to have better accuracy across a wide range of sizes. This entails a considerable saving in computations needed for inference, especially when using smaller models. Furthermore, given a fixed budget of computational and time resources (per step), we can now modify our regime along spatial and batch axes. We will explore two trade-offs: ",
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+        "type": "text",
+        "text": "• Decrease number of iterations per epoch – by enlarging $B$ at the expense of $S$ • Improve generalization per epoch – by enlarging $D$ at the expense of $S$ . ",
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+    {
+        "type": "text",
+        "text": "4 IMPROVED TRAINING PRACTICES FOR MIXSIZE ",
+        "text_level": 1,
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+        "text": "MixSize regimes continuously change the statistics of the model’s inputs, by modifying the image size as well as batch-size. This behavior may require hyper-parameter tuning and may also affect size-dependent layers such as batch normalization (Ioffe & Szegedy, 2015). To easily adapt training regimes to the use of MixSize as well as improve their final performance, we continue to describe two methods we found useful: Gradient Smoothing and Batch-norm calibration. ",
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+        "type": "text",
+        "text": "4.1 GRADIENT SMOOTHING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Training with varying batch and spatial sizes inadvertently leads to a change in the variance of the accumulated gradients. For example, in Table 1, the gradient variance is larger when computed over a small image size (unsurprisingly). This further suggests that the optimization regime should be adapted to smaller spatial sizes, in a manner similar to learning-rate adaptations that are used for large-batch training. This property was explored in previous works concerning large-batch regimes, in which a learning rate modification was suggested to compensate for the variance reduction for larger batch-sizes. Unfortunately, the nature of this modification can vary from task to task or across models (Shallue et al., 2018), with solutions such as a square-root scaling (Hoffer et al., 2017), linear scaling (Goyal et al., 2017) or a fixed norm ratio (You et al., 2017). Here we suggest changing both the spatial size as well as the batch size, which is also expected to modify the variance of gradients within each step and further complicates the choice of optimal scaling. ",
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+        "type": "text",
+        "text": "Previous works suggested methods to control the gradient norm by gradient normalization (Hazan et al., 2015) and gradient clipping (Pascanu et al., 2013). These methods explicitly disable or limit the gradient’s norm used for each optimization step, but also limit naturally occurring variations in gradient statistics. We suggest an alternative solution to previous approaches, which we refer to as ”Gradient smoothing”. Gradient smoothing mitigates the variability of gradient statistics when image sizes are constantly changing across training. ",
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+        "type": "text",
+        "text": "We introduce an exponentially moving weighted average of the gradients’ norm $\\bar { g } _ { t }$ (scalar) which is updated according to ",
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+        "img_path": "images/16fd49cc3435d9260f0664983a165e485a28eb049f58fd66485d3db2d8f4943b.jpg",
+        "text": "$$\n\\bar { g } _ { t } = \\alpha \\bar { g } _ { t - 1 } + ( 1 - \\alpha ) g _ { t }\n$$",
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+        "type": "text",
+        "text": "where ",
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+        "text": "$$\ng _ { t } = \\bigg | \\bigg | \\frac { \\partial E } { \\partial w _ { t } } \\bigg | \\bigg | _ { 2 } \\mathrm { a n d } \\bar { g } _ { 0 } = g _ { 0 } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We normalize the gradients used for each step by the smoothing coefficient, such that each consecutive step is performed with gradients of similar norm. For example, for the vanilla SGD step, we use a weight update rule of the form ",
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+        "img_path": "images/466cea2ccbb5e94eb0be33c01e0d63f748eb3d190a6b3f54dca8524cec459c3f.jpg",
+        "text": "$$\nw _ { t + 1 } = w _ { t } - \\eta \\frac { \\bar { g } _ { t } } { g _ { t } } \\frac { \\partial E } { \\partial w _ { t } } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This running estimate of gradient norm is similar to the optimizer suggested by Ginsburg et al. (2019), which keeps a per-layer estimate of gradient moments. Gradient smoothing, however, is designed to adapt globally (across all layers) to the batch and spatial size modification and can be used regardless of the optimization method used. ",
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+        "text": "We found gradient smoothing to be mostly beneficial in regimes where multiple varying batch sizes are used. Figure 5a in the Appendix demonstrates how gradient smoothing reduces the gap between gradient norms of different sizes. Measuring test error on the same model shows a slight advantage for gradient-smoothing (Appendix Figure 5b). ",
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+    {
+        "type": "text",
+        "text": "4.2 BATCH-NORM CALIBRATION FOR VARYING IMAGE SIZES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As demonstrated by Touvron et al. (2019), using a different image size at evaluation may incur a discrepancy between training and evaluation protocols, caused by using different data pre-processing. Touvron et al. (2019) suggested a post-training procedure, where a network trained on a specific fixed-size is fine-tuned on another size, later used for evaluation. Their solution required 10s of training epochs, amounting to 1000s of full forward and back-propagation computations, along with parameter updates for batch-norm and classifier layers. In contrast, we surmise that for networks trained with mixed-regimes, discrepancy issues mainly arise from the use of the batch-norm layers (Ioffe & Szegedy, 2015) and can be solved by targeting them specifically. ",
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+        "type": "text",
+        "text": "Batch-norm layers introduce a discrepancy between training and test evaluations (Ioffe, 2017), as at inference a running estimate of the mean and variance (of training data) are used instead of the actual mean and variance values. This difference is emphasized further in the use of varying image size, as changing the spatial size of an input map can significantly modify the measured variance of that map. While a fine-tuning process per image size can eliminate this discrepancy (Touvron et al., 2019), we offer a simpler alternative. For each evaluated size, we calibrate the mean and variance estimates used for that size by computing an average value over a small number of training examples. This calibration requires only a few (100s) feed-forward operations with no back-propagation or parameter update and takes only a few seconds on a single GPU. ",
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+        "type": "text",
+        "text": "Interestingly, we highlight the fact that although this process has little or no effect on models trained using a fixed-size input, it does improve our mixed-size models considerably on a wide range of image sizes. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "5.1 MIXSIZE WITH A FIXED IMAGE SIZE AT TEST-TIME: THE SPEED-ACCURACY TRADE-OFF ",
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+        "type": "text",
+        "text": "CIFAR10/100. First, we examine our method using the common visual datasets CIFAR10/100 (Krizhevsky, 2009) that consist of $3 2 \\times 3 2$ color images. We use the ResNet-44 model suggested by (He et al., 2016), Wide Resnet WRN-28-10 (Zagoruyko, 2016) and AmoebaNet (Real et al., 2019) with their original regime and batch size of 64. While for ResNet-44 we use the original augmentation protocol, we apply cutout (DeVries & Taylor, 2017) and auto-augment policies (Cubuk et al., 2018) on WRN-28-10 and AmoebaNet for both datasets (see Appendix A.1 for details). ",
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+    {
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+        "img_path": "images/b136ba7cf484e737618f6844d373518d17c7d6b2033803cccf0222271dd940bc.jpg",
+        "image_caption": [
+            "Figure 2: Training (dotted) and test accuracy vs optimization step (ResNet44, CIFAR10). We compare vanilla training with two computationally equivalent stochastic regimes: increased duplicates $\\bar { ( } D ^ { + } )$ and increased batch $( B ^ { + } )$ . $\\bar { B ^ { + } }$ regime achieves better test accuracy at a reduced number of iterations, while $D ^ { + }$ improves accuracy further at a similar computational cost. "
+        ],
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+        "type": "text",
+        "text": "As CIFAR datasets are limited in size, we consider the following balanced stochastic regime chosen: ",
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+        "img_path": "images/8a3c49504758bf154738e240183d5953b8d240a5af4b987662cdb735c7e5ef37.jpg",
+        "text": "$$\nS = \\left\\{ \\begin{array} { l l l } { 4 0 , } & { \\mathrm { w . p . } } & { p = 0 . 2 } \\\\ { 3 2 , } & { \\mathrm { w . p . } } & { p = 0 . 3 } \\\\ { 2 4 , } & { \\mathrm { w . p . } } & { p = 0 . 3 } \\\\ { 1 6 , } & { \\mathrm { w . p . } } & { p = 0 . 2 } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The regime was designed to be centered around the mean value of 28. As the original image size used for training is $3 2 \\times 3 2$ , we are now able to increase either the batch size or number of duplicates for each training step by a factor of 322S2 such that $S ^ { 2 } \\cdot B \\cdot D$ is approximately constant. We denote our modified mixed-size regimes as $B ^ { + }$ for an increased effective batch-size and $D ^ { + }$ for an increased number of BA duplicates of the same ratio. We used our sampling strategy to train and compare our regime to the baseline results. We use the original hyper-parameters without modification. For the $B ^ { + }$ regime, use our gradient smoothing method, as described in Section 4.1. For each result, we measure our final test accuracy on the original $3 2 \\times 3 2$ image size. We also perform batch-norm calibration as described in Section 4.2. From Table 2, we see that our MixSize regimes on CIFAR datasets yield two possible improvements: ",
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+    {
+        "type": "text",
+        "text": "• Reduced number of training steps to achieve a similar test accuracy using $B ^ { + }$ regime.   \n• Better test accuracy when using $D ^ { + }$ regime. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training progress on the CIFAR10 using ResNet44 is depicted in Figure 2. Interestingly, although designed only to reduce training time, we can see that our $B ^ { + }$ regime also improves accuracy in some cases. This improvement can be attributed to a regularization effect induced by changing image sizes during training, also manifested by an increase in training error throughout its progress. ",
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+    {
+        "type": "text",
+        "text": "ImageNet. We also perform large scale experiments using the ImageNet dataset (Deng et al., 2009) to confirm our findings. We used the ResNet-50 (He et al., 2016) model, with the training regime suggested by Goyal et al. (2017) that consists of base learning rate of 0.1, decreased by a factor of 10 on epochs 30, 60, 80, stopping at epoch 90. We used the base batch size of 256 over 4 devices and $L _ { 2 }$ regularization over weights of convolutional layers. We used the standard data augmentation and did not incorporate any additional regularization or augmentation techniques. ",
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+    {
+        "type": "table",
+        "img_path": "images/4e3a7c912d4a51677c2c8c31e11539b8e268dcf91ea39d6b7bebcf9b0045a728.jpg",
+        "table_caption": [
+            "Table 2: Test accuracy (Top-1) results for CIFAR and ImageNet. Each row represents models trained using the same computational and memory budget per step. Steps and accuracy are reported at the completion of a fixed epoch budget (e.g., 90 epochs for ResNet on ImageNet, 200 for ResNet on CIFAR). Accuracy is reported for model’s original size (32 for CIFAR, 224 for ImageNet). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Network</td><td rowspan=\"2\">Dataset</td><td colspan=\"3\">Steps</td><td colspan=\"3\">Accuracy</td></tr><tr><td>Baseline</td><td>B+</td><td>D+</td><td>Baseline</td><td>B+</td><td>D+</td></tr><tr><td>ResNet-44</td><td>CIFAR10</td><td>156K</td><td>109K</td><td>156K</td><td>92.84%</td><td>94.30%</td><td>94.46%</td></tr><tr><td>WRN-28-10</td><td>CIFAR10</td><td>156K</td><td>109K</td><td>156K</td><td>96.60%</td><td>97.28%</td><td>97.68%</td></tr><tr><td>AmoebaNet</td><td>CIFAR10</td><td>469K</td><td>328K</td><td>469K</td><td>98.16%</td><td>98.14%</td><td>98.32%</td></tr><tr><td>ResNet-44</td><td>CIFAR100</td><td>156K</td><td>109K</td><td>156K</td><td>70.36%</td><td>72.19%</td><td>73.10%</td></tr><tr><td>WRN-28-10</td><td>CIFAR100</td><td>156K</td><td>109K</td><td>156K</td><td>79.85%</td><td>83.08%</td><td>83.52%</td></tr><tr><td>ResNet-50</td><td>ImageNet</td><td>450K</td><td>169K</td><td>450K</td><td>76.40%</td><td>76.61%</td><td>78.04%</td></tr><tr><td>EfficientNet-B0</td><td>ImageNet</td><td>1000K</td><td>376K</td><td>1000K</td><td>76.32%</td><td>76.29%</td><td>76.53%</td></tr></table>",
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+        "type": "text",
+        "text": "Additionally, we also used the EfficientNet-B0 model suggested by Tan & Le (2019). We used the same data augmentation and regularization as the original paper, but opted for a shorter training regime with a momentum-SGD optimizer that consisted of a cosine-annealed learning rate (Loshchilov & Hutter, 2016) over 200 epochs starting from an initial base 0.1 value. ",
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+        "type": "text",
+        "text": "For the ImageNet dataset, we use the following stochastic regime found by cross-validation on several alternatives (see Appendix D): ",
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+    {
+        "type": "equation",
+        "img_path": "images/39279ab16c78cf30598e32070efbf810466cd1663f3a122c9691f08f8ef12a88.jpg",
+        "text": "$$\nS ^ { ( 1 4 4 ) } : S = { \\left\\{ \\begin{array} { l l } { 2 5 6 , } & { { \\mathrm { w . p } } p = 0 . 1 } \\\\ { 2 2 4 , } & { { \\mathrm { w . p } } p = 0 . 1 } \\\\ { 1 2 8 , } & { { \\mathrm { w . p } } p = 0 . 6 } \\\\ { 9 6 , } & { { \\mathrm { w . p } } p = 0 . 2 } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "While the original training regime consisted of images of size $2 2 4 \\times 2 2 4$ , our proposed regime makes for an average image size of $\\bar { S } \\times \\bar { S } = 1 4 4 \\times 1 4 \\bar { 4 }$ . This regime was designed so that the reduced spatial size can be used to increase the corresponding batch size or the number of BA duplicates, as described in Section 3. We are first interested in accelerating the time needed for convergence of the tested models using our $B ^ { + }$ scheme. We enlarge the batch size used for each spatial size by a factor of 2242 such that $S ^ { 2 } \\cdot B$ is kept approximately fixed. As the average batch size is larger than $B _ { o }$ , which was used with the original optimization hyper-parameters, we scale the learning rate linearly as suggested by Goyal et al. (2017) by a factor of $\\bar { \\frac { B } { B _ { o } } }$ . We note that for the proposed regimes we did not require any learning rate warm-up, due to the use of gradient smoothing. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As can be seen in Figure 3, regime $B ^ { + }$ enables training with approximately $2 . 7 \\times$ less training steps, while reaching a better-than-baseline accuracy of $7 6 . 6 1 \\%$ . As sizes were chosen to reflect in approximately equal computational cost per iteration, $B ^ { + }$ regime offers a similar improvement in total wall-clock time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we perform a similar experiment with a $D ^ { + }$ regime, where the number of BA duplicates is similarly increased with respect to $D _ { o }$ instead of the batch size. This scaling results with an average duplicates of $\\bar { D } = 3$ . ",
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+        "img_path": "images/1a84b5090d53fd568c012d510a34e8d7fc621440369594acf75a993e3fa628b9.jpg",
+        "image_caption": [
+            "Figure 3: Training (dotted) and test accuracy on ImageNet using the Baseline, $B ^ { + }$ and $D ^ { \\bar { + } }$ regimes $2 2 4 \\times 2 2 4$ evaluation size). All regimes required similar computational resources per step. $B ^ { + }$ regime required $\\approx 2 . 7 \\times$ less steps per epoch. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As the computational cost for each step remains approximately constant, as well as the number of required steps per epochs, training a model under this regime requires an equal wall-clock ",
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+    {
+        "type": "text",
+        "text": "time. However, the increased batch-augmentation improves the final test accuracy to $7 8 . 0 4 \\%$ , approximately $7 \\%$ relative improvement over the $7 6 . 4 \\%$ baseline. ",
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+    {
+        "type": "text",
+        "text": "5.2 INCREASING MODEL RESILIENCY TO TEST-TIME CHANGES IN IMAGE SIZE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we examine how MixSize affects the resulting model resiliency to changes in the image size during test-time. We evaluated the models by varying the test-time image sizes around the original 224 spatial size: $S = 2 2 4 + 3 2 \\cdot m , m \\in \\{ - \\bar { 6 } , . . . , 6 \\}$ . The common evaluation procedure for ImageNet models first scales the image to a 256 smallest dimension and crops a center $2 2 4 \\times 2 2 4$ image. We adapt this regime for other image sizes by scaling the smallest dimension to $\\lfloor { \\frac { 8 } { 7 } } S \\rfloor$ (since $\\frac { 8 } { 7 } \\cdot 2 2 4 = 2 5 6 )$ and then cropping the center $S \\times S$ patch. Models trained with a mixed regime were calibrated to a specific evaluation size by measuring batch-norm statistics for 200 batches of training samples. We note that for original fixed-size regimes this calibration procedure resulted with degraded results and so we report accuracy without calibration for these models. We did not use any fine-tuning procedure post training for any of the models. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As can be seen in Figure 4a, the baseline model trained using a fixed size, reaches $7 6 . 4 \\%$ top-1 accuracy at the same 224 spatial size it was trained on. As observed previously, the model continues to slightly improve beyond that size, to a maximum of $7 6 . 8 \\%$ accuracy. However, it is apparent that the model’s performance quickly degrades when evaluating with sizes smaller than 224. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare these results with a $D ^ { + }$ regime, trained with an average size of $\\bar { S } = 1 4 4$ . As described earlier, this model requires the same time and computational resources as the baseline model. However, due to the decreased average size, we were able to leverage more than 1 duplicates per batch on average, which improved the model’s top-1 accuracy to $7 7 . 1 4 \\%$ at size 224. Furthermore, we find that the model performs much more favorably at image sizes smaller than 224, scoring an improved (over baseline) accuracy of $7 6 . 4 3 \\%$ at only $1 6 0 \\times 1 6 0$ spatial size. We analyzed an alternative regime $S ^ { ( 2 0 8 ) }$ , where the average spatial size is larger at $2 0 8 \\times 2 0 8$ (for more details see Appendix D). The model trained with the $S ^ { ( 2 0 8 ) }$ regime offers a similar improvement in accuracy, only across a larger spatial size, as it observed an average size of $2 0 8 \\times 2 0 8$ during training. Figure 4a demonstrates that while all three models (Fixed with $S = 2 2 4$ , $S ^ { ( 1 4 4 ) }$ and $S ^ { ( 2 0 8 ) }$ ) were trained with the same compute and memory budget, mixed-size regimes offer superior accuracy over a wide range of evaluation sizes. Specifically, mixed-regime at $S = 2 0 8$ dominates the baseline fixed-size regime at all sizes, while our mixed regime at $S = 1 4 4$ achieves best results at sizes smaller than 224. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/db2b2b989b584c1ba941925f5f6a2fddc0143307bbbf6bef492dca374db7de46.jpg",
+        "image_caption": [
+            "Figure 4: Left: Test accuracy on validation set per image size, all models trained using the same computational and memory resources (regime $D ^ { + }$ ). Right: Test accuracy per billion flop (at evaluation). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also compared the classification performance across evaluated image sizes, using networks trained on a variety of fixed sizes and our mixed regimes. As a baseline, we use results obtained by Touvron et al. (2019) (trained with repeated augmentations, without fine-tuning) and compare them with mixed-regime models trained with an equal computational budget, by setting the base number of BA duplicates to $D = 2$ . As can be seen in Figure 4b, mixed-regime trained models offer a wider range of resolutions with close-to-baseline accuracy (within a $2 \\%$ change) and perform better than their fixed-size counterparts at all sizes. As the number of floating-point operations (flops) grows linearly with the number of pixels, using a mixed regime significantly improves accuracy per compute at evaluation. We further note that our $S ^ { ( 2 2 4 ) }$ model reaches a top accuracy of $7 9 . 2 7 \\%$ at a $2 8 8 \\times 2 8 8$ evaluation size. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 SUMMARY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduced and examined a performance trade-off between computational load and classification accuracy governed by the input’s spatial size. We suggested stochastic image size regimes, which randomly change the spatial dimension as well as the batch size and the number of augmentation (duplicates) in the batch. Stochastic regime benefits are threefold: (1) reduced number of training iterations; or (2) improved model accuracy (generalization) and (3) improved model robustness to changing the image size. We believe this approach may have a profound impact on the practice of training convolutional networks. Given a computational and time budget, stochastic size regimes may enable to train networks faster, with better results, as well as to target specific image sizes that will be used at test time. As the average size chosen to train is reflected in the optimal operating point for evaluation resolution, mixed regimes can be used to create networks with better performance across multiple designated use cases. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
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+        "text": "Komodakis Zagoruyko. Wide residual networks. In BMVC, 2016. ",
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+    {
+        "type": "text",
+        "text": "Appendix ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            316,
+            127
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL SETTINGS ",
+        "bbox": [
+            176,
+            150,
+            429,
+            166
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 CIFAR ",
+        "bbox": [
+            174,
+            181,
+            269,
+            196
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We used the common data augmentation technique as described by He et al. (2016). In this method, the input image is padded with 4 zero-valued pixels at each side, top and bottom. A random $3 2 \\times 3 2$ part of the padded image is then cropped and with a 0.5 probability flipped horizontally. In order to adapt to varying input scales, we add an additional augmentation step, that resizes the images using bilinear interpolation to $S \\times S$ , depending on a sampled size for each step. We note that this keeps the exact original augmentation procedure for $S = 3 2$ . ",
+        "bbox": [
+            174,
+            209,
+            825,
+            292
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B IMPACT OF GRADIENT SMOOTHING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            316,
+            500,
+            332
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Gradient norm values with and without Gradsmoothing. Solid lines are gradients for $S ~ = ~ 3 2$ while dotted lines are for $S = 1 6$ . ",
+        "bbox": [
+            179,
+            483,
+            493,
+            521
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b5a4bec38369a3a9f08bcde93793589da2f3699c013203d4024b4b0aa2b6387.jpg",
+        "image_caption": [
+            "(b) Training and test error with and with out gradient smoothing. Solid lines are test errors while dotted lines are for training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            356,
+            495,
+            477
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b880a0f1614ff8f14f091c95c236b578af67e6f91fb93f032cd8ddadba161c4b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            509,
+            354,
+            816,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 5: Impact of gradient smoothing on CIFAR10, ResNet-44. The training regime includes two image sizes: $3 2 \\times 3 2$ and $1 6 \\times 1 6$ (average size is $S = 2 4$ ). Using a $B ^ { + }$ regime creates two batch sizes: 256 and 2, 048 respectively. Gradient smoothing helps to reduce gap between gradient norms at difference batch sizes and improves final accuracy. ",
+        "bbox": [
+            173,
+            532,
+            825,
+            589
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C VARYING IMAGE-SIZE TRAINING REGIMES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            562,
+            642
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We wish to consider training regimes with varying image sizes, such that the average image size is smaller than the desired evaluation size. For example, for the height dimension $H$ , we wish to obtain an average size of $\\begin{array} { r } { \\bar { H } = \\sum _ { i } p _ { i } H _ { i } } \\end{array}$ such that $\\bar { H } < \\mathbf { \\bar { \\Gamma } } H _ { o }$ . We consider three alternatives for image size variations: ",
+        "bbox": [
+            176,
+            659,
+            825,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Increase image size from small to large, where each image size is used for number of epochs $E _ { i } = p _ { i } E _ { \\mathrm { t o t a l } }$ , where $E _ { \\mathrm { t o t a l } }$ is the total number training epochs required. • Using a random image size for each epoch, keeping the epoch number for each size at $E _ { i }$ • Sampling image size per training step at probability $p _ { i }$ ",
+        "bbox": [
+            215,
+            728,
+            823,
+            800
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As can be seen in Figure 6, we found that random sampling regimes performed better than scaling image size from small to large (Howard, 2018; Touvron et al., 2019). While sampling both at epoch and step time frames performed similarly, replacing sizes on each step seemed to converge faster and to have less noise in measured test accuracy. We note that these behaviours may partly stem from the use of batch-normalization (Ioffe & Szegedy, 2015) which is sensitive to the image size used at evaluation or insufficient hyper-parameter tuning for each specific size (e.g., spiking error at the end of the small-to-large regime). Considering these findings, we continue to perform our experiments using the third regime – sampling image size per training step. ",
+        "bbox": [
+            174,
+            813,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9d9dea62f1564c1593f40644bbbaff8dfb4165f7fd1b7a845f4ec6fddccade1.jpg",
+        "image_caption": [
+            "Figure 6: Test accuracy vs step for 3 size sampling regimes: (1) From small to large (2) Sample each Epoch (3) Sample each step. All methods reached a similar accuracy, but sampling each epoch was less noisy and did not require hyper-parameter tuning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            98,
+            818,
+            242
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D ALTERNATIVE SIZE DISTRIBUTION REGIMES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            323,
+            576,
+            338
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We used alternative size regimes balanced around 224, named $S ^ { ( 2 0 8 ) }$ and $S ^ { ( 2 2 4 ) }$ . They can be described by the following distributions: ",
+        "bbox": [
+            173,
+            351,
+            823,
+            381
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7eddb5ca723ee9ed7d782bbec1ea648e9b369db346545debcaa4034e50d712d.jpg",
+        "text": "$$\nS ^ { ( 2 0 8 ) } : ~ S = \\left\\{ \\begin{array} { l l } { 3 2 0 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\\\ { 2 8 8 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\\\ { 2 5 6 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\\\ { 2 2 4 , } & { \\mathrm { w . p } ~ p = 0 . 2 } \\\\ { 1 9 2 , } & { \\mathrm { w . p } ~ p = 0 . 2 } \\\\ { 1 6 0 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\\\ { 1 2 8 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\\\ { 9 6 , } & { \\mathrm { w . p } ~ p = 0 . 1 } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            390,
+            620,
+            523
+        ],
+        "page_idx": 12
+    }
+]

parse/train/S19eAF9ee/S19eAF9ee.md

@@ -0,0 +1,249 @@
+# STRUCTURED SEQUENCE MODELING WITH GRAPH CONVOLUTIONAL RECURRENT NETWORKS
+
+Youngjoo Seo EPFL, Switzerland youngjoo.seo@epfl.ch
+
+Michael Defferrard¨   
+EPFL, Switzerland   
+michael.defferrard@epfl.ch   
+Pierre Vandergheynst   
+EPFL, Switzerland   
+pierre.vandergheynst@epfl.ch   
+Xavier Bresson   
+EPFL, Switzerland   
+xavier.bresson@epfl.ch
+
+# ABSTRACT
+
+This paper introduces Graph Convolutional Recurrent Network (GCRN), a deep learning model able to predict structured sequences of data. Precisely, GCRN is a generalization of classical recurrent neural networks (RNN) to data structured by an arbitrary graph. Such structured sequences can represent series of frames in videos, spatio-temporal measurements on a network of sensors, or random walks on a vocabulary graph for natural language modeling. The proposed model combines convolutional neural networks (CNN) on graphs to identify spatial structures and RNN to find dynamic patterns. We study two possible architectures of GCRN, and apply the models to two practical problems: predicting moving MNIST data, and modeling natural language with the Penn Treebank dataset. Experiments show that exploiting simultaneously graph spatial and dynamic information about data can improve both precision and learning speed.
+
+# 1 INTRODUCTION
+
+Many real-world data can be cast as structured sequences, with spatio-temporal sequences being a special case. A well-studied example of spatio-temporal data are videos, where succeeding frames share temporal and spatial structures. Many works, such as Donahue et al. (2015); Karpathy & FeiFei (2015); Vinyals et al. (2015), leveraged a combination of CNN and RNN to exploit such spatial and temporal regularities. Their models are able to process possibly time-varying visual inputs for variable-length prediction. These neural network architectures consist of combining a CNN for visual feature extraction followed by a RNN for sequence learning. Such architectures have been successfully used for video activity recognition, image captioning and video description.
+
+More recently, interest has grown in properly fusing the CNN and RNN models for spatio-temporal sequence modeling. Inspired by language modeling, Ranzato et al. (2014) proposed a model to represent complex deformations and motion patterns by discovering both spatial and temporal correlations. They showed that prediction of the next video frame and interpolation of intermediate frames can be achieved by building a RNN-based language model on the visual words obtained by quantizing the image patches. Their highest-performing model, recursive CNN (rCNN), uses convolutions for both inputs and states. Shi et al. (2015) then proposed the convolutional LSTM network (convLSTM), a recurrent model for spatio-temporal sequence modeling which uses 2D-grid convolution to leverage the spatial correlations in input data. They successfully applied their model to the prediction of the evolution of radar echo maps for precipitation nowcasting.
+
+The spatial structure of many important problems may however not be as simple as regular grids. For instance, the data measured from meteorological stations lie on a irregular grid, i.e. a network of heterogeneous spatial distribution of stations. More challenging, the spatial structure of data may not even be spatial, as it is the case for social or biological networks. Eventually, the interpretation that sentences can be regarded as random walks on vocabulary graphs, a view popularized by Mikolov et al. (2013), allows us to cast language analysis problems as graph-structured sequence models.
+
+![](images/bc5c5ccd950b0931976650108dec932da5c09eea0c33f2811817daf66f4eb37c.jpg)  
+Figure 1: Illustration of the proposed GCRN model for spatio-temporal prediction of graph-structured data. The technique combines at the same time CNN on graphs and RNN. RNN can be easily exchanged with LSTM or GRU networks.
+
+![](images/a8c0f2e3882ab26eeb26dbb84bd4e7b15d885d5b9212c832e5fae885167d1d8d.jpg)  
+Figure 2: Illustration of the neighborhood on an 8-nearest-neighbor grid graph. Isotropic spectral filters of support $K$ have access to nodes at most at $K - 1$ hops.
+
+This work leverages on the recent models of Defferrard et al. (2016); Ranzato et al. (2014); Shi et al. (2015) to design the GCRN model for modeling and predicting time-varying graph-based data. The core idea is to merge CNN for graph-structured data and RNN to identify simultaneously meaningful spatial structures and dynamic patterns. A generic illustration of the proposed GCRN architecture is given by Figure 1.
+
+# 2 PRELIMINARIES
+
+# 2.1 STRUCTURED SEQUENCE MODELING
+
+Sequence modeling is the problem of predicting the most likely future length- $K$ sequence given the previous $J$ observations:
+
+$$
+\hat { x } _ { t + 1 } , \ldots , \hat { x } _ { t + K } = \operatorname * { a r g m a x } _ { x _ { t + 1 } , \ldots , x _ { t + K } } P ( x _ { t + 1 } , \ldots , x _ { t + K } | x _ { t - J + 1 } , \ldots , x _ { t } ) ,
+$$
+
+where $x _ { t } ~ \in ~ \mathbf { D }$ is an observation at time $t$ and $\mathbf { D }$ denotes the domain of the observed features. The archetypal application being the $n$ -gram language model (with $n = J + 1 ,$ ), where $P ( x _ { t + 1 } | x _ { t - J + 1 } , . . . , x _ { t } )$ models the probability of word $x _ { t + 1 }$ to appear conditioned on the past $J$ words in the sentence (Graves, 2013).
+
+In this paper, we are interested in special structured sequences, i.e. sequences where features of the observations $x _ { t }$ are not independent but linked by pairwise relationships. Such relationships are universally modeled by weighted graphs.
+
+Data $x _ { t }$ can be viewed as a graph signal, i.e. a signal defined on an undirected and weighted graph $\mathcal { G } = ( \nu , \mathcal { E } , A )$ , where $\nu$ is a finite set of $| \nu | = n$ vertices, $\mathcal { E }$ is a set of edges and $A \in \mathbb { R } ^ { n \times n }$ is a weighted adjacency matrix encoding the connection weight between two vertices. A signal $x _ { t } : \mathcal { V } \overset { \cdot } {  } \mathbb { R } ^ { d _ { x } }$ defined on the nodes of the graph may be regarded as a matrix $\boldsymbol { x } _ { t } \in \mathbb { R } ^ { n \times d _ { \boldsymbol { x } } }$ whose column $i$ is the $d _ { x }$ -dimensional value of $x _ { t }$ at the $i ^ { t h }$ node. While the number of free variables in a structured sequence of length $K$ is in principle $\mathcal { O } ( n ^ { K } d _ { x } { } ^ { K } )$ , we seek to exploit the structure of the space of possible predictions to reduce the dimensionality and hence make those problems more tractable.
+
+# 2.2 LONG SHORT-TERM MEMORY
+
+A special class of recurrent neural networks (RNN) that prevents the gradient from vanishing too quickly is the popular long short-term memory (LSTM) introduced by Hochreiter & Schmidhuber (1997). This architecture has proven stable and powerful for modeling long-range dependencies in various general-purpose sequence modeling tasks (Graves, 2013; Srivastava et al., 2015; Sutskever et al., 2014). A fully-connected LSTM (FC-LSTM) may be seen as a multivariate version of LSTM where the input $x _ { t } \in \mathbb { R } ^ { d _ { x } }$ , cell output $h _ { t } \in [ - 1 , 1 ] ^ { d _ { h } }$ and states $c _ { t } \in \mathbb { R } ^ { d _ { h } }$ are all vectors. In this paper, we follow the FC-LSTM formulation of Graves (2013), that is:
+
+$$
+\begin{array} { r l } & { i = \sigma ( W _ { x i } x _ { t } + W _ { h i } h _ { t - 1 } + w _ { c i } \odot c _ { t - 1 } + b _ { i } ) , } \\ & { f = \sigma ( W _ { x f } x _ { t } + W _ { h f } h _ { t - 1 } + w _ { c f } \odot c _ { t - 1 } + b _ { f } ) , } \\ & { c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot \operatorname { t a n h } ( W _ { x c } x _ { t } + W _ { h c } h _ { t - 1 } + b _ { c } ) , } \\ & { o = \sigma ( W _ { x o } x _ { t } + W _ { h o } h _ { t - 1 } + w _ { c o } \odot c _ { t } + b _ { o } ) , } \\ & { h _ { t } = o \odot \operatorname { t a n h } ( c _ { t } ) , } \end{array}
+$$
+
+where $\odot$ denotes the Hadamard product, $\sigma ( \cdot )$ the sigmoid function $\sigma ( x ) ~ = ~ 1 / ( 1 + e ^ { - x } )$ and $i , f , o \in [ 0 , 1 ] ^ { d _ { h } }$ are the input, forget and output gates. The weights $W _ { x }$ $\ l _ { \cdot } \in \mathbb { R } ^ { d _ { h } \times d _ { x } }$ , $\dot { W } _ { h \cdot } \in \mathbb { R } ^ { d _ { h } \times d _ { h } }$ , $w _ { c } . \in \mathbb { R } ^ { d _ { h } }$ and biases $b _ { i } , \bar { b } _ { f } , b _ { c } , \bar { b _ { o } } \in \mathbb { R } ^ { d _ { h } }$ are the model parameters.1 Such a model is called fullyconnected because the dense matrices $W _ { x }$ · and $W _ { h }$ · linearly combine all the components of $x$ and $h$ . The optional peephole connections $w _ { c }$ · $\odot c _ { t }$ , introduced by Gers & Schmidhuber (2000), have been found to improve performance on certain tasks.
+
+# 2.3 CONVOLUTIONAL NEURAL NETWORKS ON GRAPHS
+
+Generalizing convolutional neural networks (CNNs) to arbitrary graphs is a recent area of interest. Two approaches have been explored in the literature: (i) a generalization of the spatial definition of a convolution (Masci et al., 2015; Niepert et al., 2016) and (ii), a multiplication in the graph Fourier domain by the way of the convolution theorem (Bruna et al., 2014; Defferrard et al., 2016). Masci et al. (2015) introduced a spatial generalization of CNNs to 3D meshes. The authors used geodesic polar coordinates to define convolution operations on mesh patches, and formulated a deep learning architecture which allows comparison across different meshes. Hence, this method is tailored to manifolds and is not directly generalizable to arbitrary graphs. Niepert et al. (2016) proposed a spatial approach which may be decomposed in three steps: (i) select a node, (ii) construct its neighborhood and (iii) normalize the selected sub-graph, i.e. order the neighboring nodes. The extracted patches are then fed into a conventional 1D Euclidean CNN. As graphs generally do not possess a natural ordering (temporal, spatial or otherwise), a labeling procedure should be used to impose it. Bruna et al. (2014) were the first to introduce the spectral framework described below in the context of graph CNNs. The major drawback of this method is its $\mathcal { O } ( n ^ { 2 } )$ complexity, which was overcome with the technique of Defferrard et al. (2016), which offers a linear complexity $\mathcal { O } ( | \mathcal { E } | )$ and provides strictly localized filters. Kipf & Welling (2016) took a first-order approximation of the spectral filters proposed by Defferrard et al. (2016) and successfully used it for semi-supervised classification of nodes. While we focus on the framework introduced by Defferrard et al. (2016), the proposed model is agnostic to the choice of the graph convolution operator $^ { \ast _ { \mathcal { G } } }$ .
+
+As it is difficult to express a meaningful translation operator in the vertex domain (Bruna et al., 2014; Niepert et al., 2016), Defferrard et al. (2016) chose a spectral formulation for the convolution operator on graph $^ { \ast _ { \mathcal { G } } }$ . By this definition, a graph signal $x \in \overline { { \mathbb { R } ^ { n \times d _ { x } } } }$ is filtered by a non-parametric kernel $g _ { \theta } ( \Lambda ) = \mathrm { d i a g } ( \theta )$ , where $\boldsymbol { \theta } \in \mathbb { R } ^ { n }$ is a vector of Fourier coefficients, as
+
+$$
+y = g _ { \theta } * _ { \mathcal { G } } x = g _ { \theta } ( L ) x = g _ { \theta } ( U \Lambda U ^ { T } ) x = U g _ { \theta } ( \Lambda ) U ^ { T } x \in \mathbb { R } ^ { n \times d _ { x } } ,
+$$
+
+where $U \in \mathbb { R } ^ { n \times n }$ is the matrix of eigenvectors and $\boldsymbol { \Lambda } \in \mathbb { R } ^ { n \times n }$ the diagonal matrix of eigenvalues of the normalized graph Laplacian $\bar { L } \ = \ I _ { n } - D ^ { - 1 / 2 } A D ^ { - 1 / 2 } \ = U \bar { \Lambda } U ^ { T } \ \in \ \mathbb { R } ^ { n \times n }$ , where $I _ { n }$ is the identity matrix and $D \in \mathbb { R } ^ { n \times n }$ is the diagonal degree matrix with $D _ { i i } = \textstyle \sum _ { j } A _ { i j }$ (Chung, 1997). Note that the signal $x$ is filtered by $g _ { \boldsymbol { \theta } }$ with an element-wise multiplication of its graph Fourier transform $U ^ { T } x$ with $g _ { \theta }$ (Shuman et al., 2013). Evaluating (3) is however expensive, as the multiplication with $U$ is $\mathcal { O } ( n ^ { 2 } )$ . Furthermore, computing the eigendecomposition of $L$ might be prohibitively expensive for large graphs. To circumvent this problem, Defferrard et al. (2016) parametrizes $g _ { \theta }$ as a truncated expansion, up to order $K - 1$ , of Chebyshev polynomials $T _ { k }$ such that
+
+$$
+g _ { \theta } ( \Lambda ) = \sum _ { k = 0 } ^ { K - 1 } \theta _ { k } T _ { k } ( \tilde { \Lambda } ) ,
+$$
+
+where the parameter $\theta \in \mathbb { R } ^ { K }$ is a vector of Chebyshev coefficients and $T _ { k } ( \tilde { \Lambda } ) \ \in \ \mathbb { R } ^ { n \times n }$ is the Chebyshev polynomial of order $k$ evaluated at $\tilde { \Lambda } = 2 \Lambda / \lambda _ { m a x } - I _ { n }$ . The graph filtering operation can then be written as
+
+$$
+y = g _ { \theta } * _ { \mathcal { G } } x = g _ { \theta } ( L ) x = \sum _ { k = 0 } ^ { K - 1 } \theta _ { k } T _ { k } ( \tilde { L } ) x ,
+$$
+
+where $T _ { k } ( \tilde { L } ) \in \mathbb { R } ^ { n \times n }$ is the Chebyshev polynomial of order $k$ evaluated at the scaled Laplacian $\tilde { L } = 2 L / \lambda _ { m a x } - I _ { n }$ . Using the stable recurrence relation $T _ { k } ( x ) = 2 x T _ { k - 1 } ( x ) - T _ { k - 2 } ( x )$ with $T _ { 0 } = 1$ and $T _ { 1 } = x$ , one can evaluate (5) in $\mathcal { O } ( K \vert \mathcal { E } \vert )$ operations, i.e. linearly with the number of edges. Note that as the filtering operation (5) is an order $K$ polynomial of the Laplacian, it is $K$ -localized and depends only on nodes that are at maximum $K$ hops away from the central node, the $K$ -neighborhood. The reader is referred to Defferrard et al. (2016) for details and an in-depth discussion.
+
+# 3 RELATED WORKS
+
+Shi et al. (2015) introduced a model for regular grid-structured sequences, which can be seen as a special case of the proposed model where the graph is an image grid where the nodes are well ordered. Their model is essentially the classical FC-LSTM (2) where the multiplications by dense matrices $W$ have been replaced by convolutions with kernels $W$ :
+
+$$
+\begin{array} { r l } & { i = \sigma ( W _ { x i } * x _ { t } + W _ { h i } * h _ { t - 1 } + w _ { c i } \odot c _ { t - 1 } + b _ { i } ) , } \\ & { f = \sigma ( W _ { x f } * x _ { t } + W _ { h f } * h _ { t - 1 } + w _ { c f } \odot c _ { t - 1 } + b _ { f } ) , } \\ & { c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot \operatorname { t a n h } ( W _ { x c } * x _ { t } + W _ { h c } * h _ { t - 1 } + b _ { c } ) , } \\ & { o = \sigma ( W _ { x o } * x _ { t } + W _ { h o } * h _ { t - 1 } + w _ { c o } \odot c _ { t } + b _ { o } ) , } \\ & { h _ { t } = o \odot \operatorname { t a n h } ( c _ { t } ) , } \end{array}
+$$
+
+where $^ *$ denotes the 2D convolution by a set of kernels. In their setting, the input tensor $\boldsymbol { x } _ { t } ~ \in$ $\mathbb { R } ^ { n _ { r } \times n _ { c } \times d _ { x } }$ is the observation of $d _ { x }$ measurements at time $t$ of a dynamical system over a spatial region represented by a grid of $n _ { r }$ rows and $n _ { c }$ columns. The model holds spatially distributed hidden and cell states of size $d _ { h }$ given by the tensors $c _ { t } , h _ { t } \ \in \ \mathbb { R } ^ { n _ { r } \times n _ { c } \times d _ { h } }$ . The size $m$ of the convolutional kernels $W _ { h . \cdot } \in \mathbb { R } ^ { m \times \bar { m } \times d _ { h } \times \bar { d } _ { h } }$ and $W _ { x }$ · $\in \mathbb { R } ^ { m \times m \times d _ { h } \times d _ { x } }$ determines the number of parameters, which is independent of the grid size $n _ { r } \times n _ { c }$ . Earlier, Ranzato et al. (2014) proposed a similar RNN variation which uses convolutional layers instead of fully connected layers. The hidden state at time $t$ is given by
+
+$$
+h _ { t } = \operatorname { t a n h } ( \sigma ( W _ { x 2 } \ast \sigma ( W _ { x 1 } \ast x _ { t } ) ) + \sigma ( W _ { h } \ast h _ { t - 1 } ) ) ,
+$$
+
+where the convolutional kernels $W _ { h } \in \mathbb { R } ^ { d _ { h } \times d _ { h } }$ are restricted to filters of size 1x1 (effectively a fully connected layer shared across all spatial locations).
+
+Observing that natural language exhibits syntactic properties that naturally combine words into phrases, Tai et al. (2015) proposed a model for tree-structured topologies, where each LSTM has access to the states of its children. They obtained state-of-the-art results on semantic relatedness and sentiment classification. Liang et al. (2016) followed up and proposed a variant on graphs. Their sophisticated network architecture obtained state-of-the-art results for semantic object parsing on four datasets. In those models, the states are gathered from the neighborhood by way of a weighted sum with trainable weight matrices. Those weights are however not shared across the graph, which would otherwise have required some ordering of the nodes, alike any other spatial definition of graph convolution. Moreover, their formulations are limited to the one-neighborhood of the current node, with equal weight given to each neighbor.
+
+Motivated by spatio-temporal problems like modeling human motion and object interactions, Jain et al. (2016) developed a method to cast a spatio-temporal graph as a rich RNN mixture which essentially associates a RNN to each node and edge. Again, the communication is limited to directly connected nodes and edges.
+
+The closest model to our work is probably the one proposed by Li et al. (2015), which showed stat-of-the-art performance on a problem from program verification. Whereas they use the iterative procedure of the Graph Neural Networks (GNNs) model introduced by Scarselli et al. (2009) to propagate node representations until convergence, we instead use the graph CNN introduced by Defferrard et al. (2016) to diffuse information across the nodes. While their motivations are quite different, those models are related by the fact that a spectral filter defined as a polynomial of order $K$ can be implemented as a $K$ -layer GNN.2
+
+# 4 PROPOSED GCRN MODELS
+
+We propose two GCRN architectures that are quite natural, and investigate their performances in real-world applications in Section 5.
+
+Model 1. The most straightforward definition is to stack a graph CNN, defined as (5), for feature extraction and an LSTM, defined as (2), for sequence learning:
+
+$$
+\begin{array} { r l } & { x _ { t } ^ { \mathrm { { C N N } } } = { \mathrm { C N N } } _ { \mathscr { G } } ( x _ { t } ) } \\ & { \qquad i = \sigma \big ( { W _ { x i } } { x _ { t } ^ { \mathrm { { C N N } } } } + { W _ { h i } } { h _ { t - 1 } } + { w _ { c i } } \odot { c _ { t - 1 } } + { b _ { i } } \big ) , } \\ & { f = \sigma \big ( { W _ { x f } } { x _ { t } ^ { { \mathrm { C N N } } } } + { W _ { h f } } { h _ { t - 1 } } + { w _ { c f } } \odot { c _ { t - 1 } } + { b _ { f } } \big ) , } \\ & { c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot { \mathrm { t a n h } } ( { W _ { x c } } { x _ { t } ^ { \mathrm { { C N N } } } } + { W _ { h c } } { h _ { t - 1 } } + { b _ { c } } \big ) , } \\ & { o = \sigma \big ( { W _ { x o } } { x _ { t } ^ { { \mathrm { C N N } } } } + { W _ { h o } } { h _ { t - 1 } } + { w _ { c o } } \odot { c _ { t } } + { b _ { o } } \big ) , } \\ & { h _ { t } = o \odot { \mathrm { t a n h } } \big ( { c _ { t } } \big ) . } \end{array}
+$$
+
+In that setting, the input matrix $\boldsymbol { x } _ { t } \in \mathbb { R } ^ { n \times d _ { \boldsymbol { x } } }$ may represent the observation of $d _ { x }$ measurements at time $t$ of a dynamical system over a network whose organization is given by a graph $\mathcal { G }$ . $\boldsymbol { x } _ { t } ^ { \mathrm { C N N } }$ is the output of the graph CNN gate. For a proof of concept, we simply choose here $\begin{array} { r } { x _ { t } ^ { \mathrm { { C N N } } } = W ^ { \mathrm { { C N N } } } * _ { \mathcal { G } } x _ { t } . } \end{array}$ , where $W ^ { \mathrm { C N N } } \in \mathbf { \bar { \Gamma } } \mathbb { R } ^ { K \times d _ { x } \times \mathbf { \bar { d } } _ { x } }$ are the Chebyshev coefficients for the graph convolutional kernels of support $K$ . The model also holds spatially distributed hidden and cell states of size $d _ { h }$ given by the matrices $c _ { t } , h _ { t } \in \mathbb { R } ^ { n \times d _ { h } }$ . Peepholes are controlled by $w _ { c } . \in \mathbb { R } ^ { n \times d _ { h } }$ . The weights $V _ { h \cdot } \in \mathbb { R } ^ { d _ { h } \times d _ { h } }$ and $W _ { x }$ · $\mathbf { \Sigma } \in \mathbb { R } ^ { d _ { h } \times d _ { x } }$ are the parameters of the fully connected layers. An architecture such as (8) may be enough to capture the data distribution by exploiting local stationarity and compositionality properties as well as the dynamic properties.
+
+Model 2. To generalize the convLSTM model (6) to graphs we replace the Euclidean 2D convolution $^ *$ by the graph convolution $^ { \ast _ { \mathcal { G } } }$ :
+
+$$
+\begin{array} { r l } & { i = \sigma ( W _ { x i } \ast _ { \mathcal { G } } x _ { t } + W _ { h i } \ast _ { \mathcal { G } } h _ { t - 1 } + w _ { c i } \odot c _ { t - 1 } + b _ { i } ) , } \\ & { f = \sigma ( W _ { x f } \ast _ { \mathcal { G } } x _ { t } + W _ { h f } \ast _ { \mathcal { G } } h _ { t - 1 } + w _ { c f } \odot c _ { t - 1 } + b _ { f } ) , } \\ & { c _ { t } = f _ { t } \odot c _ { t - 1 } + i _ { t } \odot \operatorname { t a n h } ( W _ { x c } \ast _ { \mathcal { G } } x _ { t } + W _ { h c } \ast _ { \mathcal { G } } h _ { t - 1 } + b _ { c } ) , } \\ & { o = \sigma ( W _ { x o } \ast _ { \mathcal { G } } x _ { t } + W _ { h o } \ast _ { \mathcal { G } } h _ { t - 1 } + w _ { c o } \odot c _ { t } + b _ { o } ) , } \\ & { h _ { t } = o \odot \operatorname { t a n h } ( c _ { t } ) . } \end{array}
+$$
+
+In that setting, the support $K$ of the graph convolutional kernels defined by the Chebyshev coefficients $W _ { h }$ · $\doteq \mathbb { R } ^ { K \times d _ { h } ^ { \bullet } \times d _ { h } }$ and $W _ { x }$ · $\breve { \in \mathbb { R } ^ { K \times d _ { h } \times d _ { x } } }$ determines the number of parameters, which is independent of the number of nodes $n$ . To keep the notation simple, we write $W _ { x i } * _ { \mathcal { G } } x _ { t }$ to mean a graph convolution of $x _ { t }$ with $d _ { h } d _ { x }$ filters which are functions of the graph Laplacian $L$ parametrized by $K$ Chebyshev coefficients, as noted in (4) and (5). In a distributed computing setting, $K$ controls the communication overhead, i.e. the number of nodes any given node $i$ should exchange with in order to compute its local states.
+
+The proposed blend of RNNs and graph CNNs is not limited to LSTMs and is straightforward to apply to any kind of recursive networks. For example, a vanilla RNN $h _ { t } = \operatorname { t a n h } ( W _ { x } x _ { t } + W _ { h } h _ { t - 1 } )$
+
+<table><tr><td>Architecture</td><td>Structure</td><td>Filter size</td><td>Parameters</td><td>Runtime</td><td>Test(w/o Rot)</td><td>Test(Rot)</td></tr><tr><td>FC-LSTM</td><td>N/A</td><td>N/A</td><td>142,667,776</td><td>N/A</td><td>4832</td><td>=</td></tr><tr><td>LSTM+CNN</td><td>N/A</td><td>5×5</td><td>13,524,496</td><td>2.10</td><td>3851</td><td>4339</td></tr><tr><td>LSTM+CNN</td><td>N/A</td><td>9×9</td><td>43,802,128</td><td>6.10</td><td>3903</td><td>4208</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=3</td><td>1,629,712</td><td>0.82</td><td>3866</td><td>4367</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=5</td><td>2,711,056</td><td>1.24</td><td>3495</td><td>3932</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3400</td><td>3803</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=9</td><td>4,873,744</td><td>2.15</td><td>3395</td><td>3814</td></tr><tr><td>LSTM+GCNN</td><td>knn=4</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3446</td><td>3844</td></tr><tr><td>LSTM+GCNN</td><td>knn=16</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3578</td><td>3963</td></tr></table>
+
+Table 1: Comparison between models. Runtime is the time spent per each mini-batch in seconds. Test cross-entropies correspond to moving MNIST, and rotating and moving MNIST. LSTM $\cdot +$ GCNN is Model 2 defined in (9). Cross-entropy of FC-LSTM is taken from Shi et al. (2015).
+
+would be modified as
+
+$$
+h _ { t } = \operatorname { t a n h } ( W _ { x } * _ { \mathcal { G } } x _ { t } + W _ { h } * _ { \mathcal { G } } h _ { t - 1 } ) ,
+$$
+
+and a Gated Recurrent Unit (GRU) (Cho et al., 2014) as
+
+$$
+\begin{array} { r l } & { z = \sigma ( W _ { x z } \ast _ { \mathcal { G } } x _ { t } + W _ { h z } \ast _ { \mathcal { G } } h _ { t - 1 } ) , } \\ & { r = \sigma ( W _ { x r } \ast _ { \mathcal { G } } x _ { t } + W _ { h r } \ast _ { \mathcal { G } } h _ { t - 1 } ) , } \\ & { \tilde { h } = \operatorname { t a n h } ( W _ { x h } \ast _ { \mathcal { G } } x _ { t } + W _ { h h } \ast _ { \mathcal { G } } ( r \odot h _ { t - 1 } ) ) , } \\ & { h _ { t } = z \odot h _ { t - 1 } + ( 1 - z ) \odot \tilde { h } . } \end{array}
+$$
+
+As demonstrated by Shi et al. (2015), structure-aware LSTM cells can be stacked and used as sequence-to-sequence models using an architecture composed of an encoder, which processes the input sequence, and a decoder, which generates an output sequence. A standard practice for machine translation using RNNs (Cho et al., 2014; Sutskever et al., 2014).
+
+# 5 EXPERIMENTS
+
+# 5.1 SPATIO-TEMPORAL SEQUENCE MODELING ON MOVING-MNIST
+
+For this synthetic experiment, we use the moving-MNIST dataset generated by Shi et al. (2015). All sequences are 20 frames long (10 frames as input and 10 frames for prediction) and contain two handwritten digits bouncing inside a $6 4 \times 6 4$ patch. Following their experimental setup, all models are trained by minimizing the binary cross-entropy loss using back-propagation through time (BPTT) and RMSProp with a learning rate of $1 0 ^ { - 3 }$ and a decay rate of 0.9. We choose the best model with early-stopping on validation set. All implementations are based on their Theano code and dataset.3 The adjacency matrix $A$ is constructed as a $\mathbf { k }$ -nearest-neighbor (knn) graph with Euclidean distance and Gaussian kernel between pixel locations. For a fair comparison with Shi et al. (2015) defined in (6), all GCRN experiments are conducted with Model 2 defined in (9), which is the same architecture with the 2D convolution $^ *$ replaced by a graph convolution $^ { \ast _ { \mathcal { G } } }$ . To further explore the impact of the isotropic property of our filters, we generated a variant of the moving MNIST dataset where digits are also rotating (see Figure 4).
+
+Table 1 shows the performance of various models: (i) the baseline FC-LSTM from Shi et al. (2015), (ii) the 1-layer LSTM $^ +$ CNN from Shi et al. (2015) with different filter sizes, and (iii) the proposed LSTM+graph CNN(GCNN) defined in (9) with different supports $K$ . These results show the ability of the proposed method to capture spatio-temporal structures. Perhaps surprisingly, GCNNs can offer better performance than regular CNNs, even when the domain is a 2D grid and the data is images, the problem CNNs were initially developed for. The explanation is to be found in the differences between 2D filters and spectral graph filters. While a spectral filter of support $K = 3$ corresponds to the reach of a patch of size $5 \times 5$ (see Figure 2), the difference resides in the isotropic nature of the former and the number of parameters: $K = 3$ for the former and $5 ^ { 2 } = 2 5$ for the later.
+
+![](images/c0e894075958a2fe6ae45958014950a3aaf3f9b843faf87a98d2580afacc1d3b.jpg)  
+Figure 3: Cross-entropy on validation set: Left: performance of graph CNN with various filter support $K$ . Right: performance w.r.t. graph construction.
+
+![](images/1cc92b78c2a49f24902eeb6f096c1459249ef396fac9c72850f84cda785f4e32.jpg)  
+Figure 4: Qualitative results for moving MNIST, and rotating and moving MNIST. First row is the input sequence, second the ground truth, and third and fourth are the predictions of the $\mathrm { L S T M + C N N } ( 5 \times 5 )$ and $\mathrm { L S T M + G C N N } ( k n n = 8 , K = 7 )$ .
+
+Table 1 indeed shows that LS $\mathrm { T M } { + } \mathrm { C N N } ( 5 \times 5 )$ rivals $\mathrm { L S T M + G C N N }$ with $K = 3$ . However, when increasing the filter size to $9 \times 9$ or $K = 5$ , the GCNN variant clearly outperforms the CNN variant. This experiment demonstrates that graph spectral filters can obtain superior performance on regular domains with much less parameters thanks to their isotropic nature, a controversial property. Indeed, as the nodes are not ordered, there is no notion of an edge going up, down, on the right or on the left. All edges are treated equally, inducing some sort of rotation invariance. Additionally, Table 1 shows that the computational complexity of each model is linear with the filter size, and Figure 3 shows the learning dynamic of some of the models.
+
+# 5.2 NATURAL LANGUAGE MODELING ON PENN TREEBANK
+
+The Penn Treebank dataset has 1,036,580 words. It was pre-processed in Zaremba et al. (2014) and split4 into a training set of $9 2 9 \mathrm { k }$ words, a validation set of 73k words, and a test set of ${ 8 2 } \mathrm { k }$ words. The size of the vocabulary of this corpus is 10,000. We use the gensim library5 to compute a word2vec model (Mikolov et al., 2013) for embedding the words of the dictionary in a 200-dimensional space. Then we build the adjacency matrix of the word embedding using a 4-nearest neighbor graph with cosine distance. Figure 6 presents the computed adjacency matrix, and its 3D visualization. We used the hyperparameters of the small configuration given by the code6 based on Zaremba et al. (2014): the size of the data mini-batch is 20, the number of temporal steps to unroll is 20, the dimension of the hidden state is 200. The global learning rate is 1.0 and the norm of the gradient is bounded by 5. The learning decay function is selected to be $\mathrm { 0 . 5 ^ { m a x ( 0 , \# e p o c h - 4 ) } }$ . All experiments have 13 epochs, and dropout value is 0.75. For Zaremba et al. (2014), the input representation $x _ { t }$ can be either the 200-dim embedding vector of the word, or the 10,000-dim one-hot representation of the word. For
+
+![](images/5aea0fe6f1f13974198278cfe12fec785421338be7b8b8b7b0603d98e18d7d64.jpg)
+
+Figure 5: Learning dynamic of LSTM with and without graph structure and dropout regularization.   
+
+<table><tr><td>Architecture</td><td>Representation</td><td>Parameters</td><td>Train Perplexity</td><td>Test Perplexity</td></tr><tr><td>Zaremba et al. (2014) code6</td><td>embedding</td><td>681,800</td><td>36.96</td><td>117.29</td></tr><tr><td>Zaremba et al. (2014) code6</td><td>one-hot</td><td>34,011,600</td><td>53.89</td><td>118.82</td></tr><tr><td>LSTM</td><td>embedding</td><td>681,800</td><td>48.38</td><td>120.90</td></tr><tr><td>LSTM</td><td>one-hot</td><td>34,011,600</td><td>54.41</td><td>120.16</td></tr><tr><td>LSTM, dropout</td><td>one-hot</td><td>34,011,600</td><td>145.59</td><td>112.98</td></tr><tr><td>GCRN-M1</td><td>one-hot</td><td>42,011,602</td><td>18.49</td><td>177.14</td></tr><tr><td>GCRN-M1, dropout</td><td>one-hot</td><td>42,011,602</td><td>114.29</td><td>98.67</td></tr></table>
+
+Table 2: Comparison of models in terms of perplexity. Zaremba et al. (2014) code6 is ran as benchmark algorithm. The original Zaremba et al. (2014) code used as input representation for $x _ { t }$ the 200-dim embedding representation of words, computed here by the gensim library5. As our model runs on the 10,000-dim one-hot representation of words, we also ran Zaremba et al. (2014) code on this representation. We re-implemented Zaremba et al. (2014) code with the same architecture and hyperparameters. We remind that GCRN-M1 refers to GCRN Model 1 defined in (8).
+
+our models, the input representation is a one-hot representation of the word. This choice allows us to use the graph structure of the words.
+
+Table 2 reports the final train and test perplexity values for each investigated model and Figure 5 plots the perplexity value vs. the number of epochs for the train and test sets with and without dropout regularization. Numerical experiments show:
+
+1. Given the same experimental conditions in terms of architecture and no dropout regularization, the standalone model of LSTM is more accurate than LSTM using the spatial graph information (120.16 vs. 177.14), extracted by graph CNN with the GCRN architecture of Model 1, Eq. (8).   
+2. However, using dropout regularization, the graph LSTM model overcomes the standalone LSTM with perplexity values 98.67 vs. 112.98.   
+3. The use of spatial graph information found by graph CNN speeds up the learning process, and overfits the training dataset in the absence of dropout regularization. The graph structure likely acts a constraint on the learning system that is forced to move in the space of language topics.   
+4. We performed the same experiments with LSTM and Model 2 defined in (9). Model 1 significantly outperformed Model 2, and Model 2 did worse than standalone LSTM. This bad performance may be the result of the large increase of dimensionality in Model 2, as the dimension of the hidden and cell states changes from 200 to 10,000, the size of the vocabulary. A solution would be to downsize the data dimensionality, as done in Shi et al. (2015) in the case of image data.
+
+![](images/db4ee1673e3b3918a68b4668574358c8b6f41e4249c011cc1ed5e715b01017b4.jpg)
+
+![](images/1a0958b69c9ef0cec7f225791ed9902ac5872a3df2683f257db5466fc71df530.jpg)  
+Figure 6: Left: adjacency matrix of word embeddings. Right: 3D visualization of words’ structure.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+This work aims at learning spatio-temporal structures from graph-structured and time-varying data. In this context, the main challenge is to identify the best possible architecture that combines simultaneously recurrent neural networks like vanilla RNN, LSTM or GRU with convolutional neural networks for graph-structured data. We have investigated here two architectures, one using a stack of CNN and RNN (Model 1), and one using convLSTM that considers convolutions instead of fully connected operations in the RNN definition (Model 2). We have then considered two applications: video prediction and natural language modeling. Model 2 has shown good performances in the case of video prediction, by improving the results of Shi et al. (2015). Model 1 has also provided promising performances in the case of language modeling, particularly in terms of learning speed. It has been shown that (i) isotropic filters, maybe surprisingly, can outperform classical 2D filters on images while requiring much less parameters, and (ii) that graphs coupled with graph CNN and RNN are a versatile way of introducing and exploiting side-information, e.g. the semantic of words, by structuring a data matrix.
+
+Future work will investigate applications to data naturally structured as dynamic graph signals, for instance fMRI and sensor networks. The graph CNN model we have used is rotationally-invariant and such spatial property seems quite attractive in real situations where motion is beyond translation. We will also investigate how to benefit of the fast learning property of our system to speed up language modeling models. Eventually, it will be interesting to analyze the underlying dynamical property of generic RNN architectures in the case of graphs. Graph structures may introduce stability to RNN systems, and prevent them to express unstable dynamic behaviors.
+
+# ACKNOWLEDGMENT
+
+This research was supported in part by the European Union’s H2020 Framework Programme (H2020-MSCA-ITN-2014) under grant No. 642685 MacSeNet, and Nvidia equipment grant.
+
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+T. Mikolov, K. Chen, G. Corrado, and J. Dean. Estimation of Word Representations in Vector Space. In International Conference on Learning Representations (ICLR), 2013.
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+Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning Convolutional Neural Networks for Graphs. In International Conference on Machine Learning (ICML), 2016.
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+MarcAurelio Ranzato, Arthur Szlam, Joan Bruna, Michael Mathieu, Ronan Collobert, and Sumit Chopra. Video (language) modeling: a baseline for generative models of natural videos. arXiv:1412.6604, 2014.
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+D. Shuman, S. Narang, P. Frossard, A. Ortega, and P. Vandergheynst. The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and other Irregular Domains. IEEE Signal Processing Magazine, 2013.
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parse/train/S19eAF9ee/S19eAF9ee_content_list.json

@@ -0,0 +1,1270 @@
+[
+    {
+        "type": "text",
+        "text": "STRUCTURED SEQUENCE MODELING WITH GRAPH CONVOLUTIONAL RECURRENT NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Youngjoo Seo EPFL, Switzerland youngjoo.seo@epfl.ch ",
+        "bbox": [
+            184,
+            171,
+            382,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Michael Defferrard¨   \nEPFL, Switzerland   \nmichael.defferrard@epfl.ch   \nPierre Vandergheynst   \nEPFL, Switzerland   \npierre.vandergheynst@epfl.ch   \nXavier Bresson   \nEPFL, Switzerland   \nxavier.bresson@epfl.ch ",
+        "bbox": [
+            557,
+            170,
+            813,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            183,
+            234,
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+            276
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces Graph Convolutional Recurrent Network (GCRN), a deep learning model able to predict structured sequences of data. Precisely, GCRN is a generalization of classical recurrent neural networks (RNN) to data structured by an arbitrary graph. Such structured sequences can represent series of frames in videos, spatio-temporal measurements on a network of sensors, or random walks on a vocabulary graph for natural language modeling. The proposed model combines convolutional neural networks (CNN) on graphs to identify spatial structures and RNN to find dynamic patterns. We study two possible architectures of GCRN, and apply the models to two practical problems: predicting moving MNIST data, and modeling natural language with the Penn Treebank dataset. Experiments show that exploiting simultaneously graph spatial and dynamic information about data can improve both precision and learning speed. ",
+        "bbox": [
+            233,
+            345,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            542,
+            336,
+            559
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many real-world data can be cast as structured sequences, with spatio-temporal sequences being a special case. A well-studied example of spatio-temporal data are videos, where succeeding frames share temporal and spatial structures. Many works, such as Donahue et al. (2015); Karpathy & FeiFei (2015); Vinyals et al. (2015), leveraged a combination of CNN and RNN to exploit such spatial and temporal regularities. Their models are able to process possibly time-varying visual inputs for variable-length prediction. These neural network architectures consist of combining a CNN for visual feature extraction followed by a RNN for sequence learning. Such architectures have been successfully used for video activity recognition, image captioning and video description. ",
+        "bbox": [
+            174,
+            575,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "More recently, interest has grown in properly fusing the CNN and RNN models for spatio-temporal sequence modeling. Inspired by language modeling, Ranzato et al. (2014) proposed a model to represent complex deformations and motion patterns by discovering both spatial and temporal correlations. They showed that prediction of the next video frame and interpolation of intermediate frames can be achieved by building a RNN-based language model on the visual words obtained by quantizing the image patches. Their highest-performing model, recursive CNN (rCNN), uses convolutions for both inputs and states. Shi et al. (2015) then proposed the convolutional LSTM network (convLSTM), a recurrent model for spatio-temporal sequence modeling which uses 2D-grid convolution to leverage the spatial correlations in input data. They successfully applied their model to the prediction of the evolution of radar echo maps for precipitation nowcasting. ",
+        "bbox": [
+            174,
+            694,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The spatial structure of many important problems may however not be as simple as regular grids. For instance, the data measured from meteorological stations lie on a irregular grid, i.e. a network of heterogeneous spatial distribution of stations. More challenging, the spatial structure of data may not even be spatial, as it is the case for social or biological networks. Eventually, the interpretation that sentences can be regarded as random walks on vocabulary graphs, a view popularized by Mikolov et al. (2013), allows us to cast language analysis problems as graph-structured sequence models. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc5c5ccd950b0931976650108dec932da5c09eea0c33f2811817daf66f4eb37c.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the proposed GCRN model for spatio-temporal prediction of graph-structured data. The technique combines at the same time CNN on graphs and RNN. RNN can be easily exchanged with LSTM or GRU networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            104,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8c0f2e3882ab26eeb26dbb84bd4e7b15d885d5b9212c832e5fae885167d1d8d.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the neighborhood on an 8-nearest-neighbor grid graph. Isotropic spectral filters of support $K$ have access to nodes at most at $K - 1$ hops. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This work leverages on the recent models of Defferrard et al. (2016); Ranzato et al. (2014); Shi et al. (2015) to design the GCRN model for modeling and predicting time-varying graph-based data. The core idea is to merge CNN for graph-structured data and RNN to identify simultaneously meaningful spatial structures and dynamic patterns. A generic illustration of the proposed GCRN architecture is given by Figure 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            580,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 STRUCTURED SEQUENCE MODELING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Sequence modeling is the problem of predicting the most likely future length- $K$ sequence given the previous $J$ observations: ",
+        "bbox": [
+            174,
+            637,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aaa755830e784ee249ddca4a3a4f7d53451ec3cc4a7dba72cabdd70b500cfc9a.jpg",
+        "text": "$$\n\\hat { x } _ { t + 1 } , \\ldots , \\hat { x } _ { t + K } = \\operatorname * { a r g m a x } _ { x _ { t + 1 } , \\ldots , x _ { t + K } } P ( x _ { t + 1 } , \\ldots , x _ { t + K } | x _ { t - J + 1 } , \\ldots , x _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            670,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $x _ { t } ~ \\in ~ \\mathbf { D }$ is an observation at time $t$ and $\\mathbf { D }$ denotes the domain of the observed features. The archetypal application being the $n$ -gram language model (with $n = J + 1 ,$ ), where $P ( x _ { t + 1 } | x _ { t - J + 1 } , . . . , x _ { t } )$ models the probability of word $x _ { t + 1 }$ to appear conditioned on the past $J$ words in the sentence (Graves, 2013). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we are interested in special structured sequences, i.e. sequences where features of the observations $x _ { t }$ are not independent but linked by pairwise relationships. Such relationships are universally modeled by weighted graphs. ",
+        "bbox": [
+            174,
+            763,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data $x _ { t }$ can be viewed as a graph signal, i.e. a signal defined on an undirected and weighted graph $\\mathcal { G } = ( \\nu , \\mathcal { E } , A )$ , where $\\nu$ is a finite set of $| \\nu | = n$ vertices, $\\mathcal { E }$ is a set of edges and $A \\in \\mathbb { R } ^ { n \\times n }$ is a weighted adjacency matrix encoding the connection weight between two vertices. A signal $x _ { t } : \\mathcal { V } \\overset { \\cdot } {  } \\mathbb { R } ^ { d _ { x } }$ defined on the nodes of the graph may be regarded as a matrix $\\boldsymbol { x } _ { t } \\in \\mathbb { R } ^ { n \\times d _ { \\boldsymbol { x } } }$ whose column $i$ is the $d _ { x }$ -dimensional value of $x _ { t }$ at the $i ^ { t h }$ node. While the number of free variables in a structured sequence of length $K$ is in principle $\\mathcal { O } ( n ^ { K } d _ { x } { } ^ { K } )$ , we seek to exploit the structure of the space of possible predictions to reduce the dimensionality and hence make those problems more tractable. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 LONG SHORT-TERM MEMORY",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            423,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A special class of recurrent neural networks (RNN) that prevents the gradient from vanishing too quickly is the popular long short-term memory (LSTM) introduced by Hochreiter & Schmidhuber (1997). This architecture has proven stable and powerful for modeling long-range dependencies in various general-purpose sequence modeling tasks (Graves, 2013; Srivastava et al., 2015; Sutskever et al., 2014). A fully-connected LSTM (FC-LSTM) may be seen as a multivariate version of LSTM where the input $x _ { t } \\in \\mathbb { R } ^ { d _ { x } }$ , cell output $h _ { t } \\in [ - 1 , 1 ] ^ { d _ { h } }$ and states $c _ { t } \\in \\mathbb { R } ^ { d _ { h } }$ are all vectors. In this paper, we follow the FC-LSTM formulation of Graves (2013), that is: ",
+        "bbox": [
+            173,
+            128,
+            826,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5283c64643a0bc2bd491f8bc46a4643d9f206f0c452a1e0b30d3362102151064.jpg",
+        "text": "$$\n\\begin{array} { r l } & { i = \\sigma ( W _ { x i } x _ { t } + W _ { h i } h _ { t - 1 } + w _ { c i } \\odot c _ { t - 1 } + b _ { i } ) , } \\\\ & { f = \\sigma ( W _ { x f } x _ { t } + W _ { h f } h _ { t - 1 } + w _ { c f } \\odot c _ { t - 1 } + b _ { f } ) , } \\\\ & { c _ { t } = f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot \\operatorname { t a n h } ( W _ { x c } x _ { t } + W _ { h c } h _ { t - 1 } + b _ { c } ) , } \\\\ & { o = \\sigma ( W _ { x o } x _ { t } + W _ { h o } h _ { t - 1 } + w _ { c o } \\odot c _ { t } + b _ { o } ) , } \\\\ & { h _ { t } = o \\odot \\operatorname { t a n h } ( c _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            232,
+            679,
+            321
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ denotes the Hadamard product, $\\sigma ( \\cdot )$ the sigmoid function $\\sigma ( x ) ~ = ~ 1 / ( 1 + e ^ { - x } )$ and $i , f , o \\in [ 0 , 1 ] ^ { d _ { h } }$ are the input, forget and output gates. The weights $W _ { x }$ $\\ l _ { \\cdot } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { x } }$ , $\\dot { W } _ { h \\cdot } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { h } }$ , $w _ { c } . \\in \\mathbb { R } ^ { d _ { h } }$ and biases $b _ { i } , \\bar { b } _ { f } , b _ { c } , \\bar { b _ { o } } \\in \\mathbb { R } ^ { d _ { h } }$ are the model parameters.1 Such a model is called fullyconnected because the dense matrices $W _ { x }$ · and $W _ { h }$ · linearly combine all the components of $x$ and $h$ . The optional peephole connections $w _ { c }$ · $\\odot c _ { t }$ , introduced by Gers & Schmidhuber (2000), have been found to improve performance on certain tasks. ",
+        "bbox": [
+            174,
+            328,
+            825,
+            412
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 CONVOLUTIONAL NEURAL NETWORKS ON GRAPHS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            429,
+            571,
+            443
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generalizing convolutional neural networks (CNNs) to arbitrary graphs is a recent area of interest. Two approaches have been explored in the literature: (i) a generalization of the spatial definition of a convolution (Masci et al., 2015; Niepert et al., 2016) and (ii), a multiplication in the graph Fourier domain by the way of the convolution theorem (Bruna et al., 2014; Defferrard et al., 2016). Masci et al. (2015) introduced a spatial generalization of CNNs to 3D meshes. The authors used geodesic polar coordinates to define convolution operations on mesh patches, and formulated a deep learning architecture which allows comparison across different meshes. Hence, this method is tailored to manifolds and is not directly generalizable to arbitrary graphs. Niepert et al. (2016) proposed a spatial approach which may be decomposed in three steps: (i) select a node, (ii) construct its neighborhood and (iii) normalize the selected sub-graph, i.e. order the neighboring nodes. The extracted patches are then fed into a conventional 1D Euclidean CNN. As graphs generally do not possess a natural ordering (temporal, spatial or otherwise), a labeling procedure should be used to impose it. Bruna et al. (2014) were the first to introduce the spectral framework described below in the context of graph CNNs. The major drawback of this method is its $\\mathcal { O } ( n ^ { 2 } )$ complexity, which was overcome with the technique of Defferrard et al. (2016), which offers a linear complexity $\\mathcal { O } ( | \\mathcal { E } | )$ and provides strictly localized filters. Kipf & Welling (2016) took a first-order approximation of the spectral filters proposed by Defferrard et al. (2016) and successfully used it for semi-supervised classification of nodes. While we focus on the framework introduced by Defferrard et al. (2016), the proposed model is agnostic to the choice of the graph convolution operator $^ { \\ast _ { \\mathcal { G } } }$ . ",
+        "bbox": [
+            174,
+            454,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As it is difficult to express a meaningful translation operator in the vertex domain (Bruna et al., 2014; Niepert et al., 2016), Defferrard et al. (2016) chose a spectral formulation for the convolution operator on graph $^ { \\ast _ { \\mathcal { G } } }$ . By this definition, a graph signal $x \\in \\overline { { \\mathbb { R } ^ { n \\times d _ { x } } } }$ is filtered by a non-parametric kernel $g _ { \\theta } ( \\Lambda ) = \\mathrm { d i a g } ( \\theta )$ , where $\\boldsymbol { \\theta } \\in \\mathbb { R } ^ { n }$ is a vector of Fourier coefficients, as ",
+        "bbox": [
+            174,
+            724,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3822f4f70840b8d281bfa285deec5d7c6ba15f6433ea6dc5ddbc6d5c67ac1227.jpg",
+        "text": "$$\ny = g _ { \\theta } * _ { \\mathcal { G } } x = g _ { \\theta } ( L ) x = g _ { \\theta } ( U \\Lambda U ^ { T } ) x = U g _ { \\theta } ( \\Lambda ) U ^ { T } x \\in \\mathbb { R } ^ { n \\times d _ { x } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            787,
+            715,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $U \\in \\mathbb { R } ^ { n \\times n }$ is the matrix of eigenvectors and $\\boldsymbol { \\Lambda } \\in \\mathbb { R } ^ { n \\times n }$ the diagonal matrix of eigenvalues of the normalized graph Laplacian $\\bar { L } \\ = \\ I _ { n } - D ^ { - 1 / 2 } A D ^ { - 1 / 2 } \\ = U \\bar { \\Lambda } U ^ { T } \\ \\in \\ \\mathbb { R } ^ { n \\times n }$ , where $I _ { n }$ is the identity matrix and $D \\in \\mathbb { R } ^ { n \\times n }$ is the diagonal degree matrix with $D _ { i i } = \\textstyle \\sum _ { j } A _ { i j }$ (Chung, 1997). Note that the signal $x$ is filtered by $g _ { \\boldsymbol { \\theta } }$ with an element-wise multiplication of its graph Fourier transform $U ^ { T } x$ with $g _ { \\theta }$ (Shuman et al., 2013). Evaluating (3) is however expensive, as the multiplication with $U$ is $\\mathcal { O } ( n ^ { 2 } )$ . Furthermore, computing the eigendecomposition of $L$ might be prohibitively expensive for large graphs. To circumvent this problem, Defferrard et al. (2016) parametrizes $g _ { \\theta }$ as a truncated expansion, up to order $K - 1$ , of Chebyshev polynomials $T _ { k }$ such that ",
+        "bbox": [
+            173,
+            813,
+            825,
+            900
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            143
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca1fa376dcb83ba222b220186098e686f6ca5b2a4db0060aa601fe75f132bc17.jpg",
+        "text": "$$\ng _ { \\theta } ( \\Lambda ) = \\sum _ { k = 0 } ^ { K - 1 } \\theta _ { k } T _ { k } ( \\tilde { \\Lambda } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            140,
+            576,
+            185
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the parameter $\\theta \\in \\mathbb { R } ^ { K }$ is a vector of Chebyshev coefficients and $T _ { k } ( \\tilde { \\Lambda } ) \\ \\in \\ \\mathbb { R } ^ { n \\times n }$ is the Chebyshev polynomial of order $k$ evaluated at $\\tilde { \\Lambda } = 2 \\Lambda / \\lambda _ { m a x } - I _ { n }$ . The graph filtering operation can then be written as ",
+        "bbox": [
+            173,
+            186,
+            825,
+            231
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21ebcfd5a78b0aa995a82680ed098591dda94481e43dcd7d572a6e44709702de.jpg",
+        "text": "$$\ny = g _ { \\theta } * _ { \\mathcal { G } } x = g _ { \\theta } ( L ) x = \\sum _ { k = 0 } ^ { K - 1 } \\theta _ { k } T _ { k } ( \\tilde { L } ) x ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            227,
+            637,
+            272
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $T _ { k } ( \\tilde { L } ) \\in \\mathbb { R } ^ { n \\times n }$ is the Chebyshev polynomial of order $k$ evaluated at the scaled Laplacian $\\tilde { L } = 2 L / \\lambda _ { m a x } - I _ { n }$ . Using the stable recurrence relation $T _ { k } ( x ) = 2 x T _ { k - 1 } ( x ) - T _ { k - 2 } ( x )$ with $T _ { 0 } = 1$ and $T _ { 1 } = x$ , one can evaluate (5) in $\\mathcal { O } ( K \\vert \\mathcal { E } \\vert )$ operations, i.e. linearly with the number of edges. Note that as the filtering operation (5) is an order $K$ polynomial of the Laplacian, it is $K$ -localized and depends only on nodes that are at maximum $K$ hops away from the central node, the $K$ -neighborhood. The reader is referred to Defferrard et al. (2016) for details and an in-depth discussion. ",
+        "bbox": [
+            173,
+            273,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            393,
+            354,
+            410
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Shi et al. (2015) introduced a model for regular grid-structured sequences, which can be seen as a special case of the proposed model where the graph is an image grid where the nodes are well ordered. Their model is essentially the classical FC-LSTM (2) where the multiplications by dense matrices $W$ have been replaced by convolutions with kernels $W$ : ",
+        "bbox": [
+            174,
+            425,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d551ef75a9d0ee8af4623ee5acaffef2323d8292b37a1ef738751acab248e26d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { i = \\sigma ( W _ { x i } * x _ { t } + W _ { h i } * h _ { t - 1 } + w _ { c i } \\odot c _ { t - 1 } + b _ { i } ) , } \\\\ & { f = \\sigma ( W _ { x f } * x _ { t } + W _ { h f } * h _ { t - 1 } + w _ { c f } \\odot c _ { t - 1 } + b _ { f } ) , } \\\\ & { c _ { t } = f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot \\operatorname { t a n h } ( W _ { x c } * x _ { t } + W _ { h c } * h _ { t - 1 } + b _ { c } ) , } \\\\ & { o = \\sigma ( W _ { x o } * x _ { t } + W _ { h o } * h _ { t - 1 } + w _ { c o } \\odot c _ { t } + b _ { o } ) , } \\\\ & { h _ { t } = o \\odot \\operatorname { t a n h } ( c _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $^ *$ denotes the 2D convolution by a set of kernels. In their setting, the input tensor $\\boldsymbol { x } _ { t } ~ \\in$ $\\mathbb { R } ^ { n _ { r } \\times n _ { c } \\times d _ { x } }$ is the observation of $d _ { x }$ measurements at time $t$ of a dynamical system over a spatial region represented by a grid of $n _ { r }$ rows and $n _ { c }$ columns. The model holds spatially distributed hidden and cell states of size $d _ { h }$ given by the tensors $c _ { t } , h _ { t } \\ \\in \\ \\mathbb { R } ^ { n _ { r } \\times n _ { c } \\times d _ { h } }$ . The size $m$ of the convolutional kernels $W _ { h . \\cdot } \\in \\mathbb { R } ^ { m \\times \\bar { m } \\times d _ { h } \\times \\bar { d } _ { h } }$ and $W _ { x }$ · $\\in \\mathbb { R } ^ { m \\times m \\times d _ { h } \\times d _ { x } }$ determines the number of parameters, which is independent of the grid size $n _ { r } \\times n _ { c }$ . Earlier, Ranzato et al. (2014) proposed a similar RNN variation which uses convolutional layers instead of fully connected layers. The hidden state at time $t$ is given by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c0e6bba61d667cad5d57e19a01b68c8f4baea77f00a566816d97f20d548841cf.jpg",
+        "text": "$$\nh _ { t } = \\operatorname { t a n h } ( \\sigma ( W _ { x 2 } \\ast \\sigma ( W _ { x 1 } \\ast x _ { t } ) ) + \\sigma ( W _ { h } \\ast h _ { t - 1 } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the convolutional kernels $W _ { h } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { h } }$ are restricted to filters of size 1x1 (effectively a fully connected layer shared across all spatial locations). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observing that natural language exhibits syntactic properties that naturally combine words into phrases, Tai et al. (2015) proposed a model for tree-structured topologies, where each LSTM has access to the states of its children. They obtained state-of-the-art results on semantic relatedness and sentiment classification. Liang et al. (2016) followed up and proposed a variant on graphs. Their sophisticated network architecture obtained state-of-the-art results for semantic object parsing on four datasets. In those models, the states are gathered from the neighborhood by way of a weighted sum with trainable weight matrices. Those weights are however not shared across the graph, which would otherwise have required some ordering of the nodes, alike any other spatial definition of graph convolution. Moreover, their formulations are limited to the one-neighborhood of the current node, with equal weight given to each neighbor. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Motivated by spatio-temporal problems like modeling human motion and object interactions, Jain et al. (2016) developed a method to cast a spatio-temporal graph as a rich RNN mixture which essentially associates a RNN to each node and edge. Again, the communication is limited to directly connected nodes and edges. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The closest model to our work is probably the one proposed by Li et al. (2015), which showed stat-of-the-art performance on a problem from program verification. Whereas they use the iterative procedure of the Graph Neural Networks (GNNs) model introduced by Scarselli et al. (2009) to propagate node representations until convergence, we instead use the graph CNN introduced by Defferrard et al. (2016) to diffuse information across the nodes. While their motivations are quite different, those models are related by the fact that a spectral filter defined as a polynomial of order $K$ can be implemented as a $K$ -layer GNN.2 ",
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+    {
+        "type": "text",
+        "text": "4 PROPOSED GCRN MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose two GCRN architectures that are quite natural, and investigate their performances in real-world applications in Section 5. ",
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+        "text": "Model 1. The most straightforward definition is to stack a graph CNN, defined as (5), for feature extraction and an LSTM, defined as (2), for sequence learning: ",
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+        "type": "equation",
+        "img_path": "images/48c6ec5b9a58b44fa5b53070dc92ffee5ebe8c077532e457a3ac55ac4539eea6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { x _ { t } ^ { \\mathrm { { C N N } } } = { \\mathrm { C N N } } _ { \\mathscr { G } } ( x _ { t } ) } \\\\ & { \\qquad i = \\sigma \\big ( { W _ { x i } } { x _ { t } ^ { \\mathrm { { C N N } } } } + { W _ { h i } } { h _ { t - 1 } } + { w _ { c i } } \\odot { c _ { t - 1 } } + { b _ { i } } \\big ) , } \\\\ & { f = \\sigma \\big ( { W _ { x f } } { x _ { t } ^ { { \\mathrm { C N N } } } } + { W _ { h f } } { h _ { t - 1 } } + { w _ { c f } } \\odot { c _ { t - 1 } } + { b _ { f } } \\big ) , } \\\\ & { c _ { t } = f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot { \\mathrm { t a n h } } ( { W _ { x c } } { x _ { t } ^ { \\mathrm { { C N N } } } } + { W _ { h c } } { h _ { t - 1 } } + { b _ { c } } \\big ) , } \\\\ & { o = \\sigma \\big ( { W _ { x o } } { x _ { t } ^ { { \\mathrm { C N N } } } } + { W _ { h o } } { h _ { t - 1 } } + { w _ { c o } } \\odot { c _ { t } } + { b _ { o } } \\big ) , } \\\\ & { h _ { t } = o \\odot { \\mathrm { t a n h } } \\big ( { c _ { t } } \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In that setting, the input matrix $\\boldsymbol { x } _ { t } \\in \\mathbb { R } ^ { n \\times d _ { \\boldsymbol { x } } }$ may represent the observation of $d _ { x }$ measurements at time $t$ of a dynamical system over a network whose organization is given by a graph $\\mathcal { G }$ . $\\boldsymbol { x } _ { t } ^ { \\mathrm { C N N } }$ is the output of the graph CNN gate. For a proof of concept, we simply choose here $\\begin{array} { r } { x _ { t } ^ { \\mathrm { { C N N } } } = W ^ { \\mathrm { { C N N } } } * _ { \\mathcal { G } } x _ { t } . } \\end{array}$ , where $W ^ { \\mathrm { C N N } } \\in \\mathbf { \\bar { \\Gamma } } \\mathbb { R } ^ { K \\times d _ { x } \\times \\mathbf { \\bar { d } } _ { x } }$ are the Chebyshev coefficients for the graph convolutional kernels of support $K$ . The model also holds spatially distributed hidden and cell states of size $d _ { h }$ given by the matrices $c _ { t } , h _ { t } \\in \\mathbb { R } ^ { n \\times d _ { h } }$ . Peepholes are controlled by $w _ { c } . \\in \\mathbb { R } ^ { n \\times d _ { h } }$ . The weights $V _ { h \\cdot } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { h } }$ and $W _ { x }$ · $\\mathbf { \\Sigma } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { x } }$ are the parameters of the fully connected layers. An architecture such as (8) may be enough to capture the data distribution by exploiting local stationarity and compositionality properties as well as the dynamic properties. ",
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+        "type": "text",
+        "text": "Model 2. To generalize the convLSTM model (6) to graphs we replace the Euclidean 2D convolution $^ *$ by the graph convolution $^ { \\ast _ { \\mathcal { G } } }$ : ",
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+        "type": "equation",
+        "img_path": "images/7096a1d24cce9a1021dc9d187cb4106d4d0a0beb1b8000df88a46edc086de9a8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { i = \\sigma ( W _ { x i } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h i } \\ast _ { \\mathcal { G } } h _ { t - 1 } + w _ { c i } \\odot c _ { t - 1 } + b _ { i } ) , } \\\\ & { f = \\sigma ( W _ { x f } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h f } \\ast _ { \\mathcal { G } } h _ { t - 1 } + w _ { c f } \\odot c _ { t - 1 } + b _ { f } ) , } \\\\ & { c _ { t } = f _ { t } \\odot c _ { t - 1 } + i _ { t } \\odot \\operatorname { t a n h } ( W _ { x c } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h c } \\ast _ { \\mathcal { G } } h _ { t - 1 } + b _ { c } ) , } \\\\ & { o = \\sigma ( W _ { x o } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h o } \\ast _ { \\mathcal { G } } h _ { t - 1 } + w _ { c o } \\odot c _ { t } + b _ { o } ) , } \\\\ & { h _ { t } = o \\odot \\operatorname { t a n h } ( c _ { t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In that setting, the support $K$ of the graph convolutional kernels defined by the Chebyshev coefficients $W _ { h }$ · $\\doteq \\mathbb { R } ^ { K \\times d _ { h } ^ { \\bullet } \\times d _ { h } }$ and $W _ { x }$ · $\\breve { \\in \\mathbb { R } ^ { K \\times d _ { h } \\times d _ { x } } }$ determines the number of parameters, which is independent of the number of nodes $n$ . To keep the notation simple, we write $W _ { x i } * _ { \\mathcal { G } } x _ { t }$ to mean a graph convolution of $x _ { t }$ with $d _ { h } d _ { x }$ filters which are functions of the graph Laplacian $L$ parametrized by $K$ Chebyshev coefficients, as noted in (4) and (5). In a distributed computing setting, $K$ controls the communication overhead, i.e. the number of nodes any given node $i$ should exchange with in order to compute its local states. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The proposed blend of RNNs and graph CNNs is not limited to LSTMs and is straightforward to apply to any kind of recursive networks. For example, a vanilla RNN $h _ { t } = \\operatorname { t a n h } ( W _ { x } x _ { t } + W _ { h } h _ { t - 1 } )$ ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1f72a796efa62aa678281afc28c1ae1f8c2af391c3fb550841999e3356c792db.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>Structure</td><td>Filter size</td><td>Parameters</td><td>Runtime</td><td>Test(w/o Rot)</td><td>Test(Rot)</td></tr><tr><td>FC-LSTM</td><td>N/A</td><td>N/A</td><td>142,667,776</td><td>N/A</td><td>4832</td><td>=</td></tr><tr><td>LSTM+CNN</td><td>N/A</td><td>5×5</td><td>13,524,496</td><td>2.10</td><td>3851</td><td>4339</td></tr><tr><td>LSTM+CNN</td><td>N/A</td><td>9×9</td><td>43,802,128</td><td>6.10</td><td>3903</td><td>4208</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=3</td><td>1,629,712</td><td>0.82</td><td>3866</td><td>4367</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=5</td><td>2,711,056</td><td>1.24</td><td>3495</td><td>3932</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3400</td><td>3803</td></tr><tr><td>LSTM+GCNN</td><td>knn=8</td><td>K=9</td><td>4,873,744</td><td>2.15</td><td>3395</td><td>3814</td></tr><tr><td>LSTM+GCNN</td><td>knn=4</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3446</td><td>3844</td></tr><tr><td>LSTM+GCNN</td><td>knn=16</td><td>K=7</td><td>3,792,400</td><td>1.61</td><td>3578</td><td>3963</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1: Comparison between models. Runtime is the time spent per each mini-batch in seconds. Test cross-entropies correspond to moving MNIST, and rotating and moving MNIST. LSTM $\\cdot +$ GCNN is Model 2 defined in (9). Cross-entropy of FC-LSTM is taken from Shi et al. (2015). ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "would be modified as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d79616516db4ad1d5910e404e4644b58bff9f4005b164de6b7b572773622014c.jpg",
+        "text": "$$\nh _ { t } = \\operatorname { t a n h } ( W _ { x } * _ { \\mathcal { G } } x _ { t } + W _ { h } * _ { \\mathcal { G } } h _ { t - 1 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "and a Gated Recurrent Unit (GRU) (Cho et al., 2014) as ",
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/422b7026d5a39385da9c5717dc7ba20bfdf03c246eb59113c9e75d0d6450e48c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z = \\sigma ( W _ { x z } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h z } \\ast _ { \\mathcal { G } } h _ { t - 1 } ) , } \\\\ & { r = \\sigma ( W _ { x r } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h r } \\ast _ { \\mathcal { G } } h _ { t - 1 } ) , } \\\\ & { \\tilde { h } = \\operatorname { t a n h } ( W _ { x h } \\ast _ { \\mathcal { G } } x _ { t } + W _ { h h } \\ast _ { \\mathcal { G } } ( r \\odot h _ { t - 1 } ) ) , } \\\\ & { h _ { t } = z \\odot h _ { t - 1 } + ( 1 - z ) \\odot \\tilde { h } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As demonstrated by Shi et al. (2015), structure-aware LSTM cells can be stacked and used as sequence-to-sequence models using an architecture composed of an encoder, which processes the input sequence, and a decoder, which generates an output sequence. A standard practice for machine translation using RNNs (Cho et al., 2014; Sutskever et al., 2014). ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 SPATIO-TEMPORAL SEQUENCE MODELING ON MOVING-MNIST ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For this synthetic experiment, we use the moving-MNIST dataset generated by Shi et al. (2015). All sequences are 20 frames long (10 frames as input and 10 frames for prediction) and contain two handwritten digits bouncing inside a $6 4 \\times 6 4$ patch. Following their experimental setup, all models are trained by minimizing the binary cross-entropy loss using back-propagation through time (BPTT) and RMSProp with a learning rate of $1 0 ^ { - 3 }$ and a decay rate of 0.9. We choose the best model with early-stopping on validation set. All implementations are based on their Theano code and dataset.3 The adjacency matrix $A$ is constructed as a $\\mathbf { k }$ -nearest-neighbor (knn) graph with Euclidean distance and Gaussian kernel between pixel locations. For a fair comparison with Shi et al. (2015) defined in (6), all GCRN experiments are conducted with Model 2 defined in (9), which is the same architecture with the 2D convolution $^ *$ replaced by a graph convolution $^ { \\ast _ { \\mathcal { G } } }$ . To further explore the impact of the isotropic property of our filters, we generated a variant of the moving MNIST dataset where digits are also rotating (see Figure 4). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1 shows the performance of various models: (i) the baseline FC-LSTM from Shi et al. (2015), (ii) the 1-layer LSTM $^ +$ CNN from Shi et al. (2015) with different filter sizes, and (iii) the proposed LSTM+graph CNN(GCNN) defined in (9) with different supports $K$ . These results show the ability of the proposed method to capture spatio-temporal structures. Perhaps surprisingly, GCNNs can offer better performance than regular CNNs, even when the domain is a 2D grid and the data is images, the problem CNNs were initially developed for. The explanation is to be found in the differences between 2D filters and spectral graph filters. While a spectral filter of support $K = 3$ corresponds to the reach of a patch of size $5 \\times 5$ (see Figure 2), the difference resides in the isotropic nature of the former and the number of parameters: $K = 3$ for the former and $5 ^ { 2 } = 2 5$ for the later. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0e894075958a2fe6ae45958014950a3aaf3f9b843faf87a98d2580afacc1d3b.jpg",
+        "image_caption": [
+            "Figure 3: Cross-entropy on validation set: Left: performance of graph CNN with various filter support $K$ . Right: performance w.r.t. graph construction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/1cc92b78c2a49f24902eeb6f096c1459249ef396fac9c72850f84cda785f4e32.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative results for moving MNIST, and rotating and moving MNIST. First row is the input sequence, second the ground truth, and third and fourth are the predictions of the $\\mathrm { L S T M + C N N } ( 5 \\times 5 )$ and $\\mathrm { L S T M + G C N N } ( k n n = 8 , K = 7 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 indeed shows that LS $\\mathrm { T M } { + } \\mathrm { C N N } ( 5 \\times 5 )$ rivals $\\mathrm { L S T M + G C N N }$ with $K = 3$ . However, when increasing the filter size to $9 \\times 9$ or $K = 5$ , the GCNN variant clearly outperforms the CNN variant. This experiment demonstrates that graph spectral filters can obtain superior performance on regular domains with much less parameters thanks to their isotropic nature, a controversial property. Indeed, as the nodes are not ordered, there is no notion of an edge going up, down, on the right or on the left. All edges are treated equally, inducing some sort of rotation invariance. Additionally, Table 1 shows that the computational complexity of each model is linear with the filter size, and Figure 3 shows the learning dynamic of some of the models. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.2 NATURAL LANGUAGE MODELING ON PENN TREEBANK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The Penn Treebank dataset has 1,036,580 words. It was pre-processed in Zaremba et al. (2014) and split4 into a training set of $9 2 9 \\mathrm { k }$ words, a validation set of 73k words, and a test set of ${ 8 2 } \\mathrm { k }$ words. The size of the vocabulary of this corpus is 10,000. We use the gensim library5 to compute a word2vec model (Mikolov et al., 2013) for embedding the words of the dictionary in a 200-dimensional space. Then we build the adjacency matrix of the word embedding using a 4-nearest neighbor graph with cosine distance. Figure 6 presents the computed adjacency matrix, and its 3D visualization. We used the hyperparameters of the small configuration given by the code6 based on Zaremba et al. (2014): the size of the data mini-batch is 20, the number of temporal steps to unroll is 20, the dimension of the hidden state is 200. The global learning rate is 1.0 and the norm of the gradient is bounded by 5. The learning decay function is selected to be $\\mathrm { 0 . 5 ^ { m a x ( 0 , \\# e p o c h - 4 ) } }$ . All experiments have 13 epochs, and dropout value is 0.75. For Zaremba et al. (2014), the input representation $x _ { t }$ can be either the 200-dim embedding vector of the word, or the 10,000-dim one-hot representation of the word. For ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/5aea0fe6f1f13974198278cfe12fec785421338be7b8b8b7b0603d98e18d7d64.jpg",
+        "image_caption": [],
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+    {
+        "type": "table",
+        "img_path": "images/2fd5db6c10cb39ec3cac44638afebe278f3ba5cff9ed142b470c25c771268388.jpg",
+        "table_caption": [
+            "Figure 5: Learning dynamic of LSTM with and without graph structure and dropout regularization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>Representation</td><td>Parameters</td><td>Train Perplexity</td><td>Test Perplexity</td></tr><tr><td>Zaremba et al. (2014) code6</td><td>embedding</td><td>681,800</td><td>36.96</td><td>117.29</td></tr><tr><td>Zaremba et al. (2014) code6</td><td>one-hot</td><td>34,011,600</td><td>53.89</td><td>118.82</td></tr><tr><td>LSTM</td><td>embedding</td><td>681,800</td><td>48.38</td><td>120.90</td></tr><tr><td>LSTM</td><td>one-hot</td><td>34,011,600</td><td>54.41</td><td>120.16</td></tr><tr><td>LSTM, dropout</td><td>one-hot</td><td>34,011,600</td><td>145.59</td><td>112.98</td></tr><tr><td>GCRN-M1</td><td>one-hot</td><td>42,011,602</td><td>18.49</td><td>177.14</td></tr><tr><td>GCRN-M1, dropout</td><td>one-hot</td><td>42,011,602</td><td>114.29</td><td>98.67</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Comparison of models in terms of perplexity. Zaremba et al. (2014) code6 is ran as benchmark algorithm. The original Zaremba et al. (2014) code used as input representation for $x _ { t }$ the 200-dim embedding representation of words, computed here by the gensim library5. As our model runs on the 10,000-dim one-hot representation of words, we also ran Zaremba et al. (2014) code on this representation. We re-implemented Zaremba et al. (2014) code with the same architecture and hyperparameters. We remind that GCRN-M1 refers to GCRN Model 1 defined in (8). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "our models, the input representation is a one-hot representation of the word. This choice allows us to use the graph structure of the words. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 reports the final train and test perplexity values for each investigated model and Figure 5 plots the perplexity value vs. the number of epochs for the train and test sets with and without dropout regularization. Numerical experiments show: ",
+        "bbox": [
+            174,
+            595,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1. Given the same experimental conditions in terms of architecture and no dropout regularization, the standalone model of LSTM is more accurate than LSTM using the spatial graph information (120.16 vs. 177.14), extracted by graph CNN with the GCRN architecture of Model 1, Eq. (8).   \n2. However, using dropout regularization, the graph LSTM model overcomes the standalone LSTM with perplexity values 98.67 vs. 112.98.   \n3. The use of spatial graph information found by graph CNN speeds up the learning process, and overfits the training dataset in the absence of dropout regularization. The graph structure likely acts a constraint on the learning system that is forced to move in the space of language topics.   \n4. We performed the same experiments with LSTM and Model 2 defined in (9). Model 1 significantly outperformed Model 2, and Model 2 did worse than standalone LSTM. This bad performance may be the result of the large increase of dimensionality in Model 2, as the dimension of the hidden and cell states changes from 200 to 10,000, the size of the vocabulary. A solution would be to downsize the data dimensionality, as done in Shi et al. (2015) in the case of image data. ",
+        "bbox": [
+            209,
+            648,
+            825,
+            885
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/db4ee1673e3b3918a68b4668574358c8b6f41e4249c011cc1ed5e715b01017b4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a0958b69c9ef0cec7f225791ed9902ac5872a3df2683f257db5466fc71df530.jpg",
+        "image_caption": [
+            "Figure 6: Left: adjacency matrix of word embeddings. Right: 3D visualization of words’ structure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            114,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            324,
+            495,
+            340
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work aims at learning spatio-temporal structures from graph-structured and time-varying data. In this context, the main challenge is to identify the best possible architecture that combines simultaneously recurrent neural networks like vanilla RNN, LSTM or GRU with convolutional neural networks for graph-structured data. We have investigated here two architectures, one using a stack of CNN and RNN (Model 1), and one using convLSTM that considers convolutions instead of fully connected operations in the RNN definition (Model 2). We have then considered two applications: video prediction and natural language modeling. Model 2 has shown good performances in the case of video prediction, by improving the results of Shi et al. (2015). Model 1 has also provided promising performances in the case of language modeling, particularly in terms of learning speed. It has been shown that (i) isotropic filters, maybe surprisingly, can outperform classical 2D filters on images while requiring much less parameters, and (ii) that graphs coupled with graph CNN and RNN are a versatile way of introducing and exploiting side-information, e.g. the semantic of words, by structuring a data matrix. ",
+        "bbox": [
+            174,
+            358,
+            825,
+            539
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Future work will investigate applications to data naturally structured as dynamic graph signals, for instance fMRI and sensor networks. The graph CNN model we have used is rotationally-invariant and such spatial property seems quite attractive in real situations where motion is beyond translation. We will also investigate how to benefit of the fast learning property of our system to speed up language modeling models. Eventually, it will be interesting to analyze the underlying dynamical property of generic RNN architectures in the case of graphs. Graph structures may introduce stability to RNN systems, and prevent them to express unstable dynamic behaviors. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            643
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENT ",
+        "text_level": 1,
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+            176,
+            667,
+            346,
+            683
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This research was supported in part by the European Union’s H2020 Framework Programme (H2020-MSCA-ITN-2014) under grant No. 642685 MacSeNet, and Nvidia equipment grant. ",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
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parse/train/SJ1Xmf-Rb/SJ1Xmf-Rb.md

@@ -0,0 +1,364 @@
+# FEARNET: BRAIN-INSPIRED MODEL FOR INCREMENTAL LEARNING
+
+Ronald Kemker and Christopher Kanan∗
+
+Carlson Center for Imaging Science Rochester Institute of Technology Rochester, NY 14623, USA {rmk6217,kanan}@rit.edu
+
+# ABSTRACT
+
+Incremental class learning involves sequentially learning classes in bursts of examples from the same class. This violates the assumptions that underlie methods for training standard deep neural networks, and will cause them to suffer from catastrophic forgetting. Arguably, the best method for incremental class learning is iCaRL, but it requires storing training examples for each class, making it challenging to scale. Here, we propose FearNet for incremental class learning. FearNet is a generative model that does not store previous examples, making it memory efficient. FearNet uses a brain-inspired dual-memory system in which new memories are consolidated from a network for recent memories inspired by the mammalian hippocampal complex to a network for long-term storage inspired by medial prefrontal cortex. Memory consolidation is inspired by mechanisms that occur during sleep. FearNet also uses a module inspired by the basolateral amygdala for determining which memory system to use for recall. FearNet achieves state-of-the-art performance at incremental class learning on image (CIFAR-100, CUB-200) and audio classification (AudioSet) benchmarks.
+
+# 1 INTRODUCTION
+
+In incremental classification, an agent must sequentially learn to classify training examples, without necessarily having the ability to re-study previously seen examples. While deep neural networks (DNNs) have revolutionized machine perception (Krizhevsky et al., 2012), off-the-shelf DNNs cannot incrementally learn classes due to catastrophic forgetting. Catastrophic forgetting is a phenomenon in which a DNN completely fails to learn new data without forgetting much of its previously learned knowledge (McCloskey & Cohen, 1989). While methods have been developed to try and mitigate catastrophic forgetting, as shown in Kemker et al. (2018), these methods are not sufficient and perform poorly on larger datasets. In this paper, we propose FearNet, a brain-inspired system for incrementally learning categories that significantly outperforms previous methods.
+
+The standard way for dealing with catastrophic forgetting in DNNs is to avoid it altogether by mixing new training examples with old ones and completely re-training the model offline. For large datasets, this may require weeks of time, and it is not a scalable solution. An ideal incremental learning system would be able to assimilate new information without the need to store the entire training dataset. A major application for incremental learning includes real-time operation on-board embedded platforms that have limited computing power, storage, and memory, e.g., smart toys, smartphone applications, and robots. For example, a toy robot may need to learn to recognize objects within its local environment and of interest to its owner. Using cloud computing to overcome these resource limitations may pose privacy risks and may not be scalable to a large number of embedded devices. A better solution is on-device incremental learning, which requires the model to use less storage and computational power.
+
+In this paper, we propose an incremental learning framework called FearNet (see Fig. 1). FearNet has three brain-inspired sub-systems: 1) a recent memory system for quick recall, 2) a memory system for long-term storage, and 3) a sub-system that determines which memory system to use for a particular example. FearNet mitigates catastrophic forgetting by consolidating recent memories into long-term storage using pseudorehearsal (Robins, 1995). Pseudorehearsal allows the network to revisit previous memories during incremental training without the need to store previous training examples, which is more memory efficient.
+
+Problem Formulation: Here, incremental class learning consists of $T$ study-sessions. At time $t$ , the learner receives a batch of data $B _ { t }$ , which contains $N _ { t }$ labeled training samples, i.e., $B _ { t } ~ = ~ \{ ( \mathbf { x } _ { j } , y _ { j } ) \} _ { j = 1 } ^ { N _ { t } }$ , where $\mathbf { x } _ { j } \in \mathbb { R } ^ { d }$ is the input feature vector to be classified and $y _ { j }$ is its corresponding label. The number of training samples $N _ { t }$ may vary between sessions, and the data inside a study-session is not assumed to be independent and identically distributed (iid). During a study session, the learner only has access to its current batch, but it may use its own memory to store information from prior study sessions. We refer to the first session as the model’s “base-knowledge,” which contains exemplars from $M \geq 1$ classes. The batches learned in all subsequent sessions contain only one class, i.e., all $y _ { j }$ will be identical within those sessions.
+
+# Novel Contributions: Our contributions include:
+
+![](images/9ec0369767a96c98cf47fe8ba08e1c364ce5daf78138cb6ec61cc3ee815bb4b4.jpg)  
+Figure 1: FearNet consists of three braininspired modules based on 1) mPFC (longterm storage), 2) HC (recent storage), and 3) BLA for determining whether to use mPFC or HC for recall.
+
+1. FearNet’s architecture includes three neural networks: one inspired by the hippocampal complex (HC) for recent memories, one inspired by the medial prefrontal cortex (mPFC) for long-term storage, and one inspired by the basolateral amygdala (BLA) that determines whether to use HC or mPFC for recall.
+
+2. Motivated by memory replay during sleep, FearNet employs a generative autoencoder for pseudorehearsal, which mitigates catastrophic forgetting by generating previously learned examples that are replayed alongside novel information during consolidation. This process does not involve storing previous training data.
+
+3. FearNet achieves state-of-the-art results on large image and audio datasets with a relatively small memory footprint, demonstrating how dual-memory models can be scaled.
+
+# 2 RELATED WORK
+
+Catastrophic forgetting in DNNs occurs due to the plasticity-stability dilemma (Abraham & Robins, 2005). If the network is too plastic, older memories will quickly be overwritten; however, if the network is too stable, it is unable to learn new data. This problem was recognized almost 30 years ago (McCloskey & Cohen, 1989). In French (1999), methods developed in the 1980s and 1990s are extensively discussed, and French argued that mitigating catastrophic forgetting would require having two separate memory centers: one for the long-term storage of older memories and another to quickly process new information as it comes in. He also theorized that this type of dual-memory system would be capable of consolidating memories from the fast learning memory center to longterm storage.
+
+Catastrophic forgetting often occurs when a system is trained on non-iid data. One strategy for reducing this phenomenon is to mix old examples with new examples, which simulates iid conditions. For example, if the system learns ten classes in a study session and then needs to learn 10 new classes in a later study session, one solution could be to mix examples from the first study session into the later study session. This method is known as rehearsal, and it is one of the earliest methods for reducing catastrophic forgetting (Hetherington & Seidenberg, 1989). Rehearsal essentially uses an external memory to strengthen the model’s representations for examples learned previously, so that they are not overwritten when learning data from new classes. Rehearsal reduces forgetting, but performance is still worse than offline models. Moreover, rehearsal requires storing all of the training data. Robins (1995) argued that storing of training examples was inefficient and of “little interest,” so he introduced pseudorehearsal. Rather than replaying past training data, in pseudorehearsal, the algorithm generates new examples for a given class. In Robins (1995), this was done by creating random input vectors, having the network assign them a label, and then mixing them into the new training data. This idea was revived in Draelos et al. (2017), where a generative autoencoder was used to create pseudo-examples for unsupervised incremental learning. This method inspired FearNet’s approach to memory consolidation. Pseudorehearsal is related to memory replay that occurs in mammalian brains, which involves reactivation of recently encoded memories in HC so that they can be integrated into long-term storage in mPFC (Rasch & Born, 2013).
+
+Recently there has been renewed interest in solving catastrophic forgetting in supervised learning. Many new methods are designed to mitigate catastrophic forgetting when each study session contains a permuted version of the entire training dataset (see Goodfellow et al. (2013)). Unlike incremental class learning, all labels are contained in each study session. PathNet uses an evolutionary algorithm to find the optimal path through a large DNN, and then freezes the weights along that path (Fernando et al., 2017). It assumes all classes are seen in each study session, and it is not capable of incremental class learning. Elastic Weight Consolidation (EWC) employs a regularization scheme that redirects plasticity to the weights that are least important to previously learned study sessions (Kirkpatrick et al., 2017). After EWC learns a study session, it uses the training data to build a Fisher matrix that determines the importance of each feature to the classification task it just learned. EWC was shown to work poorly at incremental class learning in Kemker et al. (2018).
+
+The Fixed Expansion Layer (FEL) model mitigates catastrophic forgetting by using sparse updates (Coop et al., 2013). FEL uses two hidden layers, where the second hidden layer (i.e., the FEL layer) has connectivity constraints. The FEL layer is much larger than the first hidden layer, is sparsely populated with excitatory and inhibitory weights, and is not updated during training. This limits learning of dense shared representations, which reduces the risk of learning interfering with old memories. FEL requires a large number of units to work well (Kemker et al., 2018).
+
+Gepperth & Karaoguz (2016) introduced a new approach for incremental learning, which we call GeppNet. GeppNet uses a self-organizing map (SOM) to reorganize the input onto a two-dimensional lattice. This serves as a long-term memory, which is fed into a simple linear layer for classification. After the SOM is initialized, it can only be updated if the input is sufficiently novel. This prevents the model from forgetting older data too quickly. GeppNet also uses rehearsal using all previous training data. A variant of GeppNet, GeppNet+STM, uses a fixed-size memory buffer to store novel examples. When this buffer is full, it replaces the oldest example. During pre-defined intervals, the buffer is used to train the model. Gepp$\mathbf { \Gamma } _ { \mathbf { N e t + S T M } }$ is better at retaining base-knowledge since it only trains during its consolidation phase, but the STM-free version learns new data better because it updates the model on every novel labeled input.
+
+![](images/9487165ad014d2cb3f67829861104cc2ddefd93cc570c6b1da29573fe8bec15b.jpg)  
+Figure 2: iCaRL’s performance depends heavily on the number of exemplars per class (EPC) that it stores. Reducing EPC from 20 (blue) to 1 (red) severely impairs its ability to recall older information.
+
+iCaRL (Rebuffi et al., 2017) is an incremental class learning framework. Rather than directly using a DNN for classification, iCaRL uses it for supervised representation learning. During a study session, iCaRL updates a DNN using the study session’s data and a set of $J$ stored examples from earlier sessions $( J = 2 , 0 0 0$ for CIFAR-100 in their paper), which is a kind of rehearsal. After a study session, the $J$ examples retained are carefully chosen using herding. After learning the entire dataset, iCaRL has retained $J / T$ exemplars per class (e.g., $J / \bar { T } = 2 0$ for CIFAR-100). The DNN in iCaRL is then used to compute an embedding for each stored example, and then the mean embedding for each class seen is computed. To classify a new instance, the DNN is used to compute an embedding for it, and then the class with the nearest mean embedding is assigned. iCaRL’s performance is heavily influenced by the number of examples it stores, as shown in Fig. 2.
+
+# 3 MAMMALIAN MEMORY: NEUROSCIENCE AND MODELS
+
+FearNet is heavily inspired by the dual-memory model of mammalian memory (McClelland et al., 1995), which has considerable experimental support from neuroscience (Frankland et al., 2004; Takashima et al., 2006; Kitamura et al., 2017; Bontempi et al., 1999; Taupin & Gage, 2002; Gais et al., 2007). This theory proposes that HC and mPFC operate as complementary memory systems, where HC is responsible for recalling recent memories and mPFC is responsible for recalling remote (mature) memories. GeppNet is the most recent DNN to be based on this theory, but it was also independently explored in the 1990s in French (1997) and Ans & Rousset (1997). In this section, we review some of the evidence for the dual-memory model.
+
+One of the major reasons why HC is thought to be responsible for recent memories is that if HC is bilaterally destroyed, then anterograde amnesia occurs with old memories for semantic information preserved. One mechanism HC may use to facilitate creating new memories is adult neurogenesis. This occurs in HC’s dentate gyrus (Altman, 1963; Eriksson et al., 1998). The new neurons have higher initial plasticity, but it reduces as time progresses (Deng et al., 2010).
+
+In contrast, mPFC is responsible for the recall of remote (long-term) memories (Bontempi et al., 1999). Taupin & Gage (2002) and Gais et al. (2007) showed that mPFC plays a strong role in memory consolidation during REM sleep. McClelland et al. (1995) and Euston et al. (2012) theorized that, during sleep, HC reactivates recent memories to prevent forgetting which causes these recent memories to replay in mPFC as well, with dreams possibly being caused by this process. After memories are transferred from HC to mPFC, evidence suggests that corresponding memory in HC is erased (Poe, 2017).
+
+Recently, Kitamura et al. (2017) performed contextual fear conditioning (CFC) experiments in mice to trace the formation and consolidation of recent memories to long-term storage. CFC experiments involve shocking mice while subjecting them to various visual stimuli (i.e., colored lights). They found that BLA, which is responsible for regulating the brain’s fear response, would shift where it retrieved the corresponding memory from (HC or mPFC) as that memory was consolidated over time. FearNet follows the memory consolidation theory proposed by Kitamura et al. (2017).
+
+# 4 THE FEARNET MODEL
+
+FearNet has two complementary memory centers, 1) a short-term memory system that immediately learns new information for recent recall (HC) and 2) a DNN for the storage of remote memories (mPFC). FearNet also has a separate BLA network that determines which memory center contains the associated memory required for prediction. During sleep phases, FearNet uses a generative model to consolidate data from HC to mPFC through pseudorehearsal. Pseudocode for FearNet is provided in the supplemental material. Because the focus of our work is not representation learning, we use pre-trained ResNet embeddings to obtain features that are fed to FearNet.
+
+# 4.1 DUAL-MEMORY STORAGE
+
+FearNet’s HC model is a variant of a probabilistic neural network (Specht, 1990). HC computes class conditional probabilities using stored training examples. Formally, HC estimates the probability that an input feature vector $\mathbf { x }$ belongs to class $k$ as
+
+$$
+P _ { H C } \left( C = k | \mathbf { x } \right) = \frac { \beta _ { k } } { \sum _ { k ^ { \prime } } \beta _ { k ^ { \prime } } }
+$$
+
+$$
+\beta _ { k } = \left\{ \begin{array} { c c } { \big ( \epsilon + \operatorname* { m i n } _ { j } \| \mathbf x - \mathbf u _ { k , j } \| _ { 2 } \big ) ^ { - 1 } } & { \mathrm { i f ~ H C ~ c o n t a i n s ~ i n s t a n c e s ~ o f ~ c l a s s ~ } k } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+where $\epsilon > 0$ is a regularization parameter and ${ \mathbf { u } } _ { k , j }$ is the $j ^ { : }$ ’th stored exemplar in HC for class $k$ . All exemplars are removed from HC after they are consolidated into mPFC.
+
+FearNet’s mPFC is implemented using a DNN trained both to reconstruct its input using a symmetric encoder-decoder (autoencoder) and to compute $P _ { m P F C } \left( C = k | \mathbf { x } \right)$ . The autoencoder enables us to
+
+![](images/0a31f3f9e3363e7a1f2b1d9c767fa264123bab51cadfb522dfebe045d764880d.jpg)  
+Figure 3: The mPFC and BLA sub-systems in FearNet. mPFC is responsible for the long-term storage of remote memories. BLA is used during prediction time to determine if the memory should be recalled from short- or long-term memory.
+
+use pseudorehearsal, which is described in more detail in Sec. 4.2. The loss function for mPFC is
+
+$$
+\mathcal { L } _ { m P F C } = \mathcal { L } _ { c l a s s } + \mathcal { L } _ { r e c o n } ,
+$$
+
+where $\mathcal { L } _ { c l a s s }$ is the supervised classification loss and $\mathcal { L } _ { r e c o n }$ is the unsupervised reconstruction loss, as illustrated in Fig. 3(a). For $\mathcal { L } _ { c l a s s }$ , we use standard softmax loss. $\mathcal { L } _ { r e c o n }$ is the weighted sum of mean squared error (MSE) reconstruction losses from each layer, which is given by
+
+$$
+\mathcal { L } _ { r e c o n } = \sum _ { j = 0 } ^ { M } \sum _ { i = 0 } ^ { H _ { j } - 1 } \left. h _ { e n c o d e r , ( i , j ) } - h _ { d e c o d e r , ( i , j ) } \right. _ { 2 } ^ { 2 }
+$$
+
+where $M$ is the number of mPFC layers, $H _ { j }$ is the number of hidden units in layer $j$ , $h _ { e n c o d e r , ( i , j ) }$ and $h _ { d e c o d e r , ( i , j ) }$ are the outputs of the encoder/decoder at layer $j$ respectively, and $\lambda _ { j }$ is the reconstruction weight for that layer. mPFC is similar to a Ladder Network (Rasmus et al., 2015), which combines classification and reconstruction to improve regularization, especially during lowshot learning. The $\lambda _ { j }$ hyperparameters were found empirically, with $\lambda _ { 0 }$ being largest and decreasing for deeper layers (see supplementary material). This prioritizes the reconstruction task, which makes the generated pseudo-examples more realistic. When training is completed during a study session, all of the data in HC is pushed through the encoder to extract a dense feature representation of the original data, and then we compute a mean feature vector $\mu _ { c }$ and covariance matrix $\Sigma _ { c }$ for each class $c$ . These are stored and used to generate pseudo-examples during consolidation (see Sec. 4.2). We study FearNet’s performance as a function of how much data is stored in HC in Sec. 6.2.
+
+# 4.2 PSEUDOREHEARSAL FOR MEMORY CONSOLIDATION
+
+During FearNet’s sleep phase, the original inputs stored in HC are transferred to mPFC using pseudo-examples created by an autoencoder. This process is known as intrinsic replay, and it was used by Draelos et al. (2017) for unsupervised learning.
+
+Using the class statistics from the encoder, pseudo-examples for class $c$ are generated by sampling a Gaussian with mean $\mu _ { c }$ and covariance matrix $\Sigma _ { c }$ to obtain $\hat { \mathbf { x } } _ { r a n d }$ . Then, $\hat { \mathbf { x } } _ { r a n d }$ is passed through the decoder to generate a pseudo-example. To create a balanced training set, for each class that mPFC has learned, we generate $\lceil m \rceil$ pseudo-examples, where $m$ is the average number of examples per class stored in HC. The pseudo-examples are mixed with the data in HC, and the mixture is used to fine-tune mPFC using backpropagation. After consolidation, all units in HC are deleted.
+
+# 4.3 NETWORK SELECTION USING BLA
+
+During prediction, FearNet uses the BLA network (Fig. 3(b)) to determine whether to classify an input x using HC or mPFC. This can be challenging because if HC has only been trained on one class, it will put all of its probability mass on that class, whereas mPFC will likely be less confident. The output of BLA is given by $A \left( \mathbf { x } \right)$ and will be a value between 0 and 1, with a 1 indicating mPFC should be used. BLA is trained after each study session using only the data in HC and with pseudoexamples generated with mPFC, using the same procedure described in Sec. 4.2. Instead of using solely BLA to determine which network to use, we found that combining its output with those of mPFC and HC improved results. The predicted class $\hat { y }$ is computed as
+
+$$
+\hat { y } = \left\{ \begin{array} { c c } { \underset { \mathrm { a r g } } { \arg \operatorname* { m a x } } _ { k ^ { \prime } } P _ { H C } \left( C = k ^ { \prime } | \mathbf { x } \right) } & { \mathrm { i f } \ \psi > \operatorname* { m a x } _ { k } P _ { m P F C } \left( C = k | \mathbf { x } \right) } \\ { \underset { \mathrm { o t h e r w i s e } } { \arg \operatorname* { m a x } } } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+where
+
+$$
+\psi = \left( 1 - A \left( \mathbf { x } \right) \right) ^ { - 1 } \operatorname* { m a x } _ { k } P _ { H C } \left( C = k | \mathbf { x } \right) A \left( \mathbf { x } \right)
+$$
+
+$\psi$ is the probability of the class according to HC weighted by the confidence that the associated memory is actually stored in HC. BLA has the same number of layers/units as the mPFC encoder, and uses a logistic output unit. We discuss alternative BLA models in supplemental material.
+
+# 5 EXPERIMENTAL SETUP
+
+Evaluating Incremental Learning Performance. To evaluate how well the incrementally trained models perform compared to an offline model, we use the three metrics proposed in Kemker et al. (2018). After each study session $t$ in which a model learned a new class $k$ , we compute the model’s test accuracy on the new class $( \alpha _ { n e w , t } )$ , the accuracy on the base-knowledge $( \alpha _ { b a s e , t } )$ , and the accuracy of all of the test data seen to this point $( \alpha _ { a l l , t } )$ . After all $T$ study sessions are complete, a model’s ability to retain the base-knowledge is given by $\begin{array} { r } { \Omega _ { b a s e } \ = \ \frac { 1 } { T - 1 } \sum _ { t = 2 } ^ { T } \frac { \alpha _ { b a s e , t } } { \alpha _ { o f f l i n e } } } \end{array}$ where $\alpha _ { o f f l i n e }$ is the accuracy of a multi-layer perceptron (MLP) trained offline (i.e., it is given all of the training data at once). The model’s ability to immediately recall new information is measured by 1T −1 PTt=2 αnew,t. Finally, we measure how well the model does on all available test data with Ωall = T −1 P t=2 αoffline αall,t . The $\Omega _ { a l l }$ metric shows how well new memories are integrated into the model over time. For all of the metrics, higher values indicate superior performance. Both $\Omega _ { b a s e }$ and $\Omega _ { a l l }$ are relative to an offline MLP model, so a value of 1 indicates that a model has similar performance to the offline baseline. This allows results across datasets to be better compared. Note that $\Omega _ { b a s e } > 1$ and $\Omega _ { a l l } > 1$ only if the incremental learning algorithm is more accurate than the offline model, which can occur due to better regularization strategies employed by different models.
+
+Datasets. We evaluate all of the models on three benchmark datasets (Table 1): CIFAR-100, CUB-200, and AudioSet. CIFAR-100 is a popular image classification dataset containing 100 mutually-exclusive object categories, and it was used in Rebuffi et al. (2017) to evaluate iCaRL. All images are $3 2 \times 3 2$ pixels. CUB-200 is a fine-grained image classification dataset containing high resolution images of 200 different bird
+
+Table 1: Dataset Specifications   
+
+<table><tr><td></td><td>CIFAR-100</td><td>CUB-200</td><td>AudioSet</td></tr><tr><td>Classification Task</td><td>RGB Image</td><td>RGB Image</td><td>Audio</td></tr><tr><td>Classes</td><td>100</td><td>200</td><td>100</td></tr><tr><td>Feature Shape</td><td>2.048</td><td>2,048</td><td>1,280</td></tr><tr><td>Train Samples</td><td>50,000</td><td>5,994</td><td>28,779</td></tr><tr><td>Test Samples</td><td>10,000</td><td>5,794</td><td>5,523</td></tr><tr><td>Train Samples/Class</td><td>500</td><td>29-30</td><td>250-300</td></tr><tr><td>Test Samples/Class</td><td>100</td><td>11-30</td><td>43-62</td></tr></table>
+
+species (Welinder et al., 2010). We use the 2011 version of the dataset. AudioSet is an audio classification dataset (Gemmeke et al., 2017). We use the variant of AudioSet used by Kemker et al. (2018), which contains a 100 class subset such that none of the classes were super- or sub-classes of one another. Also, since the AudioSet data samples can have more than one class, the chosen samples had only one of the 100 classes chosen in this subset.
+
+For CIFAR-100 and CUB-200, we extract ResNet-50 image embeddings as the input to each of the models, where ResNet-50 was pre-trained on ImageNet (He et al., 2016). We use the output after the mean pooling layer and normalize the features to unit length. For AudioSet, we use the audio CNN embeddings produced by pre-training the model on the YouTube-8M dataset (Abu-El-Haija et al., 2016). We use the pre-extracted AudioSet feature embeddings, which represent ten second sound clips (i.e., ten 128-dimensional vectors concatenated in order).
+
+Comparison Models. We compare FearNet to FEL, GeppNet, GeppNet+STM, iCaRL, and an onenearest neighbor (1-NN). FEL, GeppNet, and GeppNet+STM were chosen due to their previously reported efficacy at incremental class learning in Kemker et al. (2018). iCARL is explicitly designed for incremental class learning, and represents the state-of-the-art on this problem. We compare against 1-NN due to its similarity to our HC model. 1-NN does not forget any previously observed examples, but it tends to have worse generalization error than parametric methods and requires storing all of the training data.
+
+![](images/8fe87b21db3278cee40742062caccd69283383f3df858d8c12bed9bd6756eed5.jpg)  
+Figure 4: Mean-class test accuracy of all classes seen so far.
+
+In each of our experiments, all models take the same feature embedding as input for a given dataset. This required modifying iCaRL by turning its CNN into a fully connected network. We performed a hyperparameter search for each model/dataset combination to tune the number of units and layers (see Supplemental Materials).
+
+Training Parameters. FearNet was implemented in Tensorflow. For mPFC and BLA, each fully connected layer uses an exponential linear unit activation function (Clevert et al., 2016). The output of the encoder also connects to a softmax output layer. Xavier initialization is used to initialize all weight layers (Glorot & Bengio, 2010), and all of the biases are initialized to one. BLA’s architecture is identical to mPFC’s encoder, except it has a logistic output unit, instead of a softmax layer.
+
+mPFC and BLA were trained using NAdam. We train mPFC on the base-knowledge set for 1,000 epochs, consolidate HC over to mPFC for 60 epochs, and train BLA for 20 epochs. Because mPFC’s decoder is vital to preserving memories, its learning rate is $1 / 1 0 0$ times lower than the encoder. We performed a hyperparameter search for each dataset and model, varying the model shape (64-1,024 units), depth (2-4 layers), and how often to sleep (see Sec. 6.2). Across datasets, mPFC and BLA performed best with two hidden layers, but the number of units per layer varied across datasets. The specific values used for each dataset are given in supplemental material. In preliminary experiments, we found no benefit to adding weight decay to mPFC, likely because the reconstruction task helps regularize the model.
+
+# 6 EXPERIMENTAL RESULTS
+
+Unless otherwise noted, each class is only seen in one unique study-session and the first baseknowledge study session contains half the classes in the dataset. We perform additional experiments to study how changing the number of base-knowledge classes affects performance in Sec. 6.2. Unless otherwise noted, FearNet sleeps every 10 study sessions across datasets.
+
+# 6.1 STATE-OF-THE-ART COMPARISON
+
+Table 2 shows incremental class learning summary results for all six methods. FearNet achieves the best $\Omega _ { b a s e }$ and $\Omega _ { a l l }$ on all three datasets. Fig. 4 shows that FearNet more closely resembles the offline MLP baseline than other methods. $\Omega _ { n e w }$ measures test accuracy on the most recently trained class. 1 For FearNet, this measures the performance of HC and BLA. $\Omega _ { n e w }$ does not account for how well the class was consolidated into mPFC which happens later during a sleep phase; however, $\Omega _ { a l l }$ does account for this. FEL achieves a high $\Omega _ { n e w }$ score because it is able to achieve nearly perfect test accuracy on every new class it learns, but this results in forgetting more quickly than FearNet. 1-NN is similar to our HC model; but on its own, it fails to generalize as well as FearNet, is memory inefficient, and is slow to make predictions. The final mean-class test accuracy for the offline MLP used to normalize the metrics is $6 9 . 9 \%$ for CIFAR-100, $5 9 . 8 \%$ for CUB-200, and $4 5 . 8 \%$ for AudioSet.
+
+Table 2: State-of-the-art comparison on CIFAR-100, CUB-200, and AudioSet. The best $\Omega _ { a l l }$ for each dataset are in bold. $\Omega _ { b a s e }$ and $\Omega _ { a l l }$ are normalized by the offline MLP baseline.   
+
+<table><tr><td>Model</td><td colspan="2">CIFAR-100 Sbase Ωnew all</td><td colspan="2">CUB-200 Snew</td><td colspan="2">AudioSet Ωnew</td><td colspan="2">Mean Sbase</td></tr><tr><td>1-Nearest Neighbor|</td><td>|0.878</td><td>0.648</td><td>Sbase |0.746</td><td>Ωaul 0.434</td><td>Sbase 10.655</td><td>0.269</td><td>Sall 0.613</td><td>Sall</td></tr><tr><td>GeppNet+STM</td><td>0.866</td><td>0.879 0.408 0.800</td><td>0.764</td><td>0.694 0.204 0.645</td><td>0.941</td><td>0.861</td><td>[0.760 0.857</td><td>0.729 0.769</td></tr><tr><td>GeppNet</td><td>0.833</td><td>0.529 0.754</td><td>0.727</td><td>0.558 0.645</td><td>0.932</td><td>0.372</td><td>0.831</td><td>0.759</td></tr><tr><td>FEL</td><td>0.707</td><td>0.999 0.619</td><td>0.702</td><td>0.976 0.641</td><td>0.491</td><td>0.499 1.000 0.456</td><td>0.879</td><td></td></tr><tr><td>iCaRL</td><td>0.746</td><td>0.807 0.749</td><td>0.942 0.547</td><td>0.864</td><td>0.740 0.487</td><td>0.733</td><td>0.633 0.801</td><td>0.572 0.782</td></tr><tr><td>FearNet</td><td>0.927</td><td>0.824 0.947</td><td>0.924</td><td>0.598 0.891</td><td>0.962</td><td>0.455 0.932</td><td>0.938</td><td>0.923</td></tr></table>
+
+<table><tr><td></td><td colspan="2">CIFAR-100</td><td colspan="2">CUB-200</td><td colspan="2">AudioSet</td></tr><tr><td></td><td>Oracle</td><td>With BLA</td><td>Oracle</td><td>With BLA</td><td>Oracle</td><td>With BLA</td></tr><tr><td>Sbase</td><td>0.965</td><td>0.927</td><td>0.968</td><td>0.924</td><td>0.970</td><td>0.962</td></tr><tr><td>Snew</td><td>0.912</td><td>0.824</td><td>0.729</td><td>0.598</td><td>0.701</td><td>0.455</td></tr><tr><td>Ωau</td><td>1.002</td><td>0.947</td><td>0.936</td><td>0.891</td><td>0.972</td><td>0.932</td></tr></table>
+
+Table 3: FearNet performance when the location of the associated memory is known using an oracle versus using BLA.
+
+# 6.2 ADDITIONAL EXPERIMENTS
+
+Novelty Detection with BLA. We evaluated the performance of BLA by comparing it to an oracle version of FearNet, i.e., a version that knew if the relevant memory was stored in either mPFC or HC. Table 3 shows that FearNet’s BLA does a good job at predicting which network to use; however, the decrease in $\Omega _ { n e w }$ suggests BLA is sometimes using mPFC when it should be using HC.
+
+When should the model sleep? To study how the frequency of memory consolidation affects FearNet’s performance, we trained FearNet on CUB-200 and varied the sleep frequency from 1-15 study sessions. When FearNet increases the number of classes it learns before sleeping (Fig. 5), it is better able to retain its base-knowledge, but this reduces its ability to recall new information. In humans, sleep deprivation is known to impair new learning (Yoo et al., 2007), and that forgetting occurs during sleep (Poe, 2017). Each time FearNet sleeps, the mPFC weights are perturbed which can cause it to gradually forget older memories. Sleeping less causes HC’s recall performance to deteriorate.
+
+![](images/2f78923b9aae75d6dcc03f23fbd92ef51d126b3d916bf7c065180de4a5bced14.jpg)  
+Figure 5: FearNet performance as the sleep frequency decreases.
+
+Multi-Modal Incremental Learning. As shown in Sec. 6.1, FearNet can incrementally learn and retain information from a single dataset, but how does it perform if new inputs differ greatly from previously learned ones? This scenario is one of the first shown to cause catastrophic forgetting in MLPs. To study this, we trained FearNet to incrementally learn CIFAR-100 and AudioSet, which after training is a 200-way classification problem. To do this, AudioSet’s features are zero-padded to make them the same length as CIFAR-100s. Table 4 shows the performance of FearNet for three separate training paradigms: 1) FearNet learns CIFAR-100 as the base
+
+<table><tr><td></td><td colspan="2">Base-Knowledge CIFAR-100 AudioSet 50/50 Mix</td></tr><tr><td>Sbase|</td><td>0.995 0.845</td><td>0.837</td></tr><tr><td>Snew</td><td>0.693 0.903</td><td>0.822</td></tr><tr><td>Saul</td><td>0.854 0.634</td><td>0.820</td></tr></table>
+
+Table 4: Multi-modal incremental learning experiment. FearNet was trained with various base-knowledge sets (column-header) and then incrementally trained on all remaining data.
+
+knowledge and then incrementally learns AudioSet; 2) FearNet learns AudioSet as the baseknowledge and then incrementally learns CIFAR-100; and 3) the base-knowledge contains a 50/50 split from both datasets with FearNet incrementally learning the remaining classes. Our results suggest FearNet is capable of incrementally learning multi-modal information, if the model has a good starting point (high base-knowledge); however, if the model starts with lower base-knowledge performance (e.g., AudioSet), the model struggles to learn new information incrementally (see Supplemental Material for detailed plots).
+
+Base-Knowledge Effect on Performance. In this section, we examine how the size of the baseknowledge (i.e., number of classes) affects FearNet’s performance on CUB-200. To do this, we varied the size of the base-knowledge from 10-150 classes, with the remaining classes learned incrementally. Detailed plots are provided in the Supplemental Material. As the base-knowledge size increases, there is a noticeable increase in overall model performance because 1) mPFC has a better learned representation from a larger quantity of data and 2) there are not as many incremental learning steps remaining for the dataset, so the base-knowledge performance is less perturbed.
+
+# 7 DISCUSSION
+
+FearNet’s mPFC is trained to both discriminate examples and also generate new examples. While the main use of mPFC’s generative abilities is to enable psuedorehearsal, this ability may also help make the model more robust to catastrophic forgetting. Gillies (1991) observed that unsupervised networks are more robust (but not immune) to catastrophic forgetting because there are no target outputs to be forgotten. Since the pseudoexample generator is learned as a unsupervised reconstruction task, this could explain why FearNet is slow to forget old information.
+
+Table 5 shows the memory requirements for each model in Sec. 6.1 for learning CIFAR-100 and a hypothetical extrapolation for learning 1,000 classes. This chart accounts for a fixed model capacity and storage of any data or class statistics. FearNet’s memory footprint is comparatively small because it only stores class statistics rather than some or all of the raw training data, which makes it better suited for deployment.
+
+<table><tr><td>Model</td><td>100 Classes</td><td>1,000 Classes</td></tr><tr><td>1-NN</td><td>4.1 GB</td><td>40.9 GB</td></tr><tr><td>GeppNet+STM</td><td>4.1 GB</td><td>41.0 GB</td></tr><tr><td>GeppNet</td><td>4.1 GB</td><td>41.0 GB</td></tr><tr><td>FEL</td><td>272.5MB</td><td>395.0 MB</td></tr><tr><td>iCaRL</td><td>17.6 MB</td><td>166.0 MB</td></tr><tr><td>FearNet</td><td>10.7MB</td><td>74.4 MB</td></tr></table>
+
+An open question is how to deal with storage and updating of class statistics if classes are seen in more than one study sessions. One possibility is to use a running update for the class means and covariances, but it may
+
+Table 5: Memory requirements to train CIFAR-100 and the amount of memory that would be required if these models were trained up to 1,000 classes.
+
+be better to favor the data from the most recent study session due to learning in the autoencoder.
+
+FearNet assumed that the output of the mPFC encoder was normally distributed for each class, which may not be the case. It would be interesting to consider modeling the classes with a more complex model, e.g., a Gaussian Mixture Model. Robins (1995) showed that pseudorehearsal worked reasonably well with randomly generated vectors because they were associated with the weights of a given class. Replaying these vectors strengthened their corresponding weights, which could be what is happening with the pseudo-examples generated by FearNet’s decoder.
+
+The largest impact on model size is the stored covariance matrix $\Sigma _ { c }$ for each class. We tested a variant of FearNet that used a diagonal $\Sigma _ { c }$ instead of a full covariance matrix. Table 6 shows that performance degrades, but FearNet still works.
+
+<table><tr><td></td><td>Full Covariance</td><td>Diagonal Covariance</td></tr><tr><td>Sbase</td><td>0.942</td><td>0.781</td></tr><tr><td>Snew</td><td>0.805</td><td>0.877</td></tr><tr><td>Ωall</td><td>0.959</td><td>0.800</td></tr><tr><td>Model Size</td><td>10.7MB</td><td>3.8 MB</td></tr></table>
+
+FearNet can be adapted to other paradigms, such as unsupervised learning and regression. For unsupervised learning, FearNet’s mPFC already does a form of it implicitly. For regression, this would require changing mPFC’s loss function and may require grouping input feature vectors into similar collections. FearNet could also be adapted to perform the supervised data permutation experiment per
+
+Table 6: Using a diagonal covariance matrix for FearNet’s class statistics instead of a full covariance matrix on CIFAR-100.
+
+formed by Goodfellow et al. (2013) and Kirkpatrick et al. (2017). This would likely require storing statistics from previous permutations and classes. FearNet would sleep between learning different permutations; however, if the number of classes was high, recent recall may suffer.
+
+# 8 CONCLUSION
+
+In this paper, we proposed a brain-inspired framework capable of incrementally learning data with different modalities and object classes. FearNet outperforms existing methods for incremental class learning on large image and audio classification benchmarks, demonstrating that FearNet is capable of recalling and consolidating recently learned information while also retaining old information. In addition, we showed that FearNet is more memory efficient, making it ideal for platforms where size, weight, and power requirements are limited. Future work will include 1) integrating BLA directly into the model (versus training it independently); 2) replacing HC with a semi-parametric model; 3) learning the feature embedding from raw inputs; and 4) replacing the pseduorehearsal mechanism with a generative model that does not require the storage of class statistics, which would be more memory efficient.
+
+# REFERENCES
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+Peter J Rousseeuw and Katrien Van Driessen. A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41(3):212–223, 1999.
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+
+# A SUPPLEMENTAL MATERIAL
+
+# A.1 MODEL HYPERPARAMETERS
+
+Table S1 shows the training parameters for the FearNet model for each dataset. We also experimented with various dropout rates, weight decay, and various activation functions; however, weight decay did not work well with FearNet’s mPFC.
+
+Table S1: FearNet Training Parameters   
+
+<table><tr><td>Hyperparameter</td><td colspan="2">Values</td></tr><tr><td>Learning Rate</td><td colspan="2">2.10-3 450 (AudioSet &amp; CIFAR-100)</td></tr><tr><td>Mini-Batch Size</td><td colspan="2">200 (CUB-200)</td></tr><tr><td rowspan="2">mPFCBase-Knowledge Epochs Memory Consolidation Epochs</td><td colspan="2">1,000</td></tr><tr><td colspan="2">60</td></tr><tr><td rowspan="2">BLA Training Epochs</td><td>20</td><td></td></tr><tr><td>CIFAR-100: [140,130]</td><td></td></tr><tr><td rowspan="2">Hidden Layer Size</td><td>CUB-200:[350,300]</td><td></td></tr><tr><td>AudioSet: [300,100]</td><td></td></tr><tr><td>Sleep Frequency</td><td colspan="2">10 (see Sec. 6.2)</td></tr><tr><td>Dropout Rate</td><td colspan="2">0.25</td></tr><tr><td>Unsupervised Loss Weights (入)</td><td colspan="2">[104,1.0,0.1]</td></tr><tr><td>Hidden Layer Activation Weight Decay</td><td colspan="2">Exponential Linear Units 0.0</td></tr></table>
+
+Table S2 shows the training parameters for the iCaRL framework used in this paper. We adapted the code from the author’s GitHub page for our own experiments. The ResNet-18 convolutional neural network was replaced with a fully-connected neural network. We experimented with various regularization strategies to increase the initial base-knowledge accuracy with weight decay working
+
+the best. The values that are given as a range of values are the hyperparameter search spaces.   
+Table S2: iCaRL Training Parameters   
+
+<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>Learning Rate</td><td>2.10-3</td></tr><tr><td>Mini-Batch Size</td><td>450</td></tr><tr><td>Exemplars per Class (EPC)</td><td>20</td></tr><tr><td>Hidden Layer Size</td><td>64-1024</td></tr><tr><td>Number of Hidden Layers</td><td>2-4</td></tr><tr><td>Dropout Rate</td><td>[0.5,0.75,1.00]</td></tr><tr><td>HiddenLayer Activation</td><td>ReLU</td></tr><tr><td>Weight Decay</td><td>[0.0,10-5,10-4,5 · 10-4]</td></tr></table>
+
+Table S3 shows the training parameters for GeppNet and GeppNet+STM. Parameters not listed here are the default parameters defined by Gepperth & Karaoguz (2016). The values that are given as a range of values are the hyperparameter search spaces.   
+Table S3: GeppNet Training Parameters   
+
+<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>SOM Lattice Shape (N)</td><td>20-36</td></tr><tr><td>Non-Linearity Suppression Threshold (0)</td><td>0.1-0.75</td></tr><tr><td>Incremental Class Learning Iterations (Tinc2 - Tinc1)</td><td>[2,000,20,000]</td></tr></table>
+
+Table S4 shows the training parameters for the Fixed Expansion Layer (FEL). The number of units in the FEL layer is given by
+
+$$
+\mathrm { F E L ~ U n i t s } = { \frac { H ^ { 2 } + H K } { K } }
+$$
+
+where $H$ is the number of units in the first hidden-layer and $K$ is the maximum number of classes in the dataset. The values that are given as a range of values are the hyperparameter search spaces.
+
+Table S4: FEL Training Parameters   
+
+<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>Hidden Layer Size (H)</td><td>64-1800</td></tr><tr><td>FEL Layer Size Number of Hidden Layers</td><td>See Equation 6</td></tr><tr><td>Mini-Batch Size</td><td>2 8</td></tr><tr><td>Initial Learning Rate</td><td>10-2</td></tr><tr><td></td><td></td></tr></table>
+
+# A.2 ICARL PERFORMANCE WITH MORE EXEMPLARS
+
+Table S5 provides additional experimental results for when there are more exemplars per class (EPC) for the iCaRL framework. Rebuffi et al. (2017) used 20 EPC in their original paper; however, we increased the number to 100 EPC to see if storing more training data helped iCaRL. Although a higher EPC does increase iCaRL performance, it still does not outperform FearNet. Note that CUB-200 only has about 30 training samples per class, so iCaRL is storing the entire training set for 100 EPC. Our main results use the default value of 20.
+
+# A.3 BLA VARIANTS
+
+Our BLA model is a classifier that determines whether a prediction should be made using HC (recent memory) or mPFC (remote memory). An alternative approach would be to use an outlier detection algorithm that determines whether the data being processed by a sub-network is an outlier for that sub-network and should therefore be processed by the other sub-network. To explore this alternative BLA formulation, we experimented with three outlier detection algorithms: 1) one-class support vector machine (SVM) (Scholkopf et al., 2001), 2) determining if the data fits into a Gaussian dis- ¨ tribution using a minimum covariance determinant estimation (i.e., elliptical envelope) (Rousseeuw
+
+Table S5: iCaRL’s performance when the stored EPC is increased from 20 to 100.   
+
+<table><tr><td>Model</td><td>CIFAR-100 Sbase Ωnew aul</td><td>CUB-200 Sbase Ωnew</td><td>Ωall</td><td>AudioSet Sbase Ωnew</td><td>Ωau</td><td>Mean Sbase</td></tr><tr><td>iCaRL (20 EPC)</td><td>0.746 0.807 0.749</td><td>0.942</td><td>0.547 0.864</td><td>0.740 0.487</td><td>0.733</td><td>Ωall [0.801</td></tr><tr><td>iCaRL (100 EPC)</td><td>0.842 0.719 0.822</td><td>0.951</td><td>0.554 0.882</td><td>0.820 0.419</td><td>0.771</td><td>0.782 0.871 0.825</td></tr><tr><td>FearNet</td><td>0.927 0.824 0.947</td><td>0.924</td><td>0.598 0.891</td><td>0.962 0.455</td><td>0.932</td><td>0.938 0.923</td></tr></table>
+
+& Driessen, 1999), and 3) the isolation forest (Liu et al., 2008). All three of these methods set a rejection criterion for if the test sample exists in HC; whereas the binary MLP reports a probability on how likely the test sample resides in HC. Table S6 compares these individual methods. Isolation Forest and Elliptic Envelope seem to prefer the data in HC, one-class SVM prefers the data in mPFC, and our binary MLP worked best at choosing the correct sub-network to use.   
+
+<table><tr><td rowspan=1 colspan=2>BLA Method</td><td rowspan=1 colspan=1>ΩbaseΩnewaul</td></tr><tr><td rowspan=4 colspan=2>Isolation ForestElliptic EnvelopeOne-Class SVMBinary MLP</td><td rowspan=1 colspan=1>0.3280.8230.368</td></tr><tr><td rowspan=1 colspan=1>elope</td><td rowspan=1 colspan=1>0.5180.8230.541</td></tr><tr><td rowspan=1 colspan=1>0.7180.4330.702</td></tr><tr><td rowspan=1 colspan=1>0.9270.9240.947</td></tr></table>
+
+Table S6: Performance of different BLA variants.
+
+# A.4 FEARNET ALGORITHM
+
+Pseudocode for FearNet’s training and prediction algorithms are given in Algorithms 1 and 2 respectively. The variables match the ones defined in the paper.
+
+<table><tr><td>Algorithm1: FearNet Training</td><td>Algorithm 2: FearNet Prediction</td></tr><tr><td>Data: X,y Classes/Study-Sessions: T; K: Sleep Frequency; Initialize mPFC with base-knowledge;</td><td>Data: X A(X) ← PBLA (C =1|X); ← maxk PHc(C=k|X)A(X). 1-A(X)</td></tr><tr><td>Store μt,Σt for each class in the base-knowledge; forc←T/2toTdo StoreX,y for class c in HC;</td><td>if  &gt; maxk PmPFc (C = k|X) then return arg maxk PHc (C = k|X);</td></tr><tr><td>if c%K==O then Fine-tune mPFC with X,y in HC and pseudo- examples generated by mPFC decoder; Update μt,∑t for all classes seen so far;</td><td>else return arg maXk PmPFc (C = k/|X);</td></tr></table>
+
+# A.5 MULTI-MODAL LEARNING EXPERIMENT
+
+Fig. S1 shows the plots for the multi-modal experiments in Sec. 6.2. The three base-knowledge experiments were 1) CIFAR-100 is the base-knowledge and AudioSet is trained incrementally, 2) AudioSet is the base-knowledge and then AudioSet is trained incrementally, and 3) the base-knowledge is a $5 0 / 5 0 ~ \mathrm { m i x }$ of the two datasets and then the remaining classes are trained incrementally. For all three base-knowledge experiments, we show the mean-class accuracy on the base-knowledge and the entire test set. FearNet works well when it adequately learns the base-knowledge (Experiment #1 and #3); however, when FearNet learns it poorly, incremental learning deteriorates.
+
+# A.6 BASE-KNOWLEDGE EFFECT ON PERFORMANCE
+
+Fig. S2 shows the effect of the base-knowledge’s size on FearNet’s performance. As expected, $\Omega _ { b a s e }$ increases because there are not as many sleep phases to overwrite existing base-knowledge. $\Omega _ { n e w }$ remains relatively even because the size of the base-knowledge has no effect on the HC model’s ability to immediately recall new information; however, there is a very slight decrease that corresponds to the BLA model erroneously favoring mPFC in a few cases. Most importantly, $\Omega _ { a l l }$ sees an increase in performance because; like $\Omega _ { b a s e }$ , there are not as many sleep phases to perturb older memories in mPFC.
+
+![](images/1ee9cfc995cd1eecefdcd49c0badb1e9c5563f08021c359856092693a85d6752.jpg)  
+Figure S1: Detailed plots for the multi-modal experiment. The top row is when the base-knowledge was CIFAR-100, the middle row is when the base-knowledge was AudioSet, and the bottom row is when the base-knowledge was a $5 0 / 5 0 ~ \mathrm { m i x }$ from the two datasets. The left column represents the mean-class accuracy on the base-knowledge test set and the right column computes mean-class accuracy on the entire test set.
+
+![](images/e8820c9ee2b911f01bafc12161f3da93d222287445581a28a5c6c5c91bd828c6.jpg)  
+Figure S2: FearNet performance as a function of base-knowledge size.

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+    {
+        "type": "text",
+        "text": "FEARNET: BRAIN-INSPIRED MODEL FOR INCREMENTAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            669,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ronald Kemker and Christopher Kanan∗ ",
+        "bbox": [
+            186,
+            170,
+            473,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Carlson Center for Imaging Science Rochester Institute of Technology Rochester, NY 14623, USA {rmk6217,kanan}@rit.edu ",
+        "bbox": [
+            184,
+            185,
+            419,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            276,
+            544,
+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Incremental class learning involves sequentially learning classes in bursts of examples from the same class. This violates the assumptions that underlie methods for training standard deep neural networks, and will cause them to suffer from catastrophic forgetting. Arguably, the best method for incremental class learning is iCaRL, but it requires storing training examples for each class, making it challenging to scale. Here, we propose FearNet for incremental class learning. FearNet is a generative model that does not store previous examples, making it memory efficient. FearNet uses a brain-inspired dual-memory system in which new memories are consolidated from a network for recent memories inspired by the mammalian hippocampal complex to a network for long-term storage inspired by medial prefrontal cortex. Memory consolidation is inspired by mechanisms that occur during sleep. FearNet also uses a module inspired by the basolateral amygdala for determining which memory system to use for recall. FearNet achieves state-of-the-art performance at incremental class learning on image (CIFAR-100, CUB-200) and audio classification (AudioSet) benchmarks. ",
+        "bbox": [
+            233,
+            309,
+            764,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            336,
+            561
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In incremental classification, an agent must sequentially learn to classify training examples, without necessarily having the ability to re-study previously seen examples. While deep neural networks (DNNs) have revolutionized machine perception (Krizhevsky et al., 2012), off-the-shelf DNNs cannot incrementally learn classes due to catastrophic forgetting. Catastrophic forgetting is a phenomenon in which a DNN completely fails to learn new data without forgetting much of its previously learned knowledge (McCloskey & Cohen, 1989). While methods have been developed to try and mitigate catastrophic forgetting, as shown in Kemker et al. (2018), these methods are not sufficient and perform poorly on larger datasets. In this paper, we propose FearNet, a brain-inspired system for incrementally learning categories that significantly outperforms previous methods. ",
+        "bbox": [
+            174,
+            579,
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+        "text": "The standard way for dealing with catastrophic forgetting in DNNs is to avoid it altogether by mixing new training examples with old ones and completely re-training the model offline. For large datasets, this may require weeks of time, and it is not a scalable solution. An ideal incremental learning system would be able to assimilate new information without the need to store the entire training dataset. A major application for incremental learning includes real-time operation on-board embedded platforms that have limited computing power, storage, and memory, e.g., smart toys, smartphone applications, and robots. For example, a toy robot may need to learn to recognize objects within its local environment and of interest to its owner. Using cloud computing to overcome these resource limitations may pose privacy risks and may not be scalable to a large number of embedded devices. A better solution is on-device incremental learning, which requires the model to use less storage and computational power. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we propose an incremental learning framework called FearNet (see Fig. 1). FearNet has three brain-inspired sub-systems: 1) a recent memory system for quick recall, 2) a memory system for long-term storage, and 3) a sub-system that determines which memory system to use for a particular example. FearNet mitigates catastrophic forgetting by consolidating recent memories into long-term storage using pseudorehearsal (Robins, 1995). Pseudorehearsal allows the network to revisit previous memories during incremental training without the need to store previous training examples, which is more memory efficient. ",
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+        "type": "text",
+        "text": "Problem Formulation: Here, incremental class learning consists of $T$ study-sessions. At time $t$ , the learner receives a batch of data $B _ { t }$ , which contains $N _ { t }$ labeled training samples, i.e., $B _ { t } ~ = ~ \\{ ( \\mathbf { x } _ { j } , y _ { j } ) \\} _ { j = 1 } ^ { N _ { t } }$ , where $\\mathbf { x } _ { j } \\in \\mathbb { R } ^ { d }$ is the input feature vector to be classified and $y _ { j }$ is its corresponding label. The number of training samples $N _ { t }$ may vary between sessions, and the data inside a study-session is not assumed to be independent and identically distributed (iid). During a study session, the learner only has access to its current batch, but it may use its own memory to store information from prior study sessions. We refer to the first session as the model’s “base-knowledge,” which contains exemplars from $M \\geq 1$ classes. The batches learned in all subsequent sessions contain only one class, i.e., all $y _ { j }$ will be identical within those sessions. ",
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+        "text": "Novel Contributions: Our contributions include: ",
+        "text_level": 1,
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+        "img_path": "images/9ec0369767a96c98cf47fe8ba08e1c364ce5daf78138cb6ec61cc3ee815bb4b4.jpg",
+        "image_caption": [
+            "Figure 1: FearNet consists of three braininspired modules based on 1) mPFC (longterm storage), 2) HC (recent storage), and 3) BLA for determining whether to use mPFC or HC for recall. "
+        ],
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+        "text": "1. FearNet’s architecture includes three neural networks: one inspired by the hippocampal complex (HC) for recent memories, one inspired by the medial prefrontal cortex (mPFC) for long-term storage, and one inspired by the basolateral amygdala (BLA) that determines whether to use HC or mPFC for recall. ",
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+        "type": "text",
+        "text": "2. Motivated by memory replay during sleep, FearNet employs a generative autoencoder for pseudorehearsal, which mitigates catastrophic forgetting by generating previously learned examples that are replayed alongside novel information during consolidation. This process does not involve storing previous training data. ",
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+        "type": "text",
+        "text": "3. FearNet achieves state-of-the-art results on large image and audio datasets with a relatively small memory footprint, demonstrating how dual-memory models can be scaled. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Catastrophic forgetting in DNNs occurs due to the plasticity-stability dilemma (Abraham & Robins, 2005). If the network is too plastic, older memories will quickly be overwritten; however, if the network is too stable, it is unable to learn new data. This problem was recognized almost 30 years ago (McCloskey & Cohen, 1989). In French (1999), methods developed in the 1980s and 1990s are extensively discussed, and French argued that mitigating catastrophic forgetting would require having two separate memory centers: one for the long-term storage of older memories and another to quickly process new information as it comes in. He also theorized that this type of dual-memory system would be capable of consolidating memories from the fast learning memory center to longterm storage. ",
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+    {
+        "type": "text",
+        "text": "Catastrophic forgetting often occurs when a system is trained on non-iid data. One strategy for reducing this phenomenon is to mix old examples with new examples, which simulates iid conditions. For example, if the system learns ten classes in a study session and then needs to learn 10 new classes in a later study session, one solution could be to mix examples from the first study session into the later study session. This method is known as rehearsal, and it is one of the earliest methods for reducing catastrophic forgetting (Hetherington & Seidenberg, 1989). Rehearsal essentially uses an external memory to strengthen the model’s representations for examples learned previously, so that they are not overwritten when learning data from new classes. Rehearsal reduces forgetting, but performance is still worse than offline models. Moreover, rehearsal requires storing all of the training data. Robins (1995) argued that storing of training examples was inefficient and of “little interest,” so he introduced pseudorehearsal. Rather than replaying past training data, in pseudorehearsal, the algorithm generates new examples for a given class. In Robins (1995), this was done by creating random input vectors, having the network assign them a label, and then mixing them into the new training data. This idea was revived in Draelos et al. (2017), where a generative autoencoder was used to create pseudo-examples for unsupervised incremental learning. This method inspired FearNet’s approach to memory consolidation. Pseudorehearsal is related to memory replay that occurs in mammalian brains, which involves reactivation of recently encoded memories in HC so that they can be integrated into long-term storage in mPFC (Rasch & Born, 2013). ",
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+    {
+        "type": "text",
+        "text": "Recently there has been renewed interest in solving catastrophic forgetting in supervised learning. Many new methods are designed to mitigate catastrophic forgetting when each study session contains a permuted version of the entire training dataset (see Goodfellow et al. (2013)). Unlike incremental class learning, all labels are contained in each study session. PathNet uses an evolutionary algorithm to find the optimal path through a large DNN, and then freezes the weights along that path (Fernando et al., 2017). It assumes all classes are seen in each study session, and it is not capable of incremental class learning. Elastic Weight Consolidation (EWC) employs a regularization scheme that redirects plasticity to the weights that are least important to previously learned study sessions (Kirkpatrick et al., 2017). After EWC learns a study session, it uses the training data to build a Fisher matrix that determines the importance of each feature to the classification task it just learned. EWC was shown to work poorly at incremental class learning in Kemker et al. (2018). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Fixed Expansion Layer (FEL) model mitigates catastrophic forgetting by using sparse updates (Coop et al., 2013). FEL uses two hidden layers, where the second hidden layer (i.e., the FEL layer) has connectivity constraints. The FEL layer is much larger than the first hidden layer, is sparsely populated with excitatory and inhibitory weights, and is not updated during training. This limits learning of dense shared representations, which reduces the risk of learning interfering with old memories. FEL requires a large number of units to work well (Kemker et al., 2018). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gepperth & Karaoguz (2016) introduced a new approach for incremental learning, which we call GeppNet. GeppNet uses a self-organizing map (SOM) to reorganize the input onto a two-dimensional lattice. This serves as a long-term memory, which is fed into a simple linear layer for classification. After the SOM is initialized, it can only be updated if the input is sufficiently novel. This prevents the model from forgetting older data too quickly. GeppNet also uses rehearsal using all previous training data. A variant of GeppNet, GeppNet+STM, uses a fixed-size memory buffer to store novel examples. When this buffer is full, it replaces the oldest example. During pre-defined intervals, the buffer is used to train the model. Gepp$\\mathbf { \\Gamma } _ { \\mathbf { N e t + S T M } }$ is better at retaining base-knowledge since it only trains during its consolidation phase, but the STM-free version learns new data better because it updates the model on every novel labeled input. ",
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+    {
+        "type": "image",
+        "img_path": "images/9487165ad014d2cb3f67829861104cc2ddefd93cc570c6b1da29573fe8bec15b.jpg",
+        "image_caption": [
+            "Figure 2: iCaRL’s performance depends heavily on the number of exemplars per class (EPC) that it stores. Reducing EPC from 20 (blue) to 1 (red) severely impairs its ability to recall older information. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "iCaRL (Rebuffi et al., 2017) is an incremental class learning framework. Rather than directly using a DNN for classification, iCaRL uses it for supervised representation learning. During a study session, iCaRL updates a DNN using the study session’s data and a set of $J$ stored examples from earlier sessions $( J = 2 , 0 0 0$ for CIFAR-100 in their paper), which is a kind of rehearsal. After a study session, the $J$ examples retained are carefully chosen using herding. After learning the entire dataset, iCaRL has retained $J / T$ exemplars per class (e.g., $J / \\bar { T } = 2 0$ for CIFAR-100). The DNN in iCaRL is then used to compute an embedding for each stored example, and then the mean embedding for each class seen is computed. To classify a new instance, the DNN is used to compute an embedding for it, and then the class with the nearest mean embedding is assigned. iCaRL’s performance is heavily influenced by the number of examples it stores, as shown in Fig. 2. ",
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+    {
+        "type": "text",
+        "text": "3 MAMMALIAN MEMORY: NEUROSCIENCE AND MODELS ",
+        "text_level": 1,
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+        "text": "FearNet is heavily inspired by the dual-memory model of mammalian memory (McClelland et al., 1995), which has considerable experimental support from neuroscience (Frankland et al., 2004; Takashima et al., 2006; Kitamura et al., 2017; Bontempi et al., 1999; Taupin & Gage, 2002; Gais et al., 2007). This theory proposes that HC and mPFC operate as complementary memory systems, where HC is responsible for recalling recent memories and mPFC is responsible for recalling remote (mature) memories. GeppNet is the most recent DNN to be based on this theory, but it was also independently explored in the 1990s in French (1997) and Ans & Rousset (1997). In this section, we review some of the evidence for the dual-memory model. ",
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+        "type": "text",
+        "text": "One of the major reasons why HC is thought to be responsible for recent memories is that if HC is bilaterally destroyed, then anterograde amnesia occurs with old memories for semantic information preserved. One mechanism HC may use to facilitate creating new memories is adult neurogenesis. This occurs in HC’s dentate gyrus (Altman, 1963; Eriksson et al., 1998). The new neurons have higher initial plasticity, but it reduces as time progresses (Deng et al., 2010). ",
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+        "type": "text",
+        "text": "In contrast, mPFC is responsible for the recall of remote (long-term) memories (Bontempi et al., 1999). Taupin & Gage (2002) and Gais et al. (2007) showed that mPFC plays a strong role in memory consolidation during REM sleep. McClelland et al. (1995) and Euston et al. (2012) theorized that, during sleep, HC reactivates recent memories to prevent forgetting which causes these recent memories to replay in mPFC as well, with dreams possibly being caused by this process. After memories are transferred from HC to mPFC, evidence suggests that corresponding memory in HC is erased (Poe, 2017). ",
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+        "type": "text",
+        "text": "Recently, Kitamura et al. (2017) performed contextual fear conditioning (CFC) experiments in mice to trace the formation and consolidation of recent memories to long-term storage. CFC experiments involve shocking mice while subjecting them to various visual stimuli (i.e., colored lights). They found that BLA, which is responsible for regulating the brain’s fear response, would shift where it retrieved the corresponding memory from (HC or mPFC) as that memory was consolidated over time. FearNet follows the memory consolidation theory proposed by Kitamura et al. (2017). ",
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+    {
+        "type": "text",
+        "text": "4 THE FEARNET MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "FearNet has two complementary memory centers, 1) a short-term memory system that immediately learns new information for recent recall (HC) and 2) a DNN for the storage of remote memories (mPFC). FearNet also has a separate BLA network that determines which memory center contains the associated memory required for prediction. During sleep phases, FearNet uses a generative model to consolidate data from HC to mPFC through pseudorehearsal. Pseudocode for FearNet is provided in the supplemental material. Because the focus of our work is not representation learning, we use pre-trained ResNet embeddings to obtain features that are fed to FearNet. ",
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+        "type": "text",
+        "text": "4.1 DUAL-MEMORY STORAGE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "FearNet’s HC model is a variant of a probabilistic neural network (Specht, 1990). HC computes class conditional probabilities using stored training examples. Formally, HC estimates the probability that an input feature vector $\\mathbf { x }$ belongs to class $k$ as ",
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+    {
+        "type": "equation",
+        "img_path": "images/e2c3ab335fbf48b2d06228c1c54eb2baaca69897d20b8d1313af78ffb2537d86.jpg",
+        "text": "$$\nP _ { H C } \\left( C = k | \\mathbf { x } \\right) = \\frac { \\beta _ { k } } { \\sum _ { k ^ { \\prime } } \\beta _ { k ^ { \\prime } } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/fdf5250852e0be7651be399d9809bb0f6acf66a44b78ca29facace7bfd947b5a.jpg",
+        "text": "$$\n\\beta _ { k } = \\left\\{ \\begin{array} { c c } { \\big ( \\epsilon + \\operatorname* { m i n } _ { j } \\| \\mathbf x - \\mathbf u _ { k , j } \\| _ { 2 } \\big ) ^ { - 1 } } & { \\mathrm { i f ~ H C ~ c o n t a i n s ~ i n s t a n c e s ~ o f ~ c l a s s ~ } k } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\epsilon > 0$ is a regularization parameter and ${ \\mathbf { u } } _ { k , j }$ is the $j ^ { : }$ ’th stored exemplar in HC for class $k$ . All exemplars are removed from HC after they are consolidated into mPFC. ",
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+    {
+        "type": "text",
+        "text": "FearNet’s mPFC is implemented using a DNN trained both to reconstruct its input using a symmetric encoder-decoder (autoencoder) and to compute $P _ { m P F C } \\left( C = k | \\mathbf { x } \\right)$ . The autoencoder enables us to ",
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+    {
+        "type": "image",
+        "img_path": "images/0a31f3f9e3363e7a1f2b1d9c767fa264123bab51cadfb522dfebe045d764880d.jpg",
+        "image_caption": [
+            "Figure 3: The mPFC and BLA sub-systems in FearNet. mPFC is responsible for the long-term storage of remote memories. BLA is used during prediction time to determine if the memory should be recalled from short- or long-term memory. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "use pseudorehearsal, which is described in more detail in Sec. 4.2. The loss function for mPFC is ",
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+    {
+        "type": "equation",
+        "img_path": "images/ad90f62032b729b0de886c36fc2533201c3b200ff65e31051b2e5f2c560fbad0.jpg",
+        "text": "$$\n\\mathcal { L } _ { m P F C } = \\mathcal { L } _ { c l a s s } + \\mathcal { L } _ { r e c o n } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { c l a s s }$ is the supervised classification loss and $\\mathcal { L } _ { r e c o n }$ is the unsupervised reconstruction loss, as illustrated in Fig. 3(a). For $\\mathcal { L } _ { c l a s s }$ , we use standard softmax loss. $\\mathcal { L } _ { r e c o n }$ is the weighted sum of mean squared error (MSE) reconstruction losses from each layer, which is given by ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4fb0b30c9bb5b5142a56cca59c9db179b0d074e64254bf766159a438f9aca0d9.jpg",
+        "text": "$$\n\\mathcal { L } _ { r e c o n } = \\sum _ { j = 0 } ^ { M } \\sum _ { i = 0 } ^ { H _ { j } - 1 } \\left. h _ { e n c o d e r , ( i , j ) } - h _ { d e c o d e r , ( i , j ) } \\right. _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $M$ is the number of mPFC layers, $H _ { j }$ is the number of hidden units in layer $j$ , $h _ { e n c o d e r , ( i , j ) }$ and $h _ { d e c o d e r , ( i , j ) }$ are the outputs of the encoder/decoder at layer $j$ respectively, and $\\lambda _ { j }$ is the reconstruction weight for that layer. mPFC is similar to a Ladder Network (Rasmus et al., 2015), which combines classification and reconstruction to improve regularization, especially during lowshot learning. The $\\lambda _ { j }$ hyperparameters were found empirically, with $\\lambda _ { 0 }$ being largest and decreasing for deeper layers (see supplementary material). This prioritizes the reconstruction task, which makes the generated pseudo-examples more realistic. When training is completed during a study session, all of the data in HC is pushed through the encoder to extract a dense feature representation of the original data, and then we compute a mean feature vector $\\mu _ { c }$ and covariance matrix $\\Sigma _ { c }$ for each class $c$ . These are stored and used to generate pseudo-examples during consolidation (see Sec. 4.2). We study FearNet’s performance as a function of how much data is stored in HC in Sec. 6.2. ",
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+    {
+        "type": "text",
+        "text": "4.2 PSEUDOREHEARSAL FOR MEMORY CONSOLIDATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "During FearNet’s sleep phase, the original inputs stored in HC are transferred to mPFC using pseudo-examples created by an autoencoder. This process is known as intrinsic replay, and it was used by Draelos et al. (2017) for unsupervised learning. ",
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+    {
+        "type": "text",
+        "text": "Using the class statistics from the encoder, pseudo-examples for class $c$ are generated by sampling a Gaussian with mean $\\mu _ { c }$ and covariance matrix $\\Sigma _ { c }$ to obtain $\\hat { \\mathbf { x } } _ { r a n d }$ . Then, $\\hat { \\mathbf { x } } _ { r a n d }$ is passed through the decoder to generate a pseudo-example. To create a balanced training set, for each class that mPFC has learned, we generate $\\lceil m \\rceil$ pseudo-examples, where $m$ is the average number of examples per class stored in HC. The pseudo-examples are mixed with the data in HC, and the mixture is used to fine-tune mPFC using backpropagation. After consolidation, all units in HC are deleted. ",
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+    {
+        "type": "text",
+        "text": "4.3 NETWORK SELECTION USING BLA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "During prediction, FearNet uses the BLA network (Fig. 3(b)) to determine whether to classify an input x using HC or mPFC. This can be challenging because if HC has only been trained on one class, it will put all of its probability mass on that class, whereas mPFC will likely be less confident. The output of BLA is given by $A \\left( \\mathbf { x } \\right)$ and will be a value between 0 and 1, with a 1 indicating mPFC should be used. BLA is trained after each study session using only the data in HC and with pseudoexamples generated with mPFC, using the same procedure described in Sec. 4.2. Instead of using solely BLA to determine which network to use, we found that combining its output with those of mPFC and HC improved results. The predicted class $\\hat { y }$ is computed as ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/88d3befb10904759c6a11b9ae16fb1bec767d6cd04a2ec0924ee914bf9c57b6f.jpg",
+        "text": "$$\n\\hat { y } = \\left\\{ \\begin{array} { c c } { \\underset { \\mathrm { a r g } } { \\arg \\operatorname* { m a x } } _ { k ^ { \\prime } } P _ { H C } \\left( C = k ^ { \\prime } | \\mathbf { x } \\right) } & { \\mathrm { i f } \\ \\psi > \\operatorname* { m a x } _ { k } P _ { m P F C } \\left( C = k | \\mathbf { x } \\right) } \\\\ { \\underset { \\mathrm { o t h e r w i s e } } { \\arg \\operatorname* { m a x } } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee123dba6e6e1954887949ef395ac365b193a08bcf15858e2f72a9f14fa6edc6.jpg",
+        "text": "$$\n\\psi = \\left( 1 - A \\left( \\mathbf { x } \\right) \\right) ^ { - 1 } \\operatorname* { m a x } _ { k } P _ { H C } \\left( C = k | \\mathbf { x } \\right) A \\left( \\mathbf { x } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$\\psi$ is the probability of the class according to HC weighted by the confidence that the associated memory is actually stored in HC. BLA has the same number of layers/units as the mPFC encoder, and uses a logistic output unit. We discuss alternative BLA models in supplemental material. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+            271,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluating Incremental Learning Performance. To evaluate how well the incrementally trained models perform compared to an offline model, we use the three metrics proposed in Kemker et al. (2018). After each study session $t$ in which a model learned a new class $k$ , we compute the model’s test accuracy on the new class $( \\alpha _ { n e w , t } )$ , the accuracy on the base-knowledge $( \\alpha _ { b a s e , t } )$ , and the accuracy of all of the test data seen to this point $( \\alpha _ { a l l , t } )$ . After all $T$ study sessions are complete, a model’s ability to retain the base-knowledge is given by $\\begin{array} { r } { \\Omega _ { b a s e } \\ = \\ \\frac { 1 } { T - 1 } \\sum _ { t = 2 } ^ { T } \\frac { \\alpha _ { b a s e , t } } { \\alpha _ { o f f l i n e } } } \\end{array}$ where $\\alpha _ { o f f l i n e }$ is the accuracy of a multi-layer perceptron (MLP) trained offline (i.e., it is given all of the training data at once). The model’s ability to immediately recall new information is measured by 1T −1 PTt=2 αnew,t. Finally, we measure how well the model does on all available test data with Ωall = T −1 P t=2 αoffline αall,t . The $\\Omega _ { a l l }$ metric shows how well new memories are integrated into the model over time. For all of the metrics, higher values indicate superior performance. Both $\\Omega _ { b a s e }$ and $\\Omega _ { a l l }$ are relative to an offline MLP model, so a value of 1 indicates that a model has similar performance to the offline baseline. This allows results across datasets to be better compared. Note that $\\Omega _ { b a s e } > 1$ and $\\Omega _ { a l l } > 1$ only if the incremental learning algorithm is more accurate than the offline model, which can occur due to better regularization strategies employed by different models. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We evaluate all of the models on three benchmark datasets (Table 1): CIFAR-100, CUB-200, and AudioSet. CIFAR-100 is a popular image classification dataset containing 100 mutually-exclusive object categories, and it was used in Rebuffi et al. (2017) to evaluate iCaRL. All images are $3 2 \\times 3 2$ pixels. CUB-200 is a fine-grained image classification dataset containing high resolution images of 200 different bird ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/7b43a068859b87a51d7d3d60c90cf1c5df884c63275c18183b9e8bf8eed8109f.jpg",
+        "table_caption": [
+            "Table 1: Dataset Specifications "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CIFAR-100</td><td>CUB-200</td><td>AudioSet</td></tr><tr><td>Classification Task</td><td>RGB Image</td><td>RGB Image</td><td>Audio</td></tr><tr><td>Classes</td><td>100</td><td>200</td><td>100</td></tr><tr><td>Feature Shape</td><td>2.048</td><td>2,048</td><td>1,280</td></tr><tr><td>Train Samples</td><td>50,000</td><td>5,994</td><td>28,779</td></tr><tr><td>Test Samples</td><td>10,000</td><td>5,794</td><td>5,523</td></tr><tr><td>Train Samples/Class</td><td>500</td><td>29-30</td><td>250-300</td></tr><tr><td>Test Samples/Class</td><td>100</td><td>11-30</td><td>43-62</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "species (Welinder et al., 2010). We use the 2011 version of the dataset. AudioSet is an audio classification dataset (Gemmeke et al., 2017). We use the variant of AudioSet used by Kemker et al. (2018), which contains a 100 class subset such that none of the classes were super- or sub-classes of one another. Also, since the AudioSet data samples can have more than one class, the chosen samples had only one of the 100 classes chosen in this subset. ",
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+    },
+    {
+        "type": "text",
+        "text": "For CIFAR-100 and CUB-200, we extract ResNet-50 image embeddings as the input to each of the models, where ResNet-50 was pre-trained on ImageNet (He et al., 2016). We use the output after the mean pooling layer and normalize the features to unit length. For AudioSet, we use the audio CNN embeddings produced by pre-training the model on the YouTube-8M dataset (Abu-El-Haija et al., 2016). We use the pre-extracted AudioSet feature embeddings, which represent ten second sound clips (i.e., ten 128-dimensional vectors concatenated in order). ",
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+    },
+    {
+        "type": "text",
+        "text": "Comparison Models. We compare FearNet to FEL, GeppNet, GeppNet+STM, iCaRL, and an onenearest neighbor (1-NN). FEL, GeppNet, and GeppNet+STM were chosen due to their previously reported efficacy at incremental class learning in Kemker et al. (2018). iCARL is explicitly designed for incremental class learning, and represents the state-of-the-art on this problem. We compare against 1-NN due to its similarity to our HC model. 1-NN does not forget any previously observed examples, but it tends to have worse generalization error than parametric methods and requires storing all of the training data. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/8fe87b21db3278cee40742062caccd69283383f3df858d8c12bed9bd6756eed5.jpg",
+        "image_caption": [
+            "Figure 4: Mean-class test accuracy of all classes seen so far. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In each of our experiments, all models take the same feature embedding as input for a given dataset. This required modifying iCaRL by turning its CNN into a fully connected network. We performed a hyperparameter search for each model/dataset combination to tune the number of units and layers (see Supplemental Materials). ",
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+    },
+    {
+        "type": "text",
+        "text": "Training Parameters. FearNet was implemented in Tensorflow. For mPFC and BLA, each fully connected layer uses an exponential linear unit activation function (Clevert et al., 2016). The output of the encoder also connects to a softmax output layer. Xavier initialization is used to initialize all weight layers (Glorot & Bengio, 2010), and all of the biases are initialized to one. BLA’s architecture is identical to mPFC’s encoder, except it has a logistic output unit, instead of a softmax layer. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "mPFC and BLA were trained using NAdam. We train mPFC on the base-knowledge set for 1,000 epochs, consolidate HC over to mPFC for 60 epochs, and train BLA for 20 epochs. Because mPFC’s decoder is vital to preserving memories, its learning rate is $1 / 1 0 0$ times lower than the encoder. We performed a hyperparameter search for each dataset and model, varying the model shape (64-1,024 units), depth (2-4 layers), and how often to sleep (see Sec. 6.2). Across datasets, mPFC and BLA performed best with two hidden layers, but the number of units per layer varied across datasets. The specific values used for each dataset are given in supplemental material. In preliminary experiments, we found no benefit to adding weight decay to mPFC, likely because the reconstruction task helps regularize the model. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Unless otherwise noted, each class is only seen in one unique study-session and the first baseknowledge study session contains half the classes in the dataset. We perform additional experiments to study how changing the number of base-knowledge classes affects performance in Sec. 6.2. Unless otherwise noted, FearNet sleeps every 10 study sessions across datasets. ",
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+    {
+        "type": "text",
+        "text": "6.1 STATE-OF-THE-ART COMPARISON ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 2 shows incremental class learning summary results for all six methods. FearNet achieves the best $\\Omega _ { b a s e }$ and $\\Omega _ { a l l }$ on all three datasets. Fig. 4 shows that FearNet more closely resembles the offline MLP baseline than other methods. $\\Omega _ { n e w }$ measures test accuracy on the most recently trained class. 1 For FearNet, this measures the performance of HC and BLA. $\\Omega _ { n e w }$ does not account for how well the class was consolidated into mPFC which happens later during a sleep phase; however, $\\Omega _ { a l l }$ does account for this. FEL achieves a high $\\Omega _ { n e w }$ score because it is able to achieve nearly perfect test accuracy on every new class it learns, but this results in forgetting more quickly than FearNet. 1-NN is similar to our HC model; but on its own, it fails to generalize as well as FearNet, is memory inefficient, and is slow to make predictions. The final mean-class test accuracy for the offline MLP used to normalize the metrics is $6 9 . 9 \\%$ for CIFAR-100, $5 9 . 8 \\%$ for CUB-200, and $4 5 . 8 \\%$ for AudioSet. ",
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+    {
+        "type": "table",
+        "img_path": "images/5077ab9e33df318c7465f84dfb058d56ebfd4e8d5ff8b148c9284ab3455b4f29.jpg",
+        "table_caption": [
+            "Table 2: State-of-the-art comparison on CIFAR-100, CUB-200, and AudioSet. The best $\\Omega _ { a l l }$ for each dataset are in bold. $\\Omega _ { b a s e }$ and $\\Omega _ { a l l }$ are normalized by the offline MLP baseline. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td colspan=\"2\">CIFAR-100 Sbase Ωnew all</td><td colspan=\"2\">CUB-200 Snew</td><td colspan=\"2\">AudioSet Ωnew</td><td colspan=\"2\">Mean Sbase</td></tr><tr><td>1-Nearest Neighbor|</td><td>|0.878</td><td>0.648</td><td>Sbase |0.746</td><td>Ωaul 0.434</td><td>Sbase 10.655</td><td>0.269</td><td>Sall 0.613</td><td>Sall</td></tr><tr><td>GeppNet+STM</td><td>0.866</td><td>0.879 0.408 0.800</td><td>0.764</td><td>0.694 0.204 0.645</td><td>0.941</td><td>0.861</td><td>[0.760 0.857</td><td>0.729 0.769</td></tr><tr><td>GeppNet</td><td>0.833</td><td>0.529 0.754</td><td>0.727</td><td>0.558 0.645</td><td>0.932</td><td>0.372</td><td>0.831</td><td>0.759</td></tr><tr><td>FEL</td><td>0.707</td><td>0.999 0.619</td><td>0.702</td><td>0.976 0.641</td><td>0.491</td><td>0.499 1.000 0.456</td><td>0.879</td><td></td></tr><tr><td>iCaRL</td><td>0.746</td><td>0.807 0.749</td><td>0.942 0.547</td><td>0.864</td><td>0.740 0.487</td><td>0.733</td><td>0.633 0.801</td><td>0.572 0.782</td></tr><tr><td>FearNet</td><td>0.927</td><td>0.824 0.947</td><td>0.924</td><td>0.598 0.891</td><td>0.962</td><td>0.455 0.932</td><td>0.938</td><td>0.923</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4e1f9a40e0b8823b06ee244e2bd219f64045a9e9ab6f5474709a28c15f84182b.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: FearNet performance when the location of the associated memory is known using an oracle versus using BLA. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">CIFAR-100</td><td colspan=\"2\">CUB-200</td><td colspan=\"2\">AudioSet</td></tr><tr><td></td><td>Oracle</td><td>With BLA</td><td>Oracle</td><td>With BLA</td><td>Oracle</td><td>With BLA</td></tr><tr><td>Sbase</td><td>0.965</td><td>0.927</td><td>0.968</td><td>0.924</td><td>0.970</td><td>0.962</td></tr><tr><td>Snew</td><td>0.912</td><td>0.824</td><td>0.729</td><td>0.598</td><td>0.701</td><td>0.455</td></tr><tr><td>Ωau</td><td>1.002</td><td>0.947</td><td>0.936</td><td>0.891</td><td>0.972</td><td>0.932</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "6.2 ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Novelty Detection with BLA. We evaluated the performance of BLA by comparing it to an oracle version of FearNet, i.e., a version that knew if the relevant memory was stored in either mPFC or HC. Table 3 shows that FearNet’s BLA does a good job at predicting which network to use; however, the decrease in $\\Omega _ { n e w }$ suggests BLA is sometimes using mPFC when it should be using HC. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When should the model sleep? To study how the frequency of memory consolidation affects FearNet’s performance, we trained FearNet on CUB-200 and varied the sleep frequency from 1-15 study sessions. When FearNet increases the number of classes it learns before sleeping (Fig. 5), it is better able to retain its base-knowledge, but this reduces its ability to recall new information. In humans, sleep deprivation is known to impair new learning (Yoo et al., 2007), and that forgetting occurs during sleep (Poe, 2017). Each time FearNet sleeps, the mPFC weights are perturbed which can cause it to gradually forget older memories. Sleeping less causes HC’s recall performance to deteriorate. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f78923b9aae75d6dcc03f23fbd92ef51d126b3d916bf7c065180de4a5bced14.jpg",
+        "image_caption": [
+            "Figure 5: FearNet performance as the sleep frequency decreases. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multi-Modal Incremental Learning. As shown in Sec. 6.1, FearNet can incrementally learn and retain information from a single dataset, but how does it perform if new inputs differ greatly from previously learned ones? This scenario is one of the first shown to cause catastrophic forgetting in MLPs. To study this, we trained FearNet to incrementally learn CIFAR-100 and AudioSet, which after training is a 200-way classification problem. To do this, AudioSet’s features are zero-padded to make them the same length as CIFAR-100s. Table 4 shows the performance of FearNet for three separate training paradigms: 1) FearNet learns CIFAR-100 as the base",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4613cabd06ec069008010343d0d65e0d6b4aafe7f22fc7b00b3698023edb4d4e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Base-Knowledge CIFAR-100 AudioSet 50/50 Mix</td></tr><tr><td>Sbase|</td><td>0.995 0.845</td><td>0.837</td></tr><tr><td>Snew</td><td>0.693 0.903</td><td>0.822</td></tr><tr><td>Saul</td><td>0.854 0.634</td><td>0.820</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Multi-modal incremental learning experiment. FearNet was trained with various base-knowledge sets (column-header) and then incrementally trained on all remaining data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "knowledge and then incrementally learns AudioSet; 2) FearNet learns AudioSet as the baseknowledge and then incrementally learns CIFAR-100; and 3) the base-knowledge contains a 50/50 split from both datasets with FearNet incrementally learning the remaining classes. Our results suggest FearNet is capable of incrementally learning multi-modal information, if the model has a good starting point (high base-knowledge); however, if the model starts with lower base-knowledge performance (e.g., AudioSet), the model struggles to learn new information incrementally (see Supplemental Material for detailed plots). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Base-Knowledge Effect on Performance. In this section, we examine how the size of the baseknowledge (i.e., number of classes) affects FearNet’s performance on CUB-200. To do this, we varied the size of the base-knowledge from 10-150 classes, with the remaining classes learned incrementally. Detailed plots are provided in the Supplemental Material. As the base-knowledge size increases, there is a noticeable increase in overall model performance because 1) mPFC has a better learned representation from a larger quantity of data and 2) there are not as many incremental learning steps remaining for the dataset, so the base-knowledge performance is less perturbed. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "FearNet’s mPFC is trained to both discriminate examples and also generate new examples. While the main use of mPFC’s generative abilities is to enable psuedorehearsal, this ability may also help make the model more robust to catastrophic forgetting. Gillies (1991) observed that unsupervised networks are more robust (but not immune) to catastrophic forgetting because there are no target outputs to be forgotten. Since the pseudoexample generator is learned as a unsupervised reconstruction task, this could explain why FearNet is slow to forget old information. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5 shows the memory requirements for each model in Sec. 6.1 for learning CIFAR-100 and a hypothetical extrapolation for learning 1,000 classes. This chart accounts for a fixed model capacity and storage of any data or class statistics. FearNet’s memory footprint is comparatively small because it only stores class statistics rather than some or all of the raw training data, which makes it better suited for deployment. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/941bde1bb3259bfb43548610b60ce1a256f30619d2d088c1f9c3da493fe7a5e3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>100 Classes</td><td>1,000 Classes</td></tr><tr><td>1-NN</td><td>4.1 GB</td><td>40.9 GB</td></tr><tr><td>GeppNet+STM</td><td>4.1 GB</td><td>41.0 GB</td></tr><tr><td>GeppNet</td><td>4.1 GB</td><td>41.0 GB</td></tr><tr><td>FEL</td><td>272.5MB</td><td>395.0 MB</td></tr><tr><td>iCaRL</td><td>17.6 MB</td><td>166.0 MB</td></tr><tr><td>FearNet</td><td>10.7MB</td><td>74.4 MB</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "An open question is how to deal with storage and updating of class statistics if classes are seen in more than one study sessions. One possibility is to use a running update for the class means and covariances, but it may ",
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+    {
+        "type": "text",
+        "text": "Table 5: Memory requirements to train CIFAR-100 and the amount of memory that would be required if these models were trained up to 1,000 classes. ",
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+    },
+    {
+        "type": "text",
+        "text": "be better to favor the data from the most recent study session due to learning in the autoencoder. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "FearNet assumed that the output of the mPFC encoder was normally distributed for each class, which may not be the case. It would be interesting to consider modeling the classes with a more complex model, e.g., a Gaussian Mixture Model. Robins (1995) showed that pseudorehearsal worked reasonably well with randomly generated vectors because they were associated with the weights of a given class. Replaying these vectors strengthened their corresponding weights, which could be what is happening with the pseudo-examples generated by FearNet’s decoder. ",
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+    },
+    {
+        "type": "text",
+        "text": "The largest impact on model size is the stored covariance matrix $\\Sigma _ { c }$ for each class. We tested a variant of FearNet that used a diagonal $\\Sigma _ { c }$ instead of a full covariance matrix. Table 6 shows that performance degrades, but FearNet still works. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/10db10d0235e7c758aa4bcb903a7aab3d4d4006fb61a06906bea9436e2ed2912.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Full Covariance</td><td>Diagonal Covariance</td></tr><tr><td>Sbase</td><td>0.942</td><td>0.781</td></tr><tr><td>Snew</td><td>0.805</td><td>0.877</td></tr><tr><td>Ωall</td><td>0.959</td><td>0.800</td></tr><tr><td>Model Size</td><td>10.7MB</td><td>3.8 MB</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "FearNet can be adapted to other paradigms, such as unsupervised learning and regression. For unsupervised learning, FearNet’s mPFC already does a form of it implicitly. For regression, this would require changing mPFC’s loss function and may require grouping input feature vectors into similar collections. FearNet could also be adapted to perform the supervised data permutation experiment per",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 6: Using a diagonal covariance matrix for FearNet’s class statistics instead of a full covariance matrix on CIFAR-100. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "formed by Goodfellow et al. (2013) and Kirkpatrick et al. (2017). This would likely require storing statistics from previous permutations and classes. FearNet would sleep between learning different permutations; however, if the number of classes was high, recent recall may suffer. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a brain-inspired framework capable of incrementally learning data with different modalities and object classes. FearNet outperforms existing methods for incremental class learning on large image and audio classification benchmarks, demonstrating that FearNet is capable of recalling and consolidating recently learned information while also retaining old information. In addition, we showed that FearNet is more memory efficient, making it ideal for platforms where size, weight, and power requirements are limited. Future work will include 1) integrating BLA directly into the model (versus training it independently); 2) replacing HC with a semi-parametric model; 3) learning the feature embedding from raw inputs; and 4) replacing the pseduorehearsal mechanism with a generative model that does not require the storage of class statistics, which would be more memory efficient. ",
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+        "text": "A SUPPLEMENTAL MATERIAL ",
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+        "text": "A.1 MODEL HYPERPARAMETERS ",
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+        "text": "Table S1 shows the training parameters for the FearNet model for each dataset. We also experimented with various dropout rates, weight decay, and various activation functions; however, weight decay did not work well with FearNet’s mPFC. ",
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+        "table_caption": [
+            "Table S1: FearNet Training Parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td colspan=\"2\">Values</td></tr><tr><td>Learning Rate</td><td colspan=\"2\">2.10-3 450 (AudioSet &amp; CIFAR-100)</td></tr><tr><td>Mini-Batch Size</td><td colspan=\"2\">200 (CUB-200)</td></tr><tr><td rowspan=\"2\">mPFCBase-Knowledge Epochs Memory Consolidation Epochs</td><td colspan=\"2\">1,000</td></tr><tr><td colspan=\"2\">60</td></tr><tr><td rowspan=\"2\">BLA Training Epochs</td><td>20</td><td></td></tr><tr><td>CIFAR-100: [140,130]</td><td></td></tr><tr><td rowspan=\"2\">Hidden Layer Size</td><td>CUB-200:[350,300]</td><td></td></tr><tr><td>AudioSet: [300,100]</td><td></td></tr><tr><td>Sleep Frequency</td><td colspan=\"2\">10 (see Sec. 6.2)</td></tr><tr><td>Dropout Rate</td><td colspan=\"2\">0.25</td></tr><tr><td>Unsupervised Loss Weights (入)</td><td colspan=\"2\">[104,1.0,0.1]</td></tr><tr><td>Hidden Layer Activation Weight Decay</td><td colspan=\"2\">Exponential Linear Units 0.0</td></tr></table>",
+        "bbox": [
+            303,
+            633,
+            694,
+            840
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table S2 shows the training parameters for the iCaRL framework used in this paper. We adapted the code from the author’s GitHub page for our own experiments. The ResNet-18 convolutional neural network was replaced with a fully-connected neural network. We experimented with various regularization strategies to increase the initial base-knowledge accuracy with weight decay working ",
+        "bbox": [
+            173,
+            868,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f546dfb66f0ea585630b26beca6f7294e314c617740dab7e07158173e53b780e.jpg",
+        "table_caption": [
+            "the best. The values that are given as a range of values are the hyperparameter search spaces. ",
+            "Table S2: iCaRL Training Parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>Learning Rate</td><td>2.10-3</td></tr><tr><td>Mini-Batch Size</td><td>450</td></tr><tr><td>Exemplars per Class (EPC)</td><td>20</td></tr><tr><td>Hidden Layer Size</td><td>64-1024</td></tr><tr><td>Number of Hidden Layers</td><td>2-4</td></tr><tr><td>Dropout Rate</td><td>[0.5,0.75,1.00]</td></tr><tr><td>HiddenLayer Activation</td><td>ReLU</td></tr><tr><td>Weight Decay</td><td>[0.0,10-5,10-4,5 · 10-4]</td></tr></table>",
+        "bbox": [
+            325,
+            127,
+            673,
+            258
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/48863b19a866539f6560fc0e33e0d39a3c3e5fa2c77190ae47e50b65cce20fb3.jpg",
+        "table_caption": [
+            "Table S3 shows the training parameters for GeppNet and GeppNet+STM. Parameters not listed here are the default parameters defined by Gepperth & Karaoguz (2016). The values that are given as a range of values are the hyperparameter search spaces. ",
+            "Table S3: GeppNet Training Parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>SOM Lattice Shape (N)</td><td>20-36</td></tr><tr><td>Non-Linearity Suppression Threshold (0)</td><td>0.1-0.75</td></tr><tr><td>Incremental Class Learning Iterations (Tinc2 - Tinc1)</td><td>[2,000,20,000]</td></tr></table>",
+        "bbox": [
+            279,
+            337,
+            720,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table S4 shows the training parameters for the Fixed Expansion Layer (FEL). The number of units in the FEL layer is given by ",
+        "bbox": [
+            168,
+            430,
+            825,
+            458
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5e73f5685a7868c3d14e025b929b9a9728b8e4a8c195152ec256afaeeefb4189.jpg",
+        "text": "$$\n\\mathrm { F E L ~ U n i t s } = { \\frac { H ^ { 2 } + H K } { K } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            457,
+            584,
+            489
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $H$ is the number of units in the first hidden-layer and $K$ is the maximum number of classes in the dataset. The values that are given as a range of values are the hyperparameter search spaces. ",
+        "bbox": [
+            173,
+            491,
+            825,
+            521
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/39b7f0e874056c8831308643934f5229027de7d5c569b5e0a1430cfe73e11b2b.jpg",
+        "table_caption": [
+            "Table S4: FEL Training Parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Values</td></tr><tr><td>Hidden Layer Size (H)</td><td>64-1800</td></tr><tr><td>FEL Layer Size Number of Hidden Layers</td><td>See Equation 6</td></tr><tr><td>Mini-Batch Size</td><td>2 8</td></tr><tr><td>Initial Learning Rate</td><td>10-2</td></tr><tr><td></td><td></td></tr></table>",
+        "bbox": [
+            364,
+            529,
+            632,
+            621
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 ICARL PERFORMANCE WITH MORE EXEMPLARS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            659,
+            560,
+            674
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table S5 provides additional experimental results for when there are more exemplars per class (EPC) for the iCaRL framework. Rebuffi et al. (2017) used 20 EPC in their original paper; however, we increased the number to 100 EPC to see if storing more training data helped iCaRL. Although a higher EPC does increase iCaRL performance, it still does not outperform FearNet. Note that CUB-200 only has about 30 training samples per class, so iCaRL is storing the entire training set for 100 EPC. Our main results use the default value of 20. ",
+        "bbox": [
+            173,
+            684,
+            825,
+            768
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 BLA VARIANTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            799,
+            330,
+            814
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our BLA model is a classifier that determines whether a prediction should be made using HC (recent memory) or mPFC (remote memory). An alternative approach would be to use an outlier detection algorithm that determines whether the data being processed by a sub-network is an outlier for that sub-network and should therefore be processed by the other sub-network. To explore this alternative BLA formulation, we experimented with three outlier detection algorithms: 1) one-class support vector machine (SVM) (Scholkopf et al., 2001), 2) determining if the data fits into a Gaussian dis- ¨ tribution using a minimum covariance determinant estimation (i.e., elliptical envelope) (Rousseeuw ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/931f35f0adc8f02ed0815e4c924a6397ad56e7fd9007ea857565546023ccfb8f.jpg",
+        "table_caption": [
+            "Table S5: iCaRL’s performance when the stored EPC is increased from 20 to 100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>CIFAR-100 Sbase Ωnew aul</td><td>CUB-200 Sbase Ωnew</td><td>Ωall</td><td>AudioSet Sbase Ωnew</td><td>Ωau</td><td>Mean Sbase</td></tr><tr><td>iCaRL (20 EPC)</td><td>0.746 0.807 0.749</td><td>0.942</td><td>0.547 0.864</td><td>0.740 0.487</td><td>0.733</td><td>Ωall [0.801</td></tr><tr><td>iCaRL (100 EPC)</td><td>0.842 0.719 0.822</td><td>0.951</td><td>0.554 0.882</td><td>0.820 0.419</td><td>0.771</td><td>0.782 0.871 0.825</td></tr><tr><td>FearNet</td><td>0.927 0.824 0.947</td><td>0.924</td><td>0.598 0.891</td><td>0.962 0.455</td><td>0.932</td><td>0.938 0.923</td></tr></table>",
+        "bbox": [
+            222,
+            102,
+            776,
+            180
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/917a5d6e0c21cc99fe67169091b535f9d81fbd545a14450e78c3cf28f186496e.jpg",
+        "table_caption": [
+            "& Driessen, 1999), and 3) the isolation forest (Liu et al., 2008). All three of these methods set a rejection criterion for if the test sample exists in HC; whereas the binary MLP reports a probability on how likely the test sample resides in HC. Table S6 compares these individual methods. Isolation Forest and Elliptic Envelope seem to prefer the data in HC, one-class SVM prefers the data in mPFC, and our binary MLP worked best at choosing the correct sub-network to use. "
+        ],
+        "table_footnote": [
+            "Table S6: Performance of different BLA variants. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>BLA Method</td><td rowspan=1 colspan=1>ΩbaseΩnewaul</td></tr><tr><td rowspan=4 colspan=2>Isolation ForestElliptic EnvelopeOne-Class SVMBinary MLP</td><td rowspan=1 colspan=1>0.3280.8230.368</td></tr><tr><td rowspan=1 colspan=1>elope</td><td rowspan=1 colspan=1>0.5180.8230.541</td></tr><tr><td rowspan=1 colspan=1>0.7180.4330.702</td></tr><tr><td rowspan=1 colspan=1>0.9270.9240.947</td></tr></table>",
+        "bbox": [
+            383,
+            300,
+            614,
+            378
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 FEARNET ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            411,
+            379,
+            425
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Pseudocode for FearNet’s training and prediction algorithms are given in Algorithms 1 and 2 respectively. The variables match the ones defined in the paper. ",
+        "bbox": [
+            173,
+            438,
+            825,
+            467
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/bc5035b02c2dc885b6dca89cf08ce7ffb3bc1e76ab72b5cc613f9ff7f1c8f622.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1: FearNet Training</td><td>Algorithm 2: FearNet Prediction</td></tr><tr><td>Data: X,y Classes/Study-Sessions: T; K: Sleep Frequency; Initialize mPFC with base-knowledge;</td><td>Data: X A(X) ← PBLA (C =1|X); ← maxk PHc(C=k|X)A(X). 1-A(X)</td></tr><tr><td>Store μt,Σt for each class in the base-knowledge; forc←T/2toTdo StoreX,y for class c in HC;</td><td>if  &gt; maxk PmPFc (C = k|X) then return arg maxk PHc (C = k|X);</td></tr><tr><td>if c%K==O then Fine-tune mPFC with X,y in HC and pseudo- examples generated by mPFC decoder; Update μt,∑t for all classes seen so far;</td><td>else return arg maXk PmPFc (C = k/|X);</td></tr></table>",
+        "bbox": [
+            173,
+            486,
+            795,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 MULTI-MODAL LEARNING EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            726,
+            503,
+            741
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Fig. S1 shows the plots for the multi-modal experiments in Sec. 6.2. The three base-knowledge experiments were 1) CIFAR-100 is the base-knowledge and AudioSet is trained incrementally, 2) AudioSet is the base-knowledge and then AudioSet is trained incrementally, and 3) the base-knowledge is a $5 0 / 5 0 ~ \\mathrm { m i x }$ of the two datasets and then the remaining classes are trained incrementally. For all three base-knowledge experiments, we show the mean-class accuracy on the base-knowledge and the entire test set. FearNet works well when it adequately learns the base-knowledge (Experiment #1 and #3); however, when FearNet learns it poorly, incremental learning deteriorates. ",
+        "bbox": [
+            173,
+            752,
+            825,
+            851
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 BASE-KNOWLEDGE EFFECT ON PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            868,
+            549,
+            883
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Fig. S2 shows the effect of the base-knowledge’s size on FearNet’s performance. As expected, $\\Omega _ { b a s e }$ increases because there are not as many sleep phases to overwrite existing base-knowledge. $\\Omega _ { n e w }$ remains relatively even because the size of the base-knowledge has no effect on the HC model’s ability to immediately recall new information; however, there is a very slight decrease that corresponds to the BLA model erroneously favoring mPFC in a few cases. Most importantly, $\\Omega _ { a l l }$ sees an increase in performance because; like $\\Omega _ { b a s e }$ , there are not as many sleep phases to perturb older memories in mPFC. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1ee9cfc995cd1eecefdcd49c0badb1e9c5563f08021c359856092693a85d6752.jpg",
+        "image_caption": [
+            "Figure S1: Detailed plots for the multi-modal experiment. The top row is when the base-knowledge was CIFAR-100, the middle row is when the base-knowledge was AudioSet, and the bottom row is when the base-knowledge was a $5 0 / 5 0 ~ \\mathrm { m i x }$ from the two datasets. The left column represents the mean-class accuracy on the base-knowledge test set and the right column computes mean-class accuracy on the entire test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            99,
+            790,
+            698
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            803,
+            825,
+            873
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e8820c9ee2b911f01bafc12161f3da93d222287445581a28a5c6c5c91bd828c6.jpg",
+        "image_caption": [
+            "Figure S2: FearNet performance as a function of base-knowledge size. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            336,
+            404,
+            655,
+            592
+        ],
+        "page_idx": 15
+    }
+]

parse/train/n-219jrTht/n-219jrTht.md

@@ -0,0 +1,472 @@
+# On the Explanatory Power of Decision Trees
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Decision trees have long been recognized as models of choice in sensitive applica  
+2 tions where interpretability is of paramount importance. In this paper, we examine   
+3 the computational ability of Boolean decision trees in deriving, minimizing, and   
+4 counting sufficient reasons and contrastive explanations. We prove that the set   
+5 of all sufficient reasons of minimal size for an instance given a decision tree can   
+6 be exponentially larger than the size of the input (the instance and the decision   
+7 tree). Therefore, generating the full set of sufficient reasons can be out of reach. In   
+8 addition, computing a single sufficient reason does not prove enough in general;   
+9 indeed, two sufficient reasons for the same instance may differ on many features.   
+10 To deal with this issue and generate synthetic views of the set of all sufficient   
+11 reasons, we introduce the notions of relevant features and of necessary features that   
+12 characterize the (possibly negated) features appearing in at least one or in every   
+13 sufficient reason, and we show that they can be computed in polynomial time. We   
+14 also introduce the notion of explanatory importance, that indicates how frequent   
+15 each (possibly negated) feature is in the set of all sufficient reasons. We show how   
+16 the explanatory importance of a feature and the number of sufficient reasons can be   
+17 obtained via a model counting operation, which turns out to be practical in many   
+18 cases. We also explain how to enumerate sufficient reasons of minimal size. We   
+19 finally show that, unlike sufficient reasons, the set of all contrastive explanations   
+20 for an instance given a decision tree can be derived, minimized and counted in   
+21 polynomial time.
+
+# 22 1 Introduction
+
+23 In essence, explaining a decision to a person is to give the details or reasons that help a person   
+24 (the explainee) understand why the decision has been made. This is a significant issue especially   
+25 when decisions are made by Machine Learning (ML) models, such as random forests, Markov   
+26 networks, support vector machines, and deep neural networks. Actually, with the growing number   
+27 of applications that rely on ML techniques, researches on eXplainable AI (XAI) have become   
+28 increasingly important, by providing efficient methods for interpreting ML models, and explaining   
+29 their decisions (see for instance [10, 11, 12, 13, 16, 19, 22, 23, 24, 28, 30]).   
+30 When dealing with Boolean classifiers, which is what we do in this paper, two decisions are possible,   
+31 only: 1 for the instances classified as positive instances, and 0 for the remaining ones (the negative   
+32 instances). Whatever the way $_ { \textbf { \em x } }$ has been classified, an explainee may seek for explanations from   
+33 two distinct types [23]. On the one hand, abductive explanations for $_ { \textbf { \em x } }$ are intended to explain why $_ { \textbf { \em x } }$   
+34 has been classified in the way it has been classified by the ML model (thus, addressing the “Why?”   
+35 question). On the other hand, the purpose of contrastive (also known as counterfactual) explanations   
+36 for $_ { \textbf { \em x } }$ is to explain why $_ { \textbf { \em x } }$ has not been classified by the ML model as the explainee expected it (thus,   
+37 addressing the “Why not?” question). In both cases, explanations that are as simple as possible are   
+38 preferred (where simplicity is modeled as irredundancy, or even as size minimality).   
+39 Although there is no formal notion of interpretability [21], for classification problems, decision trees   
+40 [3, 26] are arguably among the most interpretable ML models. Because of their interpretability,   
+41 decision trees are often considered as target models for distilling a black-box model into a compre  
+42 hensible one [4, 10]. Furthermore, decision trees are often the components of choice for building   
+43 (less interpretable, but potentially more accurate) ensemble classifiers, such as random forests [2] and   
+44 gradient boosted decision trees [5].   
+45 The interpretability of decision trees is endowed with two key characteristics. On the one hand,   
+46 decision trees are transparent: each node in a decision tree has some meaning, and the principles used   
+47 for generating all nodes can be explained. On the other hand, decision trees are locally explainable:   
+48 by construction of a decision tree $T$ , any input instance $_ { \textbf { \em x } }$ is mapped to a unique root-to-leaf path   
+49 that yields to a decision label. The subset of (positive and negative) features $t _ { x } ^ { T }$ occurring in the   
+50 path used to find the right label 1 or 0 for $_ { \textbf { \em x } }$ in the decision tree $T$ can be viewed as a “direct reason”   
+51 for classifying $_ { \textbf { \em x } }$ as a positive instance or as a negative instance. $t _ { x } ^ { T }$ is an abductive explanation for   
+52 $_ { \textbf { \em x } }$ given $T$ , which explains why $_ { \textbf { \em x } }$ has been classified by $T$ as it has been classified. Indeed, every   
+53 instance $\mathbf { x } ^ { \prime }$ that coincides with $_ { \textbf { \em x } }$ on $t _ { x } ^ { T }$ is classified by $T$ in the same way as $_ { \textbf { \em x } }$ . However, such   
+54 “direct reasons” can contain arbitrarily many redundant features [17]. This motivates to take account   
+55 for other types of abductive explanations in the case of decision trees, namely, sufficient reasons [7]   
+56 (also known as prime implicant explanations [29]), that are irredundant abductive explanations, and   
+57 minimal sufficient reasons (i.e., those sufficient reasons of minimal size).   
+58 In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing   
+59 and counting sufficient reasons and contrastive explanations. We prove that the set of all sufficient   
+60 reasons of minimal size for an instance given a decision tree can be exponentially larger than the size   
+61 of the input. When this is the case, generating the full set of sufficient reasons (i.e., the complete   
+62 reason for the instance [7]) is typically out of reach. In addition, computing a single sufficient reason   
+63 does not prove enough in general; indeed; two sufficient reasons for the same instance may differ on   
+64 many features. To deal with this issue and generate synthetic views of the set of all sufficient reasons,   
+65 we introduce the notions of relevant features and of necessary features that characterize the (possibly   
+66 negated) features appearing in at least one or in every sufficient reason, and we show that they can be   
+67 computed in polynomial time. We also introduce the notion of explanatory importance, that indicates   
+68 how frequent each (possibly negated) feature is in the set of all sufficient reasons. Though deriving   
+69 the explanatory importance of a feature in the set of sufficient reasons and determining the cardinality   
+70 of this set are two computationally demanding tasks, we show how they can be achieved thanks to   
+71 model counting operation, which turns out to be practical in many cases. We also explain how to   
+72 enumerate sufficient reasons of minimal size, which is a way to count them when they are not too   
+73 numerous. We finally show that, from a computational standpoint, contrastive explanations highly   
+74 depart from sufficient reasons. Indeed, the set of all contrastive explanations for an instance given a   
+75 decision tree can be computed in polynomial time. As a consequence, such explanations can also be   
+76 minimized and counted in polynomial time.   
+77 The rest of the paper is organized as follows. Preliminaries about decision trees, abductive reasons,   
+78 and contrastive explanations are given in Section 2. The computation of all sufficient reasons is   
+79 considered in Section 3. Necessary and relevant features are presented in this section, as well as   
+80 the approach for assessing the explanatory importance of a feature and for counting the number of   
+81 sufficient reasons. We also explain there how minimal sufficient reasons can be enumerated. An   
+82 algorithm for computing all the contrastive explanations for the instance given the decision tree is   
+83 presented in Section 4. Experimental results are reported in Section 5. Finally, Section 6 concludes   
+84 the paper. All the proofs and additional empirical results are reported as a supplementary material.
+
+# 85 2 Decision Trees, Abductive and Contrastive Explanations
+
+86 For an integer $n$ , let $[ n ]$ be the set $\{ 1 , \cdots , n \}$ . By ${ \mathcal { F } } _ { n }$ we denote the class of all Boolean functions   
+87 from $\{ 0 , 1 \bar  \} ^ { n }$ to $\{ 0 , 1 \}$ , and we use $X _ { n } = \{ x _ { 1 } , \cdot \cdot \cdot , x _ { n } \}$ to denote the set of input Boolean variables,   
+88 corresponding to the features under consideration. Any assignment $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ is called an instance.   
+89 If $f ( { \pmb x } ) = 1$ for some $f \in \mathcal { F } _ { n }$ , then $_ { \textbf { \em x } }$ is called a model of $f$ . $_ { \textbf { \em x } }$ is a positive instance when $f ( { \pmb x } ) = 1$   
+90 and a negative instance when $f ( { \pmb x } ) = 0$ .   
+91 We refer to $f$ as a propositional formula when it is described using the Boolean connectives $\wedge$   
+92 (conjunction), $\vee$ (disjunction) and $\neg$ (negation), together with the Boolean constants 1 (true) and 0   
+93 (false). As usual, a literal $\ell$ is a variable $x _ { i }$ (a positive literal) or its negation $\neg x _ { i }$ , also denoted $\overline { { x } } _ { i }$ (a   
+94 negative literal). A positive literal $x _ { i }$ is associated with a positive feature (i.e., $x _ { i }$ is set to 1), while a   
+95 negative literal $\overline { { x } } _ { i }$ is associated with a negative feature (i.e., $x _ { i }$ is set to 0). A term (or monomial) $t$ is   
+96 a conjunction of literals, and a clause $c$ is a disjunction of literals. A DNF formula is a disjunction   
+97 of terms and a CNF formula is a conjunction of clauses. The set of variables occurring in a formula   
+98 $f$ is denoted $V a r ( f )$ . A formula $f$ is consistent if and only if it has a model. A CNF formula is   
+99 monotone whenever every occurrence of a literal in the formula has the same polarity (i.e., if a literal   
+100 occurs positively (resp. negatively) in the formula, then it does not have any negative (resp. positive)   
+101 occurrence in the formula). A formula $f _ { 1 }$ implies a formula $f _ { 2 }$ , noted $f _ { 1 } \models f _ { 2 }$ , if and only if every   
+102 model of $f _ { 1 }$ is a model of $f _ { 2 }$ . Two formulae $f _ { 1 }$ and $f _ { 2 }$ are equivalent, noted $f _ { 1 } \equiv f _ { 2 }$ whenever they   
+103 have the same models. The conditioning of a formula $f$ by a literal $\ell$ , denoted $f \mid \ell$ , is the formula   
+104 obtained from $f$ by replacing each occurrence of $x _ { i }$ with 1 (resp. 0) and each occurrence of $\overline { { x } } _ { i }$ with 0   
+105 (resp. 1) if $\ell = x _ { i }$ (resp. $\ell = \overline { { x } } _ { i }$ ).   
+106 In what follows, we shall often treat assignments as terms, and terms and clauses as sets of literals.   
+107 Given an assignment $z \in \{ 0 , 1 \} ^ { n }$ , the corresponding term is defined as
+
+![](images/06b4efff3f201077dfe01cba37433417efc5861eb79b21fc130ae6485bebde8d.jpg)  
+Figure 1: A decision tree $T$ for recognizing Cattleya orchids. The left (resp. right) child of any decision node labelled by $x _ { i }$ corresponds to the assignment of $x _ { i }$ to 0 (resp. 1).
+
+$$
+t _ { z } = \bigwedge _ { i = 1 } ^ { n } x _ { i } ^ { z _ { i } } { \mathrm { ~ w h e r e ~ } } x _ { i } ^ { 0 } = { \overline { { x } } } _ { i } { \mathrm { ~ a n d ~ } } x _ { i } ^ { 1 } = x _ { i }
+$$
+
+108 A term $t$ covers an assignment $_ z$ if $t \subseteq t _ { z }$ . An implicant of a Boolean function $f$ is a term that implies   
+109 $f$ . A prime implicant of $f$ is an implicant $t$ of $f$ such that no proper subset of $t$ is an implicant of $f$ .   
+110 Dually, an implicate of a Boolean function $f$ is a clause that is implied by $f$ , and a prime implicate of   
+111 $f$ is an implicate $c$ of $f$ such that no proper subset of $c$ is an implicate of $f$ .   
+112 With these basic notions in hand, we shall focus on the following representation class of Boolean   
+113 functions:   
+114 Definition 1 (Decision Tree). A (Boolean) decision tree is a binary tree $T$ , each of whose internal   
+115 nodes is labeled with one of $n$ input Boolean variables, and whose leaves are labeled 0 or 1. Every   
+116 variable is assumed (without loss of generality) to appear at most once on any root-to-leaf path   
+117 (read-once property). The value $T ( \pmb { x } ) \in \{ 0 , 1 \}$ of $T$ on an input instance $_ { \textbf { \em x } }$ is given by the label of   
+118 the leaf reached from the root as follows: at each node, go to the left or right child depending on   
+119 whether the input value of the corresponding variable is 0 or 1, respectively. The size of $T$ , denoted   
+120 $| T |$ , is given by the number of its nodes.   
+121 The class of decision trees over $X _ { n }$ is denoted $\mathbb { D } \mathbb { T } _ { n }$ . It is well-known that any decision tree $T \in \mathsf { D T } _ { n }$   
+122 can be transformed in linear time into an equivalent disjunction of terms, denoted $\tt D N F ( T )$ , where   
+123 each term corresponds to a path from the root to a leaf labeled with 1. Dually, $T$ can be transformed   
+124 in linear time into a conjunction of clauses, denoted $\mathrm { C N F } ( T )$ , where each clause is the negation of the   
+125 term describing a path from the root to a leaf labeled with 0.
+
+126 For illustration, the following toy example will be used throughout the paper as a running example:
+
+127 Example 1. The decision tree in Figure 1 separates Cattleya orchids from other orchids using the   
+128 following features: $x _ { 1 }$ : “has fragrant flowers”, $x _ { 2 }$ : “has one or two leaves”, $x _ { 3 }$ : “has large flowers”,   
+129 and $x _ { 4 }$ : “is sympodial”.   
+30 As a salient characteristic, decision trees convey a single explicit abductive explanation for classifying   
+31 any input instance:
+
+Definition 2 (Direct Reason). Let 32 $T \in \mathsf { D T } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ . The direct reason for $_ { \textbf { \em x } }$ given $T$ is the term, denoted 33 $t _ { x } ^ { T }$ , corresponding to the unique root-to-leaf path of $T$ that is compatible with $_ { \textbf { \em x } }$ .
+
+34 Another important notion of abductive explanations is the following concept of sufficient reason[7],   
+35 that, unlike the notion of direct reason, is not specific to decision trees:
+
+Definition 3 (Sufficient Reason). Let 136 $f \in \mathcal { F } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ such that $f ( { \pmb x } ) = 1$ (resp. $f ( { \pmb x } ) = 0 ,$ ).   
+137 A sufficient reason for $_ { \textbf { \em x } }$ given $f$ is a prime implicant $t$ of $f$ (resp. $\neg f$ ) that covers $_ { \textbf { \em x } }$ . $s r ( { \pmb x } , f )$   
+138 denotes the set of sufficient reasons for $_ { \textbf { \em x } }$ given $f$ .
+
+139 Thus, a sufficient reason [7] (also known as prime implicant explanation [29]) for an instance $_ { \textbf { \em x } }$ given   
+140 a class described by a Boolean function $f$ is a subset $t$ of the characteristics of $_ { \textbf { \em x } }$ that is minimal w.r.t.   
+141 set inclusion such that any instance $\mathbf { x } ^ { \prime }$ sharing this set $t$ of characteristics is classified by $f$ as $_ { \textbf { \em x } }$ is.   
+142 Thus, when $f ( { \pmb x } ) = 1$ , $t$ is a sufficient reason for $_ { \textbf { \em x } }$ given $f$ if and only if $t$ is a prime implicant of $f$   
+143 such that $_ { \textbf { \em x } }$ implies $t$ , and when $f ( { \pmb x } ) = 0$ , $t$ is a sufficient reason for $_ { \textbf { \em x } }$ given $f$ if and only if $t$ is a   
+144 prime implicant of $\neg f$ such that $t$ covers $_ { \textbf { \em x } }$ . Accordingly, sufficient reasons are suited to explain why   
+145 the instance at hand $_ { \textbf { \em x } }$ has been classified by $f$ as it has been classified. Unlike direct reasons [17],   
+146 sufficient reasons do not contain any redundant feature.   
+47 When considering the sufficient reasons of the input instance, one may be interested in focusing on   
+48 the shortest ones, alias the minimal sufficient reasons. Those reasons are valuable since conciseness   
+49 is often a desirable property of explanations (Occam’s razor). Formally:
+
+Definition 4 (Minimal Sufficient Reason). Let 50 $f \in \mathcal { F } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ . A minimal sufficient reason 51 for $_ { \textbf { \em x } }$ given $f$ is a sufficient reason for $_ { \textbf { \em x } }$ given $f$ that contains a minimal number of literals.
+
+152 Finally, unlike direct and (possibly minimal) sufficient reasons that aim to explain the classification   
+153 of the instance $_ { \textbf { \em x } }$ under consideration as achieved by the classifier $f$ , contrastive explanations are   
+154 valuable when $_ { \textbf { \em x } }$ has not been classified by $f$ as expected by the explainee. In this case, one looks for   
+155 minimal subsets of the features that when switched in $_ { \textbf { \em x } }$ are enough to get instances that are classified   
+156 positively (resp. negatively) by $f$ if $_ { \textbf { \em x } }$ is classified negatively (resp. positively) by $f$ . Formally, a   
+157 contrastive explanation for $_ { \textbf { \em x } }$ given $f$ [15] is a subset $t$ of the characteristics of $_ { \textbf { \em x } }$ that is minimal   
+158 w.r.t. set inclusion among those such that at least one instance $\mathbf { x } ^ { \prime }$ that coincides with $_ { \textbf { \em x } }$ except on the   
+159 characteristics from $t$ is not classified by $f$ as $_ { \textbf { \em x } }$ is.
+
+Definition 5 (Contrastive Explanation). Let $f \in \mathcal { F } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ such that $f ( { \pmb x } ) = 1$ (resp. $f ( { \pmb x } ) = 0 ,$ ). A contrastive explanation for $_ { \textbf { \em x } }$ given $f$ is a term t over $X _ { n }$ such that $t \subseteq t _ { x }$ , $t _ { x } \wedge t$ is not an implicant of $f$ (resp. $\neg f )$ ), and for every $\ell \in t$ , $t \setminus \{ \ell \}$ does not satisfy this last condition.
+
+163 Example 2. Based on our running example, we can observe that $T ( \pmb { x } ) = 1$ for the instance   
+164 $\pmb { x } = ( 1 , 1 , 1 , 1 )$ . The direct reason for $_ { \textbf { \em x } }$ given $T$ is the term $t _ { x } ^ { T } = x _ { 1 } \wedge x _ { 2 } \wedge x _ { 3 } \wedge x _ { 4 }$ . $x _ { 1 } \wedge x _ { 4 }$ and   
+165 $x _ { 2 } \wedge x _ { 3 } \wedge x _ { 4 }$ are the sufficient reasons for $_ { \textbf { \em x } }$ given $T$ . $x _ { 1 } \wedge x _ { 4 }$ is the unique minimal sufficient reason   
+166 for $_ { \textbf { \em x } }$ given $T$ . $x _ { 4 }$ , $x _ { 1 } \wedge x _ { 2 }$ , and $x _ { 1 } \wedge x _ { 3 }$ are the contrastive explanations for $_ { \textbf { \em x } }$ given $T$ . Thus, the   
+167 instance $( 1 , 1 , 1 , 0 )$ that differs with $_ { \textbf { \em x } }$ only on $x _ { 4 }$ is not classified by $T$ as $_ { \textbf { \em x } }$ is $( ( 1 , 1 , 1 , 0 )$ is classified   
+168 as a negative instance).   
+169 We mention in passing that when dealing with decision trees $T$ , we could have focused only on   
+170 explanations for the positive instances $_ { \textbf { \em x } }$ given $T$ . This comes from the fact that $\mathbb { D } \mathbb { T } _ { n }$ is closed under   
+171 negation, in the sense that for any $T \in \mathsf { D T } _ { n }$ , $\neg T$ can be obtained by just replacing from $T$ the label   
+172 of each leaf with its complement. So, for any instance $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ , a direct reason (resp. sufficient   
+173 reason, minimal sufficient reason, contrastive explanation) explaining why $T ( { \pmb x } ) = 0$ is precisely the   
+174 same as a direct reason (resp. sufficient reason, minimal sufficient reason, contrastive explanation)   
+175 explaining why $( \neg T ) ( { \pmb x } ) = { \bar { 1 } }$ . Considering $T$ or its negation $\neg T$ has no computational impact since   
+176 $\neg T$ can be computed in time linear in the size of $T$ .
+
+# 177 3 Computing All Sufficient Reasons
+
+178 Sufficient reasons can be exponentially numerous. When switching from the direct reason for   
+179 an instance (that is unique but not always redundancy-free) to its sufficient reasons, a main obstacle   
+180 to be dealt with lies in the number of reasons to be considered. Indeed, even for the restricted class   
+181 of decision trees with logarithmic depth, an input instance can have exponentially many sufficient   
+182 reasons:
+
+![](images/db2a5437074379661004bed2173191f91491967548948fa2cda660b68aa58dc2.jpg)  
+Figure 2: Two sufficient reasons for an mnist instance (top), and an explanatory heat map and the explanatory features for an mnist instance (bottom).
+
+Proposition 1. There is a decision tree 183 $T \in \mathsf { D T } _ { n }$ of depth $\log _ { 2 } ( n + 1 )$ such that for any $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ , the number of sufficient reasons for 184 $_ { \textbf { \em x } }$ given $T$ is at least $\lfloor { \frac { 3 } { 2 } } ^ { \frac { n + 1 } { 2 } } \rfloor$ .
+
+By definition, the minimal sufficient reasons for $_ { \textbf { \em x } }$ given $T$ cannot be more numerous than its sufficient reasons. However, focusing on minimal sufficient reasons does not solve the problem since an instance can also have exponentially many minimal sufficient reasons:
+
+Propositisuch that 2. For econtains $n \in \mathbb N$ such that des and th $n$ is odd, there is e is an instance e s $T \in \mathsf { D T } _ { n }$ of depth the numb $\textstyle { \frac { n + 1 } { 2 } }$ $T$ $2 n + 1$ $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ minimal sufficient reasons for x given T is equal to 2 n−1.
+
+191 In many practical cases, the number of sufficient reasons for an instance given a decision tree can   
+192 be very large. Figure 2 (top) shows an mnist instance (the leftmost subfigure) that has 482 185 073   
+193 664 sufficient reasons. Among them there are very dissimilar sufficient reasons. As an illustration,   
+194 the two rightmost subfigures present two sufficient reasons for this instance, and they differ on many   
+195 features (blue (resp. red) dots correspond to pixels on (resp. off)).   
+196 For such datasets, computing the set of all the sufficient reasons for a given instance is not always   
+197 feasible. Furthermore, if the computation succeeds but the number of sufficient reasons is huge, their   
+198 (disjunctively interpreted) set, alias the complete reason for the instance [7], can hardly be considered   
+199 as intelligible by the explainee. Finally, due to the number of sufficient reasons and their diversity,   
+200 deriving one of them is not informative enough. Thus, one needs to design approaches to synthesizing   
+201 their set while avoiding the two pitfalls (the computational one and the informational one).
+
+Synthesizing the set of sufficient reasons. In this objective, the following notions of necessary / (ir)relevant features appear useful. These notions of necessity and relevance echo the ones that have been considered in [9] for logic-based abduction.
+
+Definition 6 (Explanatory Features). Let $f \in \mathcal { F } _ { n }$ , and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ be an instance. Let e be an explanation type.
+
+• A literal \` over $X _ { n }$ is a necessary feature for the family e of explanations for $_ { \textbf { \em x } }$ given $f$ if and only if \` belongs to every explanation t for $_ { \textbf { \em x } }$ given $f$ such that $t$ is of type e. $N e c _ { e } ( { \pmb x } , f )$ denotes the set of all necessary features for the family e of explanations for $_ { \textbf { \em x } }$ given $f$ .
+
+• A literal $\ell$ over $X _ { n }$ is $a$ relevant feature for the family e of explanations for $_ { \textbf { \em x } }$ given $f$ if and only if \` belongs to at least one explanation $t$ for $_ { \textbf { \em x } }$ given $f$ such that $t$ is of type e. $R e l _ { e } ( { \pmb x } , f )$ denotes the set of all relevant features for the family e of explanations for $_ { \textbf { \em x } }$ given $f$ . $I r r _ { e } ( { \pmb x } , f )$ , which is the complement of $R e l _ { e } ( { \pmb x } , f )$ in the set of all literals over $X _ { n }$ , denotes the set of all irrelevant features for the family e of explanations for $_ { \textbf { \em x } }$ given $f$ .
+
+15 The necessary (resp. irrelevant) features for the family $s$ of sufficient reasons for $_ { \textbf { \em x } }$ given $f$ are the   
+16 most (resp. less) important features for explaining the classification of $_ { \textbf { \em x } }$ by $f$ , since they belong to   
+17 every (resp. no) sufficient reason for $_ { \textbf { \em x } }$ given $f$ .
+
+When a single sufficient reason $t$ for $_ { \textbf { \em x } }$ given $f$ has been computed, the cardinality of $t$ deprived from the features of $N e c _ { s } ( { \pmb x } , f )$ is small, and the cardinality of the symmetric difference between $t$ and $R e l _ { s } ( { \pmb x } , f )$ is small as well, $t$ can be viewed as a good representative of the complete reason for $_ { \textbf { \em x } }$ given $f$ in the sense that a sufficient reason $t ^ { \prime }$ for $_ { \textbf { \em x } }$ given $f$ that differs a lot from $t$ cannot exist.
+
+In the case when $f$ is a decision tree $T$ , though the set of all sufficient reasons for $_ { \textbf { \em x } }$ given $T$ cannot be generated when it is too large, $N e c _ { s } ( { \pmb x } , f )$ , $R e l _ { s } ( { \pmb x } , f )$ , and $I r r _ { s } ( { \pmb x } , f )$ can be derived efficiently:
+
+Proposition 3. Let $T \in \mathsf { D T } _ { n }$ , and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ . Computing $N e c _ { s } ( { \pmb x } , T )$ , $R e l _ { s } ( { \pmb x } , f )$ , and $I r r _ { s } ( { \pmb x } , T )$ can be done in $\mathcal { O } ( ( n + | T | ) \times | T | )$ time.
+
+226 Going a step further consists in evaluating the explanatory importance of every (positive or negative)   
+227 feature:
+
+Definition 7 (Explanatory Importance). Let $f \in { \mathcal { F } } _ { n } ,$ , and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ be an instance. Let e be an explanation type, and $E _ { e } ( x , f )$ the set of all explanations for $_ { \textbf { \em x } }$ given $f$ that are of type e. The explanatory importance of a literal $\ell$ over $X _ { n }$ for $_ { \textbf { \em x } }$ given $f$ w.r.t. e is given by
+
+$$
+I m p _ { e } ( \ell , \pmb { x } , f ) = \frac { \# ( \{ t \in E _ { e } ( \pmb { x } , f ) : \ell \in t \} ) } { \# ( E _ { e } ( \pmb { x } , f ) ) } .
+$$
+
+228 Example 3. On the running example, we have $N e c _ { s } ( { \pmb x } , T ) = \{ x _ { 4 } \}$ , and ${ R e l _ { s } } ( { \pmb x } , T ) = \{ { x } _ { 1 } , { x } _ { 2 } , { x } _ { 3 } .$ ,   
+229 $x _ { 4 } \}$ . We also have $I m p _ { s } ( x _ { 4 } , { \pmb x } , T ) = 1$ $, I m p _ { s } ( x _ { 1 } , { \pmb x } , T ) = I m p _ { s } ( x _ { 2 } , { \pmb x } , T ) = I m p _ { s } ( x _ { 3 } , { \pmb x } , T ) = \frac { 1 } { 2 }$ ,   
+230 and $I m p _ { s } ( \ell , { \pmb x } , T ) = 0$ for every other literal $\ell$ (the negative ones over $\{ x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \} ,$ ).   
+231 The notion of explanatory importance must not be confused with the notions of feature importance   
+232 (which can be defined and assessed in many different ways): the former is local (i.e., relative to an   
+233 instance) and not global, it concerns literals and not variables (polarity matters), and it is about the   
+234 explanation task, not the prediction one.   
+235 In order to compute the explanatory importance of a literal, a straightforward approach consists in   
+236 enumerating the explanations of $E _ { e } ( { \pmb x } , \bar { f } )$ . This is feasible when this set is not too large, which is not   
+237 always the case for sufficient reasons even when $f$ is a decision tree $T$ . Thus, for dealing with the   
+238 remaining case, an alternative approach must be looked for.   
+239 We designed such an approach for computing $I m p _ { s } ( \boldsymbol { \ell } , \pmb { x } , T )$ . We know that $s r ( \pmb { x } , T )$ is by construc  
+240 tion the set of prime implicants of $g = \{ c \cap t _ { \pmb { x } } : c \in \mathbb { C } \mathbb { N } \mathbb { F } ( T ) \}$ . Thus, we exploited the translation   
+241 presented in [18] showing how to associate in polynomial time with a given CNF formula (here,   
+242 $g$ ) another formula (over a distinct set of variables), let us say $h$ , such that the models of $h$ are   
+243 in one-to-one correspondence with the prime implicants of $g$ . In our case, the translation can be   
+244 simplified because $g$ is a monotone CNF formula. Since $h$ is not primarily a CNF formula, leveraging   
+245 Tseitin transformation [31], we turned $h$ in linear time into a query-equivalent CNF formula $i$ . Note   
+246 that every auxiliary variable that is introduced in $i$ is defined from the other variables (those occurring   
+247 in $h$ ), so that the number of models of $i$ is the same as the number of models of $h$ . Finally, we took   
+248 advantage of the compilation-based model counter D4 [20] to compile $i$ into a d-DNNF circuit [6],   
+249 and this enabled us to compute in time polynomial in the size of $i$ both the number of sufficient   
+250 reasons and the explanatory importance of every literal (indeed, the d-DNNF language supports in   
+251 polytime the model counting query and the conditioning transformation [8]). We show in Section   
+252 5 that, despite a high complexity in the worst case (the size of $i$ can be exponential in $| T | )$ , this   
+253 approach based on knowledge compilation proves quite efficient in practice.
+
+Clearly enough, when $I m p _ { e } ( \ell , \pmb { x } , T )$ has been computed for every $\ell$ , one can easily generate explanatory heat maps. Figure 2 (bottom) shows an mnist instance (the leftmost subfigure) that has 19 115 685 sufficient reasons, 6 necessary literals, and 94 relevant literals. The central subfigure is the corresponding heat map. Blue (resp. red) pixels correspond to positive (resp. negative) literals in the instance, and the intensity of the color aims to reflect the explanatory importance of the corresponding literal. The rightmost subfigure gives the explanatory features (dark pixels are associated with necessary literals, and light pixels to relevant literals).
+
+261 Enumerating the minimal sufficient reasons. An approach to synthesizing the set of sufficient   
+262 reasons consists in focusing on the minimal ones. Indeed, though the set of minimal sufficient reasons   
+263 for an instance given a decision tree can be exponentially large, the number of minimal sufficient   
+264 reasons cannot exceed the number of sufficient reasons, and it can be significantly lower in practice.
+
+However, unlike sufficient reasons that can be generated in polynomial time using a greedy algorithm (see e.g., [17]), computing minimal reasons is not an easy task:
+
+Proposition 4. Let $T \in \mathsf { D T } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ . Computing a minimal sufficient reason for $_ { \textbf { \em x } }$ given $T$ is NP-hard.
+
+269 Despite this intractability result, minimal sufficient reasons can be generated in many practical cases.   
+270 A common approach for handling NP-optimization problems is to rely on modern constraint solvers.   
+271 One follows this direction here and casts the task of finding minimal sufficient reasons as a Boolean   
+272 constraint optimization problem. We first need to recall that a PARTIAL MAXSAT problem consists   
+273 of a pair $( C _ { \mathrm { s o f t } } , C _ { \mathrm { h a r d } } )$ where $C _ { \mathrm { s o f t } }$ and $C _ { \mathrm { h a r d } }$ are (finite) set of clauses. The goal is to find a Boolean   
+274 assignment that maximizes the number of clauses $c$ in $C _ { \mathrm { s o f t } }$ that are satisfied, while satisfying all   
+275 clauses in $C _ { \mathrm { h a r d } }$ .
+
+Proposition 5. Let $T$ be a decision tree in $\mathbb { D } \mathbb { T } _ { n }$ and $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ be an instance such that $T ( \pmb { x } ) = 1$ . Let $( C _ { \mathrm { s o f t } } , C _ { \mathrm { h a r d } } )$ be an instance of the PARTIAL MAXSAT problem such that:
+
+The intersection of 276 $t _ { x }$ with $t _ { x ^ { * } }$ where $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } } ^ { * }$ is an optimal solution of $( C _ { \mathrm { h a r d } } , C _ { \mathrm { s o f t } } )$ , is a minimal 277 sufficient reason for $_ { \textbf { \em x } }$ given $T$ .
+
+78 Clearly enough, if $_ { \textbf { \em x } }$ is such that $T ( { \pmb x } ) = 0$ , then it is enough to consider the same instance of   
+79 PARTIAL MAXSAT as above, except that $C _ { \mathrm { h a r d } } = \{ c \cap t _ { x } : c \in \mathbb { C } \mathbb { N } \mathbb { F } ( \neg T ) \}$ .   
+80 Finally, one can take advantage of this PARTIAL MAXSAT characterization for generating a preset   
+number of minimal sufficient reasons (basically, one generates a first reason $t$ , then one adds to $C _ { \mathrm { h a r d } }$   
+2 the negation of $t$ as a clause as well as a CNF encoding of a cardinality constraint for ensuring that the   
+83 next reasons to be generated have the same size as the one of $t$ , and we resume until the bound is   
+84 reached or no solution exists).
+
+# 85 4 Computing All Contrastive Explanations
+
+Interestingly, it has been shown that sufficient reasons and contrastive explanations are connected by a minimal hitting set duality [15]. This duality can be leveraged to derive one of the two sets of explanations from the other one using algorithms for computing minimal hitting sets [27, 32].
+
+However, in the case of decision trees, a more direct and much more efficient approach to derive all the contrastive explanations for $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ given $T \in \mathbb { D T } _ { n }$ can be designed. Indeed, unlike what happens for sufficient reasons (see Section 3), the set of all contrastive explanations for $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ given a decision tree $T \in \mathsf { D T } _ { n }$ can be computed in polynomial time from $_ { \textbf { \em x } }$ and $T$ :
+
+Proposition 6. The set of all contrastive explanations for $\pmb { x } \in \{ 0 , 1 \} ^ { n }$ given a decision tree $T \in \mathsf { D T } _ { n }$ can be computed in time polynomial in $n + | T |$ as min( $\{ c \cap t _ { x } : c \in \mathrm { C N F } ( f ) \} , \subseteq )$ .
+
+Example 4. On the running example, we have $\mathsf { F } ( T ) = \{ x _ { 1 } \vee x _ { 2 } , x _ { 1 } \vee \overline { { x _ { 2 } } } \vee x _ { 3 } , x _ { 1 } \vee \overline { { x _ { 2 } } } \vee \overline { { x _ { 3 } } } \vee x _ { 4 } ,$ $\overline { { x _ { 1 } } } \vee x _ { 2 } \vee x _ { 3 } \vee x _ { 4 }$ , $\overline { { x _ { 1 } } } \vee x _ { 2 } \vee \overline { { x _ { 3 } } } \vee x _ { 4 }$ , $\overline { { x _ { 1 } } } \vee \overline { { x _ { 2 } } } \vee x _ { 3 } \vee x _ { 4 } , \overline { { x _ { 1 } } } \vee \overline { { x _ { 2 } } } \vee \overline { { x _ { 3 } } } \vee x _ { 4 } \Big \}$ . Thus, with $\pmb { x } = ( 1 , 1 , 1 , 1 )$ , we have mi $n \big ( \{ c \cap t _ { \pmb { x } } : c \in \mathtt { C N F } ( f ) \} , \underline { { \mathsf { C } } } \big ) = \{ x _ { 1 } \vee x _ { 2 } , x _ { 1 } \vee x _ { 3 } , x _ { 4 } \}$ , which corresponds to the contrastive explanations $x _ { 1 } \wedge x _ { 2 }$ , $x _ { 1 } \wedge x _ { 3 }$ , $x _ { 4 }$ for $_ { \textbf { \em x } }$ given $T$ (viewing clauses and terms as sets of literals).
+
+As straightforward consequences of Proposition 6, computing necessary $/$ relevant features and computing the explanatory importance of features w.r.t. contrastive explanations can be achieved in time polynomial in $n + | T |$ . Similarly, statistics about the size of contrastive explanations can be easily established, and contrastive explanations can be easily minimized and counted.
+
+# 5 Experiments
+
+Empirical setting. We have considered 90 datasets, which are standard benchmarks from the wellknown repositories Kaggle (www.kaggle.com), OpenML (www.openml.org), and UCI (archive.
+
+Table 1: Empirical results based on 12 datasets.   
+
+<table><tr><td></td><td colspan="3">Decision Tree</td><td colspan="2">ISufficientl</td><td colspan="2">IMinimall</td><td colspan="2">#Nec.Features</td><td colspan="2">#Rel. Features</td></tr><tr><td>Dataset</td><td>%A</td><td>#N</td><td>#B</td><td>med</td><td>max</td><td>med</td><td>max</td><td>med</td><td>max</td><td>med</td><td>max</td></tr><tr><td>recidivism</td><td>63.41</td><td>13828.80</td><td>147.60</td><td>14</td><td>22</td><td>13</td><td>22</td><td>6</td><td>19</td><td>60</td><td>98</td></tr><tr><td>adult</td><td>81.36</td><td>12934.00</td><td>2974.80</td><td>16</td><td>36</td><td>16</td><td>36</td><td>7</td><td>22</td><td>263</td><td>543</td></tr><tr><td>bank marketing</td><td>87.40</td><td>6656.40</td><td>1432.60</td><td>14</td><td>21</td><td>14</td><td>21</td><td>3</td><td>16</td><td>247</td><td>398</td></tr><tr><td>bank</td><td>88.99</td><td>5523.60</td><td>977.80</td><td>13</td><td>24</td><td>13</td><td>24</td><td>4</td><td>15</td><td>200</td><td>330</td></tr><tr><td>lending loan</td><td>73.49</td><td>2610.40</td><td>1131.40</td><td>16</td><td>31</td><td>16</td><td>31</td><td>8</td><td>25</td><td>226</td><td>442</td></tr><tr><td>contraceptive</td><td>50.44</td><td>1252.20</td><td>88.60</td><td>11</td><td>20</td><td>11</td><td>20</td><td>8</td><td>17</td><td>25</td><td>47</td></tr><tr><td>compas</td><td>65.98</td><td>1230.00</td><td>46.20</td><td>6</td><td>14</td><td>6</td><td>14</td><td>3</td><td>12</td><td>16</td><td>33</td></tr><tr><td>christine</td><td>63.36</td><td>853.20</td><td>426</td><td>12</td><td>47</td><td>12</td><td>47</td><td>8</td><td>41</td><td>92</td><td>202</td></tr><tr><td>farm-ads</td><td>86.75</td><td>544.80</td><td>264.60</td><td>20</td><td>99</td><td>20</td><td>99</td><td>16</td><td>92</td><td>73</td><td>192</td></tr><tr><td>mnist49</td><td>95.47</td><td>539.60</td><td>267.90</td><td>22</td><td>30</td><td>22</td><td>30</td><td>9</td><td>19</td><td>91</td><td>166</td></tr><tr><td>spambase</td><td>91.94</td><td>536.40</td><td>264.80</td><td>15</td><td>29</td><td>15</td><td>29</td><td>9</td><td>24</td><td>68</td><td>146</td></tr><tr><td>mnist38</td><td>96.07</td><td>506.60</td><td>251.40</td><td>19</td><td>28</td><td>19</td><td>28</td><td>8</td><td>20</td><td>93.50</td><td>157</td></tr><tr><td></td><td colspan="4">#Sufficient</td><td colspan="2"></td><td colspan="2">#Contrastive</td><td colspan="2">#Minimal</td></tr><tr><td>Dataset</td><td colspan="2">med</td><td colspan="2">max</td><td></td><td>med</td><td>max</td><td>med</td><td></td><td></td><td>max</td></tr><tr><td>recidivism</td><td colspan="2">10387</td><td colspan="2">9734080</td><td></td><td>54</td><td>145</td><td>3</td><td>16</td><td>2</td><td>144</td></tr><tr><td>adult</td><td colspan="2"></td><td colspan="2">≥ 1573835722607300000000000</td><td></td><td>201</td><td>470</td><td>4</td><td>16</td><td>3</td><td>256</td></tr><tr><td>bank marketing</td><td colspan="2"></td><td colspan="2">≥7460375213484350000000</td><td></td><td>189</td><td>337</td><td>4</td><td>13</td><td>8</td><td>432</td></tr><tr><td>bank</td><td colspan="2"></td><td colspan="2">≥7433951979018500000</td><td></td><td>150</td><td>277</td><td>4</td><td>13</td><td>4</td><td>168</td></tr><tr><td>lending loan</td><td colspan="2">459258918095775</td><td colspan="2">943243242816203000000000000000</td><td></td><td>157</td><td>311</td><td>3</td><td>12</td><td>3</td><td>192</td></tr><tr><td>contraceptive</td><td colspan="2">20.50</td><td colspan="2"></td><td>4272</td><td>21</td><td>52</td><td>2</td><td>11</td><td>2</td><td>48</td></tr><tr><td>compas</td><td colspan="2">16</td><td colspan="2"></td><td>444</td><td>13</td><td>33</td><td>2</td><td>11</td><td>2</td><td>21</td></tr><tr><td>christine</td><td colspan="2">63108</td><td colspan="2">2167735434744</td><td></td><td>71</td><td>151</td><td>3</td><td>8</td><td>2</td><td>4096</td></tr><tr><td>farm-ads</td><td colspan="2">1177,50 7392384</td><td colspan="2">921895392 715892613696000</td><td></td><td>59 61</td><td>166 106</td><td>2 2</td><td>10 12</td><td></td><td></td></tr><tr><td>mnist49 spambase</td><td colspan="2">15712</td><td colspan="2">2535069312</td><td></td><td>50</td><td>107</td><td>2</td><td>11</td><td>4</td><td>10000</td></tr><tr><td>mnist38</td><td colspan="2">14849376</td><td colspan="2">16922386736640</td><td></td><td>62</td><td>107</td><td>3</td><td>11</td><td></td><td>384</td></tr><tr><td></td><td colspan="2"></td><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td>32</td><td>3072</td></tr></table>
+
+6 ics.uci.edu/ml/). mnist38 and mnist49 are subsets of the mnist dataset, restricted to the 7 instances of 3 and 8 (resp. 4 and 9) digits. Because some datasets are suited to the multi-label 8 classification task, we used the standard “one versus all” policy to deal with them: all the classes but 09 the target one are considered as the complementary class of the target. Categorical features have been treated as arbitrary numbers (the scale is nominal). As to numeric features, no data preprocessing has taken place: these features have been binarized on-the-fly by the decision tree learning algorithm that 12 has been used.
+
+313 For every benchmark $b$ , a 10-fold cross validation process has been achieved. Namely, a set of 10   
+314 decision trees $T _ { b }$ have been computed and evaluated from the labelled instances of $b$ , partitioned into   
+315 10 parts. One part was used as the test set and the remaining 9 parts as the training set for generating   
+316 a decision tree. This tree is thus in 1-to-1 correspondence with the test set chosen within the whole   
+317 dataset $b$ . The classification performance for $b$ was measured as the mean accuracy obtained over the   
+318 10 decision trees generated from $b$ . The CART algorithm, and more specifically its implementation   
+319 provided by the Scikit-Learn library [25] has been used to learn decision trees. All hyper-parameters   
+320 of the learning algorithm have been set to their default value. Notably, decision trees have been   
+321 learned using the Gini criterion, and without any maximal depth or any other manual limitation.
+
+For each benchmark $b$ , each decision tree $T _ { b }$ , and a subset of at most 100 instances $_ { \textbf { \em x } }$ picked up at random in the test set following a uniform distribution, we computed a sufficient reason for $_ { \textbf { \em x } }$ given $T _ { b }$ (using the standard greedy algorithm run on the direct reason $t _ { x } ^ { T _ { b } }$ ), and a minimal sufficient reason for $_ { \textbf { \em x } }$ given $T _ { b }$ using the PARTIAL MAXSAT encoding presented in Proposition 5. This enabled us to draw some statistics (median, maximum) about the sizes of the reasons that have been generated. Using the algorithm presented in the proof of Proposition 3, we also derived the necessary and relevant explanatory features for each $_ { \textbf { \em x } }$ , and again drew some statistics about them. Exploiting the model counter D4, we computed the number of sufficient reasons for $_ { \textbf { \em x } }$ given $T _ { b }$ , as well as the explanatory importance of every feature. Taking advantage of the algorithm given in Proposition 4, we computed the number of contrastive explanations for $_ { \textbf { \em x } }$ given $T _ { b }$ , and drew some statistics about those numbers and about the sizes of the contrastive explanations. Finally, using the approach described in Section 3, we enumerated all the minimal sufficient reasons for $_ { \textbf { \em x } }$ given $T _ { b }$ up to a limit of $1 0 0 0 0$ , and again drew some statistics about the numbers of minimal sufficient reasons. Of course, for each computation, we measured the corresponding runtimes since this is fundamental to determine the extent to which the algorithms are practical (details are provided as a supplementary material).
+
+337 All the experiments have been conducted on a computer equipped with Intel(R) XEON E5-2637 CPU   
+338 $\textcircled { \omega } 3 . 5 \ : \mathrm { G H z }$ and $1 2 8 \mathrm { G i B }$ of memory. D4 [20] was run with its default parameters. For computing   
+339 minimal reasons, we used the Pysat library [14], which provides the implementation of the RC2   
+340 PARTIAL MAXSAT solver. This solver was run using the parameters corresponding to the “Glucose”   
+341 setting. A time-out of 100s per instance was set for D4.
+
+Results. Table 1 (top and bottom) reports an excerpt of our results, focusing on 12 benchmarks out of 90 (the selected datasets are among those containing many instances and/or many features). The leftmost column gives the name of the dataset $b$ . Columns $\% { \dot { A } }$ , $\% N$ , and $\# B$ give, respectively, the mean accuracy over the 10 decision trees, the average number of nodes in those trees, and the average number of binary features they are based on. The next columns give statistics (median, maximum) about, respectively, the size of the sufficient reasons (|Sufficient|) and of the minimal sufficient reasons (|Minimal|) that have been computed, as well as about the number of necessary (#Nec. Features) and relevant $( \# \mathsf { R e l }$ . Features) features that appear in the full set of sufficient reasons for the instance. Table 1 (bottom) give statistics (median, maximum) about, respectively, the number of sufficient reasons (#Sufficient), the number of contrastive explanations (#Contrastive) and their sizes (|Contrastive|), and finally the number of minimal sufficient reasons ( $\#$ Minimal).
+
+353 As to the computation times, it turns out that all the algorithms described in the previous sections   
+354 proved as efficient in practice. This is not surprising for those algorithms having a polytime worst-case   
+355 complexity (the greedy algorithm for computing a sufficient reason, the one for deriving explanatory   
+356 features, and the one for computing all the contrastive explanations). It was less obvious at first   
+357 sight for the algorithms used for counting the number of sufficient reasons and for computing the   
+358 explanatory importance of features. However, all the computations that have been run have terminated   
+359 in due time, except for 3 datasets out of 90, namely adult, bank_marketing, and bank. For these   
+360 datasets, the time limit of 100s has been reached for, respectively, 203, 150, and 336 instances out of   
+361 1000 (in this case, the median number of sufficient reasons has not been reported). Notably, for all   
+362 the 90 datasets but those 3, the median time required for counting the number of sufficient reasons   
+363 and computing the explanatory importance of features never exceeded 1s. Computing a minimal   
+364 sufficient reason, and more generally all such reasons looked challenging as well, due to both the   
+365 intrinsic complexity of computing a minimal sufficient reason and to their number. Nevertheless,   
+366 our enumeration algorithm succeeded in deriving all the minimal sufficient reasons for every dataset   
+367 except 3 out of 90, namely farm-ads, mnist49, and gisette. For these datasets, the limit of 10   
+368 000 reasons has been reached for, respectively, 5, 16, and 3 instances out of 1000. Interestingly,   
+369 the median time needed to derive all the minimal sufficient reasons for the instances for which the   
+370 computation has been successful exceeded 1s only for 2 datasets (adult and bank_marketing).   
+371 Beyond providing evidence that the number of reasons can be huge, our experiments have highlighted   
+372 that the greedy algorithm for deriving a sufficient reason computes in practice a minimal sufficient   
+373 reason in many cases. They have also shown that the number of explanatory relevant features for an   
+374 instance is typically much lower than the number of binary features used to describe it, and that the   
+375 number of explanatory necessary features is also significantly lower than the number of explanatory   
+376 relevant features. The gap between the two explains the possibly enormous number of sufficient   
+377 reasons. When considering the full set of reasons, a considerable difference between the number of   
+378 sufficient reasons and the number of minimal sufficient reasons can also be observed. Finally, like   
+379 minimal sufficient reasons, the number of contrastive explanations appears in many cases not very   
+380 large, which is a good point from an intelligibility perspective.
+
+# 81 6 Conclusion
+
+In light of our results, it turns out that the explanatory power of decision trees goes far beyond its ability to generate direct reasons. From a decision tree, the explanatory importance of features and the minimal sufficient reasons for an instance can be computed efficiently most of the time. For decision trees, fully addressing the “Why not?” question also appears as easier than fully addressing the “Why?” question: computing the full set of sufficient reasons for the instance at hand is typically out of reach, while computing its full set of contrastive explanations is tractable.
+
+388 Accordingly, the language of decision trees appears not only as appealing for the learning purpose,   
+389 but also as a good target when one needs to reason on the various forms of explanations (abductive   
+390 and contrastive ones) associated with the predictions made. This coheres with (and completes) the   
+391 results reported in [1], showing that many other explanation and verification tasks are tractable for   
+392 decision tree classifiers.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [No] One cannot expect any negative impact (the paper is about explaining predictions).   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] As a supplementary material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] But the results we obtained have been averaged over a number of trials.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] The pieces of software we used are furnished as a supplementary material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] This issue is irrelevant for this paper.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] The datasets we used are anonymized and do not contain personally identifiable information or offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] We did not use crowdsourcing or conducted research with human subjects.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] We did not use crowdsourcing or conducted research with human subjects.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] We did not use crowdsourcing or conducted research with human subjects.

parse/train/n-219jrTht/n-219jrTht_content_list.json

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+        "type": "text",
+        "text": "On the Explanatory Power of Decision Trees ",
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+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    },
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+        "text": "1 Decision trees have long been recognized as models of choice in sensitive applica  \n2 tions where interpretability is of paramount importance. In this paper, we examine   \n3 the computational ability of Boolean decision trees in deriving, minimizing, and   \n4 counting sufficient reasons and contrastive explanations. We prove that the set   \n5 of all sufficient reasons of minimal size for an instance given a decision tree can   \n6 be exponentially larger than the size of the input (the instance and the decision   \n7 tree). Therefore, generating the full set of sufficient reasons can be out of reach. In   \n8 addition, computing a single sufficient reason does not prove enough in general;   \n9 indeed, two sufficient reasons for the same instance may differ on many features.   \n10 To deal with this issue and generate synthetic views of the set of all sufficient   \n11 reasons, we introduce the notions of relevant features and of necessary features that   \n12 characterize the (possibly negated) features appearing in at least one or in every   \n13 sufficient reason, and we show that they can be computed in polynomial time. We   \n14 also introduce the notion of explanatory importance, that indicates how frequent   \n15 each (possibly negated) feature is in the set of all sufficient reasons. We show how   \n16 the explanatory importance of a feature and the number of sufficient reasons can be   \n17 obtained via a model counting operation, which turns out to be practical in many   \n18 cases. We also explain how to enumerate sufficient reasons of minimal size. We   \n19 finally show that, unlike sufficient reasons, the set of all contrastive explanations   \n20 for an instance given a decision tree can be derived, minimized and counted in   \n21 polynomial time. ",
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+    {
+        "type": "text",
+        "text": "22 1 Introduction ",
+        "text_level": 1,
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+            657
+        ],
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+    },
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+        "text": "23 In essence, explaining a decision to a person is to give the details or reasons that help a person   \n24 (the explainee) understand why the decision has been made. This is a significant issue especially   \n25 when decisions are made by Machine Learning (ML) models, such as random forests, Markov   \n26 networks, support vector machines, and deep neural networks. Actually, with the growing number   \n27 of applications that rely on ML techniques, researches on eXplainable AI (XAI) have become   \n28 increasingly important, by providing efficient methods for interpreting ML models, and explaining   \n29 their decisions (see for instance [10, 11, 12, 13, 16, 19, 22, 23, 24, 28, 30]).   \n30 When dealing with Boolean classifiers, which is what we do in this paper, two decisions are possible,   \n31 only: 1 for the instances classified as positive instances, and 0 for the remaining ones (the negative   \n32 instances). Whatever the way $_ { \\textbf { \\em x } }$ has been classified, an explainee may seek for explanations from   \n33 two distinct types [23]. On the one hand, abductive explanations for $_ { \\textbf { \\em x } }$ are intended to explain why $_ { \\textbf { \\em x } }$   \n34 has been classified in the way it has been classified by the ML model (thus, addressing the “Why?”   \n35 question). On the other hand, the purpose of contrastive (also known as counterfactual) explanations   \n36 for $_ { \\textbf { \\em x } }$ is to explain why $_ { \\textbf { \\em x } }$ has not been classified by the ML model as the explainee expected it (thus,   \n37 addressing the “Why not?” question). In both cases, explanations that are as simple as possible are   \n38 preferred (where simplicity is modeled as irredundancy, or even as size minimality).   \n39 Although there is no formal notion of interpretability [21], for classification problems, decision trees   \n40 [3, 26] are arguably among the most interpretable ML models. Because of their interpretability,   \n41 decision trees are often considered as target models for distilling a black-box model into a compre  \n42 hensible one [4, 10]. Furthermore, decision trees are often the components of choice for building   \n43 (less interpretable, but potentially more accurate) ensemble classifiers, such as random forests [2] and   \n44 gradient boosted decision trees [5].   \n45 The interpretability of decision trees is endowed with two key characteristics. On the one hand,   \n46 decision trees are transparent: each node in a decision tree has some meaning, and the principles used   \n47 for generating all nodes can be explained. On the other hand, decision trees are locally explainable:   \n48 by construction of a decision tree $T$ , any input instance $_ { \\textbf { \\em x } }$ is mapped to a unique root-to-leaf path   \n49 that yields to a decision label. The subset of (positive and negative) features $t _ { x } ^ { T }$ occurring in the   \n50 path used to find the right label 1 or 0 for $_ { \\textbf { \\em x } }$ in the decision tree $T$ can be viewed as a “direct reason”   \n51 for classifying $_ { \\textbf { \\em x } }$ as a positive instance or as a negative instance. $t _ { x } ^ { T }$ is an abductive explanation for   \n52 $_ { \\textbf { \\em x } }$ given $T$ , which explains why $_ { \\textbf { \\em x } }$ has been classified by $T$ as it has been classified. Indeed, every   \n53 instance $\\mathbf { x } ^ { \\prime }$ that coincides with $_ { \\textbf { \\em x } }$ on $t _ { x } ^ { T }$ is classified by $T$ in the same way as $_ { \\textbf { \\em x } }$ . However, such   \n54 “direct reasons” can contain arbitrarily many redundant features [17]. This motivates to take account   \n55 for other types of abductive explanations in the case of decision trees, namely, sufficient reasons [7]   \n56 (also known as prime implicant explanations [29]), that are irredundant abductive explanations, and   \n57 minimal sufficient reasons (i.e., those sufficient reasons of minimal size).   \n58 In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing   \n59 and counting sufficient reasons and contrastive explanations. We prove that the set of all sufficient   \n60 reasons of minimal size for an instance given a decision tree can be exponentially larger than the size   \n61 of the input. When this is the case, generating the full set of sufficient reasons (i.e., the complete   \n62 reason for the instance [7]) is typically out of reach. In addition, computing a single sufficient reason   \n63 does not prove enough in general; indeed; two sufficient reasons for the same instance may differ on   \n64 many features. To deal with this issue and generate synthetic views of the set of all sufficient reasons,   \n65 we introduce the notions of relevant features and of necessary features that characterize the (possibly   \n66 negated) features appearing in at least one or in every sufficient reason, and we show that they can be   \n67 computed in polynomial time. We also introduce the notion of explanatory importance, that indicates   \n68 how frequent each (possibly negated) feature is in the set of all sufficient reasons. Though deriving   \n69 the explanatory importance of a feature in the set of sufficient reasons and determining the cardinality   \n70 of this set are two computationally demanding tasks, we show how they can be achieved thanks to   \n71 model counting operation, which turns out to be practical in many cases. We also explain how to   \n72 enumerate sufficient reasons of minimal size, which is a way to count them when they are not too   \n73 numerous. We finally show that, from a computational standpoint, contrastive explanations highly   \n74 depart from sufficient reasons. Indeed, the set of all contrastive explanations for an instance given a   \n75 decision tree can be computed in polynomial time. As a consequence, such explanations can also be   \n76 minimized and counted in polynomial time.   \n77 The rest of the paper is organized as follows. Preliminaries about decision trees, abductive reasons,   \n78 and contrastive explanations are given in Section 2. The computation of all sufficient reasons is   \n79 considered in Section 3. Necessary and relevant features are presented in this section, as well as   \n80 the approach for assessing the explanatory importance of a feature and for counting the number of   \n81 sufficient reasons. We also explain there how minimal sufficient reasons can be enumerated. An   \n82 algorithm for computing all the contrastive explanations for the instance given the decision tree is   \n83 presented in Section 4. Experimental results are reported in Section 5. Finally, Section 6 concludes   \n84 the paper. All the proofs and additional empirical results are reported as a supplementary material. ",
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+        "text": "85 2 Decision Trees, Abductive and Contrastive Explanations ",
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+        "text": "86 For an integer $n$ , let $[ n ]$ be the set $\\{ 1 , \\cdots , n \\}$ . By ${ \\mathcal { F } } _ { n }$ we denote the class of all Boolean functions   \n87 from $\\{ 0 , 1 \\bar  \\} ^ { n }$ to $\\{ 0 , 1 \\}$ , and we use $X _ { n } = \\{ x _ { 1 } , \\cdot \\cdot \\cdot , x _ { n } \\}$ to denote the set of input Boolean variables,   \n88 corresponding to the features under consideration. Any assignment $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ is called an instance.   \n89 If $f ( { \\pmb x } ) = 1$ for some $f \\in \\mathcal { F } _ { n }$ , then $_ { \\textbf { \\em x } }$ is called a model of $f$ . $_ { \\textbf { \\em x } }$ is a positive instance when $f ( { \\pmb x } ) = 1$   \n90 and a negative instance when $f ( { \\pmb x } ) = 0$ .   \n91 We refer to $f$ as a propositional formula when it is described using the Boolean connectives $\\wedge$   \n92 (conjunction), $\\vee$ (disjunction) and $\\neg$ (negation), together with the Boolean constants 1 (true) and 0   \n93 (false). As usual, a literal $\\ell$ is a variable $x _ { i }$ (a positive literal) or its negation $\\neg x _ { i }$ , also denoted $\\overline { { x } } _ { i }$ (a   \n94 negative literal). A positive literal $x _ { i }$ is associated with a positive feature (i.e., $x _ { i }$ is set to 1), while a   \n95 negative literal $\\overline { { x } } _ { i }$ is associated with a negative feature (i.e., $x _ { i }$ is set to 0). A term (or monomial) $t$ is   \n96 a conjunction of literals, and a clause $c$ is a disjunction of literals. A DNF formula is a disjunction   \n97 of terms and a CNF formula is a conjunction of clauses. The set of variables occurring in a formula   \n98 $f$ is denoted $V a r ( f )$ . A formula $f$ is consistent if and only if it has a model. A CNF formula is   \n99 monotone whenever every occurrence of a literal in the formula has the same polarity (i.e., if a literal   \n100 occurs positively (resp. negatively) in the formula, then it does not have any negative (resp. positive)   \n101 occurrence in the formula). A formula $f _ { 1 }$ implies a formula $f _ { 2 }$ , noted $f _ { 1 } \\models f _ { 2 }$ , if and only if every   \n102 model of $f _ { 1 }$ is a model of $f _ { 2 }$ . Two formulae $f _ { 1 }$ and $f _ { 2 }$ are equivalent, noted $f _ { 1 } \\equiv f _ { 2 }$ whenever they   \n103 have the same models. The conditioning of a formula $f$ by a literal $\\ell$ , denoted $f \\mid \\ell$ , is the formula   \n104 obtained from $f$ by replacing each occurrence of $x _ { i }$ with 1 (resp. 0) and each occurrence of $\\overline { { x } } _ { i }$ with 0   \n105 (resp. 1) if $\\ell = x _ { i }$ (resp. $\\ell = \\overline { { x } } _ { i }$ ).   \n106 In what follows, we shall often treat assignments as terms, and terms and clauses as sets of literals.   \n107 Given an assignment $z \\in \\{ 0 , 1 \\} ^ { n }$ , the corresponding term is defined as ",
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+            "Figure 1: A decision tree $T$ for recognizing Cattleya orchids. The left (resp. right) child of any decision node labelled by $x _ { i }$ corresponds to the assignment of $x _ { i }$ to 0 (resp. 1). "
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+        "text": "$$\nt _ { z } = \\bigwedge _ { i = 1 } ^ { n } x _ { i } ^ { z _ { i } } { \\mathrm { ~ w h e r e ~ } } x _ { i } ^ { 0 } = { \\overline { { x } } } _ { i } { \\mathrm { ~ a n d ~ } } x _ { i } ^ { 1 } = x _ { i }\n$$",
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+        "text": "108 A term $t$ covers an assignment $_ z$ if $t \\subseteq t _ { z }$ . An implicant of a Boolean function $f$ is a term that implies   \n109 $f$ . A prime implicant of $f$ is an implicant $t$ of $f$ such that no proper subset of $t$ is an implicant of $f$ .   \n110 Dually, an implicate of a Boolean function $f$ is a clause that is implied by $f$ , and a prime implicate of   \n111 $f$ is an implicate $c$ of $f$ such that no proper subset of $c$ is an implicate of $f$ .   \n112 With these basic notions in hand, we shall focus on the following representation class of Boolean   \n113 functions:   \n114 Definition 1 (Decision Tree). A (Boolean) decision tree is a binary tree $T$ , each of whose internal   \n115 nodes is labeled with one of $n$ input Boolean variables, and whose leaves are labeled 0 or 1. Every   \n116 variable is assumed (without loss of generality) to appear at most once on any root-to-leaf path   \n117 (read-once property). The value $T ( \\pmb { x } ) \\in \\{ 0 , 1 \\}$ of $T$ on an input instance $_ { \\textbf { \\em x } }$ is given by the label of   \n118 the leaf reached from the root as follows: at each node, go to the left or right child depending on   \n119 whether the input value of the corresponding variable is 0 or 1, respectively. The size of $T$ , denoted   \n120 $| T |$ , is given by the number of its nodes.   \n121 The class of decision trees over $X _ { n }$ is denoted $\\mathbb { D } \\mathbb { T } _ { n }$ . It is well-known that any decision tree $T \\in \\mathsf { D T } _ { n }$   \n122 can be transformed in linear time into an equivalent disjunction of terms, denoted $\\tt D N F ( T )$ , where   \n123 each term corresponds to a path from the root to a leaf labeled with 1. Dually, $T$ can be transformed   \n124 in linear time into a conjunction of clauses, denoted $\\mathrm { C N F } ( T )$ , where each clause is the negation of the   \n125 term describing a path from the root to a leaf labeled with 0. ",
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+        "text": "126 For illustration, the following toy example will be used throughout the paper as a running example: ",
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+        "text": "127 Example 1. The decision tree in Figure 1 separates Cattleya orchids from other orchids using the   \n128 following features: $x _ { 1 }$ : “has fragrant flowers”, $x _ { 2 }$ : “has one or two leaves”, $x _ { 3 }$ : “has large flowers”,   \n129 and $x _ { 4 }$ : “is sympodial”.   \n30 As a salient characteristic, decision trees convey a single explicit abductive explanation for classifying   \n31 any input instance: ",
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+        "text": "Definition 2 (Direct Reason). Let 32 $T \\in \\mathsf { D T } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ . The direct reason for $_ { \\textbf { \\em x } }$ given $T$ is the term, denoted 33 $t _ { x } ^ { T }$ , corresponding to the unique root-to-leaf path of $T$ that is compatible with $_ { \\textbf { \\em x } }$ . ",
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+        "text": "34 Another important notion of abductive explanations is the following concept of sufficient reason[7],   \n35 that, unlike the notion of direct reason, is not specific to decision trees: ",
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+        "text": "Definition 3 (Sufficient Reason). Let 136 $f \\in \\mathcal { F } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ such that $f ( { \\pmb x } ) = 1$ (resp. $f ( { \\pmb x } ) = 0 ,$ ).   \n137 A sufficient reason for $_ { \\textbf { \\em x } }$ given $f$ is a prime implicant $t$ of $f$ (resp. $\\neg f$ ) that covers $_ { \\textbf { \\em x } }$ . $s r ( { \\pmb x } , f )$   \n138 denotes the set of sufficient reasons for $_ { \\textbf { \\em x } }$ given $f$ . ",
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+        "text": "139 Thus, a sufficient reason [7] (also known as prime implicant explanation [29]) for an instance $_ { \\textbf { \\em x } }$ given   \n140 a class described by a Boolean function $f$ is a subset $t$ of the characteristics of $_ { \\textbf { \\em x } }$ that is minimal w.r.t.   \n141 set inclusion such that any instance $\\mathbf { x } ^ { \\prime }$ sharing this set $t$ of characteristics is classified by $f$ as $_ { \\textbf { \\em x } }$ is.   \n142 Thus, when $f ( { \\pmb x } ) = 1$ , $t$ is a sufficient reason for $_ { \\textbf { \\em x } }$ given $f$ if and only if $t$ is a prime implicant of $f$   \n143 such that $_ { \\textbf { \\em x } }$ implies $t$ , and when $f ( { \\pmb x } ) = 0$ , $t$ is a sufficient reason for $_ { \\textbf { \\em x } }$ given $f$ if and only if $t$ is a   \n144 prime implicant of $\\neg f$ such that $t$ covers $_ { \\textbf { \\em x } }$ . Accordingly, sufficient reasons are suited to explain why   \n145 the instance at hand $_ { \\textbf { \\em x } }$ has been classified by $f$ as it has been classified. Unlike direct reasons [17],   \n146 sufficient reasons do not contain any redundant feature.   \n47 When considering the sufficient reasons of the input instance, one may be interested in focusing on   \n48 the shortest ones, alias the minimal sufficient reasons. Those reasons are valuable since conciseness   \n49 is often a desirable property of explanations (Occam’s razor). Formally: ",
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+        "text": "Definition 4 (Minimal Sufficient Reason). Let 50 $f \\in \\mathcal { F } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ . A minimal sufficient reason 51 for $_ { \\textbf { \\em x } }$ given $f$ is a sufficient reason for $_ { \\textbf { \\em x } }$ given $f$ that contains a minimal number of literals. ",
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+        "text": "152 Finally, unlike direct and (possibly minimal) sufficient reasons that aim to explain the classification   \n153 of the instance $_ { \\textbf { \\em x } }$ under consideration as achieved by the classifier $f$ , contrastive explanations are   \n154 valuable when $_ { \\textbf { \\em x } }$ has not been classified by $f$ as expected by the explainee. In this case, one looks for   \n155 minimal subsets of the features that when switched in $_ { \\textbf { \\em x } }$ are enough to get instances that are classified   \n156 positively (resp. negatively) by $f$ if $_ { \\textbf { \\em x } }$ is classified negatively (resp. positively) by $f$ . Formally, a   \n157 contrastive explanation for $_ { \\textbf { \\em x } }$ given $f$ [15] is a subset $t$ of the characteristics of $_ { \\textbf { \\em x } }$ that is minimal   \n158 w.r.t. set inclusion among those such that at least one instance $\\mathbf { x } ^ { \\prime }$ that coincides with $_ { \\textbf { \\em x } }$ except on the   \n159 characteristics from $t$ is not classified by $f$ as $_ { \\textbf { \\em x } }$ is. ",
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+        "text": "Definition 5 (Contrastive Explanation). Let $f \\in \\mathcal { F } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ such that $f ( { \\pmb x } ) = 1$ (resp. $f ( { \\pmb x } ) = 0 ,$ ). A contrastive explanation for $_ { \\textbf { \\em x } }$ given $f$ is a term t over $X _ { n }$ such that $t \\subseteq t _ { x }$ , $t _ { x } \\wedge t$ is not an implicant of $f$ (resp. $\\neg f )$ ), and for every $\\ell \\in t$ , $t \\setminus \\{ \\ell \\}$ does not satisfy this last condition. ",
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+        "text": "163 Example 2. Based on our running example, we can observe that $T ( \\pmb { x } ) = 1$ for the instance   \n164 $\\pmb { x } = ( 1 , 1 , 1 , 1 )$ . The direct reason for $_ { \\textbf { \\em x } }$ given $T$ is the term $t _ { x } ^ { T } = x _ { 1 } \\wedge x _ { 2 } \\wedge x _ { 3 } \\wedge x _ { 4 }$ . $x _ { 1 } \\wedge x _ { 4 }$ and   \n165 $x _ { 2 } \\wedge x _ { 3 } \\wedge x _ { 4 }$ are the sufficient reasons for $_ { \\textbf { \\em x } }$ given $T$ . $x _ { 1 } \\wedge x _ { 4 }$ is the unique minimal sufficient reason   \n166 for $_ { \\textbf { \\em x } }$ given $T$ . $x _ { 4 }$ , $x _ { 1 } \\wedge x _ { 2 }$ , and $x _ { 1 } \\wedge x _ { 3 }$ are the contrastive explanations for $_ { \\textbf { \\em x } }$ given $T$ . Thus, the   \n167 instance $( 1 , 1 , 1 , 0 )$ that differs with $_ { \\textbf { \\em x } }$ only on $x _ { 4 }$ is not classified by $T$ as $_ { \\textbf { \\em x } }$ is $( ( 1 , 1 , 1 , 0 )$ is classified   \n168 as a negative instance).   \n169 We mention in passing that when dealing with decision trees $T$ , we could have focused only on   \n170 explanations for the positive instances $_ { \\textbf { \\em x } }$ given $T$ . This comes from the fact that $\\mathbb { D } \\mathbb { T } _ { n }$ is closed under   \n171 negation, in the sense that for any $T \\in \\mathsf { D T } _ { n }$ , $\\neg T$ can be obtained by just replacing from $T$ the label   \n172 of each leaf with its complement. So, for any instance $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ , a direct reason (resp. sufficient   \n173 reason, minimal sufficient reason, contrastive explanation) explaining why $T ( { \\pmb x } ) = 0$ is precisely the   \n174 same as a direct reason (resp. sufficient reason, minimal sufficient reason, contrastive explanation)   \n175 explaining why $( \\neg T ) ( { \\pmb x } ) = { \\bar { 1 } }$ . Considering $T$ or its negation $\\neg T$ has no computational impact since   \n176 $\\neg T$ can be computed in time linear in the size of $T$ . ",
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+        "text": "177 3 Computing All Sufficient Reasons ",
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+        "text": "178 Sufficient reasons can be exponentially numerous. When switching from the direct reason for   \n179 an instance (that is unique but not always redundancy-free) to its sufficient reasons, a main obstacle   \n180 to be dealt with lies in the number of reasons to be considered. Indeed, even for the restricted class   \n181 of decision trees with logarithmic depth, an input instance can have exponentially many sufficient   \n182 reasons: ",
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+            "Figure 2: Two sufficient reasons for an mnist instance (top), and an explanatory heat map and the explanatory features for an mnist instance (bottom). "
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+    {
+        "type": "text",
+        "text": "Proposition 1. There is a decision tree 183 $T \\in \\mathsf { D T } _ { n }$ of depth $\\log _ { 2 } ( n + 1 )$ such that for any $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ , the number of sufficient reasons for 184 $_ { \\textbf { \\em x } }$ given $T$ is at least $\\lfloor { \\frac { 3 } { 2 } } ^ { \\frac { n + 1 } { 2 } } \\rfloor$ . ",
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+    {
+        "type": "text",
+        "text": "By definition, the minimal sufficient reasons for $_ { \\textbf { \\em x } }$ given $T$ cannot be more numerous than its sufficient reasons. However, focusing on minimal sufficient reasons does not solve the problem since an instance can also have exponentially many minimal sufficient reasons: ",
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+    {
+        "type": "text",
+        "text": "Propositisuch that 2. For econtains $n \\in \\mathbb N$ such that des and th $n$ is odd, there is e is an instance e s $T \\in \\mathsf { D T } _ { n }$ of depth the numb $\\textstyle { \\frac { n + 1 } { 2 } }$ $T$ $2 n + 1$ $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ minimal sufficient reasons for x given T is equal to 2 n−1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "191 In many practical cases, the number of sufficient reasons for an instance given a decision tree can   \n192 be very large. Figure 2 (top) shows an mnist instance (the leftmost subfigure) that has 482 185 073   \n193 664 sufficient reasons. Among them there are very dissimilar sufficient reasons. As an illustration,   \n194 the two rightmost subfigures present two sufficient reasons for this instance, and they differ on many   \n195 features (blue (resp. red) dots correspond to pixels on (resp. off)).   \n196 For such datasets, computing the set of all the sufficient reasons for a given instance is not always   \n197 feasible. Furthermore, if the computation succeeds but the number of sufficient reasons is huge, their   \n198 (disjunctively interpreted) set, alias the complete reason for the instance [7], can hardly be considered   \n199 as intelligible by the explainee. Finally, due to the number of sufficient reasons and their diversity,   \n200 deriving one of them is not informative enough. Thus, one needs to design approaches to synthesizing   \n201 their set while avoiding the two pitfalls (the computational one and the informational one). ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Synthesizing the set of sufficient reasons. In this objective, the following notions of necessary / (ir)relevant features appear useful. These notions of necessity and relevance echo the ones that have been considered in [9] for logic-based abduction. ",
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+    {
+        "type": "text",
+        "text": "Definition 6 (Explanatory Features). Let $f \\in \\mathcal { F } _ { n }$ , and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ be an instance. Let e be an explanation type. ",
+        "bbox": [
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+        "type": "text",
+        "text": "• A literal \\` over $X _ { n }$ is a necessary feature for the family e of explanations for $_ { \\textbf { \\em x } }$ given $f$ if and only if \\` belongs to every explanation t for $_ { \\textbf { \\em x } }$ given $f$ such that $t$ is of type e. $N e c _ { e } ( { \\pmb x } , f )$ denotes the set of all necessary features for the family e of explanations for $_ { \\textbf { \\em x } }$ given $f$ . ",
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+    {
+        "type": "text",
+        "text": "• A literal $\\ell$ over $X _ { n }$ is $a$ relevant feature for the family e of explanations for $_ { \\textbf { \\em x } }$ given $f$ if and only if \\` belongs to at least one explanation $t$ for $_ { \\textbf { \\em x } }$ given $f$ such that $t$ is of type e. $R e l _ { e } ( { \\pmb x } , f )$ denotes the set of all relevant features for the family e of explanations for $_ { \\textbf { \\em x } }$ given $f$ . $I r r _ { e } ( { \\pmb x } , f )$ , which is the complement of $R e l _ { e } ( { \\pmb x } , f )$ in the set of all literals over $X _ { n }$ , denotes the set of all irrelevant features for the family e of explanations for $_ { \\textbf { \\em x } }$ given $f$ . ",
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+        "type": "text",
+        "text": "15 The necessary (resp. irrelevant) features for the family $s$ of sufficient reasons for $_ { \\textbf { \\em x } }$ given $f$ are the   \n16 most (resp. less) important features for explaining the classification of $_ { \\textbf { \\em x } }$ by $f$ , since they belong to   \n17 every (resp. no) sufficient reason for $_ { \\textbf { \\em x } }$ given $f$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "When a single sufficient reason $t$ for $_ { \\textbf { \\em x } }$ given $f$ has been computed, the cardinality of $t$ deprived from the features of $N e c _ { s } ( { \\pmb x } , f )$ is small, and the cardinality of the symmetric difference between $t$ and $R e l _ { s } ( { \\pmb x } , f )$ is small as well, $t$ can be viewed as a good representative of the complete reason for $_ { \\textbf { \\em x } }$ given $f$ in the sense that a sufficient reason $t ^ { \\prime }$ for $_ { \\textbf { \\em x } }$ given $f$ that differs a lot from $t$ cannot exist. ",
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+    },
+    {
+        "type": "text",
+        "text": "In the case when $f$ is a decision tree $T$ , though the set of all sufficient reasons for $_ { \\textbf { \\em x } }$ given $T$ cannot be generated when it is too large, $N e c _ { s } ( { \\pmb x } , f )$ , $R e l _ { s } ( { \\pmb x } , f )$ , and $I r r _ { s } ( { \\pmb x } , f )$ can be derived efficiently: ",
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. Let $T \\in \\mathsf { D T } _ { n }$ , and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ . Computing $N e c _ { s } ( { \\pmb x } , T )$ , $R e l _ { s } ( { \\pmb x } , f )$ , and $I r r _ { s } ( { \\pmb x } , T )$ can be done in $\\mathcal { O } ( ( n + | T | ) \\times | T | )$ time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "226 Going a step further consists in evaluating the explanatory importance of every (positive or negative)   \n227 feature: ",
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+    {
+        "type": "text",
+        "text": "Definition 7 (Explanatory Importance). Let $f \\in { \\mathcal { F } } _ { n } ,$ , and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ be an instance. Let e be an explanation type, and $E _ { e } ( x , f )$ the set of all explanations for $_ { \\textbf { \\em x } }$ given $f$ that are of type e. The explanatory importance of a literal $\\ell$ over $X _ { n }$ for $_ { \\textbf { \\em x } }$ given $f$ w.r.t. e is given by ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f0409e8d45e6589ebe1bae5663879f60f156725ca2cb02269ba892aaddfe7e28.jpg",
+        "text": "$$\nI m p _ { e } ( \\ell , \\pmb { x } , f ) = \\frac { \\# ( \\{ t \\in E _ { e } ( \\pmb { x } , f ) : \\ell \\in t \\} ) } { \\# ( E _ { e } ( \\pmb { x } , f ) ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "228 Example 3. On the running example, we have $N e c _ { s } ( { \\pmb x } , T ) = \\{ x _ { 4 } \\}$ , and ${ R e l _ { s } } ( { \\pmb x } , T ) = \\{ { x } _ { 1 } , { x } _ { 2 } , { x } _ { 3 } .$ ,   \n229 $x _ { 4 } \\}$ . We also have $I m p _ { s } ( x _ { 4 } , { \\pmb x } , T ) = 1$ $, I m p _ { s } ( x _ { 1 } , { \\pmb x } , T ) = I m p _ { s } ( x _ { 2 } , { \\pmb x } , T ) = I m p _ { s } ( x _ { 3 } , { \\pmb x } , T ) = \\frac { 1 } { 2 }$ ,   \n230 and $I m p _ { s } ( \\ell , { \\pmb x } , T ) = 0$ for every other literal $\\ell$ (the negative ones over $\\{ x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \\} ,$ ).   \n231 The notion of explanatory importance must not be confused with the notions of feature importance   \n232 (which can be defined and assessed in many different ways): the former is local (i.e., relative to an   \n233 instance) and not global, it concerns literals and not variables (polarity matters), and it is about the   \n234 explanation task, not the prediction one.   \n235 In order to compute the explanatory importance of a literal, a straightforward approach consists in   \n236 enumerating the explanations of $E _ { e } ( { \\pmb x } , \\bar { f } )$ . This is feasible when this set is not too large, which is not   \n237 always the case for sufficient reasons even when $f$ is a decision tree $T$ . Thus, for dealing with the   \n238 remaining case, an alternative approach must be looked for.   \n239 We designed such an approach for computing $I m p _ { s } ( \\boldsymbol { \\ell } , \\pmb { x } , T )$ . We know that $s r ( \\pmb { x } , T )$ is by construc  \n240 tion the set of prime implicants of $g = \\{ c \\cap t _ { \\pmb { x } } : c \\in \\mathbb { C } \\mathbb { N } \\mathbb { F } ( T ) \\}$ . Thus, we exploited the translation   \n241 presented in [18] showing how to associate in polynomial time with a given CNF formula (here,   \n242 $g$ ) another formula (over a distinct set of variables), let us say $h$ , such that the models of $h$ are   \n243 in one-to-one correspondence with the prime implicants of $g$ . In our case, the translation can be   \n244 simplified because $g$ is a monotone CNF formula. Since $h$ is not primarily a CNF formula, leveraging   \n245 Tseitin transformation [31], we turned $h$ in linear time into a query-equivalent CNF formula $i$ . Note   \n246 that every auxiliary variable that is introduced in $i$ is defined from the other variables (those occurring   \n247 in $h$ ), so that the number of models of $i$ is the same as the number of models of $h$ . Finally, we took   \n248 advantage of the compilation-based model counter D4 [20] to compile $i$ into a d-DNNF circuit [6],   \n249 and this enabled us to compute in time polynomial in the size of $i$ both the number of sufficient   \n250 reasons and the explanatory importance of every literal (indeed, the d-DNNF language supports in   \n251 polytime the model counting query and the conditioning transformation [8]). We show in Section   \n252 5 that, despite a high complexity in the worst case (the size of $i$ can be exponential in $| T | )$ , this   \n253 approach based on knowledge compilation proves quite efficient in practice. ",
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+    {
+        "type": "text",
+        "text": "Clearly enough, when $I m p _ { e } ( \\ell , \\pmb { x } , T )$ has been computed for every $\\ell$ , one can easily generate explanatory heat maps. Figure 2 (bottom) shows an mnist instance (the leftmost subfigure) that has 19 115 685 sufficient reasons, 6 necessary literals, and 94 relevant literals. The central subfigure is the corresponding heat map. Blue (resp. red) pixels correspond to positive (resp. negative) literals in the instance, and the intensity of the color aims to reflect the explanatory importance of the corresponding literal. The rightmost subfigure gives the explanatory features (dark pixels are associated with necessary literals, and light pixels to relevant literals). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "261 Enumerating the minimal sufficient reasons. An approach to synthesizing the set of sufficient   \n262 reasons consists in focusing on the minimal ones. Indeed, though the set of minimal sufficient reasons   \n263 for an instance given a decision tree can be exponentially large, the number of minimal sufficient   \n264 reasons cannot exceed the number of sufficient reasons, and it can be significantly lower in practice. ",
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "However, unlike sufficient reasons that can be generated in polynomial time using a greedy algorithm (see e.g., [17]), computing minimal reasons is not an easy task: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. Let $T \\in \\mathsf { D T } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ . Computing a minimal sufficient reason for $_ { \\textbf { \\em x } }$ given $T$ is NP-hard. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "269 Despite this intractability result, minimal sufficient reasons can be generated in many practical cases.   \n270 A common approach for handling NP-optimization problems is to rely on modern constraint solvers.   \n271 One follows this direction here and casts the task of finding minimal sufficient reasons as a Boolean   \n272 constraint optimization problem. We first need to recall that a PARTIAL MAXSAT problem consists   \n273 of a pair $( C _ { \\mathrm { s o f t } } , C _ { \\mathrm { h a r d } } )$ where $C _ { \\mathrm { s o f t } }$ and $C _ { \\mathrm { h a r d } }$ are (finite) set of clauses. The goal is to find a Boolean   \n274 assignment that maximizes the number of clauses $c$ in $C _ { \\mathrm { s o f t } }$ that are satisfied, while satisfying all   \n275 clauses in $C _ { \\mathrm { h a r d } }$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 5. Let $T$ be a decision tree in $\\mathbb { D } \\mathbb { T } _ { n }$ and $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ be an instance such that $T ( \\pmb { x } ) = 1$ . Let $( C _ { \\mathrm { s o f t } } , C _ { \\mathrm { h a r d } } )$ be an instance of the PARTIAL MAXSAT problem such that: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The intersection of 276 $t _ { x }$ with $t _ { x ^ { * } }$ where $\\mathbf { \\nabla } _ { \\mathbf { \\mathcal { X } } } ^ { * }$ is an optimal solution of $( C _ { \\mathrm { h a r d } } , C _ { \\mathrm { s o f t } } )$ , is a minimal 277 sufficient reason for $_ { \\textbf { \\em x } }$ given $T$ . ",
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+    {
+        "type": "text",
+        "text": "78 Clearly enough, if $_ { \\textbf { \\em x } }$ is such that $T ( { \\pmb x } ) = 0$ , then it is enough to consider the same instance of   \n79 PARTIAL MAXSAT as above, except that $C _ { \\mathrm { h a r d } } = \\{ c \\cap t _ { x } : c \\in \\mathbb { C } \\mathbb { N } \\mathbb { F } ( \\neg T ) \\}$ .   \n80 Finally, one can take advantage of this PARTIAL MAXSAT characterization for generating a preset   \nnumber of minimal sufficient reasons (basically, one generates a first reason $t$ , then one adds to $C _ { \\mathrm { h a r d } }$   \n2 the negation of $t$ as a clause as well as a CNF encoding of a cardinality constraint for ensuring that the   \n83 next reasons to be generated have the same size as the one of $t$ , and we resume until the bound is   \n84 reached or no solution exists). ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "85 4 Computing All Contrastive Explanations ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, it has been shown that sufficient reasons and contrastive explanations are connected by a minimal hitting set duality [15]. This duality can be leveraged to derive one of the two sets of explanations from the other one using algorithms for computing minimal hitting sets [27, 32]. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "However, in the case of decision trees, a more direct and much more efficient approach to derive all the contrastive explanations for $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ given $T \\in \\mathbb { D T } _ { n }$ can be designed. Indeed, unlike what happens for sufficient reasons (see Section 3), the set of all contrastive explanations for $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ given a decision tree $T \\in \\mathsf { D T } _ { n }$ can be computed in polynomial time from $_ { \\textbf { \\em x } }$ and $T$ : ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 6. The set of all contrastive explanations for $\\pmb { x } \\in \\{ 0 , 1 \\} ^ { n }$ given a decision tree $T \\in \\mathsf { D T } _ { n }$ can be computed in time polynomial in $n + | T |$ as min( $\\{ c \\cap t _ { x } : c \\in \\mathrm { C N F } ( f ) \\} , \\subseteq )$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Example 4. On the running example, we have $\\mathsf { F } ( T ) = \\{ x _ { 1 } \\vee x _ { 2 } , x _ { 1 } \\vee \\overline { { x _ { 2 } } } \\vee x _ { 3 } , x _ { 1 } \\vee \\overline { { x _ { 2 } } } \\vee \\overline { { x _ { 3 } } } \\vee x _ { 4 } ,$ $\\overline { { x _ { 1 } } } \\vee x _ { 2 } \\vee x _ { 3 } \\vee x _ { 4 }$ , $\\overline { { x _ { 1 } } } \\vee x _ { 2 } \\vee \\overline { { x _ { 3 } } } \\vee x _ { 4 }$ , $\\overline { { x _ { 1 } } } \\vee \\overline { { x _ { 2 } } } \\vee x _ { 3 } \\vee x _ { 4 } , \\overline { { x _ { 1 } } } \\vee \\overline { { x _ { 2 } } } \\vee \\overline { { x _ { 3 } } } \\vee x _ { 4 } \\Big \\}$ . Thus, with $\\pmb { x } = ( 1 , 1 , 1 , 1 )$ , we have mi $n \\big ( \\{ c \\cap t _ { \\pmb { x } } : c \\in \\mathtt { C N F } ( f ) \\} , \\underline { { \\mathsf { C } } } \\big ) = \\{ x _ { 1 } \\vee x _ { 2 } , x _ { 1 } \\vee x _ { 3 } , x _ { 4 } \\}$ , which corresponds to the contrastive explanations $x _ { 1 } \\wedge x _ { 2 }$ , $x _ { 1 } \\wedge x _ { 3 }$ , $x _ { 4 }$ for $_ { \\textbf { \\em x } }$ given $T$ (viewing clauses and terms as sets of literals). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As straightforward consequences of Proposition 6, computing necessary $/$ relevant features and computing the explanatory importance of features w.r.t. contrastive explanations can be achieved in time polynomial in $n + | T |$ . Similarly, statistics about the size of contrastive explanations can be easily established, and contrastive explanations can be easily minimized and counted. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Empirical setting. We have considered 90 datasets, which are standard benchmarks from the wellknown repositories Kaggle (www.kaggle.com), OpenML (www.openml.org), and UCI (archive. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/3e3f66ba37184ccdfb0a68b3158d34832e5765b8bb4fa548059879ddccbbf79c.jpg",
+        "table_caption": [
+            "Table 1: Empirical results based on 12 datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Decision Tree</td><td colspan=\"2\">ISufficientl</td><td colspan=\"2\">IMinimall</td><td colspan=\"2\">#Nec.Features</td><td colspan=\"2\">#Rel. Features</td></tr><tr><td>Dataset</td><td>%A</td><td>#N</td><td>#B</td><td>med</td><td>max</td><td>med</td><td>max</td><td>med</td><td>max</td><td>med</td><td>max</td></tr><tr><td>recidivism</td><td>63.41</td><td>13828.80</td><td>147.60</td><td>14</td><td>22</td><td>13</td><td>22</td><td>6</td><td>19</td><td>60</td><td>98</td></tr><tr><td>adult</td><td>81.36</td><td>12934.00</td><td>2974.80</td><td>16</td><td>36</td><td>16</td><td>36</td><td>7</td><td>22</td><td>263</td><td>543</td></tr><tr><td>bank marketing</td><td>87.40</td><td>6656.40</td><td>1432.60</td><td>14</td><td>21</td><td>14</td><td>21</td><td>3</td><td>16</td><td>247</td><td>398</td></tr><tr><td>bank</td><td>88.99</td><td>5523.60</td><td>977.80</td><td>13</td><td>24</td><td>13</td><td>24</td><td>4</td><td>15</td><td>200</td><td>330</td></tr><tr><td>lending loan</td><td>73.49</td><td>2610.40</td><td>1131.40</td><td>16</td><td>31</td><td>16</td><td>31</td><td>8</td><td>25</td><td>226</td><td>442</td></tr><tr><td>contraceptive</td><td>50.44</td><td>1252.20</td><td>88.60</td><td>11</td><td>20</td><td>11</td><td>20</td><td>8</td><td>17</td><td>25</td><td>47</td></tr><tr><td>compas</td><td>65.98</td><td>1230.00</td><td>46.20</td><td>6</td><td>14</td><td>6</td><td>14</td><td>3</td><td>12</td><td>16</td><td>33</td></tr><tr><td>christine</td><td>63.36</td><td>853.20</td><td>426</td><td>12</td><td>47</td><td>12</td><td>47</td><td>8</td><td>41</td><td>92</td><td>202</td></tr><tr><td>farm-ads</td><td>86.75</td><td>544.80</td><td>264.60</td><td>20</td><td>99</td><td>20</td><td>99</td><td>16</td><td>92</td><td>73</td><td>192</td></tr><tr><td>mnist49</td><td>95.47</td><td>539.60</td><td>267.90</td><td>22</td><td>30</td><td>22</td><td>30</td><td>9</td><td>19</td><td>91</td><td>166</td></tr><tr><td>spambase</td><td>91.94</td><td>536.40</td><td>264.80</td><td>15</td><td>29</td><td>15</td><td>29</td><td>9</td><td>24</td><td>68</td><td>146</td></tr><tr><td>mnist38</td><td>96.07</td><td>506.60</td><td>251.40</td><td>19</td><td>28</td><td>19</td><td>28</td><td>8</td><td>20</td><td>93.50</td><td>157</td></tr><tr><td></td><td colspan=\"4\">#Sufficient</td><td colspan=\"2\"></td><td colspan=\"2\">#Contrastive</td><td colspan=\"2\">#Minimal</td></tr><tr><td>Dataset</td><td colspan=\"2\">med</td><td colspan=\"2\">max</td><td></td><td>med</td><td>max</td><td>med</td><td></td><td></td><td>max</td></tr><tr><td>recidivism</td><td colspan=\"2\">10387</td><td colspan=\"2\">9734080</td><td></td><td>54</td><td>145</td><td>3</td><td>16</td><td>2</td><td>144</td></tr><tr><td>adult</td><td colspan=\"2\"></td><td colspan=\"2\">≥ 1573835722607300000000000</td><td></td><td>201</td><td>470</td><td>4</td><td>16</td><td>3</td><td>256</td></tr><tr><td>bank marketing</td><td colspan=\"2\"></td><td colspan=\"2\">≥7460375213484350000000</td><td></td><td>189</td><td>337</td><td>4</td><td>13</td><td>8</td><td>432</td></tr><tr><td>bank</td><td colspan=\"2\"></td><td colspan=\"2\">≥7433951979018500000</td><td></td><td>150</td><td>277</td><td>4</td><td>13</td><td>4</td><td>168</td></tr><tr><td>lending loan</td><td colspan=\"2\">459258918095775</td><td colspan=\"2\">943243242816203000000000000000</td><td></td><td>157</td><td>311</td><td>3</td><td>12</td><td>3</td><td>192</td></tr><tr><td>contraceptive</td><td colspan=\"2\">20.50</td><td colspan=\"2\"></td><td>4272</td><td>21</td><td>52</td><td>2</td><td>11</td><td>2</td><td>48</td></tr><tr><td>compas</td><td colspan=\"2\">16</td><td colspan=\"2\"></td><td>444</td><td>13</td><td>33</td><td>2</td><td>11</td><td>2</td><td>21</td></tr><tr><td>christine</td><td colspan=\"2\">63108</td><td colspan=\"2\">2167735434744</td><td></td><td>71</td><td>151</td><td>3</td><td>8</td><td>2</td><td>4096</td></tr><tr><td>farm-ads</td><td colspan=\"2\">1177,50 7392384</td><td colspan=\"2\">921895392 715892613696000</td><td></td><td>59 61</td><td>166 106</td><td>2 2</td><td>10 12</td><td></td><td></td></tr><tr><td>mnist49 spambase</td><td colspan=\"2\">15712</td><td colspan=\"2\">2535069312</td><td></td><td>50</td><td>107</td><td>2</td><td>11</td><td>4</td><td>10000</td></tr><tr><td>mnist38</td><td colspan=\"2\">14849376</td><td colspan=\"2\">16922386736640</td><td></td><td>62</td><td>107</td><td>3</td><td>11</td><td></td><td>384</td></tr><tr><td></td><td colspan=\"2\"></td><td colspan=\"2\"></td><td></td><td></td><td></td><td></td><td></td><td>32</td><td>3072</td></tr></table>",
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+        "text": "6 ics.uci.edu/ml/). mnist38 and mnist49 are subsets of the mnist dataset, restricted to the 7 instances of 3 and 8 (resp. 4 and 9) digits. Because some datasets are suited to the multi-label 8 classification task, we used the standard “one versus all” policy to deal with them: all the classes but 09 the target one are considered as the complementary class of the target. Categorical features have been treated as arbitrary numbers (the scale is nominal). As to numeric features, no data preprocessing has taken place: these features have been binarized on-the-fly by the decision tree learning algorithm that 12 has been used. ",
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+        "text": "313 For every benchmark $b$ , a 10-fold cross validation process has been achieved. Namely, a set of 10   \n314 decision trees $T _ { b }$ have been computed and evaluated from the labelled instances of $b$ , partitioned into   \n315 10 parts. One part was used as the test set and the remaining 9 parts as the training set for generating   \n316 a decision tree. This tree is thus in 1-to-1 correspondence with the test set chosen within the whole   \n317 dataset $b$ . The classification performance for $b$ was measured as the mean accuracy obtained over the   \n318 10 decision trees generated from $b$ . The CART algorithm, and more specifically its implementation   \n319 provided by the Scikit-Learn library [25] has been used to learn decision trees. All hyper-parameters   \n320 of the learning algorithm have been set to their default value. Notably, decision trees have been   \n321 learned using the Gini criterion, and without any maximal depth or any other manual limitation. ",
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+        "text": "For each benchmark $b$ , each decision tree $T _ { b }$ , and a subset of at most 100 instances $_ { \\textbf { \\em x } }$ picked up at random in the test set following a uniform distribution, we computed a sufficient reason for $_ { \\textbf { \\em x } }$ given $T _ { b }$ (using the standard greedy algorithm run on the direct reason $t _ { x } ^ { T _ { b } }$ ), and a minimal sufficient reason for $_ { \\textbf { \\em x } }$ given $T _ { b }$ using the PARTIAL MAXSAT encoding presented in Proposition 5. This enabled us to draw some statistics (median, maximum) about the sizes of the reasons that have been generated. Using the algorithm presented in the proof of Proposition 3, we also derived the necessary and relevant explanatory features for each $_ { \\textbf { \\em x } }$ , and again drew some statistics about them. Exploiting the model counter D4, we computed the number of sufficient reasons for $_ { \\textbf { \\em x } }$ given $T _ { b }$ , as well as the explanatory importance of every feature. Taking advantage of the algorithm given in Proposition 4, we computed the number of contrastive explanations for $_ { \\textbf { \\em x } }$ given $T _ { b }$ , and drew some statistics about those numbers and about the sizes of the contrastive explanations. Finally, using the approach described in Section 3, we enumerated all the minimal sufficient reasons for $_ { \\textbf { \\em x } }$ given $T _ { b }$ up to a limit of $1 0 0 0 0$ , and again drew some statistics about the numbers of minimal sufficient reasons. Of course, for each computation, we measured the corresponding runtimes since this is fundamental to determine the extent to which the algorithms are practical (details are provided as a supplementary material). ",
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+        "text": "337 All the experiments have been conducted on a computer equipped with Intel(R) XEON E5-2637 CPU   \n338 $\\textcircled { \\omega } 3 . 5 \\ : \\mathrm { G H z }$ and $1 2 8 \\mathrm { G i B }$ of memory. D4 [20] was run with its default parameters. For computing   \n339 minimal reasons, we used the Pysat library [14], which provides the implementation of the RC2   \n340 PARTIAL MAXSAT solver. This solver was run using the parameters corresponding to the “Glucose”   \n341 setting. A time-out of 100s per instance was set for D4. ",
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+        "text": "Results. Table 1 (top and bottom) reports an excerpt of our results, focusing on 12 benchmarks out of 90 (the selected datasets are among those containing many instances and/or many features). The leftmost column gives the name of the dataset $b$ . Columns $\\% { \\dot { A } }$ , $\\% N$ , and $\\# B$ give, respectively, the mean accuracy over the 10 decision trees, the average number of nodes in those trees, and the average number of binary features they are based on. The next columns give statistics (median, maximum) about, respectively, the size of the sufficient reasons (|Sufficient|) and of the minimal sufficient reasons (|Minimal|) that have been computed, as well as about the number of necessary (#Nec. Features) and relevant $( \\# \\mathsf { R e l }$ . Features) features that appear in the full set of sufficient reasons for the instance. Table 1 (bottom) give statistics (median, maximum) about, respectively, the number of sufficient reasons (#Sufficient), the number of contrastive explanations (#Contrastive) and their sizes (|Contrastive|), and finally the number of minimal sufficient reasons ( $\\#$ Minimal). ",
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+        "text": "353 As to the computation times, it turns out that all the algorithms described in the previous sections   \n354 proved as efficient in practice. This is not surprising for those algorithms having a polytime worst-case   \n355 complexity (the greedy algorithm for computing a sufficient reason, the one for deriving explanatory   \n356 features, and the one for computing all the contrastive explanations). It was less obvious at first   \n357 sight for the algorithms used for counting the number of sufficient reasons and for computing the   \n358 explanatory importance of features. However, all the computations that have been run have terminated   \n359 in due time, except for 3 datasets out of 90, namely adult, bank_marketing, and bank. For these   \n360 datasets, the time limit of 100s has been reached for, respectively, 203, 150, and 336 instances out of   \n361 1000 (in this case, the median number of sufficient reasons has not been reported). Notably, for all   \n362 the 90 datasets but those 3, the median time required for counting the number of sufficient reasons   \n363 and computing the explanatory importance of features never exceeded 1s. Computing a minimal   \n364 sufficient reason, and more generally all such reasons looked challenging as well, due to both the   \n365 intrinsic complexity of computing a minimal sufficient reason and to their number. Nevertheless,   \n366 our enumeration algorithm succeeded in deriving all the minimal sufficient reasons for every dataset   \n367 except 3 out of 90, namely farm-ads, mnist49, and gisette. For these datasets, the limit of 10   \n368 000 reasons has been reached for, respectively, 5, 16, and 3 instances out of 1000. Interestingly,   \n369 the median time needed to derive all the minimal sufficient reasons for the instances for which the   \n370 computation has been successful exceeded 1s only for 2 datasets (adult and bank_marketing).   \n371 Beyond providing evidence that the number of reasons can be huge, our experiments have highlighted   \n372 that the greedy algorithm for deriving a sufficient reason computes in practice a minimal sufficient   \n373 reason in many cases. They have also shown that the number of explanatory relevant features for an   \n374 instance is typically much lower than the number of binary features used to describe it, and that the   \n375 number of explanatory necessary features is also significantly lower than the number of explanatory   \n376 relevant features. The gap between the two explains the possibly enormous number of sufficient   \n377 reasons. When considering the full set of reasons, a considerable difference between the number of   \n378 sufficient reasons and the number of minimal sufficient reasons can also be observed. Finally, like   \n379 minimal sufficient reasons, the number of contrastive explanations appears in many cases not very   \n380 large, which is a good point from an intelligibility perspective. ",
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+        "type": "text",
+        "text": "81 6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In light of our results, it turns out that the explanatory power of decision trees goes far beyond its ability to generate direct reasons. From a decision tree, the explanatory importance of features and the minimal sufficient reasons for an instance can be computed efficiently most of the time. For decision trees, fully addressing the “Why not?” question also appears as easier than fully addressing the “Why?” question: computing the full set of sufficient reasons for the instance at hand is typically out of reach, while computing its full set of contrastive explanations is tractable. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "388 Accordingly, the language of decision trees appears not only as appealing for the learning purpose,   \n389 but also as a good target when one needs to reason on the various forms of explanations (abductive   \n390 and contrastive ones) associated with the predictions made. This coheres with (and completes) the   \n391 results reported in [1], showing that many other explanation and verification tasks are tractable for   \n392 decision tree classifiers. ",
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+    {
+        "type": "text",
+        "text": "representations. In Proc. of KR’20, pages 838–849, 2020. [2] L. Breiman. Random forests. Machine Learning, 45(1):5–32, 2001.   \n[3] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Wadsworth, 1984.   \n[4] L. Breiman and N. Shang. Born again trees. Technical report, https://www.stat.berkeley.edu/ breiman/BAtrees.pdf, 1996.   \n[5] T. Chen and C. Guestrin. XGBoost: A scalable tree boosting system. In Proc. of KDD’16, page 785–794, 2016.   \n[6] A. Darwiche. Decomposable negation normal form. Journal of the Association for Computing Machinery, 48(4):608–647, 2001.   \n[7] A. Darwiche and A. Hirth. On the reasons behind decisions. In Proc. of ECAI’20, pages 712–720, 2020.   \n[8] A. Darwiche and P. Marquis. A knowledge compilation map. Journal of Artificial Intelligence Research, 17:229–264, 2002.   \n[9] Th. Eiter and G. Gottlob. The complexity of logic-based abduction. Journal of the Association for Computing Machinery, 42(1):3–42, 1995.   \n[10] N. Frosst and G. E. Hinton. Distilling a neural network into a soft decision tree. In Proc. of the First International Workshop on Comprehensibility and Explanation in AI and ML, volume 2071 of CEUR Workshop Proceedings. CEUR-WS.org, 2017.   \n[11] R. Guidotti, A. Monreale, S. Ruggieri, F. Turini, F. Giannotti, and D. Pedreschi. A survey of methods for explaining black box models. ACM Computing Surveys, 51(5):93:1–93:42, 2019.   \n[12] S. Hooker, D. Erhan, P-J. Kindermans, and B. Kim. A benchmark for interpretability methods in deep neural networks. In Proc. of NeurIPS’19, pages 9737–9748, 2019.   \n[13] J. Huysmans, K. Dejaeger, C. Mues, J. Vanthienen, and B. Baesens. An empirical evaluation of the comprehensibility of decision table, tree and rule based predictive models. Decis. Support Syst., 51(1):141–154, 2011.   \n[14] A. Ignatiev, A. Morgado, and J. Marques-Silva. PySAT: A Python toolkit for prototyping with SAT oracles. In Proc. of SAT’18, pages 428–437, 2018.   \n[15] A. Ignatiev, N. Narodytska, N. Asher, and J. Marques-Silva. On relating ’why?’ and ’why not?’ explanations. CoRR, abs/2012.11067, 2020.   \n[16] A. Ignatiev, N. Narodytska, and J. Marques-Silva. Abduction-based explanations for machine learning models. In Proc. of AAAI’19, pages 1511–1519, 2019.   \n[17] Y. Izza, A. Ignatiev, and J. Marques-Silva. On explaining decision trees. CoRR, abs/2010.11034, 2020.   \n[18] S. Jabbour, J. Marques-Silva, L. Sais, and Y. Salhi. Enumerating prime implicants of propositional formulae in conjunctive normal form. In Proc. of JELIA’14, pages 152–165, 2014.   \n[19] B. Kim, M. Wattenberg, J. Gilmer, C. Cai, J. Wexler, F. Viegas, and R. Sayres. Interpretability beyond feature attribution: Quantitative testing with concept activation vectors (TCAV). In Proc. of ICML’18, pages 2668–2677, 2018.   \n[20] J.-M. Lagniez and P. Marquis. An Improved Decision-DNNF Compiler. In Proc. of IJCAI’17, pages 667–673, 2017.   \n[21] Z. C. Lipton. The mythos of model interpretability. Communications of the ACM, 61(10):36–43, 2018.   \n[22] S. Lundberg and S-I. Lee. A unified approach to interpreting model predictions. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Proc. of NIPS’17, pages 4765–4774, 2017.   \n[23] T. Miller. Explanation in artificial intelligence: Insights from the social sciences. Artificial Intelligence, 267:1–38, 2019. ",
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+        "text": "443 [24] Ch. Molnar. Interpretable Machine Learning - A Guide for Making Black Box Models Explain  \n444 able. Leanpub, 2019.   \n445 [25] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel,   \n446 P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher,   \n447 M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine   \n448 Learning Research, 12:2825–2830, 2011.   \n449 [26] J. R. Quinlan. Induction of decision trees. Machine Learning, 1(1):81–106, 1986.   \n450 [27] R. Reiter. A theory of diagnosis from first principles. Artificial Intelligence, 32:57–95, 1987.   \n451 [28] M. Ribeiro, S. Singh, and C. Guestrin. “Why should I trust you?”: Explaining the predictions   \n452 of any classifier. In Proc. of KDD’16, pages 97–101, 2016.   \n453 [29] A. Shih, A. Choi, and A. Darwiche. A symbolic approach to explaining Bayesian network   \n454 classifiers. In Proc. of IJCAI’18, pages 5103–5111, 2018.   \n455 [30] A. Shih, A. Darwiche, and A. Choi. Verifying binarized neural networks by Angluin-style   \n456 learning. In Proc. of SAT’19, pages 354–370, 2019.   \n457 [31] G.S. Tseitin. On the complexity of derivation in propositional calculus, chapter Structures in   \n458 Constructive Mathematics and Mathematical Logic, pages 115–125. Steklov Mathematical   \n459 Institute, 1968.   \n460 [32] F. Wotawa. A variant of Reiter’s hitting-set algorithm. Inf. Process. Lett., 79(1):45–51, 2001. ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [No] One cannot expect any negative impact (the paper is about explaining predictions).   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] As a supplementary material. ",
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+        "text": "(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]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] But the results we obtained have been averaged over a number of trials.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] The pieces of software we used are furnished as a supplementary material.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] This issue is irrelevant for this paper.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] The datasets we used are anonymized and do not contain personally identifiable information or offensive content. ",
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+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] We did not use crowdsourcing or conducted research with human subjects.   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] We did not use crowdsourcing or conducted research with human subjects.   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] We did not use crowdsourcing or conducted research with human subjects. ",
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