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+# THE EFFICACY OF $L _ { 1 }$ REGULARIZATION IN NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+A crucial problem in neural networks is to select the most appropriate number of hidden neurons and obtain tight statistical risk bounds. In this work, we present a new perspective towards the bias-variance tradeoff in neural networks. As an alternative to selecting the number of neurons, we theoretically show that $L _ { 1 }$ regularization can control the generalization error and sparsify the input dimension. In particular, with an appropriate $L _ { 1 }$ regularization on the output layer, the network can produce a statistical risk that is near minimax optimal. Moreover, an appropriate $L _ { 1 }$ regularization on the input layer leads to a risk bound that does not involve the input data dimension. Our analysis is based on a new amalgamation of dimension-based and norm-based complexity analysis to bound the generalization error. A consequent observation from our results is that an excessively large number of neurons do not necessarily inflate generalization errors under a suitable regularization.
+
+# 1 INTRODUCTION
+
+Neural networks have been successfully applied in modeling nonlinear regression functions in various domains of applications. A critical evaluation metric for a predictive learning model is to measure its statistical risk bound. For example, the $L _ { 1 }$ or $L _ { 2 }$ risks of typical parametric models such as linear regressions are at the order of $( d / n ) ^ { 1 / 2 }$ for small $d$ (Seber & Lee, 2012), where $d$ and $n$ denote respectively the input dimension and number of observations. Obtaining the risk bound for a nonparametric regression model such as neural networks is highly nontrivial. It involves an approximation error (or bias) term as well as a generalization error (or variance) term. The standard analysis of generalization error bounds may not be sufficient to describe the overall predictive performance of a model class unless the data is assumed to be generated from it. For the model class of two-layer feedforward networks and a rather general data-generating process, Barron (1993; 1994) proved an approximation error bound of $O ( r ^ { - 1 / 2 } )$ where $r$ denotes the number of neurons. The author further developed a statistical risk error bound of $O ( ( d / n ) ^ { 1 / 4 } )$ , which is the tightest statistical risk bound for the class of two-layer neural networks up to the authors’ knowledge (for $d < n$ ). This risk bound is based on an optimal bias-variance tradeoff involving an deliberate choice of $r$ . Note that the risk is at a convergence rate much slower than the classical parametric rate. We will tackle the same problem from a different perspective, and obtain a much tighter risk bound.
+
+A practical challenge closely related to statistical risks is to select the most appropriate neural network architecture for a particular data domain (Ding et al., 2018). For two-layer neural networks, this is equivalent to selecting the number of hidden neurons $r$ . While a small $r$ tends to underfit, researchers have observed that the network is not overfitting even for moderately large $r$ . Nevertheless, recent research has also shown that an overly large $r$ (e.g., when $r > n$ ) does cause overfitting with high probability (Zhang et al., 2016). It can be shown under some non-degeneracy conditions that a two-layer neural network with more than $n$ hidden neurons can perfectly fit $n$ arbitrary data, even in the presence of noise, which inevitably leads to overfitting. A theoretical choice of $r$ suggested by the asymptotic analysis in (Barron, 1994) is at the order of $( n / d ) ^ { 1 / 2 }$ , and a practical choice of $r$ is often from cross-validation with an appropriate splitting ratio (Ding et al., 2018). An alternative perspective that we advocate is to learn from a single neural network with sufficiently many neurons and an appropriate $L _ { 1 }$ regularization on the neuron coefficients, instead of performing a selection from multiple candidate neural models. A potential benefit of this approach is easier hardware implementation and computation since we do not need to implement multiple models separately. Perhaps more importantly, this perspective of training enables much tighter risk bounds, as we will demonstrate. In this work, we focus on the model class of two-layer feedforward neural networks.
+
+Our main contributions are summarized below. First, we prove that $L _ { 1 }$ regularization on the coefficients of the output layer can produce a risk bound $O ( ( d / n ) ^ { 1 / 2 } )$ (up to a logarithmic factor) under the $L _ { 1 }$ training loss, which approaches the minimax optimal rate. Such a rate has not been established under the $L _ { 2 }$ training loss so far. The result indicates a potential benefit of using $L _ { 1 }$ regularization for training a neural network, instead of selecting from a number of neurons. Additionally, a key ingredient of our result is a unique amalgamation of dimension-based and norm-based risk analysis, which may be interesting on its own right. The technique leads to an interesting observation that an excessively large $r$ can reduce approximation error while not increasing generalization error under $L _ { 1 }$ regularizations. This implies that an explicit regularization can eliminate overfitting even when the specified number of neurons is enormous. Moreover, we prove that the $L _ { 1 }$ regularization on the input layer can induce sparsity by producing a risk bound that does not involve $d$ , where $d$ may be much larger compared with the true number of significant variables.
+
+Related work on neural network analysis. Despite the practical success of neural networks, a systematic understanding of their theoretical limit remains an ongoing challenge and has motivated research from various perspectives. Cybenko (1989) showed that any continuous function could be approximated arbitrarily well by a two-layer perceptron with sigmoid activation functions. Barron (1993; 1994) established an approximation error bound of using two-layer neural networks to fit arbitrary smooth functions and their statistical risk bounds. A dimension-free Rademacher complexity for deep ReLU neural networks was recently developed (Golowich et al., 2017; Barron & Klusowski, 2019). Based on a contraction lemma, a series of norm-based complexities and their corresponding generalization errors are developed (Neyshabur et al., 2015, and the references therein). Another perspective is to assume that the data are generated by a neural network and convert its parameter estimation into a tensor decomposition problem through the score function of the known or estimated input distribution (Anandkumar et al., 2014; Janzamin et al., 2015; Ge et al., 2017; Mondelli & Montanari, 2018). Also, tight error bounds have been established recently by assuming that neural networks of parsimonious structures generate the data. In this direction, Schmidt-Hieber (2017) proved that specific deep neural networks with few non-zero network parameters can achieve minimax rates of convergence. Bauer & Kohler (2019) developed an error bound that is free from the input dimension, by assuming a generalized hierarchical interaction model.
+
+Related work on $L _ { 1 }$ regularization. The use of $L _ { 1 }$ regularization has been widely studied in linear regression problems (Hastie et al., 2009, Chapter 3). The use of $L _ { 1 }$ regularization for training neural networks has been recently advocated in deep learning practice. A prominent use of $L _ { 1 }$ regularization was to empirically sparsify weight coefficients and thus compress a network that requires intensive memory usage (Cheng et al., 2017). The extension of $L _ { 1 }$ regularization to group$L _ { 1 }$ regularization (Yuan & Lin, 2006) has also been extensively used in learning various neural networks (Han et al., 2015; Zhao et al., 2015; Wen et al., 2016; Scardapane et al., 2017). Despite the above practice, the efficacy of $L _ { 1 }$ regularization in neural networks deserves more theoretical study. In the context of two-layer neural networks, we will show that the $L _ { 1 }$ regularizations in the output and input layers play two different roles: the former for reducing generalization error caused by excessive neurons while the latter for sparsifying input signals in the presence of substantial redundancy. Unlike previous theoretical work, we consider the $L _ { 1 }$ loss, which ranks among the most popular loss functions in, e.g., learning from ordinal data (Pedregosa et al., 2017) or imaging data (Zhao et al., 2016), and for which the statistical risk has not been studied previously. In practice, the use of $L _ { 1 }$ loss for training has been implemented in prevalent computational frameworks such as Tensorflow (Google, 2016), Pytorch (Ketkar, 2017), and Keras (Gulli & Pal, 2017).
+
+# 2 PROBLEM FORMULATION
+
+# 2.1 MODEL ASSUMPTION AND EVALUATION
+
+Suppose we have $n$ labeled observations $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 , \dots , n }$ , where $y _ { i }$ ’s are continuously-valued responses or labels. We assume that the underlying data generating model is $y _ { i } = f _ { * } ( \bar { x } _ { i } ) + \varepsilon _ { i }$ for some unknown function $f _ { * } ( \cdot )$ , where $x _ { i }$ ’ $\mathbf { \Phi } _ { \mathbf { \bar { \nu } } } \in \dot { \mathbb { X } } \subset \mathbb { R } ^ { d }$ are independent and identically distributed,
+
+![](images/3dd926fb4b401afb96027dbbfddca0613aaf3b232ec959dc44b5508fc45a8ed2.jpg)  
+Figure 1: A graph showing the two-layer neural network model considered in (2).
+
+and $\varepsilon _ { i }$ ’s are independent and identically distributed that is symmetric at zero and
+
+$$
+\mathbb { E } \left( \varepsilon _ { i } ^ { 2 } \mid x _ { i } \right) \leq \tau ^ { 2 } .
+$$
+
+Here, $\mathbb { X }$ is a bounded set that contains zero, for example $\{ x : \| x \| _ { \infty } \leq M \}$ for some constant $M$ . Our goal is learn a regression model ${ \hat { f } } _ { n } : x \mapsto { \hat { f } } _ { n } ( x )$ for prediction. The $\hat { f } _ { n }$ is obtained from the following form of neural networks
+
+$$
+\sum _ { j = 1 } ^ { r } a _ { j } \sigma ( w _ { j } ^ { \top } x + b _ { j } ) + a _ { 0 } ,
+$$
+
+where $a _ { 0 } , a _ { j } , b _ { j } \in \mathbb { R } , w _ { j } \in \mathbb { R } ^ { d }$ , $\textit { j } = \ 1 , \ldots , r$ , are parameters to estimate. We let $a =$   
+$[ a _ { 0 } , a _ { 1 } , \ldots , \bar { a } _ { r } ] ^ { \mathrm { { \scriptscriptstyle T } } }$ denote the output layer coefficients. An illustray accomplished by minimizing the empirical risk plus a regularization term. We first consider th The esti-, for somethe output $\textstyle n ^ { - 1 } \sum _ { i = 1 } ^ { \bar { n } } \ell ( y _ { i } , { \overline { { f ( x _ { i } ) } } } )$ $l ( \cdot )$ $L _ { 1 }$   
+layer. In particular, we search for such $f$ by the empirical risk minimization from the function class
+
+$$
+\mathcal { F } _ { V } = \left\{ f : \mathbb { R } ^ { d } \to \mathbb { R } \Big | f ( x ) = \sum _ { j = 1 } ^ { r } a _ { j } \sigma ( w _ { j } ^ { \top } x + b _ { j } ) + a _ { 0 } , \| a \| _ { 1 } \leq V \right\}
+$$
+
+where $V$ is a constant. The following statistical risk measures the predictive performance of a learned model $f$ :
+
+$$
+\begin{array} { r } { \mathcal { R } ( f ) \overset { \Delta } { = } \mathbb { E } \ell ( y , f ( x ) ) - \mathbb { E } \ell ( y , f _ { * } ( x ) ) . } \end{array}
+$$
+
+The loss function $\ell ( \cdot )$ is pre-determined by data analysts, usually the $L _ { 1 }$ loss defined by $\ell ( y , \tilde { y } ) =$ $| y - \tilde { y } |$ or the $L _ { 2 }$ loss defined by $\ell _ { 2 } ( y , \tilde { y } ) = ( y - \bar { y } ) ^ { 2 }$ . Under the $L _ { 1 }$ loss, the risk is $\textstyle { \mathcal { R } } ( f ) =$ $\mathbb { E } \left| f _ { * } ( x ) + \varepsilon - f ( x ) \right| - \mathbb { E } \left| \varepsilon \right|$ , which is nonnegative for symmetric random variables $\varepsilon$ . It is typical to use the same loss function for both training and evaluation.
+
+# 2.2 NOTATION
+
+Throughout the paper, we use $n , d , k , r$ to denote the number of observations, the number of input variables or input dimension, the number of significant input variables or sparsity level, the number of neurons (or hidden dimension), respectively. We write $a _ { n } \gtrsim b _ { n }$ , $b _ { n } \lesssim a _ { n }$ , or $b _ { n } = O ( a _ { n } )$ , if $\left| b _ { n } / a _ { n } \right| < c$ for some constant $c$ for all sufficiently large $n$ . We write $a _ { n } \asymp b _ { n }$ if $a _ { n } \gtrsim b _ { n }$ as well as $a _ { n } \lesssim b _ { n }$ . Let $\mathcal { N } ( \boldsymbol { \mu } , V )$ denote Gaussian distribution with mean $\pmb { \mu }$ and covariance $V$ . Let $\| \cdot \| _ { 1 }$ and $\| \cdot \| _ { 2 }$ denote the common $L _ { 1 }$ and $L _ { 2 }$ vector norms, respectively. Let $\mathbb { X }$ denote the essential support of $X$ . For any vector $z \in \mathbb { R } ^ { d }$ , we define $\begin{array} { r } { \| \boldsymbol { z } \| _ { \mathbb { X } } \triangleq \operatorname* { s u p } _ { \boldsymbol { x } \in \mathbb { X } } | \boldsymbol { x } ^ { \top } \boldsymbol { z } | } \end{array}$ , which may or may not be infinity. If $\mathbb { X } = \{ x : \| x \| _ { \infty } \leq M \}$ , $\| \boldsymbol { z } \| _ { \mathbb { X } }$ is equivalent to $M \| z \| _ { 1 }$ . Throughout the paper, $\hat { f } _ { n }$ denotes the estimated regression function with $n$ being the number of observations.
+
+# 2.3 ASSUMPTIONS AND CLASSICAL RESULTS
+
+We introduce some technical assumptions necessary for our analysis, and state-of-the-art statistical risk bounds built through dimension-based complexity analysis.
+
+Assumption 1. The activation function $\sigma ( \cdot )$ is a bounded function on the real line satisfying $\sigma ( x ) $ 1 as $x \to \infty$ and $\sigma ( x )  0$ as $x \to - \infty$ , and it is $L$ -Lipschitz for some constant $L$ .
+
+Assumption 2. The regularization constant $V$ is larger than $2 C + f _ { * } ( 0 )$ , where $C$ is any constant such that the Fourier transform of $f _ { * }$ , denoted by $F$ , satisfies
+
+$$
+\int _ { \mathbb { R } ^ { d } } \| \omega \| _ { \mathbb { X } } F ( d \omega ) \leq C .
+$$
+
+Assumption 3. $\sigma ( x )$ approaches its limits at least polynomially fast, meaning that $| \sigma ( x ) - \mathbf { 1 } \{ x >$ $0 \} | < \varepsilon$ for all $| x | > x _ { \varepsilon }$ where $x _ { \varepsilon }$ is a polynomial of $1 / \varepsilon$ . Also, the value of $\eta \triangleq \operatorname* { s u p } _ { j } \| w _ { j } \| _ { \mathbb { X } }$ scales with $n$ polynomially meaning that $\log \eta = O ( \log n )$ as $n \to \infty$ .
+
+Assumption 4. There exists a constant $c > 0$ and a bounded subset ${ \mathcal { S } } \subset \mathbb { R }$ such that $\mathbb { P } ( X \in { \mathcal { S } } ) > c$ and $\mathrm { i n f } _ { x \in S } \sigma ^ { \prime } ( x ) > c$ for $X \sim \mathcal { N } ( 0 , 1 )$ .
+
+We explain each assumption below. The above notation of $C , V$ follow those in (Barron, 1993; 1994). Assumption 1 specifies the class of the activation functions we consider. A specific case is the popular activation function $\sigma ( x ) = 1 / \{ 1 { + } \exp ( - x ) \}$ . Assumption 2, first introduced in (Barron, 1993), specifies the smoothness condition for $f _ { * }$ to ensure the approximation property of neural networks (see Theorem 2.1). In Assumption 3, the condition for $w$ is for technical convenience. It could also be replaced with the following alternative condition: There exists a constant $c > 0$ such that the distribution of $x$ satisfies
+
+$$
+\operatorname* { s u p } _ { w : \| w \| _ { 2 } = 1 } \mathbb { P } \big ( \log ( | w ^ { \top } x | ) < c \log \varepsilon \big ) < \varepsilon
+$$
+
+for any $\varepsilon \in ( 0 , 1 )$ . Simply speaking, the input data $x$ is not too small with high probability. This condition is rather mild. For example, it holds when each component of $x$ has a a bounded density function. This alternative condition ensures that for some small constant $\varepsilon > 0$ and any $w \in \mathbb { R } ^ { \tilde { d } }$ , there exists a surrogate of $w$ $, \hat { w } \in \mathbb R ^ { d }$ with $\log \| \hat { w } \| _ { 2 } = O ( - \log \varepsilon )$ , such that
+
+$$
+\mathbb { P } ( | \sigma ( w ^ { \top } x ) - \sigma ( \hat { w } ^ { \top } x ) | > \varepsilon ) < \varepsilon .
+$$
+
+And this can be used to surrogate the assumption of $w$ in Assumption 3 throughout the proofs in the appendix. Assumption 4 means that $\sigma ( \cdot )$ is not a nearly-constant function. This condition is only used to bound the minimax lower bound in Theorem 3.2.
+
+Theorem 2.1 (Approximation error bound (Barron, 1993)). Suppose that Assumptions 1, 2, 3 hold. We have
+
+$$
+\operatorname* { i n f } _ { f \in \mathcal { F } _ { V } } \left\{ \int _ { \mathbb { X } } ( f ( x ) - f _ { * } ( x ) ) ^ { 2 } \mu ( d x ) \right\} ^ { 1 / 2 } \leq 2 C \left( \frac { 1 } { \sqrt { r } } + \delta _ { \eta } \right) ,
+$$
+
+where $\mu$ denotes a probability measure on $\mathbb { X } ,$
+
+$$
+\delta _ { \eta } = \operatorname* { i n f } _ { 0 < \varepsilon < 1 / 2 } \biggl \{ 2 \varepsilon + \operatorname* { s u p } _ { | x | > \varepsilon } \bigl | \sigma ( \eta x ) - \mathbf { 1 } \bigl \{ x > 0 \bigr \} \bigr | \biggr \} ,
+$$
+
+$\eta$ is defined in Assumption 3, and $C$ is defined in (4).
+
+Theorem 2.2 (Statistical risk bound (Barron, 1994)). Suppose that Assumptions $^ { l }$ , 2, 3 hold. Then the $L _ { 2 }$ estimator ${ \hat { f } } _ { n }$ in $\mathcal { F } _ { V }$ satisfies $\mathbb { E } \left\{ \hat { f } _ { n } ( x ) - f _ { * } ( x ) \right\} ^ { 2 } \lesssim V ^ { 2 } / r + ( r d \log n ) / n$ . In particular, $i f$ we choose $r \asymp V { \sqrt { n / ( d \log n ) } }$ , then ${ \mathbb E } \left\{ \hat { f } _ { n } ( x ) - f _ { * } ( x ) \right\} ^ { 2 } \lesssim V \sqrt { ( d \log n ) / n }$ .
+
+It is known that a typical parametric rate under the $L _ { 2 }$ loss is at the order of $O ( d / n )$ , much faster than the above result. This gap is mainly due to excessive model complexity in bounding generalization errors. We will show in Section 3 that the gap in the rate of convergence can be filled when using $L _ { 1 }$ loss. Our technique will be based on the machinery of Rademacher complexity, and we bound this complexity through a joint analysis of the norm of coefficients (‘norm-based’) as well as dimension of parameters (‘dimension-based’).
+
+# 2.4 MODEL COMPLEXITY AND GENERALIZATION ERROR
+
+The statistical risk consists of two parts. The first part is an approximation error term non-increasing in the number of neurons $r$ , and the second part describes generalization errors. The key issue for risk analysis is to bound the second term using a suitable model complexity and then tradeoff with the first term. We will develop our theory based on the following measure of complexity.
+
+Let $\mathcal { F }$ denote a class of functions each mapping from $\mathbb { X }$ to $\mathbb { R }$ , and $x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \in \mathbb { X }$ . Following a similar terminology as in (Neyshabur et al., 2015), the Rademacher complexity, or simply ‘complexity’, of a function class $\mathcal { F }$ is defined by $\begin{array} { r } { \mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } } | n ^ { - 1 } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( x _ { i } ) | } \end{array}$ , where $\xi _ { i } , i = 1 , 2 , \ldots , n$ are independent symmetric Bernoulli random variables.
+
+Lemma 2.3 (Rademacher complexity of $\mathcal { F } _ { V }$ ). Suppose that Assumptions $^ { l }$ , 3 hold. Then for the Rademacher complexity of $\mathcal { F } _ { V }$ , we have
+
+$$
+\mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } _ { V } } \left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( x _ { i } ) \right| \lesssim \frac { V \sqrt { d \log n } } { \sqrt { n } } .
+$$
+
+The proof is included in Appendix A.1. The bound in (6) is derived from an amalgamation of dimension-based and norm-based analysis elaborated in the appendix. It is somewhat surprising that the bound does not explicitly involve the approximation error part (that depends on $r$ and $\eta$ ). This Rademacher complexity bound enables us to derive tight statistical risk bounds in the following section.
+
+# 3 MAIN RESULTS
+
+3.1 STATISTICAL RISK BOUND FOR THE $L _ { 1 }$ REGULARIZED NETWORKS IN (3)
+
+Theorem 3.1 (Statistical risk bound). Suppose that Assumptions 1, 2, 3 hold. Then the constrained $L _ { 1 }$ estimator $\hat { f } _ { n }$ over $\mathcal { F } _ { V }$ satisfies
+
+$$
+\mathcal { R } ( \hat { f } _ { n } ) \lesssim \bigg ( \frac { 1 } { \sqrt { r } } + \delta _ { \eta } \bigg ) C + \frac { V \sqrt { d \log n } + \tau } { \sqrt { n } } ,
+$$
+
+where $\delta _ { \eta }$ is defined in (5), and $\tau$ was introduced in $( l )$ . Moreover, choosing the parameters $r , \eta$ large enough, we have
+
+$$
+\mathcal { R } ( \hat { f } _ { n } ) \lesssim \frac { V \sqrt { d \log n } + \tau } { \sqrt { n } } .
+$$
+
+The proof is in Appendix A.2. We briefly explain our main idea in deriving the risk bound (7). A standard statistical risk bound contains two parts which correspond to the approximation error and generalization error, respectively. The approximation error part in (7) is the first term, which involves the hidden dimension $r$ and the norm of input coefficients through $\eta$ . This observation motivates us to use the norm of output-layer coefficients through $V$ and the input dimension $d$ to derive a generalization error bound. In this way, the generalization error term does not involve $r$ already used for bounding the approximation error, and thus a bias-variance tradeoff through $r$ is avoided. This thought leads to the generalization error part in (7), which is the second term involving $V$ and $d$ . Its proof combines the machinery of both dimension-based and norm-based complexity analysis. From our analysis, the error bound in Theorem 3.1 is a consequence of the $L _ { 1 }$ loss function and the employed $L _ { 1 }$ regularization. In comparison with the previous result of Theorem 2.2, the bound obtained in Theorem 3.1 is tight and it approaches the parametric rate $\sqrt { d / n }$ for the $d < n$ regime. Though we can only prove for $L _ { 1 }$ loss in this work, we conjecture that the same rate is achieved using $L _ { 2 }$ loss.
+
+In the following, we further show that the above risk bound is minimax optimal. The minimax optimality indicates that deep neural networks with more than two layers will not perform much better than shallow neural networks when the underlying regression function belongs to $\mathcal { F } _ { V }$ .
+
+Theorem 3.2 (Minimax risk bound). Suppose that Assumptions 1 and 4 hold, and $x _ { 1 } , x _ { 2 } , \dotsc , x _ { n } \stackrel { i : } { \sim }$ id∼ $\mathcal { N } ( 0 , \pmb { I } _ { d } )$ , then $\operatorname { i n f } _ { \hat { f } _ { n } }$ $\begin{array} { r } { \operatorname* { s u p } _ { f \in \mathcal { F } _ { V } } \mathcal { R } ( \hat { f } _ { n } ( x ) ) \gtrsim V \sqrt { d / n } } \end{array}$ .
+
+Here the $\mathcal { F } _ { V }$ is the same one as defined in (3). All the smooth functions $f _ { * } ( \cdot )$ that satisfy $V >$ $2 C + f _ { * } ( 0 )$ and (4) belong to $\mathcal { F } _ { V }$ according to Theorem 2.1. The proof is included in Appendix A.3.
+
+# 3.2 ADAPTIVENESS TO THE INPUT SPARSITY
+
+It is common to input a large dimensional signal to a neural network, while only few components are genuinely significant for prediction. For example, in environmental science, high dimensional weather signals are input for prediction while few are physically related (Shi et al., 2015). In image processing, the image label is relevant to few background pixels (Han et al., 2015). In natural language processing, a large number of redundant sentences sourced from Wikipedia articles are input for language prediction (Diao et al., 2019). The practice motivates our next results to provide a tight risk bound for neural networks whose input signals are highly sparse.
+
+Assumption 5. There exists a positive integer $k \leq d$ and an index set $S \subset \{ 1 , \ldots , d \}$ with card $( S ) =$ $k$ , such that $f _ { \ast } ( x ) = g _ { \ast } ( x _ { S } )$ for some function $g _ { * } ( \cdot )$ with probability one.
+
+The subset $S$ is generally unknown to data analysts. Nevertheless, if we know $k$ , named the sparsity level, the risk bound could be further improved by a suitable regularization on the input coefficients. We have the following result where $d$ is replaced with $k$ in the risk bound of Theorem 3.1.
+
+Proposition 3.3. Suppose that that Assumptions 1, 2, 3, 5 hold. Suppose that $\hat { f } _ { n }$ is the $L _ { 1 }$ estimator over the following function class
+
+$$
+\bigg \{ f : \mathbb { R } ^ { d } \to \mathbb { R } \Big | f ( x ) = \sum _ { j = 1 } ^ { r } a _ { j } \sigma ( w _ { j } ^ { \top } x + b _ { j } ) + a _ { 0 } , \| a \| _ { 1 } \leq V , \operatorname* { s u p } _ { j } \| w _ { j } \| _ { 0 } \leq k \bigg \} .
+$$
+
+Then $\mathcal { R } ( \hat { f } _ { n } ) \lesssim \sqrt { \{ k \log ( d n ) \} / n }$ .
+
+The proof is included in Appendix A.4. The above statistical risk bound is also minimax optimal according to a similar argument in Theorem 3.2. From a practical point of view, the above $L _ { 0 }$ constraint is usually difficult to implement, especially for a large input dimension $d$ . Alternatively, one may impose an $L _ { 1 }$ constraint instead of an $L _ { 0 }$ constraint on the input coefficients. Our next result is concerned with the risk bound when the model is learned from a joint regularization on the output and input layers. For technical convenience, we will assume that $\mathbb { X }$ is a bounded set.
+
+Theorem 3.4. Consider the following function class of two-layer neural networks
+
+$$
+\mathcal { F } _ { V , \eta } = \Bigg \{ f : \mathbb { R } ^ { d }  \mathbb { R } \Big | f ( x ) = \sum _ { j = 1 } ^ { r } a _ { j } \sigma ( w _ { j } ^ { \top } x + b _ { j } ) + a _ { 0 } , \| a \| _ { 1 } \leq V , \operatorname* { s u p } _ { 1 \leq j \leq r } ( \| w _ { j } \| _ { 1 } + | b _ { j } | ) \leq \eta \Bigg \} .
+$$
+
+Suppose that $V \gtrsim C$ , where $C$ is defined in (4). Then the constrained $L _ { 1 }$ estimator $\hat { f } _ { n }$ over $\mathcal { F } _ { V , \eta }$ satisfies
+
+$$
+\mathcal { R } ( \hat { f } _ { n } ) \lesssim C \left( \frac { 1 } { \sqrt { r } } + \delta _ { \eta } \right) + \frac { V \eta + \tau } { \sqrt { n } } ,
+$$
+
+where $\delta _ { \eta }$ is defined in (5). In particular, choosing $r$ large enough, we have
+
+$$
+\mathcal { R } ( \hat { f } _ { n } ) \lesssim C \delta _ { \eta } + \frac { V \eta + \tau } { \sqrt { n } }
+$$
+
+which does not involve the input dimension $d$ and the number of hidden neurons $r$ . Moreover,
+
+$$
+\sigma ( x ) = 1 / ( 1 + e ^ { - x } ) , \quad \eta \asymp \biggl ( n \log ^ { 2 } n \biggr ) ^ { 1 / 3 } , t h e n \mathscr { R } ( \hat { f } _ { n } ) \lesssim V \{ ( \log n ) / n \} ^ { 1 / 3 } .
+$$
+
+The proof is included in Appendix A.5. In the above result, the risk bound is at the order of $O ( n ^ { - 1 / 3 } )$ , which is slower than the $O ( n ^ { - 1 / 2 } )$ in the previous Theorem 3.1 and Proposition 3.3 if ignoring $d$ and logarithmic factors of $n$ . However, for a large input dimension $d$ that is even much larger than $n$ , the bound can be much tighter than the previous bounds since it is dimension-free.
+
+# 4 CONCLUSION AND FURTHER REMARKS
+
+We studied the tradeoff between model complexity and statistical risk in two-layer neural networks from the explicit regularization perspective. We end our paper with two future problems. First, in Theorem 3.4, For a small $d$ , the order of $n ^ { - 1 / 3 }$ seems to be an artifact resulting from our technical arguments. We conjecture that in the small $d$ regime, this risk bound could be improved to $O ( n ^ { - 1 / 2 } )$ by certain adaptive regularizations. Second, it would be interesting to emulate the current approach to yield similarly tight risk bounds for deep forward neural networks.
+
+REFERENCES   
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+Andrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory, 39(3):930–945, 1993.   
+Andrew R Barron. Approximation and estimation bounds for artificial neural networks. Machine learning, 14(1):115–133, 1994.   
+Andrew R Barron and Jason M Klusowski. Complexity, statistical risk, and metric entropy of deep nets using total path variation. arXiv preprint arXiv:1902.00800, 2019.   
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+Yu Cheng, Duo Wang, Pan Zhou, and Tao Zhang. A survey of model compression and acceleration for deep neural networks. arXiv preprint arXiv:1710.09282, 2017.   
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+Enmao Diao, Jie Ding, and Vahid Tarokh. Restricted recurrent neural networks. 2019 IEEE Conf. on Big Data, 2019.   
+Jie Ding, Vahid Tarokh, and Yuhong Yang. Model selection techniques: An overview. IEEE Signal Process. Mag., 35(6):16–34, 2018.   
+Rong Ge, Jason D Lee, and Tengyu Ma. Learning one-hidden-layer neural networks with landscape design. arXiv preprint arXiv:1711.00501, 2017.   
+Noah Golowich, Alexander Rakhlin, and Ohad Shamir. Size-independent sample complexity of neural networks. arXiv preprint arXiv:1712.06541, 2017.   
+Research Team Google. Tensorflow: A system for large-scale machine learning. Proc. 12th Symp. Operating Syst. Des. Implementation, pp. 265–283, 2016.   
+Antonio Gulli and Sujit Pal. Deep Learning with Keras. Packt Publishing Ltd, 2017.   
+Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. Advance. Neural Inf. Process. Sys., pp. 1135–1143, 2015.   
+Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements of statistical learning: data mining, inference, and prediction. Springer Science & Business Media, 2009.   
+Majid Janzamin, Hanie Sedghi, and Anima Anandkumar. Beating the perils of non-convexity: Guaranteed training of neural networks using tensor methods. arXiv preprint arXiv:1506.08473, 2015.   
+Nikhil Ketkar. Introduction to pytorch. Deep learning with python, pp. 195–208, 2017.   
+Marco Mondelli and Andrea Montanari. On the connection between learning two-layers neural networks and tensor decomposition. arXiv preprint arXiv:1802.07301, 2018.   
+Behnam Neyshabur, Ryota Tomioka, and Nathan Srebro. Norm-based capacity control in neural networks. Conf. Learning Theory, pp. 1376–1401, 2015.   
+Fabian Pedregosa, Francis Bach, and Alexandre Gramfort. On the consistency of ordinal regression methods. J. Mach. Learn. Res., 18(1):1769–1803, 2017.   
+Simone Scardapane, Danilo Comminiello, Amir Hussain, and Aurelio Uncini. Group sparse regularization for deep neural networks. Neurocomputing, 241:81–89, 2017.   
+Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. arXiv preprint arXiv:1708.06633, 2017.   
+George AF Seber and Alan J Lee. Linear regression analysis, volume 329. John Wiley & Sons, 2012.   
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+Lei Zhao, Qinghua Hu, and Wenwu Wang. Heterogeneous feature selection with multi-modal deep neural networks and sparse group LASSO. IEEE Trans. Multimed., 17(11):1936–1948, 2015.
+
+# A APPENDIX
+
+# A.1 PROOF OF LEMMA 2.3
+
+We first prove (6), which uses an amalgamation of dimension-based and norm-based analysis. For the output layer, we use the following norm-based analysis
+
+$$
+\begin{array} { r l } & { \mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \left| \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } f ( z _ { i } ) \right| = \mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \left| \langle a , \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( W ^ { \top } z _ { i } + b ) \rangle \right| } \\ & { \leq \operatorname* { s u p } \left\| a \right\| _ { 1 } \mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \left\| \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( W ^ { \top } z _ { i } + b ) \right\| _ { \infty } \leq V \mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \underset { j } { \operatorname* { m a x } } \bigg | \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w _ { j } ^ { \top } z _ { i } + b _ { j } ) \bigg | } \\ & { \leq V \mathbb { E } \underset { w \in \mathbb { R } ^ { d } } { \operatorname* { s u p } } \bigg | \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w ^ { \top } z _ { i } + b ) \bigg | . } \end{array}
+$$
+
+For notational convenience, we define $w _ { 0 } = 0 , b _ { 0 } = 0$ , and $a _ { 0 } = \sigma ( 0 ) ^ { - 1 } a _ { 0 } \sigma ( w _ { 0 } ^ { \top } z + b _ { 0 } )$ so that $a _ { 0 }$ can be treated in a similar manner as other $a _ { i }$ ’s. Without loss of generality, we do not separately consider $a _ { 0 }$ in the following proofs.
+
+Next, we prove that
+
+$$
+\mathbb { E } \operatorname* { s u p } _ { w \in \mathbb { R } ^ { d } } \Big | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w ^ { \top } z _ { i } + b ) \Big | \lesssim \sqrt { \frac { d \log n } { n } } ,
+$$
+
+and thus conclude the proof. The proof will be based on an $\varepsilon$ -net argument together with the union bound. For any $\varepsilon$ , let $\bar { W } _ { \varepsilon } \subset \mathbb { R } ^ { d }$ denote the subset
+
+$$
+W _ { \varepsilon } = \left\{ w = \frac { \varepsilon } { 2 d } ( i _ { 1 } , i _ { 2 } , \ldots , i _ { d } ) : i _ { j } \in \mathbb { Z } , \| w \| _ { 1 } \leq \eta _ { n } \right\} .
+$$
+
+Then, for any $w , b$ , there exists some element $\hat { w } \in W _ { \varepsilon }$ such that
+
+$$
+\begin{array} { r l } & { \underset { \tau \in \mathbb X } { \operatorname* { s u p } } | \sigma ( w ^ { \top } z + b ) - \sigma ( \hat { w } ^ { \top } z + \hat { b } ) | \leq \underset { z } { \operatorname* { s u p } } | ( w ^ { \top } z + b ) - ( \hat { w } ^ { \top } z + \hat { b } ) | \leq \underset { z } { \operatorname* { s u p } } | ( w - \hat { w } ) ^ { \top } z | + | b - \hat { b } | } \\ & { \qquad \leq \| w - \hat { w } \| _ { 1 } \underset { z } { \operatorname* { s u p } } \| z \| _ { \infty } + | b - \hat { b } | \leq \varepsilon , } \end{array}
+$$
+
+where $\hat { b } = \left( \varepsilon / 2 d \right) \lfloor \left( 2 d b / \varepsilon \right) \rfloor$ and $\lfloor \cdot \rfloor$ is the floor function. By Bernstein’s Inequality, for any $w , b$
+
+$$
+\mathbb { P } \bigg ( \vert \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w ^ { \top } z _ { i } + b ) \vert > t \bigg ) \le 2 \exp \Biggl \{ - \frac { n t ^ { 2 } } { 2 ( 1 + t / 3 ) } \Biggr \} .
+$$
+
+By taking the union bound over $W _ { \varepsilon }$ , and use the fact that log card $\lfloor ( W _ { \varepsilon } ) \lesssim d \log ( n d / \varepsilon )$ , we obtain
+
+$$
+\operatorname* { s u p } _ { w \in \mathbb { R } ^ { d } } \left. \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w ^ { \top } z _ { i } + b ) \right. \lesssim \varepsilon + \sqrt { \frac { d } { n } \log \frac { n d } { \varepsilon } \log \frac { 1 } { \delta } } ,
+$$
+
+with probability at least $1 - \delta$ . Then the desired result is obtained by taking $\varepsilon \sim \sqrt { ( d \log n ) / n }$
+
+# A.2 PROOF OF THEOREM 3.1
+
+The proof is based on the following contraction lemma used in (Neyshabur et al., 2015).
+
+Lemma A.1 (Contraction Lemma). Suppose that $g$ is $L$ -Lipschitz and $g ( 0 ) = 0$ . Then for any function class $\mathcal { F }$ mapping from $\mathbb { X }$ to $\mathbb { R }$ and any set $\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { n } \}$ , we have
+
+$$
+\mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } } \biggl | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } g ( f ( x _ { i } ) ) \biggr | \leq 2 L \mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } } \biggl | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( x _ { i } ) \biggr | .
+$$
+
+With the above lemma, we have the following result.
+
+Lemma A.2. The constrained $L _ { 1 }$ estimator ${ \hat { f } } _ { n }$ over $\mathcal { F }$ satisfies
+
+$$
+\mathcal { R } ( \hat { f } _ { n } ) \leq \operatorname* { m i n } _ { f \in \mathcal { F } } \mathbb { E } | f ( x ) - f _ { * } ( x ) | + 2 \mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } } | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( z _ { i } ) | + 2 \sqrt { \frac { \mathbb { E } y ^ { 2 } } { n } } .
+$$
+
+Proof. Define the empirical risk as:
+
+$$
+\mathcal { R } _ { n } ( f ) = \mathbb { E } \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } | f _ { * } ( x _ { i } ) + \varepsilon _ { i } - f ( x _ { i } ) | \right) - \mathbb { E } | \varepsilon | .
+$$
+
+Since $\hat { f } _ { n }$ minimizes $\begin{array} { r } { n ^ { - 1 } \sum _ { i = 1 } ^ { n } | f _ { * } ( x _ { i } ) + \varepsilon _ { i } - f ( x _ { i } ) | } \end{array}$ in $\mathcal { F }$ , we have
+
+$$
+\begin{array} { r } { \mathcal { R } ( \hat { f } _ { n } ) \leq \mathcal { R } ( \hat { f } _ { n } ) - \{ \mathcal { R } _ { n } ( \hat { f } _ { n } ) - \mathcal { R } _ { n } ( \hat { f } ) \} = \{ \mathcal { R } ( \hat { f } _ { n } ) - \mathcal { R } _ { n } ( \hat { f } _ { n } ) \} + \mathcal { R } _ { n } ( f _ { 0 } ) , } \end{array}
+$$
+
+where $f _ { 0 } = \arg \operatorname* { m i n } _ { f \in \mathcal { F } } \mathcal { R } ( f )$ . We also have
+
+$$
+\mathcal { R } _ { n } ( f _ { 0 } ) = \mathcal { R } ( f _ { 0 } ) = \operatorname* { m i n } _ { f \in \mathcal { F } } \mathbb { E } \left( | f _ { * } ( x ) + \varepsilon - f ( x _ { i } ) | - | \varepsilon | \right) \leq \operatorname* { m i n } _ { f \in \mathcal { F } } \mathbb { E } \left| f ( x ) - f _ { * } ( x ) \right| .
+$$
+
+In the following, we will analyze the term $\mathcal { R } ( \hat { f } _ { n } ) - \mathcal { R } _ { n } ( \hat { f } _ { n } )$ in (14). Let $z _ { i }$ ’s denote independent and identically distributed copies of $x _ { i }$ ’s.
+
+$$
+\begin{array} { r l } & { \mathcal { R } \big ( \hat { f } _ { n } \big ) - \mathcal { R } _ { n } \big ( \hat { f } _ { n } \big ) = \mathbb { E } \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \bigg \{ | \hat { f } _ { n } ( z _ { i } ) - f _ { * } ( z _ { i } ) - \varepsilon _ { i } | - | \hat { f } _ { n } ( x _ { i } ) - f _ { * } ( x _ { i } ) - \varepsilon _ { i } | \bigg \} } \\ & { \qquad \leq \mathbb { E } \displaystyle \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \bigg \{ | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \varepsilon _ { i } | - | f ( x _ { i } ) - f _ { * } ( x _ { i } ) - \varepsilon _ { i } | \bigg \} } \\ & { \qquad \leq 2 \mathbb { E } \displaystyle \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \varepsilon _ { i } | , } \end{array}
+$$
+
+where $\xi _ { 1 } , \ldots , \xi _ { n }$ are independent and identically distributed symmetric Bernoulli random variables that are independent with $z _ { i }$ ’s. According to Lemma A.1, since $g ( x ) \ = \ | x |$ is 1-Lipschitz and $g ( 0 ) = 0$ , we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \big | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \varepsilon _ { i } \big | \leq 2 \mathbb { E } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \Big | \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } \big ( f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \varepsilon _ { i } \big ) \Big | } \\ & { \qquad \leq 2 \mathbb { E } \underset { f \in \mathcal { F } } { \operatorname* { s u p } } \bigg | \frac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \xi _ { i } f ( z _ { i } ) \bigg | + 2 \sqrt { \frac { \mathbb { E } y ^ { 2 } } { n } } . } \end{array}
+$$
+
+Combining this and (15), we conclude the proof of Lemma A.2.
+
+Proof of Theorem 3.1. The proof of (7) is a direct consequence of Lemma 2.3, Lemma A.2, Theorem 2.1 and the fact that the first moment is no more than the second moment. The proof of (8) follows from the fact that $\delta ( \eta ) \to 0$ as $\eta  \infty$ .
+
+# A.3 PROOF OF THEOREM 3.2
+
+Define a subclass of $\mathcal { F } _ { V }$ by
+
+$$
+\mathcal { F } _ { 0 } = \Bigg \{ f : \mathbb { R } ^ { d }  \mathbb { R } \Big | f ( x ) = V \sigma ( w ^ { \top } x ) , \| w \| _ { 2 } = 1 \Bigg \} .
+$$
+
+In the following, we will prove the minimax bound for $\mathcal { F } _ { V }$ by analyzing $\mathcal { F } _ { 0 }$ . Notice that
+
+$$
+\begin{array} { r } { \mathbb { E } | \sigma ( w _ { 1 } ^ { \top } x ) - \sigma ( w _ { 2 } ^ { \top } x ) | \geq \mathbb { E } \operatorname* { i n f } _ { u } \sigma ^ { \prime } ( u ) \cdot | w _ { 1 } ^ { \top } x - w _ { 2 } ^ { \top } x | \cdot \mathbb { I } ( w _ { 1 } ^ { \top } x , w _ { 2 } ^ { \top } x \in S ) \gtrsim \| w _ { 1 } - w _ { 2 } \| _ { 2 } . } \end{array}
+$$
+
+Let $M _ { 1 } ( \varepsilon )$ denote the packing $\varepsilon$ -entropy of $\mathcal { F } _ { 0 }$ with $L _ { 1 }$ distance, then $M _ { 1 } ( \varepsilon )$ is greater than the packing $\varepsilon$ -entropy of $\mathbb { B } _ { 1 } ^ { \bar { d } }$ with $L _ { 2 }$ distance, which means $M _ { 1 } ( \varepsilon ) \gtrsim d$ . Let $V _ { k } ( \varepsilon )$ denote the covering $\varepsilon$ -entropy of $\mathcal { F } _ { 0 }$ with the square root Kullback-Leibler divergence, then according to its relation with the $L _ { 2 }$ distance shown in (Yang & Barron, 1999), we have
+
+$$
+V _ { k } ( \varepsilon ) \leq M _ { 2 } ( \sqrt { 2 } \varepsilon ) \lesssim d \log \frac { 1 } { \varepsilon } ,
+$$
+
+where $M _ { 2 } ( \varepsilon )$ denote the packing $\varepsilon$ -entropy of $\mathcal { F } _ { V }$ with $L _ { 2 }$ loss function. The second inequality is proved in a similar way to the proof of Lemma 2.3, which is omitted here for brevity. Hence, according to (Yang & Barron, 1999, Theorem 1),
+
+$$
+\operatorname* { i n f } _ { \hat { f } _ { n } } \operatorname* { s u p } _ { f \in { \mathcal { F } } _ { V } } { \mathcal { R } } ( \hat { f } _ { n } ( x ) ) \geq \operatorname* { i n f } _ { \hat { f } _ { n } } \operatorname* { s u p } _ { f \in { \mathcal { F } } _ { 0 } } { \mathcal { R } } ( \hat { f } _ { n } ( x ) ) \gtrsim V { \sqrt { \frac { d } { n } } } ,
+$$
+
+This concludes the proof.
+
+# A.4 PROOF OF PROPOSITION 3.3
+
+To prove the proposition, it is sufficient to verify the following Rademacher complexity bound
+
+$$
+\mathbb { E } \operatorname { s u p } \bigg | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w ^ { \top } z _ { i } + b ) \bigg | \lesssim \sqrt { k \log d \log n } ,
+$$
+
+which can be derived easily by adjusting the proof in Lemma 2.3. Then the result follows with a similar analysis as in Theorem 3.1.
+
+# A.5 PROOF OF THEOREM 3.4
+
+It can be verified from the identity (9) that
+
+$$
+\mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( x _ { i } ) \right| \leq \sum _ { j = 0 } ^ { r } \mathbb { E } \underset { f \in \mathcal { F } _ { V } } { \operatorname* { s u p } } \left| a _ { j } \right| \left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w _ { j } ^ { \top } x _ { i } + b _ { j } ) \right| .
+$$
+
+Then according to Lemma A.1, we have
+
+$$
+\mathbb { E } \operatorname* { s u p } _ { f \in { \mathcal F } _ { V } } \left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } \sigma ( w _ { j } ^ { \top } x _ { i } + b _ { j } ) \right| \lesssim \sqrt { \frac { \log n } { n } } ( \| w _ { j } \| _ { \mathbb { X } } + | b _ { j } | ) .
+$$
+
+Combining (16) and (17), we obtain the following lemma that may be interesting on its own right.
+
+# Lemma A.3. We have
+
+$$
+\mathbb { E } \operatorname* { s u p } _ { f \in \mathcal { F } _ { V } } \left| \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \xi _ { i } f ( x _ { i } ) \right| \lesssim \sqrt { \frac { \log n } { n } } \sum _ { j = 0 } ^ { r } | a _ { j } | ( \| w _ { j } \| _ { \mathfrak { X } } + | b _ { j } | ) \lesssim V \sqrt { \frac { \log n } { n } } \operatorname* { m a x } _ { j } \| w _ { j } \| _ { \mathfrak { X } } .
+$$
+
+Since $\| w \| _ { \mathbb { X } } \lesssim \| w \| _ { 1 }$ and $\{ w : \| w \| _ { \mathbb { X } } \lesssim \eta \} \subset \{ w : \| w \| _ { 1 } \lesssim \eta \}$ , the $\| \cdot \| _ { \mathbb { X } }$ can be replaced with $\| \cdot \| _ { 1 }$ in the bounds in Lemmas A.3 and A.2. Then, with a similar argument as in the proof of Theorem 3.1, we conclude the proof of Theorem 3.4.

parse/train/6MaBrlQ5JM/6MaBrlQ5JM_content_list.json

@@ -0,0 +1,1651 @@
+[
+    {
+        "type": "text",
+        "text": "THE EFFICACY OF $L _ { 1 }$ REGULARIZATION IN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A crucial problem in neural networks is to select the most appropriate number of hidden neurons and obtain tight statistical risk bounds. In this work, we present a new perspective towards the bias-variance tradeoff in neural networks. As an alternative to selecting the number of neurons, we theoretically show that $L _ { 1 }$ regularization can control the generalization error and sparsify the input dimension. In particular, with an appropriate $L _ { 1 }$ regularization on the output layer, the network can produce a statistical risk that is near minimax optimal. Moreover, an appropriate $L _ { 1 }$ regularization on the input layer leads to a risk bound that does not involve the input data dimension. Our analysis is based on a new amalgamation of dimension-based and norm-based complexity analysis to bound the generalization error. A consequent observation from our results is that an excessively large number of neurons do not necessarily inflate generalization errors under a suitable regularization. ",
+        "bbox": [
+            233,
+            267,
+            764,
+            448
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            336,
+            496
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks have been successfully applied in modeling nonlinear regression functions in various domains of applications. A critical evaluation metric for a predictive learning model is to measure its statistical risk bound. For example, the $L _ { 1 }$ or $L _ { 2 }$ risks of typical parametric models such as linear regressions are at the order of $( d / n ) ^ { 1 / 2 }$ for small $d$ (Seber & Lee, 2012), where $d$ and $n$ denote respectively the input dimension and number of observations. Obtaining the risk bound for a nonparametric regression model such as neural networks is highly nontrivial. It involves an approximation error (or bias) term as well as a generalization error (or variance) term. The standard analysis of generalization error bounds may not be sufficient to describe the overall predictive performance of a model class unless the data is assumed to be generated from it. For the model class of two-layer feedforward networks and a rather general data-generating process, Barron (1993; 1994) proved an approximation error bound of $O ( r ^ { - 1 / 2 } )$ where $r$ denotes the number of neurons. The author further developed a statistical risk error bound of $O ( ( d / n ) ^ { 1 / 4 } )$ , which is the tightest statistical risk bound for the class of two-layer neural networks up to the authors’ knowledge (for $d < n$ ). This risk bound is based on an optimal bias-variance tradeoff involving an deliberate choice of $r$ . Note that the risk is at a convergence rate much slower than the classical parametric rate. We will tackle the same problem from a different perspective, and obtain a much tighter risk bound. ",
+        "bbox": [
+            174,
+            508,
+            825,
+            734
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A practical challenge closely related to statistical risks is to select the most appropriate neural network architecture for a particular data domain (Ding et al., 2018). For two-layer neural networks, this is equivalent to selecting the number of hidden neurons $r$ . While a small $r$ tends to underfit, researchers have observed that the network is not overfitting even for moderately large $r$ . Nevertheless, recent research has also shown that an overly large $r$ (e.g., when $r > n$ ) does cause overfitting with high probability (Zhang et al., 2016). It can be shown under some non-degeneracy conditions that a two-layer neural network with more than $n$ hidden neurons can perfectly fit $n$ arbitrary data, even in the presence of noise, which inevitably leads to overfitting. A theoretical choice of $r$ suggested by the asymptotic analysis in (Barron, 1994) is at the order of $( n / d ) ^ { 1 / 2 }$ , and a practical choice of $r$ is often from cross-validation with an appropriate splitting ratio (Ding et al., 2018). An alternative perspective that we advocate is to learn from a single neural network with sufficiently many neurons and an appropriate $L _ { 1 }$ regularization on the neuron coefficients, instead of performing a selection from multiple candidate neural models. A potential benefit of this approach is easier hardware implementation and computation since we do not need to implement multiple models separately. Perhaps more importantly, this perspective of training enables much tighter risk bounds, as we will demonstrate. In this work, we focus on the model class of two-layer feedforward neural networks. ",
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+        "text": "Our main contributions are summarized below. First, we prove that $L _ { 1 }$ regularization on the coefficients of the output layer can produce a risk bound $O ( ( d / n ) ^ { 1 / 2 } )$ (up to a logarithmic factor) under the $L _ { 1 }$ training loss, which approaches the minimax optimal rate. Such a rate has not been established under the $L _ { 2 }$ training loss so far. The result indicates a potential benefit of using $L _ { 1 }$ regularization for training a neural network, instead of selecting from a number of neurons. Additionally, a key ingredient of our result is a unique amalgamation of dimension-based and norm-based risk analysis, which may be interesting on its own right. The technique leads to an interesting observation that an excessively large $r$ can reduce approximation error while not increasing generalization error under $L _ { 1 }$ regularizations. This implies that an explicit regularization can eliminate overfitting even when the specified number of neurons is enormous. Moreover, we prove that the $L _ { 1 }$ regularization on the input layer can induce sparsity by producing a risk bound that does not involve $d$ , where $d$ may be much larger compared with the true number of significant variables. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Related work on neural network analysis. Despite the practical success of neural networks, a systematic understanding of their theoretical limit remains an ongoing challenge and has motivated research from various perspectives. Cybenko (1989) showed that any continuous function could be approximated arbitrarily well by a two-layer perceptron with sigmoid activation functions. Barron (1993; 1994) established an approximation error bound of using two-layer neural networks to fit arbitrary smooth functions and their statistical risk bounds. A dimension-free Rademacher complexity for deep ReLU neural networks was recently developed (Golowich et al., 2017; Barron & Klusowski, 2019). Based on a contraction lemma, a series of norm-based complexities and their corresponding generalization errors are developed (Neyshabur et al., 2015, and the references therein). Another perspective is to assume that the data are generated by a neural network and convert its parameter estimation into a tensor decomposition problem through the score function of the known or estimated input distribution (Anandkumar et al., 2014; Janzamin et al., 2015; Ge et al., 2017; Mondelli & Montanari, 2018). Also, tight error bounds have been established recently by assuming that neural networks of parsimonious structures generate the data. In this direction, Schmidt-Hieber (2017) proved that specific deep neural networks with few non-zero network parameters can achieve minimax rates of convergence. Bauer & Kohler (2019) developed an error bound that is free from the input dimension, by assuming a generalized hierarchical interaction model. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Related work on $L _ { 1 }$ regularization. The use of $L _ { 1 }$ regularization has been widely studied in linear regression problems (Hastie et al., 2009, Chapter 3). The use of $L _ { 1 }$ regularization for training neural networks has been recently advocated in deep learning practice. A prominent use of $L _ { 1 }$ regularization was to empirically sparsify weight coefficients and thus compress a network that requires intensive memory usage (Cheng et al., 2017). The extension of $L _ { 1 }$ regularization to group$L _ { 1 }$ regularization (Yuan & Lin, 2006) has also been extensively used in learning various neural networks (Han et al., 2015; Zhao et al., 2015; Wen et al., 2016; Scardapane et al., 2017). Despite the above practice, the efficacy of $L _ { 1 }$ regularization in neural networks deserves more theoretical study. In the context of two-layer neural networks, we will show that the $L _ { 1 }$ regularizations in the output and input layers play two different roles: the former for reducing generalization error caused by excessive neurons while the latter for sparsifying input signals in the presence of substantial redundancy. Unlike previous theoretical work, we consider the $L _ { 1 }$ loss, which ranks among the most popular loss functions in, e.g., learning from ordinal data (Pedregosa et al., 2017) or imaging data (Zhao et al., 2016), and for which the statistical risk has not been studied previously. In practice, the use of $L _ { 1 }$ loss for training has been implemented in prevalent computational frameworks such as Tensorflow (Google, 2016), Pytorch (Ketkar, 2017), and Keras (Gulli & Pal, 2017). ",
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+    {
+        "type": "text",
+        "text": "2 PROBLEM FORMULATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 MODEL ASSUMPTION AND EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Suppose we have $n$ labeled observations $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 , \\dots , n }$ , where $y _ { i }$ ’s are continuously-valued responses or labels. We assume that the underlying data generating model is $y _ { i } = f _ { * } ( \\bar { x } _ { i } ) + \\varepsilon _ { i }$ for some unknown function $f _ { * } ( \\cdot )$ , where $x _ { i }$ ’ $\\mathbf { \\Phi } _ { \\mathbf { \\bar { \\nu } } } \\in \\dot { \\mathbb { X } } \\subset \\mathbb { R } ^ { d }$ are independent and identically distributed, ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3dd926fb4b401afb96027dbbfddca0613aaf3b232ec959dc44b5508fc45a8ed2.jpg",
+        "image_caption": [
+            "Figure 1: A graph showing the two-layer neural network model considered in (2). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and $\\varepsilon _ { i }$ ’s are independent and identically distributed that is symmetric at zero and ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/7d4a3abef251bd0d230037c0ac821592fbe2f44b12017f911da31a477bddb950.jpg",
+        "text": "$$\n\\mathbb { E } \\left( \\varepsilon _ { i } ^ { 2 } \\mid x _ { i } \\right) \\leq \\tau ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "Here, $\\mathbb { X }$ is a bounded set that contains zero, for example $\\{ x : \\| x \\| _ { \\infty } \\leq M \\}$ for some constant $M$ . Our goal is learn a regression model ${ \\hat { f } } _ { n } : x \\mapsto { \\hat { f } } _ { n } ( x )$ for prediction. The $\\hat { f } _ { n }$ is obtained from the following form of neural networks ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3dc66402aedc7feb5599692b244286cd0ef9e6381386d8aa2991a0d41826d466.jpg",
+        "text": "$$\n\\sum _ { j = 1 } ^ { r } a _ { j } \\sigma ( w _ { j } ^ { \\top } x + b _ { j } ) + a _ { 0 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $a _ { 0 } , a _ { j } , b _ { j } \\in \\mathbb { R } , w _ { j } \\in \\mathbb { R } ^ { d }$ , $\\textit { j } = \\ 1 , \\ldots , r$ , are parameters to estimate. We let $a =$   \n$[ a _ { 0 } , a _ { 1 } , \\ldots , \\bar { a } _ { r } ] ^ { \\mathrm { { \\scriptscriptstyle T } } }$ denote the output layer coefficients. An illustray accomplished by minimizing the empirical risk plus a regularization term. We first consider th The esti-, for somethe output $\\textstyle n ^ { - 1 } \\sum _ { i = 1 } ^ { \\bar { n } } \\ell ( y _ { i } , { \\overline { { f ( x _ { i } ) } } } )$ $l ( \\cdot )$ $L _ { 1 }$   \nlayer. In particular, we search for such $f$ by the empirical risk minimization from the function class ",
+        "bbox": [
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+        "page_idx": 2
+    },
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+        "type": "equation",
+        "img_path": "images/8a1bd0bba94e0d13acf5c2ef9fed755855f1f3f1c52fc60cd66cf3ebc99e8dcd.jpg",
+        "text": "$$\n\\mathcal { F } _ { V } = \\left\\{ f : \\mathbb { R } ^ { d } \\to \\mathbb { R } \\Big | f ( x ) = \\sum _ { j = 1 } ^ { r } a _ { j } \\sigma ( w _ { j } ^ { \\top } x + b _ { j } ) + a _ { 0 } , \\| a \\| _ { 1 } \\leq V \\right\\}\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $V$ is a constant. The following statistical risk measures the predictive performance of a learned model $f$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a81b96cf36fa8616be33ef258a91e37842ad4741909a934bbdd1908350a96db3.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { R } ( f ) \\overset { \\Delta } { = } \\mathbb { E } \\ell ( y , f ( x ) ) - \\mathbb { E } \\ell ( y , f _ { * } ( x ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The loss function $\\ell ( \\cdot )$ is pre-determined by data analysts, usually the $L _ { 1 }$ loss defined by $\\ell ( y , \\tilde { y } ) =$ $| y - \\tilde { y } |$ or the $L _ { 2 }$ loss defined by $\\ell _ { 2 } ( y , \\tilde { y } ) = ( y - \\bar { y } ) ^ { 2 }$ . Under the $L _ { 1 }$ loss, the risk is $\\textstyle { \\mathcal { R } } ( f ) =$ $\\mathbb { E } \\left| f _ { * } ( x ) + \\varepsilon - f ( x ) \\right| - \\mathbb { E } \\left| \\varepsilon \\right|$ , which is nonnegative for symmetric random variables $\\varepsilon$ . It is typical to use the same loss function for both training and evaluation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 NOTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Throughout the paper, we use $n , d , k , r$ to denote the number of observations, the number of input variables or input dimension, the number of significant input variables or sparsity level, the number of neurons (or hidden dimension), respectively. We write $a _ { n } \\gtrsim b _ { n }$ , $b _ { n } \\lesssim a _ { n }$ , or $b _ { n } = O ( a _ { n } )$ , if $\\left| b _ { n } / a _ { n } \\right| < c$ for some constant $c$ for all sufficiently large $n$ . We write $a _ { n } \\asymp b _ { n }$ if $a _ { n } \\gtrsim b _ { n }$ as well as $a _ { n } \\lesssim b _ { n }$ . Let $\\mathcal { N } ( \\boldsymbol { \\mu } , V )$ denote Gaussian distribution with mean $\\pmb { \\mu }$ and covariance $V$ . Let $\\| \\cdot \\| _ { 1 }$ and $\\| \\cdot \\| _ { 2 }$ denote the common $L _ { 1 }$ and $L _ { 2 }$ vector norms, respectively. Let $\\mathbb { X }$ denote the essential support of $X$ . For any vector $z \\in \\mathbb { R } ^ { d }$ , we define $\\begin{array} { r } { \\| \\boldsymbol { z } \\| _ { \\mathbb { X } } \\triangleq \\operatorname* { s u p } _ { \\boldsymbol { x } \\in \\mathbb { X } } | \\boldsymbol { x } ^ { \\top } \\boldsymbol { z } | } \\end{array}$ , which may or may not be infinity. If $\\mathbb { X } = \\{ x : \\| x \\| _ { \\infty } \\leq M \\}$ , $\\| \\boldsymbol { z } \\| _ { \\mathbb { X } }$ is equivalent to $M \\| z \\| _ { 1 }$ . Throughout the paper, $\\hat { f } _ { n }$ denotes the estimated regression function with $n$ being the number of observations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.3 ASSUMPTIONS AND CLASSICAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We introduce some technical assumptions necessary for our analysis, and state-of-the-art statistical risk bounds built through dimension-based complexity analysis. ",
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+    {
+        "type": "text",
+        "text": "Assumption 1. The activation function $\\sigma ( \\cdot )$ is a bounded function on the real line satisfying $\\sigma ( x ) $ 1 as $x \\to \\infty$ and $\\sigma ( x )  0$ as $x \\to - \\infty$ , and it is $L$ -Lipschitz for some constant $L$ . ",
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+    {
+        "type": "text",
+        "text": "Assumption 2. The regularization constant $V$ is larger than $2 C + f _ { * } ( 0 )$ , where $C$ is any constant such that the Fourier transform of $f _ { * }$ , denoted by $F$ , satisfies ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3e9c7cbdfe260f12bbdb941785e2d679d6bbc07a6fb0a103a23bb2a981c280a2.jpg",
+        "text": "$$\n\\int _ { \\mathbb { R } ^ { d } } \\| \\omega \\| _ { \\mathbb { X } } F ( d \\omega ) \\leq C .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3. $\\sigma ( x )$ approaches its limits at least polynomially fast, meaning that $| \\sigma ( x ) - \\mathbf { 1 } \\{ x >$ $0 \\} | < \\varepsilon$ for all $| x | > x _ { \\varepsilon }$ where $x _ { \\varepsilon }$ is a polynomial of $1 / \\varepsilon$ . Also, the value of $\\eta \\triangleq \\operatorname* { s u p } _ { j } \\| w _ { j } \\| _ { \\mathbb { X } }$ scales with $n$ polynomially meaning that $\\log \\eta = O ( \\log n )$ as $n \\to \\infty$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4. There exists a constant $c > 0$ and a bounded subset ${ \\mathcal { S } } \\subset \\mathbb { R }$ such that $\\mathbb { P } ( X \\in { \\mathcal { S } } ) > c$ and $\\mathrm { i n f } _ { x \\in S } \\sigma ^ { \\prime } ( x ) > c$ for $X \\sim \\mathcal { N } ( 0 , 1 )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We explain each assumption below. The above notation of $C , V$ follow those in (Barron, 1993; 1994). Assumption 1 specifies the class of the activation functions we consider. A specific case is the popular activation function $\\sigma ( x ) = 1 / \\{ 1 { + } \\exp ( - x ) \\}$ . Assumption 2, first introduced in (Barron, 1993), specifies the smoothness condition for $f _ { * }$ to ensure the approximation property of neural networks (see Theorem 2.1). In Assumption 3, the condition for $w$ is for technical convenience. It could also be replaced with the following alternative condition: There exists a constant $c > 0$ such that the distribution of $x$ satisfies ",
+        "bbox": [
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+        "img_path": "images/cbc570daed8c8d56b5c5b311903e4d7794f20b21cdb6e27d29ce2c45e06ed35f.jpg",
+        "text": "$$\n\\operatorname* { s u p } _ { w : \\| w \\| _ { 2 } = 1 } \\mathbb { P } \\big ( \\log ( | w ^ { \\top } x | ) < c \\log \\varepsilon \\big ) < \\varepsilon\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for any $\\varepsilon \\in ( 0 , 1 )$ . Simply speaking, the input data $x$ is not too small with high probability. This condition is rather mild. For example, it holds when each component of $x$ has a a bounded density function. This alternative condition ensures that for some small constant $\\varepsilon > 0$ and any $w \\in \\mathbb { R } ^ { \\tilde { d } }$ , there exists a surrogate of $w$ $, \\hat { w } \\in \\mathbb R ^ { d }$ with $\\log \\| \\hat { w } \\| _ { 2 } = O ( - \\log \\varepsilon )$ , such that ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b6b4177ff5026c9271e2938aa536a8942647c9f671ed18620addcac64b8e90de.jpg",
+        "text": "$$\n\\mathbb { P } ( | \\sigma ( w ^ { \\top } x ) - \\sigma ( \\hat { w } ^ { \\top } x ) | > \\varepsilon ) < \\varepsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "And this can be used to surrogate the assumption of $w$ in Assumption 3 throughout the proofs in the appendix. Assumption 4 means that $\\sigma ( \\cdot )$ is not a nearly-constant function. This condition is only used to bound the minimax lower bound in Theorem 3.2. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Theorem 2.1 (Approximation error bound (Barron, 1993)). Suppose that Assumptions 1, 2, 3 hold. We have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d7bbb3065be36a4ac967b550472a15da7f8c4f8cdefc2e53dcf2d74923db069b.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { f \\in \\mathcal { F } _ { V } } \\left\\{ \\int _ { \\mathbb { X } } ( f ( x ) - f _ { * } ( x ) ) ^ { 2 } \\mu ( d x ) \\right\\} ^ { 1 / 2 } \\leq 2 C \\left( \\frac { 1 } { \\sqrt { r } } + \\delta _ { \\eta } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $\\mu$ denotes a probability measure on $\\mathbb { X } ,$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d65e70b17e2d02f86cf98cfc7e20bd932cd8bb1cc9be53f303bc181c42a02bca.jpg",
+        "text": "$$\n\\delta _ { \\eta } = \\operatorname* { i n f } _ { 0 < \\varepsilon < 1 / 2 } \\biggl \\{ 2 \\varepsilon + \\operatorname* { s u p } _ { | x | > \\varepsilon } \\bigl | \\sigma ( \\eta x ) - \\mathbf { 1 } \\bigl \\{ x > 0 \\bigr \\} \\bigr | \\biggr \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\eta$ is defined in Assumption 3, and $C$ is defined in (4). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2.2 (Statistical risk bound (Barron, 1994)). Suppose that Assumptions $^ { l }$ , 2, 3 hold. Then the $L _ { 2 }$ estimator ${ \\hat { f } } _ { n }$ in $\\mathcal { F } _ { V }$ satisfies $\\mathbb { E } \\left\\{ \\hat { f } _ { n } ( x ) - f _ { * } ( x ) \\right\\} ^ { 2 } \\lesssim V ^ { 2 } / r + ( r d \\log n ) / n$ . In particular, $i f$ we choose $r \\asymp V { \\sqrt { n / ( d \\log n ) } }$ , then ${ \\mathbb E } \\left\\{ \\hat { f } _ { n } ( x ) - f _ { * } ( x ) \\right\\} ^ { 2 } \\lesssim V \\sqrt { ( d \\log n ) / n }$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "It is known that a typical parametric rate under the $L _ { 2 }$ loss is at the order of $O ( d / n )$ , much faster than the above result. This gap is mainly due to excessive model complexity in bounding generalization errors. We will show in Section 3 that the gap in the rate of convergence can be filled when using $L _ { 1 }$ loss. Our technique will be based on the machinery of Rademacher complexity, and we bound this complexity through a joint analysis of the norm of coefficients (‘norm-based’) as well as dimension of parameters (‘dimension-based’). ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "2.4 MODEL COMPLEXITY AND GENERALIZATION ERROR",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The statistical risk consists of two parts. The first part is an approximation error term non-increasing in the number of neurons $r$ , and the second part describes generalization errors. The key issue for risk analysis is to bound the second term using a suitable model complexity and then tradeoff with the first term. We will develop our theory based on the following measure of complexity. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Let $\\mathcal { F }$ denote a class of functions each mapping from $\\mathbb { X }$ to $\\mathbb { R }$ , and $x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } \\in \\mathbb { X }$ . Following a similar terminology as in (Neyshabur et al., 2015), the Rademacher complexity, or simply ‘complexity’, of a function class $\\mathcal { F }$ is defined by $\\begin{array} { r } { \\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } } | n ^ { - 1 } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( x _ { i } ) | } \\end{array}$ , where $\\xi _ { i } , i = 1 , 2 , \\ldots , n$ are independent symmetric Bernoulli random variables. ",
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+    {
+        "type": "text",
+        "text": "Lemma 2.3 (Rademacher complexity of $\\mathcal { F } _ { V }$ ). Suppose that Assumptions $^ { l }$ , 3 hold. Then for the Rademacher complexity of $\\mathcal { F } _ { V }$ , we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe99c05d0ea067a2ff102e99101e64508f97ae32bb1299529ef7eb871088d736.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } _ { V } } \\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( x _ { i } ) \\right| \\lesssim \\frac { V \\sqrt { d \\log n } } { \\sqrt { n } } .\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proof is included in Appendix A.1. The bound in (6) is derived from an amalgamation of dimension-based and norm-based analysis elaborated in the appendix. It is somewhat surprising that the bound does not explicitly involve the approximation error part (that depends on $r$ and $\\eta$ ). This Rademacher complexity bound enables us to derive tight statistical risk bounds in the following section. ",
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+    {
+        "type": "text",
+        "text": "3 MAIN RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 STATISTICAL RISK BOUND FOR THE $L _ { 1 }$ REGULARIZED NETWORKS IN (3)",
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+        "type": "text",
+        "text": "Theorem 3.1 (Statistical risk bound). Suppose that Assumptions 1, 2, 3 hold. Then the constrained $L _ { 1 }$ estimator $\\hat { f } _ { n }$ over $\\mathcal { F } _ { V }$ satisfies ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/c2cf109dff3bf123ffdb6c79ad05b576a95290e91e8f70d8eab1d2a089d16e01.jpg",
+        "text": "$$\n\\mathcal { R } ( \\hat { f } _ { n } ) \\lesssim \\bigg ( \\frac { 1 } { \\sqrt { r } } + \\delta _ { \\eta } \\bigg ) C + \\frac { V \\sqrt { d \\log n } + \\tau } { \\sqrt { n } } ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta _ { \\eta }$ is defined in (5), and $\\tau$ was introduced in $( l )$ . Moreover, choosing the parameters $r , \\eta$ large enough, we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2d72c07724f45b6f9ce582c917e3a4908f09aac82fdcf51aa40cad7035bc3a8.jpg",
+        "text": "$$\n\\mathcal { R } ( \\hat { f } _ { n } ) \\lesssim \\frac { V \\sqrt { d \\log n } + \\tau } { \\sqrt { n } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proof is in Appendix A.2. We briefly explain our main idea in deriving the risk bound (7). A standard statistical risk bound contains two parts which correspond to the approximation error and generalization error, respectively. The approximation error part in (7) is the first term, which involves the hidden dimension $r$ and the norm of input coefficients through $\\eta$ . This observation motivates us to use the norm of output-layer coefficients through $V$ and the input dimension $d$ to derive a generalization error bound. In this way, the generalization error term does not involve $r$ already used for bounding the approximation error, and thus a bias-variance tradeoff through $r$ is avoided. This thought leads to the generalization error part in (7), which is the second term involving $V$ and $d$ . Its proof combines the machinery of both dimension-based and norm-based complexity analysis. From our analysis, the error bound in Theorem 3.1 is a consequence of the $L _ { 1 }$ loss function and the employed $L _ { 1 }$ regularization. In comparison with the previous result of Theorem 2.2, the bound obtained in Theorem 3.1 is tight and it approaches the parametric rate $\\sqrt { d / n }$ for the $d < n$ regime. Though we can only prove for $L _ { 1 }$ loss in this work, we conjecture that the same rate is achieved using $L _ { 2 }$ loss. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the following, we further show that the above risk bound is minimax optimal. The minimax optimality indicates that deep neural networks with more than two layers will not perform much better than shallow neural networks when the underlying regression function belongs to $\\mathcal { F } _ { V }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (Minimax risk bound). Suppose that Assumptions 1 and 4 hold, and $x _ { 1 } , x _ { 2 } , \\dotsc , x _ { n } \\stackrel { i : } { \\sim }$ id∼ $\\mathcal { N } ( 0 , \\pmb { I } _ { d } )$ , then $\\operatorname { i n f } _ { \\hat { f } _ { n } }$ $\\begin{array} { r } { \\operatorname* { s u p } _ { f \\in \\mathcal { F } _ { V } } \\mathcal { R } ( \\hat { f } _ { n } ( x ) ) \\gtrsim V \\sqrt { d / n } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here the $\\mathcal { F } _ { V }$ is the same one as defined in (3). All the smooth functions $f _ { * } ( \\cdot )$ that satisfy $V >$ $2 C + f _ { * } ( 0 )$ and (4) belong to $\\mathcal { F } _ { V }$ according to Theorem 2.1. The proof is included in Appendix A.3. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 ADAPTIVENESS TO THE INPUT SPARSITY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "It is common to input a large dimensional signal to a neural network, while only few components are genuinely significant for prediction. For example, in environmental science, high dimensional weather signals are input for prediction while few are physically related (Shi et al., 2015). In image processing, the image label is relevant to few background pixels (Han et al., 2015). In natural language processing, a large number of redundant sentences sourced from Wikipedia articles are input for language prediction (Diao et al., 2019). The practice motivates our next results to provide a tight risk bound for neural networks whose input signals are highly sparse. ",
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+    {
+        "type": "text",
+        "text": "Assumption 5. There exists a positive integer $k \\leq d$ and an index set $S \\subset \\{ 1 , \\ldots , d \\}$ with card $( S ) =$ $k$ , such that $f _ { \\ast } ( x ) = g _ { \\ast } ( x _ { S } )$ for some function $g _ { * } ( \\cdot )$ with probability one. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The subset $S$ is generally unknown to data analysts. Nevertheless, if we know $k$ , named the sparsity level, the risk bound could be further improved by a suitable regularization on the input coefficients. We have the following result where $d$ is replaced with $k$ in the risk bound of Theorem 3.1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.3. Suppose that that Assumptions 1, 2, 3, 5 hold. Suppose that $\\hat { f } _ { n }$ is the $L _ { 1 }$ estimator over the following function class ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c06532e64b4f0abc5fb979fac8b7eceb728cbc58b2d6fe5fd5d9695f09c4779.jpg",
+        "text": "$$\n\\bigg \\{ f : \\mathbb { R } ^ { d } \\to \\mathbb { R } \\Big | f ( x ) = \\sum _ { j = 1 } ^ { r } a _ { j } \\sigma ( w _ { j } ^ { \\top } x + b _ { j } ) + a _ { 0 } , \\| a \\| _ { 1 } \\leq V , \\operatorname* { s u p } _ { j } \\| w _ { j } \\| _ { 0 } \\leq k \\bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then $\\mathcal { R } ( \\hat { f } _ { n } ) \\lesssim \\sqrt { \\{ k \\log ( d n ) \\} / n }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proof is included in Appendix A.4. The above statistical risk bound is also minimax optimal according to a similar argument in Theorem 3.2. From a practical point of view, the above $L _ { 0 }$ constraint is usually difficult to implement, especially for a large input dimension $d$ . Alternatively, one may impose an $L _ { 1 }$ constraint instead of an $L _ { 0 }$ constraint on the input coefficients. Our next result is concerned with the risk bound when the model is learned from a joint regularization on the output and input layers. For technical convenience, we will assume that $\\mathbb { X }$ is a bounded set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3.4. Consider the following function class of two-layer neural networks ",
+        "bbox": [
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+        "page_idx": 5
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+        "type": "equation",
+        "img_path": "images/f21ecd906092d63f80c2d70228d2af2b026295733a38dccb2195254b1c3dc1be.jpg",
+        "text": "$$\n\\mathcal { F } _ { V , \\eta } = \\Bigg \\{ f : \\mathbb { R } ^ { d }  \\mathbb { R } \\Big | f ( x ) = \\sum _ { j = 1 } ^ { r } a _ { j } \\sigma ( w _ { j } ^ { \\top } x + b _ { j } ) + a _ { 0 } , \\| a \\| _ { 1 } \\leq V , \\operatorname* { s u p } _ { 1 \\leq j \\leq r } ( \\| w _ { j } \\| _ { 1 } + | b _ { j } | ) \\leq \\eta \\Bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Suppose that $V \\gtrsim C$ , where $C$ is defined in (4). Then the constrained $L _ { 1 }$ estimator $\\hat { f } _ { n }$ over $\\mathcal { F } _ { V , \\eta }$ satisfies ",
+        "bbox": [
+            176,
+            555,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b5acf4a6cdbca67390440cde444b57a2d2bd4384970fd532b9f75abca450d9bc.jpg",
+        "text": "$$\n\\mathcal { R } ( \\hat { f } _ { n } ) \\lesssim C \\left( \\frac { 1 } { \\sqrt { r } } + \\delta _ { \\eta } \\right) + \\frac { V \\eta + \\tau } { \\sqrt { n } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            585,
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+            621
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta _ { \\eta }$ is defined in (5). In particular, choosing $r$ large enough, we have ",
+        "bbox": [
+            173,
+            621,
+            661,
+            636
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1807c6e1de6d2051697250d5271ae73819ec631b87f89b2ab8c5af1eefdd8def.jpg",
+        "text": "$$\n\\mathcal { R } ( \\hat { f } _ { n } ) \\lesssim C \\delta _ { \\eta } + \\frac { V \\eta + \\tau } { \\sqrt { n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            637,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which does not involve the input dimension $d$ and the number of hidden neurons $r$ . Moreover, ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc6b12fbf8700df497c3aaa1949bdceed735bfedec9a678b1b36ca6c32159c81.jpg",
+        "text": "$$\n\\sigma ( x ) = 1 / ( 1 + e ^ { - x } ) , \\quad \\eta \\asymp \\biggl ( n \\log ^ { 2 } n \\biggr ) ^ { 1 / 3 } , t h e n \\mathscr { R } ( \\hat { f } _ { n } ) \\lesssim V \\{ ( \\log n ) / n \\} ^ { 1 / 3 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The proof is included in Appendix A.5. In the above result, the risk bound is at the order of $O ( n ^ { - 1 / 3 } )$ , which is slower than the $O ( n ^ { - 1 / 2 } )$ in the previous Theorem 3.1 and Proposition 3.3 if ignoring $d$ and logarithmic factors of $n$ . However, for a large input dimension $d$ that is even much larger than $n$ , the bound can be much tighter than the previous bounds since it is dimension-free. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION AND FURTHER REMARKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We studied the tradeoff between model complexity and statistical risk in two-layer neural networks from the explicit regularization perspective. We end our paper with two future problems. First, in Theorem 3.4, For a small $d$ , the order of $n ^ { - 1 / 3 }$ seems to be an artifact resulting from our technical arguments. We conjecture that in the small $d$ regime, this risk bound could be improved to $O ( n ^ { - 1 / 2 } )$ by certain adaptive regularizations. Second, it would be interesting to emulate the current approach to yield similarly tight risk bounds for deep forward neural networks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nAnimashree Anandkumar, Rong Ge, Daniel Hsu, Sham M Kakade, and Matus Telgarsky. Tensor decompositions for learning latent variable models. J. Mach. Learn. Res., 15:2773–2832, 2014.   \nAndrew R Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory, 39(3):930–945, 1993.   \nAndrew R Barron. Approximation and estimation bounds for artificial neural networks. Machine learning, 14(1):115–133, 1994.   \nAndrew R Barron and Jason M Klusowski. Complexity, statistical risk, and metric entropy of deep nets using total path variation. arXiv preprint arXiv:1902.00800, 2019.   \nBenedikt Bauer and Michael Kohler. On deep learning as a remedy for the curse of dimensionality in nonparametric regression. Ann. Stat., 47(4):2261–2285, 2019.   \nYu Cheng, Duo Wang, Pan Zhou, and Tao Zhang. A survey of model compression and acceleration for deep neural networks. arXiv preprint arXiv:1710.09282, 2017.   \nGeorge Cybenko. Approximations by superpositions of a sigmoidal function. Math. Control Signals Syst., 2:183–192, 1989.   \nEnmao Diao, Jie Ding, and Vahid Tarokh. Restricted recurrent neural networks. 2019 IEEE Conf. on Big Data, 2019.   \nJie Ding, Vahid Tarokh, and Yuhong Yang. Model selection techniques: An overview. IEEE Signal Process. Mag., 35(6):16–34, 2018.   \nRong Ge, Jason D Lee, and Tengyu Ma. Learning one-hidden-layer neural networks with landscape design. arXiv preprint arXiv:1711.00501, 2017.   \nNoah Golowich, Alexander Rakhlin, and Ohad Shamir. Size-independent sample complexity of neural networks. arXiv preprint arXiv:1712.06541, 2017.   \nResearch Team Google. Tensorflow: A system for large-scale machine learning. Proc. 12th Symp. Operating Syst. Des. Implementation, pp. 265–283, 2016.   \nAntonio Gulli and Sujit Pal. Deep Learning with Keras. Packt Publishing Ltd, 2017.   \nSong Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. Advance. Neural Inf. Process. Sys., pp. 1135–1143, 2015.   \nTrevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements of statistical learning: data mining, inference, and prediction. Springer Science & Business Media, 2009.   \nMajid Janzamin, Hanie Sedghi, and Anima Anandkumar. Beating the perils of non-convexity: Guaranteed training of neural networks using tensor methods. arXiv preprint arXiv:1506.08473, 2015.   \nNikhil Ketkar. Introduction to pytorch. Deep learning with python, pp. 195–208, 2017.   \nMarco Mondelli and Andrea Montanari. On the connection between learning two-layers neural networks and tensor decomposition. arXiv preprint arXiv:1802.07301, 2018.   \nBehnam Neyshabur, Ryota Tomioka, and Nathan Srebro. Norm-based capacity control in neural networks. Conf. Learning Theory, pp. 1376–1401, 2015.   \nFabian Pedregosa, Francis Bach, and Alexandre Gramfort. On the consistency of ordinal regression methods. J. Mach. Learn. Res., 18(1):1769–1803, 2017.   \nSimone Scardapane, Danilo Comminiello, Amir Hussain, and Aurelio Uncini. Group sparse regularization for deep neural networks. Neurocomputing, 241:81–89, 2017.   \nJohannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. arXiv preprint arXiv:1708.06633, 2017.   \nGeorge AF Seber and Alan J Lee. Linear regression analysis, volume 329. John Wiley & Sons, 2012.   \nXingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-kin Wong, and Wang-chun Woo. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. In Advance. Neural Inf. Process. Sys., pp. 802–810, 2015.   \nWei Wen, Chunpeng Wu, Yandan Wang, Yiran Chen, and Hai Li. Learning structured sparsity in deep neural networks. Advance. Neural Inf. Process. Sys., pp. 2074–2082, 2016.   \nYuhong Yang and Andrew Barron. Information-theoretic determination of minimax rates of convergence. Ann. Stat., pp. 1564–1599, 1999.   \nMing Yuan and Yi Lin. Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B Methodol., 68(1):49–67, 2006.   \nChiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.   \nHang Zhao, Orazio Gallo, Iuri Frosio, and Jan Kautz. Loss functions for image restoration with neural networks. IEEE Trans. Comput., 3(1):47–57, 2016.   \nLei Zhao, Qinghua Hu, and Wenwu Wang. Heterogeneous feature selection with multi-modal deep neural networks and sparse group LASSO. IEEE Trans. Multimed., 17(11):1936–1948, 2015. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A.1 PROOF OF LEMMA 2.3 ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We first prove (6), which uses an amalgamation of dimension-based and norm-based analysis. For the output layer, we use the following norm-based analysis ",
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+        "type": "equation",
+        "img_path": "images/4aa6bd22f103ea31c7a5cd400e433ee3efb46db8adfccb7a5ad1a6bdc844e091.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\left| \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( z _ { i } ) \\right| = \\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\left| \\langle a , \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( W ^ { \\top } z _ { i } + b ) \\rangle \\right| } \\\\ & { \\leq \\operatorname* { s u p } \\left\\| a \\right\\| _ { 1 } \\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\left\\| \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( W ^ { \\top } z _ { i } + b ) \\right\\| _ { \\infty } \\leq V \\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\underset { j } { \\operatorname* { m a x } } \\bigg | \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w _ { j } ^ { \\top } z _ { i } + b _ { j } ) \\bigg | } \\\\ & { \\leq V \\mathbb { E } \\underset { w \\in \\mathbb { R } ^ { d } } { \\operatorname* { s u p } } \\bigg | \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w ^ { \\top } z _ { i } + b ) \\bigg | . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            518,
+            800,
+            646
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For notational convenience, we define $w _ { 0 } = 0 , b _ { 0 } = 0$ , and $a _ { 0 } = \\sigma ( 0 ) ^ { - 1 } a _ { 0 } \\sigma ( w _ { 0 } ^ { \\top } z + b _ { 0 } )$ so that $a _ { 0 }$ can be treated in a similar manner as other $a _ { i }$ ’s. Without loss of generality, we do not separately consider $a _ { 0 }$ in the following proofs. ",
+        "bbox": [
+            174,
+            651,
+            823,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Next, we prove that ",
+        "bbox": [
+            174,
+            702,
+            305,
+            715
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e36029514ed0faacc8632c993965e7167911069bb3e03f64fc3d0290b7dac4b.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname* { s u p } _ { w \\in \\mathbb { R } ^ { d } } \\Big | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w ^ { \\top } z _ { i } + b ) \\Big | \\lesssim \\sqrt { \\frac { d \\log n } { n } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            722,
+            647,
+            763
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "and thus conclude the proof. The proof will be based on an $\\varepsilon$ -net argument together with the union bound. For any $\\varepsilon$ , let $\\bar { W } _ { \\varepsilon } \\subset \\mathbb { R } ^ { d }$ denote the subset ",
+        "bbox": [
+            171,
+            768,
+            823,
+            797
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/442b3db49e37c39462844d81521269e848dd4d159ee2ce11440ed8a2eb97adbf.jpg",
+        "text": "$$\nW _ { \\varepsilon } = \\left\\{ w = \\frac { \\varepsilon } { 2 d } ( i _ { 1 } , i _ { 2 } , \\ldots , i _ { d } ) : i _ { j } \\in \\mathbb { Z } , \\| w \\| _ { 1 } \\leq \\eta _ { n } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            804,
+            679,
+            838
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Then, for any $w , b$ , there exists some element $\\hat { w } \\in W _ { \\varepsilon }$ such that ",
+        "bbox": [
+            174,
+            843,
+            593,
+            859
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0efb5099b4818d1351b50e8c291b72fd1244ee6979a95ef6c7c7c91a190e9485.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { \\tau \\in \\mathbb X } { \\operatorname* { s u p } } | \\sigma ( w ^ { \\top } z + b ) - \\sigma ( \\hat { w } ^ { \\top } z + \\hat { b } ) | \\leq \\underset { z } { \\operatorname* { s u p } } | ( w ^ { \\top } z + b ) - ( \\hat { w } ^ { \\top } z + \\hat { b } ) | \\leq \\underset { z } { \\operatorname* { s u p } } | ( w - \\hat { w } ) ^ { \\top } z | + | b - \\hat { b } | } \\\\ & { \\qquad \\leq \\| w - \\hat { w } \\| _ { 1 } \\underset { z } { \\operatorname* { s u p } } \\| z \\| _ { \\infty } + | b - \\hat { b } | \\leq \\varepsilon , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            864,
+            823,
+            922
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { b } = \\left( \\varepsilon / 2 d \\right) \\lfloor \\left( 2 d b / \\varepsilon \\right) \\rfloor$ and $\\lfloor \\cdot \\rfloor$ is the floor function. By Bernstein’s Inequality, for any $w , b$ ",
+        "bbox": [
+            171,
+            102,
+            810,
+            119
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7dfdcb964d768962db20a93b13eafb99621db07c580484cd6b4d59e1c61df98e.jpg",
+        "text": "$$\n\\mathbb { P } \\bigg ( \\vert \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w ^ { \\top } z _ { i } + b ) \\vert > t \\bigg ) \\le 2 \\exp \\Biggl \\{ - \\frac { n t ^ { 2 } } { 2 ( 1 + t / 3 ) } \\Biggr \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            122,
+            692,
+            161
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "By taking the union bound over $W _ { \\varepsilon }$ , and use the fact that log card $\\lfloor ( W _ { \\varepsilon } ) \\lesssim d \\log ( n d / \\varepsilon )$ , we obtain ",
+        "bbox": [
+            173,
+            162,
+            808,
+            179
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f6a012155c35a2ee63647559f38e6f9f11cf5b21b4abe6dd9e3b558623f3f3b.jpg",
+        "text": "$$\n\\operatorname* { s u p } _ { w \\in \\mathbb { R } ^ { d } } \\left. \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w ^ { \\top } z _ { i } + b ) \\right. \\lesssim \\varepsilon + \\sqrt { \\frac { d } { n } \\log \\frac { n d } { \\varepsilon } \\log \\frac { 1 } { \\delta } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            181,
+            678,
+            222
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "with probability at least $1 - \\delta$ . Then the desired result is obtained by taking $\\varepsilon \\sim \\sqrt { ( d \\log n ) / n }$ ",
+        "bbox": [
+            171,
+            226,
+            795,
+            242
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROOF OF THEOREM 3.1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            253,
+            388,
+            268
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The proof is based on the following contraction lemma used in (Neyshabur et al., 2015). ",
+        "bbox": [
+            173,
+            276,
+            751,
+            291
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.1 (Contraction Lemma). Suppose that $g$ is $L$ -Lipschitz and $g ( 0 ) = 0$ . Then for any function class $\\mathcal { F }$ mapping from $\\mathbb { X }$ to $\\mathbb { R }$ and any set $\\{ x _ { 1 } , x _ { 2 } , \\ldots , x _ { n } \\}$ , we have ",
+        "bbox": [
+            169,
+            292,
+            823,
+            321
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b1cc395cb6cc88063ed29b25cc5a705bdd8cb6a913e1360d43b433d4674853a.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\biggl | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } g ( f ( x _ { i } ) ) \\biggr | \\leq 2 L \\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\biggl | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( x _ { i } ) \\biggr | .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            324,
+            679,
+            366
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "With the above lemma, we have the following result. ",
+        "bbox": [
+            173,
+            375,
+            519,
+            388
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.2. The constrained $L _ { 1 }$ estimator ${ \\hat { f } } _ { n }$ over $\\mathcal { F }$ satisfies ",
+        "bbox": [
+            173,
+            392,
+            591,
+            409
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14a1781b192fd8f0e7c53aaf7bc0c5ec5a9b5980ed12a82b73079c00f2b33c3c.jpg",
+        "text": "$$\n\\mathcal { R } ( \\hat { f } _ { n } ) \\leq \\operatorname* { m i n } _ { f \\in \\mathcal { F } } \\mathbb { E } | f ( x ) - f _ { * } ( x ) | + 2 \\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } } | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( z _ { i } ) | + 2 \\sqrt { \\frac { \\mathbb { E } y ^ { 2 } } { n } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            411,
+            733,
+            452
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Proof. Define the empirical risk as: ",
+        "bbox": [
+            174,
+            465,
+            408,
+            479
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04d086e51df04241e1a46b566b563e73a667025b994d205c2c8e04416407883b.jpg",
+        "text": "$$\n\\mathcal { R } _ { n } ( f ) = \\mathbb { E } \\left( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } | f _ { * } ( x _ { i } ) + \\varepsilon _ { i } - f ( x _ { i } ) | \\right) - \\mathbb { E } | \\varepsilon | .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            481,
+            673,
+            522
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Since $\\hat { f } _ { n }$ minimizes $\\begin{array} { r } { n ^ { - 1 } \\sum _ { i = 1 } ^ { n } | f _ { * } ( x _ { i } ) + \\varepsilon _ { i } - f ( x _ { i } ) | } \\end{array}$ in $\\mathcal { F }$ , we have ",
+        "bbox": [
+            174,
+            525,
+            620,
+            544
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d230b5f55c93b298435aa1231111bec9daf883149a9ef9dba0ad0437588e7ba8.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { R } ( \\hat { f } _ { n } ) \\leq \\mathcal { R } ( \\hat { f } _ { n } ) - \\{ \\mathcal { R } _ { n } ( \\hat { f } _ { n } ) - \\mathcal { R } _ { n } ( \\hat { f } ) \\} = \\{ \\mathcal { R } ( \\hat { f } _ { n } ) - \\mathcal { R } _ { n } ( \\hat { f } _ { n } ) \\} + \\mathcal { R } _ { n } ( f _ { 0 } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            545,
+            745,
+            565
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { 0 } = \\arg \\operatorname* { m i n } _ { f \\in \\mathcal { F } } \\mathcal { R } ( f )$ . We also have ",
+        "bbox": [
+            176,
+            566,
+            472,
+            582
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3482ffcb4cfb0e800a413f84ac0f2d3a84537213f49c65818bb1f315f03fd7e.jpg",
+        "text": "$$\n\\mathcal { R } _ { n } ( f _ { 0 } ) = \\mathcal { R } ( f _ { 0 } ) = \\operatorname* { m i n } _ { f \\in \\mathcal { F } } \\mathbb { E } \\left( | f _ { * } ( x ) + \\varepsilon - f ( x _ { i } ) | - | \\varepsilon | \\right) \\leq \\operatorname* { m i n } _ { f \\in \\mathcal { F } } \\mathbb { E } \\left| f ( x ) - f _ { * } ( x ) \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            584,
+            764,
+            609
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In the following, we will analyze the term $\\mathcal { R } ( \\hat { f } _ { n } ) - \\mathcal { R } _ { n } ( \\hat { f } _ { n } )$ in (14). Let $z _ { i }$ ’s denote independent and identically distributed copies of $x _ { i }$ ’s. ",
+        "bbox": [
+            173,
+            613,
+            825,
+            643
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dfaf409abffeea9a0ff02ad23f495e6771ff85bbb51dffde05e38d1a678d4f3b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { R } \\big ( \\hat { f } _ { n } \\big ) - \\mathcal { R } _ { n } \\big ( \\hat { f } _ { n } \\big ) = \\mathbb { E } \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\bigg \\{ | \\hat { f } _ { n } ( z _ { i } ) - f _ { * } ( z _ { i } ) - \\varepsilon _ { i } | - | \\hat { f } _ { n } ( x _ { i } ) - f _ { * } ( x _ { i } ) - \\varepsilon _ { i } | \\bigg \\} } \\\\ & { \\qquad \\leq \\mathbb { E } \\displaystyle \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\bigg \\{ | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \\varepsilon _ { i } | - | f ( x _ { i } ) - f _ { * } ( x _ { i } ) - \\varepsilon _ { i } | \\bigg \\} } \\\\ & { \\qquad \\leq 2 \\mathbb { E } \\displaystyle \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \\varepsilon _ { i } | , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            646,
+            777,
+            772
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\xi _ { 1 } , \\ldots , \\xi _ { n }$ are independent and identically distributed symmetric Bernoulli random variables that are independent with $z _ { i }$ ’s. According to Lemma A.1, since $g ( x ) \\ = \\ | x |$ is 1-Lipschitz and $g ( 0 ) = 0$ , we have ",
+        "bbox": [
+            174,
+            772,
+            825,
+            814
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a77822824ba417e405481d635eae2aecaabff7f8bcb51e07cd323ae63f698596.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\big | f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \\varepsilon _ { i } \\big | \\leq 2 \\mathbb { E } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\Big | \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\big ( f ( z _ { i } ) - f _ { * } ( z _ { i } ) - \\varepsilon _ { i } \\big ) \\Big | } \\\\ & { \\qquad \\leq 2 \\mathbb { E } \\underset { f \\in \\mathcal { F } } { \\operatorname* { s u p } } \\bigg | \\frac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( z _ { i } ) \\bigg | + 2 \\sqrt { \\frac { \\mathbb { E } y ^ { 2 } } { n } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            816,
+            764,
+            901
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Combining this and (15), we conclude the proof of Lemma A.2. ",
+        "bbox": [
+            173,
+            909,
+            593,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 3.1. The proof of (7) is a direct consequence of Lemma 2.3, Lemma A.2, Theorem 2.1 and the fact that the first moment is no more than the second moment. The proof of (8) follows from the fact that $\\delta ( \\eta ) \\to 0$ as $\\eta  \\infty$ . ",
+        "bbox": [
+            173,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A.3 PROOF OF THEOREM 3.2 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            159,
+            390,
+            172
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Define a subclass of $\\mathcal { F } _ { V }$ by ",
+        "bbox": [
+            173,
+            181,
+            354,
+            195
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4882094fdc8e7914bc12d785558c229d042e92132cf8b53a8856b177a401e669.jpg",
+        "text": "$$\n\\mathcal { F } _ { 0 } = \\Bigg \\{ f : \\mathbb { R } ^ { d }  \\mathbb { R } \\Big | f ( x ) = V \\sigma ( w ^ { \\top } x ) , \\| w \\| _ { 2 } = 1 \\Bigg \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            200,
+            674,
+            236
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the following, we will prove the minimax bound for $\\mathcal { F } _ { V }$ by analyzing $\\mathcal { F } _ { 0 }$ . Notice that ",
+        "bbox": [
+            178,
+            239,
+            754,
+            256
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b536815ed14c54a88f24a461696883838e912c5dc9149e4d0a517489f5974ab7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } | \\sigma ( w _ { 1 } ^ { \\top } x ) - \\sigma ( w _ { 2 } ^ { \\top } x ) | \\geq \\mathbb { E } \\operatorname* { i n f } _ { u } \\sigma ^ { \\prime } ( u ) \\cdot | w _ { 1 } ^ { \\top } x - w _ { 2 } ^ { \\top } x | \\cdot \\mathbb { I } ( w _ { 1 } ^ { \\top } x , w _ { 2 } ^ { \\top } x \\in S ) \\gtrsim \\| w _ { 1 } - w _ { 2 } \\| _ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
+            258,
+            799,
+            284
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Let $M _ { 1 } ( \\varepsilon )$ denote the packing $\\varepsilon$ -entropy of $\\mathcal { F } _ { 0 }$ with $L _ { 1 }$ distance, then $M _ { 1 } ( \\varepsilon )$ is greater than the packing $\\varepsilon$ -entropy of $\\mathbb { B } _ { 1 } ^ { \\bar { d } }$ with $L _ { 2 }$ distance, which means $M _ { 1 } ( \\varepsilon ) \\gtrsim d$ . Let $V _ { k } ( \\varepsilon )$ denote the covering $\\varepsilon$ -entropy of $\\mathcal { F } _ { 0 }$ with the square root Kullback-Leibler divergence, then according to its relation with the $L _ { 2 }$ distance shown in (Yang & Barron, 1999), we have ",
+        "bbox": [
+            173,
+            290,
+            826,
+            347
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3cc66943152e9261383f1fe77c9091ff53f5ea58a57aee39ac4c9bc5582edf50.jpg",
+        "text": "$$\nV _ { k } ( \\varepsilon ) \\leq M _ { 2 } ( \\sqrt { 2 } \\varepsilon ) \\lesssim d \\log \\frac { 1 } { \\varepsilon } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            351,
+            599,
+            381
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $M _ { 2 } ( \\varepsilon )$ denote the packing $\\varepsilon$ -entropy of $\\mathcal { F } _ { V }$ with $L _ { 2 }$ loss function. The second inequality is proved in a similar way to the proof of Lemma 2.3, which is omitted here for brevity. Hence, according to (Yang & Barron, 1999, Theorem 1), ",
+        "bbox": [
+            174,
+            386,
+            825,
+            429
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/adff543b3f7019c3e94781e4a72c81332c15361b299c5058cb5efc10f16f63b9.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\hat { f } _ { n } } \\operatorname* { s u p } _ { f \\in { \\mathcal { F } } _ { V } } { \\mathcal { R } } ( \\hat { f } _ { n } ( x ) ) \\geq \\operatorname* { i n f } _ { \\hat { f } _ { n } } \\operatorname* { s u p } _ { f \\in { \\mathcal { F } } _ { 0 } } { \\mathcal { R } } ( \\hat { f } _ { n } ( x ) ) \\gtrsim V { \\sqrt { \\frac { d } { n } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            434,
+            669,
+            472
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This concludes the proof. ",
+        "bbox": [
+            174,
+            477,
+            341,
+            492
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A.4 PROOF OF PROPOSITION 3.3 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            505,
+            415,
+            520
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To prove the proposition, it is sufficient to verify the following Rademacher complexity bound ",
+        "bbox": [
+            173,
+            526,
+            790,
+            542
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f76ce8f65afdbadc182693218847da331d1826cb0387448a85b9124e80ae547c.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname { s u p } \\bigg | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w ^ { \\top } z _ { i } + b ) \\bigg | \\lesssim \\sqrt { k \\log d \\log n } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            547,
+            656,
+            588
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "which can be derived easily by adjusting the proof in Lemma 2.3. Then the result follows with a similar analysis as in Theorem 3.1. ",
+        "bbox": [
+            174,
+            592,
+            823,
+            622
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A.5 PROOF OF THEOREM 3.4 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            633,
+            390,
+            648
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "It can be verified from the identity (9) that ",
+        "bbox": [
+            174,
+            656,
+            452,
+            671
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52135cd563220dee21c731bc8b1cafd6a83e80c2708812f4d5b7856b0cc2d7fe.jpg",
+        "text": "$$\n\\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( x _ { i } ) \\right| \\leq \\sum _ { j = 0 } ^ { r } \\mathbb { E } \\underset { f \\in \\mathcal { F } _ { V } } { \\operatorname* { s u p } } \\left| a _ { j } \\right| \\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w _ { j } ^ { \\top } x _ { i } + b _ { j } ) \\right| .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            676,
+            723,
+            719
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Then according to Lemma A.1, we have ",
+        "bbox": [
+            174,
+            724,
+            441,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/757e557b35a64840d1e45f887db281807c2d050ae865eae3d353252e606e8de4.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname* { s u p } _ { f \\in { \\mathcal F } _ { V } } \\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } \\sigma ( w _ { j } ^ { \\top } x _ { i } + b _ { j } ) \\right| \\lesssim \\sqrt { \\frac { \\log n } { n } } ( \\| w _ { j } \\| _ { \\mathbb { X } } + | b _ { j } | ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            744,
+            696,
+            785
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Combining (16) and (17), we obtain the following lemma that may be interesting on its own right. ",
+        "bbox": [
+            173,
+            790,
+            815,
+            805
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.3. We have ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            808,
+            325,
+            821
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/950f80ff61cfb487e3d57d650e4c5624130a8086a84ebc8d372e6aeb36fa1017.jpg",
+        "text": "$$\n\\mathbb { E } \\operatorname* { s u p } _ { f \\in \\mathcal { F } _ { V } } \\left| \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\xi _ { i } f ( x _ { i } ) \\right| \\lesssim \\sqrt { \\frac { \\log n } { n } } \\sum _ { j = 0 } ^ { r } | a _ { j } | ( \\| w _ { j } \\| _ { \\mathfrak { X } } + | b _ { j } | ) \\lesssim V \\sqrt { \\frac { \\log n } { n } } \\operatorname* { m a x } _ { j } \\| w _ { j } \\| _ { \\mathfrak { X } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            825,
+            779,
+            868
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Since $\\| w \\| _ { \\mathbb { X } } \\lesssim \\| w \\| _ { 1 }$ and $\\{ w : \\| w \\| _ { \\mathbb { X } } \\lesssim \\eta \\} \\subset \\{ w : \\| w \\| _ { 1 } \\lesssim \\eta \\}$ , the $\\| \\cdot \\| _ { \\mathbb { X } }$ can be replaced with $\\| \\cdot \\| _ { 1 }$ in the bounds in Lemmas A.3 and A.2. Then, with a similar argument as in the proof of Theorem 3.1, we conclude the proof of Theorem 3.4. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 9
+    }
+]

parse/train/7J-fKoXiReA/7J-fKoXiReA.md

@@ -0,0 +1,266 @@
+# Does Knowledge Distillation Really Work?
+
+Samuel Stanton NYU
+
+Pavel Izmailov NYU
+
+Polina Kirichenko NYU
+
+Alexander A. Alemi Google Research
+
+Andrew Gordon Wilson NYU
+
+# Abstract
+
+Knowledge distillation is a popular technique for training a small student network to emulate a larger teacher model, such as an ensemble of networks. We show that while knowledge distillation can improve student generalization, it does not typically work as it is commonly understood: there often remains a surprisingly large discrepancy between the predictive distributions of the teacher and the student, even in cases when the student has the capacity to perfectly match the teacher. We identify difficulties in optimization as a key reason for why the student is unable to match the teacher. We also show how the details of the dataset used for distillation play a role in how closely the student matches the teacher — and that more closely matching the teacher paradoxically does not always lead to better student generalization.
+
+# 1 Introduction
+
+Large, deep networks can learn representations that generalize well. While smaller, more efficient networks lack the inductive biases to find these representations from training data alone, they may have the capacity to represent these solutions [e.g., 2, 18, 32, 45]. Influential work on knowledge distillation $\overline { { \| 2 2 } }$ argues that Bucila et al. ˘ [5] “demonstrate convincingly that the knowledge acquired by a large ensemble of models [the teacher] can be transferred to a single small model [the student]”. Indeed this quote encapsulates the conventional narrative of knowledge distillation: a student model learns a high-fidelity representation of a larger teacher, enabled by the teacher’s soft labels.
+
+Conversely, in Figure $\perp$ we show that with modern architectures knowledge distillation can lead to students with very different predictions from their teachers, even when the student has the capacity to perfectly match the teacher. Indeed, it is becoming well-known that in self-distillation the student fails to match the teacher and, paradoxically, student generalization improves as a result [14, 40]. However, when the teacher is a large model (e.g. a deep ensemble) improvements in fidelity translate into improvements in generalization, as we show in Figure $1 ( \mathsf { b } )$ . For these large models there is still a significant accuracy gap between student and teacher, so fidelity is aligned with generalization.
+
+We will distinguish between fidelity, the ability of a student to match a teacher’s predictions, and generalization, the performance of a student in predicting unseen, in-distribution data. We show that in many cases it is surprisingly difficult to obtain good student fidelity. In Section 5 we investigate the hypothesis that low fidelity is an identifiability problem that can be solved by augmenting the distillation dataset. In Section $\boxed { 6 }$ we investigate the hypothesis that low fidelity is an optimization problem resulting in a failure of the student to match the teacher even on the original training dataset. We present a summary of our conclusions in Section 7.
+
+Does knowledge distillation really work? In short: Yes, in the sense that it often improves student generalization. No, in that knowledge distillation often fails to live up to its name, transferring very limited knowledge from teacher to student.
+
+![](images/b16f411144a106e66a8b6b067f80e336493e1572c29f9021f516793d8553eb0c.jpg)  
+Figure 1: Evaluating the fidelity of knowledge distillation. The effect of enlarging the CIFAR-100 distillation dataset with GAN-generated samples. (a): The student and teacher are both single ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases. (b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. Student fidelity again increases as the dataset grows, but test accuracy now slightly increases. The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials.
+
+# 2 Related Work
+
+Knowledge distillation can improve model efficiency [38, 45], unsupervised domain adaptation [37], improved object detection $\pmb { \Vert }$ , model transparency $\lVert \rVert \bigotimes \rVert$ , and adversarial robustness [15, 42].
+
+Seminal work by Bucila et al. ˘ [5] showed that teacher-ensembles with thousands of simple components could be compressed into a single shallow network that matched or outperformed its teacher. Other early work proposed distilling ensembles of shallow networks into a single network [55], an idea which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53]. Recently Fakoor et al. [13] developed a data-augmentation scheme for the distillation of large ensembles of simple models for tabular data, achieving impressive results on a wide range of tabular benchmarks. Malinin et al. [35] proposed a method to model the implicit distribution over predictive distributions from which the ensemble component predictive distributions are drawn, rather than just the ensemble model average.
+
+Our work focuses explicitly on student fidelity, decoupling our understanding of good fidelity from good generalization. We show that achieving good fidelity is extremely difficult, even with a variety of interventions, and seek to understand, by systematically considering several hypotheses, why knowledge distillation does not produce high fidelity students for modern architectures and datasets. In contrast, the distillation literature focuses largely on improving student generalization, without particularly distinguishing between fidelity and generalization.
+
+For example, concurrent work by Beyer et al. [4] does not carefully distinguish generalization and fidelity metrics, but they assert that high student fidelity is conceptually desirable and apparently difficult to achieve when measured as the gap between teacher and student accuracy. As a result their work focuses most heavily on practical modifications to the distillation procedure for the best student top-1 accuracy. In this paper we investigate many of the same prescriptions, including careful treatment of data augmentation (such as showing the teacher and student the exact same input images), the addition of MixUp, and extended training duration. We also find that such interventions do improve student accuracy, but there still remains a large discrepancy between the predictive distributions of the teacher and the student. We also investigate multiple optimizers. While we do not pursue Shampoo $\mathbb { I Z } \mathbb { I I }$ specifically, Beyer et al. $\pmb { \Vert 4 \Vert }$ find similar qualitative results for Shampoo and Adam, besides faster convergence for Shampoo.
+
+# 3 Preliminaries
+
+We will focus on the supervised classification setting, with input space $\mathcal { X }$ and label space $\mathcal { V }$ , where $| { \mathcal { V } } | = c$ . Let $f : \mathcal { X } \times \Theta  \mathbb { R } ^ { c }$ be a classifier parameterized by $\theta \in \Theta$ whose outputs define a categorical predictive distribution over $\mathcal { V }$ , $\hat { p } ( y = i | \mathbf { x } ) = \sigma _ { i } ( f ( \mathbf { x } , \theta ) )$ , where $\sigma _ { i } ( { \bf z } ) : = \dot { \exp ( z _ { i } ) } / \sum _ { j } \exp \bar { ( } z _ { j } )$ is the softmax link function. We will often refer to the outputs of a classifier $\mathbf { z } : = f ( \mathbf { x } , \theta )$ as logits. For convenience, we will use $t$ and $s$ as shorthand for $f _ { \mathrm { t e a c h e r } }$ and $f _ { \mathrm { s t u d e n t } }$ , respectively. When the teacher is an $m$ -component ensemble, the component logits $\left( \mathbf { z } _ { 1 } , \ldots , \mathbf { z } _ { m } \right)$ , where $\mathbf { z } _ { i } = f _ { i } ( \mathbf { x } , \theta _ { i } )$ , are combined to form the teacher logits: $\begin{array} { r } { \mathbf { z } _ { t } = \log \bar { ( \sum _ { i = 1 } ^ { m } \sigma ( \mathbf { \bar { z } } _ { i } ) / m ) } } \end{array}$ . These combined logits correspond to the predictive distribution of the ensemble model average. The experiments in the main text consider $m \in \{ 1 , 3 , 5 \}$ , and we include results up to $m = 1 2$ in Appendix B.2.1
+
+# 3.1 Knowledge Distillation
+
+Hinton et al. $[ [ 2 2 ] ]$ proposed a simple approach to knowledge distillation. The student minimizes a weighted combination of two objectives, $\mathcal { L } _ { s } : = \alpha \mathcal { L } _ { \mathrm { N L L } } + ( 1 - \alpha ) \mathcal { L } _ { \mathrm { K D } }$ , where $\alpha \in [ 0 , 1 )$ . Specifically,
+
+$$
+\mathcal { L } _ { \mathrm { N L L } } ( \mathbf { z } _ { s } , \mathbf { y } ) : = - \sum _ { j = 1 } ^ { c } y _ { j } \log \sigma _ { j } ( \mathbf { z } _ { s } ) , ~ \mathcal { L } _ { \mathrm { K D } } ( \mathbf { z } _ { s } , \mathbf { z } _ { t } ) : = - \tau ^ { 2 } \sum _ { j = 1 } ^ { c } \sigma _ { j } \left( \frac { \mathbf { z } _ { t } } { \tau } \right) \log \sigma _ { j } \left( \frac { \mathbf { z } _ { s } } { \tau } \right) .
+$$
+
+$\mathcal { L } _ { \mathrm { N L L } }$ is the usual supervised cross-entropy between the student logits $\mathbf { z } _ { s }$ and the one-hot labels $\mathbf { y }$ . Recalling that $\begin{array} { r } { \mathrm { K L } ( p | | q ) = \sum _ { j } p _ { j } ( \log q _ { j } - \log p _ { j } ) } \end{array}$ , we see that $\mathcal { L } _ { \mathrm { N L L } }$ is equivalent (up to a constant) to the KL from the empirical data distribution to the student predictive distribution $( \hat { p } _ { s } )$ . ${ \mathcal { L } } _ { \mathrm { K D } }$ is the added knowledge distillation term that encourages the student to match the teacher. It is the cross-entropy between the teacher and student predictive distributions $\hat { p } _ { t } = \sigma ( \mathbf { z } _ { t } )$ and $\hat { p } _ { s } = \sigma ( { \bf z } _ { s } )$ , both scaled by a temperature hyperparameter $\tau > 0$ . If $\tau = 1$ then ${ \mathcal { L } } _ { \mathrm { K D } }$ is similarly equivalent to the KL from the teacher to the student, $\mathrm { K L } ( \hat { p } _ { t } | | \hat { p } _ { s } )$ . Since we focus on distillation fidelity, we choose $\alpha = 0$ for all experiments in the main text to avoid any confounding from true labels, but we also include a limited ablation of $\alpha$ in Figure $\boxed { 1 4 }$ in Appendix $C . 5$ for the curious reader.
+
+As $\tau \to + \infty$ , $\nabla _ { \mathbf { z } _ { s } } \mathcal { L } _ { \mathrm { K D } } ( \mathbf { z } _ { s } , \mathbf { z } _ { t } ) \approx \mathbf { z } _ { t } - \mathbf { z } _ { s }$ , and thus in the limit $\nabla _ { \mathbf { z } _ { s } } \mathcal { L } _ { \mathrm { K D } }$ is approximately equivalent to $\nabla _ { \mathbf { z } _ { s } } | | \mathbf { z } _ { t } - \mathbf { z } _ { s } | | _ { 2 } ^ { 2 } / 2$ , assigning equal significance to every class logit, regardless of its contribution to the predictive distribution. In other words $\tau$ determines the “softness” of the teacher labels, which in turn determines the allocation of student capacity. If the student is much smaller than the teacher, the student capacity can be focused on matching the teacher’s top- $k$ predictions, rather than matching the full teacher distribution by choosing a moderate value (e.g. $\tau = 4$ ). In Appendix ${ \bf B . l }$ we include further discussion on the interplay of teacher ensemble size, teacher network capacity, and distillation temperature on the student labels.
+
+The teacher and student often share at least some training data. It is also common to enlarge the student training data in some way (e.g. incorporating unlabeled examples as in Ba and Caruana $\pmb { \mathbb { D } } \mathbf { l }$ ). When there is a possibility of confusion, we will refer to the student’s training data as the distillation data to distinguish it from the teacher’s training data.
+
+# 3.2 Metrics and Evaluation
+
+To measure generalization, we report top-1 accuracy, negative log-likelihood (NLL) and expected calibration error (ECE) $\boxed { 1 1 6 }$ . To measure fidelity, we report the following:
+
+$$
+\begin{array} { r l } & { \displaystyle \mathrm { A v e r a g e ~ T o p - 1 ~ A g r e e m e n t : } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } 1 \{ \mathrm { a r g m a x } \sigma _ { j } ( \mathbf { z } _ { t , i } ) = \underset { j } { \mathrm { a r g m a x } } \sigma _ { j } ( \mathbf { z } _ { s , i } ) \} , } \\ & { \displaystyle \mathrm { A v e r a g e ~ P r e d i c t i v e ~ K L : } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathrm { K L } \left( \hat { p } _ { t } ( \mathbf { y } | \mathbf { x } _ { i } ) \parallel \hat { p } _ { s } ( \mathbf { y } | \mathbf { x } _ { i } ) \right) , } \end{array}
+$$
+
+Eqn. $( 2 )$ is the average agreement between the student and teacher’s top-1 label. Eqn. $\textcircled{3}$ is the average KL divergence from the predictive distribution of the teacher to that of the student, a measure of fidelity sensitive to all of the labels.
+
+While improvements in generalization metrics are relatively easy to understand, interpreting fidelity metrics requires some care. For example, suppose we have three independent models: $f _ { 1 } , f _ { 2 }$ , and $f _ { 3 }$ that respectively achieve $55 \%$ , $7 5 \%$ , and $9 5 \%$ test accuracy. $f _ { 1 }$ and $f _ { 3 }$ can agree on at most $60 \%$ of points, whereas $f _ { 2 }$ and $f _ { 3 }$ agree on at least $70 \%$ , but it would obviously be incorrect to make any claim about $f _ { 2 }$ being a better distillation of $f _ { 3 }$ since each model was trained completely independently. To account for such confounding when evaluating the distillation of a student $s$ from a teacher $t$ , we also evaluate another student $s ^ { \prime }$ distilled through an identical procedure from an independent teacher.
+
+By comparing the fidelity of $( t , s )$ and $( t , s ^ { \prime } )$ we can distinguish between a generic improvement in generalization and an improvement specifically to fidelity. If $s$ and $s ^ { \prime }$ have comparable fidelity, then the students agree with the teacher at many points because they generalize well, and not the reverse.
+
+# 4 Knowledge Distillation Transfers Knowledge Poorly
+
+In this section, we present evidence that we are not able to distill large networks such as a ResNet-56 with high fidelity, and discuss why high fidelity is an important objective.
+
+# 4.1 When is knowledge transfer successful?
+
+We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as a motivating example. We train the teacher on a random subset of 200 examples from the MNIST training set for 100 epochs, resulting in a $8 4 \%$ to $8 6 \%$ teacher test accuracy across different subsets.2 We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as $2 5 \%$ , $50 \%$ , and $100 \%$ of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images.
+
+In Figure $2$ we see that knowledge distillation works as expected. With enough examples the student learns to make the same predictions as the teacher (over $9 9 \%$ top-1 test agreement). Notably, in this case, self-distillation does not improve generalization, since the slight difference between the teacher and student accuracy is explained by variance between trials.
+
+Now we consider a more challenging task: distilling a ResNet-56 teacher trained on CIFAR-100 into an identical student network (Figure $^ { 1 , }$ left). Since no dataset drawn from the same distribution as CIFAR-100 is publicly available, to augment the distillation data, we instead combined samples from an SN-GAN $\textcircled { \ 3 9 }$ pre-trained on CIFAR-100 with the original CIFAR-100 train dataset. Appendix A.3 details the hyperparameters and training procedure for the GAN, teacher, and student.
+
+Like the MNIST experiment, as we enlarge the distillation dataset the student fidelity improves. However, in this case the improvement is modest, with the fidelity reaching nowhere near $9 9 \%$ test agreement. Since a ResNet-56 has many more parameters than a LeNet-5, it is possible that the student simply has not seen enough examples to perfectly emulate the teacher, a hypothesis we discuss in more detail in Section $\underline { { \boldsymbol { \mathsf { F . 1 } } } } \big \| .$ Also, like the MNIST experiment, as the distillation dataset grows the student accuracy approaches the teacher’s. Unlike the MNIST experiment, the student test accuracy is higher than the teacher’s when the distillation dataset is small, so increasing fidelity decreases student generalization.
+
+![](images/346544e04a6a5855d2eed11bdd0190beeb1adbb82c1cd34a0f17fe19958b3f37.jpg)  
+Figure 2: LeNet-5 self-distillation on MNIST with additional distillation data. The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials.
+
+# 4.2 What can self-distillation tell us about knowledge distillation in general?
+
+We have seen in Figure $\mathbb { U } ( { \mathrm { a } } )$ that with self-distillation the student can exceed the teacher performance, in accordance with Furlanello et al. [14]. This result is only possible by virtue of failing at the distillation procedure: if the student matched the teacher perfectly then the student could not outperform the teacher. On the other hand, if the teacher generalizes significantly better than an independently trained student, we would expect the benefits of fidelity to dominate other regularization effects associated with not matching the teacher. This setting reflects the original motivation for knowledge distillation, where we wish to faithfully transfer the representation discovered by a large model or ensemble of models into a more efficient student.
+
+In Figure $\mathbb { M } ( { \mathsf { b } } )$ we see that if we move from self-distillation to the distillation of a 3 ResNet-56 teacher ensemble, fidelity becomes positively correlated with generalization. But there is still a significant gap in fidelity, even after the distillation set is enlarged with $5 0 k$ GAN samples. In practice, the gap remains large enough that higher fidelity students do not always have better generalization, and the regularization effects we see in self-distillation do play a role for more broadly understanding student generalization. We will indeed show in Section $\boxed { 5 }$ that higher fidelity students do not always generalize better, even if the teacher generalizes much better than the student.
+
+![](images/a6142c0a9a06255373aeb1f38774818c84e68662d3c4d427eee5d2f1eff00a17.jpg)  
+Figure 3: Data augmentation and distillation: Test accuracy and teacher-student agreement when distilling a 5-component ResNet-56 teacher ensemble into a ResNet-56 student on CIFAR-100 with varying augmentation policies. The best performing policy is shown in green, results averaged over 3 runs. Additional metrics are reported in Figure $1 \bar { 1 }$ in Appendix $\mathbf { C } .$ Mixup and GAN augmentation provide the best generalization, and Mixup $\tau = 4$ ) provides the best fidelity. The baseline policy (crops and flips) with $\tau = 4$ is a surprisingly strong baseline. The error bars indicate $\pm \sigma$ .
+
+# 4.3 If distillation already improves generalization, why care about fidelity?
+
+While knowledge distillation does often improve generalization, understanding the relationship between fidelity and generalization, and how to maximize fidelity, is important for several reasons — including better generalization!
+
+Better generalization in distilling large teacher models and ensembles. Knowledge distillation was initially motivated as a means to deploy powerful models to small devices or low-latency controllers [e.g., 10, 21, 26, 52, 54]. While in self-distillation generalization and fidelity are in tension, there is often a significant disparity in generalization between large teacher models, including ensembles, and smaller students. We have seen this disparity in Figure $\bar { \mathbb { M } } ( { \mathfrak { b } } )$ . We additionally show in Figure 10 in Appendix $\underline { { \overline { { \mathbf { B . l } } } } } ]$ that as we increase the number of ensemble components, the generalization disparity between teacher and distilled student increases. Improving student fidelity is the most obvious way to close the generalization disparity between student and teacher in these settings. Even if one exclusively cares about student accuracy, fidelity is a key consideration outside self-distillation.
+
+Interpretability and reliability. Knowledge distillation has been identified as a means to transfer representations discovered by large black-box models into simpler more interpretable models, for example to provide insights into medical diagnostics, or discovering rules for understanding sentiment in text [e.g., 23, 24, 6, 33, 8]. The ability to perform this transfer could have extraordinary scientific consequences: large models can often discover structure in data that we would not have anticipated a priori. Moreover, we often want to transfer properties such as well-calibrated uncertainties or robustness, which have been well-established for larger models, so that we can safely deploy more efficient models in their place. In both cases, achieving good distillation fidelity is crucial.
+
+Understanding. The name knowledge distillation implies we are transferring knowledge from the teacher to the student. For this reason, improved student generalization as a consequence of a distillation procedure is sometimes conflated with fidelity. Decoupling fidelity and generalization, and explicitly studying fidelity, is foundational to understanding how knowledge distillation works and how we can make it more useful across a variety of applications.
+
+# 4.4 Possible causes of low distillation fidelity
+
+If we are able to match the student model to the teacher on a comprehensive distillation dataset, we expect it to match on the test data as well, achieving high distillation fidelity3. Possible causes of the poor distillation fidelity in our CIFAR-100 experiments include:
+
+![](images/8a2a6126b32fff96f219ed1e9e4115bce416ee0fae8ece2522be6c23d9e96dca.jpg)  
+Figure 4: Data recycling and distillation: results on subsampled CIFAR-100. Top: We fix the temperature $( \tau = 4$ ) and vary the number of ensemble components $( m )$ , comparing students distilled on the same dataset as the teacher $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 0 } )$ , a reserved dataset $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 1 } )$ , or both $( \mathcal { D } _ { 0 } / \mathcal { D } _ { 0 } \cup \mathcal { D } _ { 1 } )$ . Distilling on both produces the best result, while distilling on $\mathcal { D } _ { 0 }$ increases accuracy and decreases fidelity, relative to $\mathcal { D } _ { 1 }$ . Bottom: We repeat the experiment, but fix $m = 3$ and vary $\tau$ . The shaded region corresponds to $\mu \pm \sigma$ , estimated over 3 trials.
+
+Student capacity – We observe low fidelity even in the self-distillation setting, so we can rule out student capacity as a primary cause, but we also confirm in Figure 12 in Appendix $\mathbb { E . l }$ that increasing the student capacity has very little effect on fidelity in the ensemble-distillation setting.
+
+Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2.
+
+Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing that our findings apply to datasets of larger scale and complexity.
+
+Data domain – Similarly in Section $\boxed { C . 4 }$ we observe low distillation fidelity in the context of text classification (sentiment analysis on the IMDB dataset), showing our results are relevant beyond image classification.
+
+Identifiability (Section $5$ ) – the distillation data is insufficient to distinguish high-fidelity and lowfidelity students. In other words, matching the teacher predictions on the distillation dataset does not lead to matching predictions on the test data.
+
+Optimization (Section 6) – we are unable to solve the distillation optimization problem sufficiently well. The student does not agree with the teacher on test because it does not even agree on train.
+
+# 5 Identifiability: Are We Using the Right Distillation Dataset?
+
+We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student enough of the right input-teacher label pairs to define the solution we want?
+
+# 5.1 Should we do more data augmentation?
+
+Data augmentation is a simple and practical method to increase the support of the distillation data distribution. If identifiability is a primary cause of poor distillation fidelity, using a more extensive data augmentation strategy during distillation should improve fidelity.
+
+To test this hypothesis, we evaluated the effect of several augmentation strategies on student fidelity and generalization. In Figure $^ { 3 , }$ the teacher is a 5-component ensemble of ResNet-56 networks trained on CIFAR-100 with the Baseline augmentation strategy: horizontal flips and random crops.
+
+We report the student accuracy and teacher-student agreement for each augmentation strategy, and also include results for Baseline with $\tau = 1$ and $\tau = 4$ to demonstrate the effect of logit tempering.
+
+We first observe that the best augmentation policies for generalization, $M i x U p$ , and $G A N \mathbb { H }$ are not the best policies for fidelity. Furthermore, although many augmentation strategies enable slightly higher distillation fidelity compared to Baseline $\tau = 1 .$ ), even the best augmentation policy, Mixup $\tau = 4 ,$ ), only achieves a modest $86 \%$ test agreement. In fact the Baseline $\tau = 4 ,$ ) policy is quite competitive, achieving $8 4 . 5 \%$ test agreement. Many of the augmentation strategies also slightly improve teacher-student KL relative to Baseline $\tau = 4$ ) (see Figure $^ { 1 1 ) }$
+
+In Figure 11 in Appendix ${ \bf B } . 3$ we report all generalization and fidelity metrics for a range of ensemble sizes, as well as the results for the independent student baseline discussed in Section $\underline { { \bar { 3 . 2 } } }$ Often these independent students, taught how to mimic a completely different model, have nearly as good test agreement with the teacher as the student explicitly trained to emulate it. See Appendix $\mathbf { \bar { A } } . 1$ for a detailed description of the augmentation procedures.
+
+Should data augmentation be close to the data distribution? In theory, any data augmentation should help with identifiability: if a student matches a teacher on more data, it is more likely to match the teacher elsewhere. However, the Noise and $O O D$ augmentation strategies based on noise and outof-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice, data augmentation has an effect beyond improving identifiability — it has a regularizing effect, making optimization more challenging. We explore this facet of data augmentation in Section 6.
+
+The slight improvements to fidelity with extensive augmentations suggest that increasing the support of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so small compared to heuristics like logit tempering (which does not modify the support at all), it is very unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity.
+
+# 5.2 The data recycling hypothesis
+
+If simply showing the student more labels does not always significantly improve fidelity, perhaps we are not showing the student the right labels. Additional data augmentation during distillation does give the student more teacher labels to match, but also introduces a distribution shift between the images the teacher was trained on and the images the student is distilling on. Even when the teacher and student have the same augmentation policy, reusing the teacher’s training data for distillation violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an independent draw from the true joint distribution over images and teacher labels. What if there was no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test distribution over images and teacher labels?
+
+To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100 training dataset $\mathcal { D }$ into two equal parts, $\mathcal { D } _ { 0 }$ and $\mathcal { D } _ { 1 }$ . We train teacher ResNet-56 ensembles on $\mathcal { D } _ { 0 }$ , and then compare $s _ { 0 }$ , a student distilled on the original $\mathcal { D } _ { 0 }$ , $s _ { 1 }$ , a student distilled on the unseen $\mathcal { D } _ { 1 }$ , and $s _ { 0 \cup 1 }$ , a student distilled on both: $\mathcal { D } _ { 0 } \cup \mathcal { D } _ { 1 }$ . Note that the students cannot access the true labels, only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in the top row and the logit temperature in the bottom row.
+
+Surprisingly, $s _ { 0 }$ attains higher test accuracy than $s _ { 1 }$ , while showing worse ECE and lower fidelity (measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis that $s _ { 1 }$ should be a higher fidelity distillation of the teacher than $s _ { 0 }$ does hold, but the gain in fidelity does not result in $s _ { 1 }$ best replicating the teacher’s accuracy. The best attributes of $s _ { 0 }$ and $s _ { 1 }$ are combined by $s _ { 0 \cup 1 }$ , which coincides with how unlabeled data is typically used in practice $\pmb { \left. 2 \right. }$ . The reason for this puzzling observation is simply that for the larger teachers fidelity has not improved enough to also improve generalization. In fact, the best teacher-student agreement is only around $8 5 \%$ , no improvement when compared to the results from extensive data augmentation in the last section. We again find that modifying the distillation data can slightly improve fidelity, but the evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data.
+
+![](images/ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg)  
+Figure 5: The train agreement for teacher ensembles $( m \in \{ 1 , 3 , 5 \} )$ ) and student on the distillation data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing the softness of the teacher labels by adding examples not in the teacher train data makes distillation more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle: agreement from subsampled CIFAR-100 experiment in Figure $\sharp$ Right: agreement for some of the augmentation policies in Figure 3. The shaded region is not visible because the variance is very low.
+
+# 6 Optimization: Does the Student Match the Teacher on Distillation Data?
+
+If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data, perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does the student match the teacher on the data it is trained to match it on?
+
+# 6.1 More distillation data lowers train agreement
+
+In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved teacher-student agreement on test, but the agreement remains relatively low (below $8 0 \%$ ) even for the largest distillation dataset that we considered. In Figure $5$ (left panel), we report the teacher-student agreement for the same experiment, but now on the distillation dataset. We now observe the opposite trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match the teacher. Even when the student has identical capacity to the teacher, the student only achieves $9 5 \%$ agreement with the teacher when we use $5 0 k$ synthetic images for distillation.
+
+The drop in train agreement is even more pronounced when we use extensive data augmentation. In Figure $\checkmark$ right panel, we report the teacher-student agreement on the train set with data augmentation for a subset of augmentation strategies presented in Section $\underline { { \boldsymbol { \mathsf { F . 1 } } } } \big \| .$ We use the CIFAR-100 dataset and the ResNet-56 model for the teachers and the students (for details, see Section $5 . 1 )$ . In each case, we measure agreement on the augmented training set that was used during distillation. While for the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color Jitter augmentations, the agreement is between $8 0 \%$ and $9 0 \%$ for all the considered teacher sizes. For Combined Augs, the combination of these three augmentation strategies, the agreement drops even further, to just $6 0 \%$ in self-distillation!
+
+Our intuition about how knowledge distillation should work largely hinges on the assumption that after distillation the student matches the teacher on the distillation set. However, the results presented in this section suggest that in practice the optimization method is unable to achieve high fidelity even on the distillation dataset when extensive data augmentation or synthetic data is used. The inability to solve the optimization problem undermines distillation: in order to find a student that would match the teacher on all inputs, we need to at least be able to find a student that would match the teacher on all of the distillation data.
+
+Optimization and the train-test fidelity gap. Notably, despite having the lowest train agreement, the Combined Augs policy results in better test agreement than other polices with better train agreement (Figure $3 )$ ). This result highlights a fundamental trade-off in knowledge distillation: the student needs many teacher labels match the teacher on test, but introducing examples not in the teacher train data makes matching the teacher on the distillation data very difficult.
+
+![](images/1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg)  
+Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves agreement, but it remains below $8 5 \%$ even after $5 k$ epochs. (b): Final train loss and agreement when the initialization is a convex combination of teacher and random weights, $\theta _ { s } = \lambda \theta _ { t } + \mathbf { \bar { ( } 1 - } \lambda ) \theta _ { r }$ . (c): Projections of the distillation loss surface on the plane intersecting $\theta _ { t }$ , the initial student weights, and the final student weights for different $\lambda$ . When $\lambda$ is small, the student converges to a suboptimal solution with low agreement. The uncertainty regions correspond to $\mu \pm \sigma$ , estimated over 3 trials.
+
+# 6.2 Why is train agreement so low?
+
+A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also replace the BatchNorm layers $\mathbb { \left. \overline { { 2 5 } } \right. }$ in ResNet-20 with LayerNorm $\pmb { \left[ \sqrt { 3 } \right] }$ , because we found that with BatchNorm layers even when the teacher and the student have identical weights, they can make different predictions due to differences in the activation statistics accumulated by the BatchNorm layers. Layer normalization does not collect any activation statistics, so the student will match the teacher as long as the weights coincide.
+
+Can we solve the optimization problem better? We verify that the distillation fidelity cannot be significantly improved by training longer or with a different optimizer. By default, in our experiments we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a weight decay value of $1 0 ^ { - 4 }$ . In Figure $\boxed { 6 }$ we report the results for the SGD and Adam $\mathbb { \ Z } \mathbb { 1 }$ optimizers run for $1 k$ and $5 k$ epochs without weight decay. Switching from SGD to Adam only reduced fidelity.
+
+For both optimizers, training for more epochs does slightly improve train agreement. In particular, with SGD we achieve $8 3 . 3 \%$ agreement when training for $5 k$ epochs compared to $7 8 . 9 5 \%$ when training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train agreement could reach $1 0 0 \%$ . However, training for $5 k$ epochs is significantly longer than what is typically done in practice (100 to 500 epochs). Furthermore, the improvement from $1 k$ to $5 k$ epochs is only about $2 \%$ , suggesting that we would need to train for tens of thousands of epochs, even in the optimistic case that agreement improves linearly, in order to get close to $1 0 0 \%$ train agreement.
+
+The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100 with any of the interventions we have discussed so far, we now ask if there is any modification of the problem that can produce a high-fidelity student.
+
+In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the distillation loss — the teacher’s own weights $\theta _ { t }$ . Letting $\theta _ { r }$ be a random weight initialization, in Figure $\boxed { 6 }$ (a) we examine the effect of choosing the student initialization to be a convex combination of the teacher and random weights, $\theta _ { s } = \lambda \bar { \theta _ { t } } + ( 1 - \lambda ) \theta _ { r }$ . After being initialized in this way, the student was trained as before. In other words $\lambda = 0$ corresponds to a random initialization and $\lambda = 1$ corresponds to initializing the student weights at the final teacher weights.
+
+We find that if the student is initialized far from the teacher $\lambda \leq 0 . 2 5 )$ , the optimizer converges to a sub-optimal value of the distillation loss, producing a student that significantly disagrees with the teacher. However at $\lambda = 0 . 3 7 5$ there is a sudden change. The final train loss drops to the optimal value and the agreement drastically increases, and the behavior continues for $\lambda > 0 . 3 7 5$ . To further investigate, in Figure $6 ( \mathrm { c ) }$ we visualize the distillation loss surface for $\lambda \in \{ 0 , 0 . 2 5 , 0 . 3 7 5 \}$ projected on the 2D subspace intersecting $\theta _ { t }$ , the initial student weights, and the final student weights. If the student is initialized far from the teacher $( \lambda \in \{ 0 , 0 . 2 5 \} )$ , it converges to a distinct, sub-optimal basin of the loss surface. On the other hand, when initialized close to the teacher $\lambda = 0 . 3 7 5$ ), the student converges to the same basin as the teacher, achieving nearly $100 \%$ agreement.
+
+Table 1: We examine whether fidelity can be improved in the context of ResNet-20 self-distillation on CIFAR-100 if the teacher and student share the same weight initialization. All metrics are computed on the test set. A shared initialization does make the student slightly more similar to the teacher in activation space (measured by CKA), but in function space the results are indistinguishable from randomly initialized students. We report the mean and standard deviation, estimated from 10 trials. The average teacher accuracy was 70.522 (0.412).   
+
+<table><tr><td colspan="3"></td><td colspan="3">CKA (1)</td></tr><tr><td>Init.</td><td>Agree. (↑)</td><td>KL (↓）</td><td>Stage 1</td><td> Stage 2</td><td>Stage 3</td></tr><tr><td>Rand.</td><td>77.174 (0.352)</td><td>0.836 (0.016)</td><td>0.939 (0.017)</td><td>0.925 (0.027)</td><td>0.885 (0.011)</td></tr><tr><td>Teach.</td><td>77.098 (0.238)</td><td>0.838 (0.020)</td><td>0.951 (0.017)</td><td>0.937 (0.020)</td><td>0.890 (0.015)</td></tr></table>
+
+Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained by initializing the student near the final teacher weights, it is possible that similar results could be obtained by initializing the student at the initial teacher weights. In Table $^ 1$ we compare students distilled from random initializations with those initialized at the initial teacher weights. In addition to the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA) $\pmb { \Vert 2 8 \Vert }$ of the preactivations of each of the teacher and student networks. There is a small increase in CKA, indicating that sharing an initialization between teacher and student does increase alignment in activation space, but functionally the students are identical to their randomly initialized counterparts – there is no observable change in accuracy, agreement, or predictive KL when compared to random initialization.
+
+To summarize, we have at last identified a root cause of the ineffectiveness of all our previous interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge to optimal student parameters, even when we know a solution and give the initialization a small head start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match the teacher on all inputs, the student has to at least match the teacher on the data used for distillation, and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to poor distillation fidelity.
+
+# 7 Discussion
+
+Our work provides several new key findings about knowledge distillation:
+
+• Good student accuracy does not imply good distillation fidelity: even outside of selfdistillation, the models with the best generalization do not always achieve the best fidelity. • Student fidelity is correlated with calibration when distilling ensembles: although the highest-fidelity student is not always the most accurate, it is always the best calibrated. • Optimization is challenging in knowledge distillation: even in cases when the student has sufficient capacity to match the teacher on the distillation data, it is unable to do so. • There is a trade-off between optimization complexity and distillation data quality: Enlarging the distillation dataset beyond the teacher training data makes it easier for the student to identify the correct solution, but also makes an already difficult optimization problem harder.
+
+In standard deep learning, we are saved by not needing to solve the optimization problem well: while it true that our training loss is highly multimodal, properties such as the flatness of good solutions, the inductive biases of the network, and the implicit biases of SGD, often enable good generalization in practice. In knowledge distillation, however, good fidelity is directly aligned with solving what turns out to be an exceptionally difficult optimization problem.
+
+# Acknowledgements
+
+The authors would like to thank Gregory Benton, Marc Finzi, Sanae Lotfi, Nate Gruver, and Ben Poole for helpful feedback. This research is supported by an Amazon Research Award, NSF I-DISRE 193471, NIH R01DA048764-01A1, NSF IIS-1910266, and NSF 1922658NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science. Samuel Stanton is also supported by a United States Department of Defense NDSEG fellowship.
+
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parse/train/7J-fKoXiReA/7J-fKoXiReA_content_list.json

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+    {
+        "type": "text",
+        "text": "Does Knowledge Distillation Really Work? ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Samuel Stanton NYU ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pavel Izmailov NYU ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Polina Kirichenko NYU ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alexander A. Alemi Google Research ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andrew Gordon Wilson NYU ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation is a popular technique for training a small student network to emulate a larger teacher model, such as an ensemble of networks. We show that while knowledge distillation can improve student generalization, it does not typically work as it is commonly understood: there often remains a surprisingly large discrepancy between the predictive distributions of the teacher and the student, even in cases when the student has the capacity to perfectly match the teacher. We identify difficulties in optimization as a key reason for why the student is unable to match the teacher. We also show how the details of the dataset used for distillation play a role in how closely the student matches the teacher — and that more closely matching the teacher paradoxically does not always lead to better student generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Large, deep networks can learn representations that generalize well. While smaller, more efficient networks lack the inductive biases to find these representations from training data alone, they may have the capacity to represent these solutions [e.g., 2, 18, 32, 45]. Influential work on knowledge distillation $\\overline { { \\| 2 2 } }$ argues that Bucila et al. ˘ [5] “demonstrate convincingly that the knowledge acquired by a large ensemble of models [the teacher] can be transferred to a single small model [the student]”. Indeed this quote encapsulates the conventional narrative of knowledge distillation: a student model learns a high-fidelity representation of a larger teacher, enabled by the teacher’s soft labels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Conversely, in Figure $\\perp$ we show that with modern architectures knowledge distillation can lead to students with very different predictions from their teachers, even when the student has the capacity to perfectly match the teacher. Indeed, it is becoming well-known that in self-distillation the student fails to match the teacher and, paradoxically, student generalization improves as a result [14, 40]. However, when the teacher is a large model (e.g. a deep ensemble) improvements in fidelity translate into improvements in generalization, as we show in Figure $1 ( \\mathsf { b } )$ . For these large models there is still a significant accuracy gap between student and teacher, so fidelity is aligned with generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will distinguish between fidelity, the ability of a student to match a teacher’s predictions, and generalization, the performance of a student in predicting unseen, in-distribution data. We show that in many cases it is surprisingly difficult to obtain good student fidelity. In Section 5 we investigate the hypothesis that low fidelity is an identifiability problem that can be solved by augmenting the distillation dataset. In Section $\\boxed { 6 }$ we investigate the hypothesis that low fidelity is an optimization problem resulting in a failure of the student to match the teacher even on the original training dataset. We present a summary of our conclusions in Section 7. ",
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+    {
+        "type": "text",
+        "text": "Does knowledge distillation really work? In short: Yes, in the sense that it often improves student generalization. No, in that knowledge distillation often fails to live up to its name, transferring very limited knowledge from teacher to student. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/b16f411144a106e66a8b6b067f80e336493e1572c29f9021f516793d8553eb0c.jpg",
+        "image_caption": [
+            "Figure 1: Evaluating the fidelity of knowledge distillation. The effect of enlarging the CIFAR-100 distillation dataset with GAN-generated samples. (a): The student and teacher are both single ResNet-56 networks. Student fidelity increases as the dataset grows, but test accuracy decreases. (b): The student is a single ResNet-56 network and the teacher is a 3-component ensemble. Student fidelity again increases as the dataset grows, but test accuracy now slightly increases. The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation can improve model efficiency [38, 45], unsupervised domain adaptation [37], improved object detection $\\pmb { \\Vert }$ , model transparency $\\lVert \\rVert \\bigotimes \\rVert$ , and adversarial robustness [15, 42]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Seminal work by Bucila et al. ˘ [5] showed that teacher-ensembles with thousands of simple components could be compressed into a single shallow network that matched or outperformed its teacher. Other early work proposed distilling ensembles of shallow networks into a single network [55], an idea which resonates with more recent work on the distillation of deep ensembles [2, 7, 46, 50, 53]. Recently Fakoor et al. [13] developed a data-augmentation scheme for the distillation of large ensembles of simple models for tabular data, achieving impressive results on a wide range of tabular benchmarks. Malinin et al. [35] proposed a method to model the implicit distribution over predictive distributions from which the ensemble component predictive distributions are drawn, rather than just the ensemble model average. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work focuses explicitly on student fidelity, decoupling our understanding of good fidelity from good generalization. We show that achieving good fidelity is extremely difficult, even with a variety of interventions, and seek to understand, by systematically considering several hypotheses, why knowledge distillation does not produce high fidelity students for modern architectures and datasets. In contrast, the distillation literature focuses largely on improving student generalization, without particularly distinguishing between fidelity and generalization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For example, concurrent work by Beyer et al. [4] does not carefully distinguish generalization and fidelity metrics, but they assert that high student fidelity is conceptually desirable and apparently difficult to achieve when measured as the gap between teacher and student accuracy. As a result their work focuses most heavily on practical modifications to the distillation procedure for the best student top-1 accuracy. In this paper we investigate many of the same prescriptions, including careful treatment of data augmentation (such as showing the teacher and student the exact same input images), the addition of MixUp, and extended training duration. We also find that such interventions do improve student accuracy, but there still remains a large discrepancy between the predictive distributions of the teacher and the student. We also investigate multiple optimizers. While we do not pursue Shampoo $\\mathbb { I Z } \\mathbb { I I }$ specifically, Beyer et al. $\\pmb { \\Vert 4 \\Vert }$ find similar qualitative results for Shampoo and Adam, besides faster convergence for Shampoo. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will focus on the supervised classification setting, with input space $\\mathcal { X }$ and label space $\\mathcal { V }$ , where $| { \\mathcal { V } } | = c$ . Let $f : \\mathcal { X } \\times \\Theta  \\mathbb { R } ^ { c }$ be a classifier parameterized by $\\theta \\in \\Theta$ whose outputs define a categorical predictive distribution over $\\mathcal { V }$ , $\\hat { p } ( y = i | \\mathbf { x } ) = \\sigma _ { i } ( f ( \\mathbf { x } , \\theta ) )$ , where $\\sigma _ { i } ( { \\bf z } ) : = \\dot { \\exp ( z _ { i } ) } / \\sum _ { j } \\exp \\bar { ( } z _ { j } )$ is the softmax link function. We will often refer to the outputs of a classifier $\\mathbf { z } : = f ( \\mathbf { x } , \\theta )$ as logits. For convenience, we will use $t$ and $s$ as shorthand for $f _ { \\mathrm { t e a c h e r } }$ and $f _ { \\mathrm { s t u d e n t } }$ , respectively. When the teacher is an $m$ -component ensemble, the component logits $\\left( \\mathbf { z } _ { 1 } , \\ldots , \\mathbf { z } _ { m } \\right)$ , where $\\mathbf { z } _ { i } = f _ { i } ( \\mathbf { x } , \\theta _ { i } )$ , are combined to form the teacher logits: $\\begin{array} { r } { \\mathbf { z } _ { t } = \\log \\bar { ( \\sum _ { i = 1 } ^ { m } \\sigma ( \\mathbf { \\bar { z } } _ { i } ) / m ) } } \\end{array}$ . These combined logits correspond to the predictive distribution of the ensemble model average. The experiments in the main text consider $m \\in \\{ 1 , 3 , 5 \\}$ , and we include results up to $m = 1 2$ in Appendix B.2.1 ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.1 Knowledge Distillation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hinton et al. $[ [ 2 2 ] ]$ proposed a simple approach to knowledge distillation. The student minimizes a weighted combination of two objectives, $\\mathcal { L } _ { s } : = \\alpha \\mathcal { L } _ { \\mathrm { N L L } } + ( 1 - \\alpha ) \\mathcal { L } _ { \\mathrm { K D } }$ , where $\\alpha \\in [ 0 , 1 )$ . Specifically, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb71c932a12c9ab68952314d721b8775d82ec4a2743705ae6677175ea85b298b.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { N L L } } ( \\mathbf { z } _ { s } , \\mathbf { y } ) : = - \\sum _ { j = 1 } ^ { c } y _ { j } \\log \\sigma _ { j } ( \\mathbf { z } _ { s } ) , ~ \\mathcal { L } _ { \\mathrm { K D } } ( \\mathbf { z } _ { s } , \\mathbf { z } _ { t } ) : = - \\tau ^ { 2 } \\sum _ { j = 1 } ^ { c } \\sigma _ { j } \\left( \\frac { \\mathbf { z } _ { t } } { \\tau } \\right) \\log \\sigma _ { j } \\left( \\frac { \\mathbf { z } _ { s } } { \\tau } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { L } _ { \\mathrm { N L L } }$ is the usual supervised cross-entropy between the student logits $\\mathbf { z } _ { s }$ and the one-hot labels $\\mathbf { y }$ . Recalling that $\\begin{array} { r } { \\mathrm { K L } ( p | | q ) = \\sum _ { j } p _ { j } ( \\log q _ { j } - \\log p _ { j } ) } \\end{array}$ , we see that $\\mathcal { L } _ { \\mathrm { N L L } }$ is equivalent (up to a constant) to the KL from the empirical data distribution to the student predictive distribution $( \\hat { p } _ { s } )$ . ${ \\mathcal { L } } _ { \\mathrm { K D } }$ is the added knowledge distillation term that encourages the student to match the teacher. It is the cross-entropy between the teacher and student predictive distributions $\\hat { p } _ { t } = \\sigma ( \\mathbf { z } _ { t } )$ and $\\hat { p } _ { s } = \\sigma ( { \\bf z } _ { s } )$ , both scaled by a temperature hyperparameter $\\tau > 0$ . If $\\tau = 1$ then ${ \\mathcal { L } } _ { \\mathrm { K D } }$ is similarly equivalent to the KL from the teacher to the student, $\\mathrm { K L } ( \\hat { p } _ { t } | | \\hat { p } _ { s } )$ . Since we focus on distillation fidelity, we choose $\\alpha = 0$ for all experiments in the main text to avoid any confounding from true labels, but we also include a limited ablation of $\\alpha$ in Figure $\\boxed { 1 4 }$ in Appendix $C . 5$ for the curious reader. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As $\\tau \\to + \\infty$ , $\\nabla _ { \\mathbf { z } _ { s } } \\mathcal { L } _ { \\mathrm { K D } } ( \\mathbf { z } _ { s } , \\mathbf { z } _ { t } ) \\approx \\mathbf { z } _ { t } - \\mathbf { z } _ { s }$ , and thus in the limit $\\nabla _ { \\mathbf { z } _ { s } } \\mathcal { L } _ { \\mathrm { K D } }$ is approximately equivalent to $\\nabla _ { \\mathbf { z } _ { s } } | | \\mathbf { z } _ { t } - \\mathbf { z } _ { s } | | _ { 2 } ^ { 2 } / 2$ , assigning equal significance to every class logit, regardless of its contribution to the predictive distribution. In other words $\\tau$ determines the “softness” of the teacher labels, which in turn determines the allocation of student capacity. If the student is much smaller than the teacher, the student capacity can be focused on matching the teacher’s top- $k$ predictions, rather than matching the full teacher distribution by choosing a moderate value (e.g. $\\tau = 4$ ). In Appendix ${ \\bf B . l }$ we include further discussion on the interplay of teacher ensemble size, teacher network capacity, and distillation temperature on the student labels. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The teacher and student often share at least some training data. It is also common to enlarge the student training data in some way (e.g. incorporating unlabeled examples as in Ba and Caruana $\\pmb { \\mathbb { D } } \\mathbf { l }$ ). When there is a possibility of confusion, we will refer to the student’s training data as the distillation data to distinguish it from the teacher’s training data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 Metrics and Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To measure generalization, we report top-1 accuracy, negative log-likelihood (NLL) and expected calibration error (ECE) $\\boxed { 1 1 6 }$ . To measure fidelity, we report the following: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a559a29bcef38ef5367b475572a19dc8ef744e6ac20aecd0747505152909e9c2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathrm { A v e r a g e ~ T o p - 1 ~ A g r e e m e n t : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } 1 \\{ \\mathrm { a r g m a x } \\sigma _ { j } ( \\mathbf { z } _ { t , i } ) = \\underset { j } { \\mathrm { a r g m a x } } \\sigma _ { j } ( \\mathbf { z } _ { s , i } ) \\} , } \\\\ & { \\displaystyle \\mathrm { A v e r a g e ~ P r e d i c t i v e ~ K L : } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathrm { K L } \\left( \\hat { p } _ { t } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\parallel \\hat { p } _ { s } ( \\mathbf { y } | \\mathbf { x } _ { i } ) \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Eqn. $( 2 )$ is the average agreement between the student and teacher’s top-1 label. Eqn. $\\textcircled{3}$ is the average KL divergence from the predictive distribution of the teacher to that of the student, a measure of fidelity sensitive to all of the labels. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While improvements in generalization metrics are relatively easy to understand, interpreting fidelity metrics requires some care. For example, suppose we have three independent models: $f _ { 1 } , f _ { 2 }$ , and $f _ { 3 }$ that respectively achieve $55 \\%$ , $7 5 \\%$ , and $9 5 \\%$ test accuracy. $f _ { 1 }$ and $f _ { 3 }$ can agree on at most $60 \\%$ of points, whereas $f _ { 2 }$ and $f _ { 3 }$ agree on at least $70 \\%$ , but it would obviously be incorrect to make any claim about $f _ { 2 }$ being a better distillation of $f _ { 3 }$ since each model was trained completely independently. To account for such confounding when evaluating the distillation of a student $s$ from a teacher $t$ , we also evaluate another student $s ^ { \\prime }$ distilled through an identical procedure from an independent teacher. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "By comparing the fidelity of $( t , s )$ and $( t , s ^ { \\prime } )$ we can distinguish between a generic improvement in generalization and an improvement specifically to fidelity. If $s$ and $s ^ { \\prime }$ have comparable fidelity, then the students agree with the teacher at many points because they generalize well, and not the reverse. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Knowledge Distillation Transfers Knowledge Poorly ",
+        "text_level": 1,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In this section, we present evidence that we are not able to distill large networks such as a ResNet-56 with high fidelity, and discuss why high fidelity is an important objective. ",
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+    {
+        "type": "text",
+        "text": "4.1 When is knowledge transfer successful? ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We first consider the easy task of distilling a LeNet-5 teacher into an identical student network as a motivating example. We train the teacher on a random subset of 200 examples from the MNIST training set for 100 epochs, resulting in a $8 4 \\%$ to $8 6 \\%$ teacher test accuracy across different subsets.2 We then distill the teacher using the full MNIST train dataset with 60,000 examples, as well as $2 5 \\%$ , $50 \\%$ , and $100 \\%$ of the EMNIST train dataset [11]. The EMNIST train set contains 697,932 images. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In Figure $2$ we see that knowledge distillation works as expected. With enough examples the student learns to make the same predictions as the teacher (over $9 9 \\%$ top-1 test agreement). Notably, in this case, self-distillation does not improve generalization, since the slight difference between the teacher and student accuracy is explained by variance between trials. ",
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+    {
+        "type": "text",
+        "text": "Now we consider a more challenging task: distilling a ResNet-56 teacher trained on CIFAR-100 into an identical student network (Figure $^ { 1 , }$ left). Since no dataset drawn from the same distribution as CIFAR-100 is publicly available, to augment the distillation data, we instead combined samples from an SN-GAN $\\textcircled { \\ 3 9 }$ pre-trained on CIFAR-100 with the original CIFAR-100 train dataset. Appendix A.3 details the hyperparameters and training procedure for the GAN, teacher, and student. ",
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+    {
+        "type": "text",
+        "text": "Like the MNIST experiment, as we enlarge the distillation dataset the student fidelity improves. However, in this case the improvement is modest, with the fidelity reaching nowhere near $9 9 \\%$ test agreement. Since a ResNet-56 has many more parameters than a LeNet-5, it is possible that the student simply has not seen enough examples to perfectly emulate the teacher, a hypothesis we discuss in more detail in Section $\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .$ Also, like the MNIST experiment, as the distillation dataset grows the student accuracy approaches the teacher’s. Unlike the MNIST experiment, the student test accuracy is higher than the teacher’s when the distillation dataset is small, so increasing fidelity decreases student generalization. ",
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+        "img_path": "images/346544e04a6a5855d2eed11bdd0190beeb1adbb82c1cd34a0f17fe19958b3f37.jpg",
+        "image_caption": [
+            "Figure 2: LeNet-5 self-distillation on MNIST with additional distillation data. The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. "
+        ],
+        "image_footnote": [],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 What can self-distillation tell us about knowledge distillation in general? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have seen in Figure $\\mathbb { U } ( { \\mathrm { a } } )$ that with self-distillation the student can exceed the teacher performance, in accordance with Furlanello et al. [14]. This result is only possible by virtue of failing at the distillation procedure: if the student matched the teacher perfectly then the student could not outperform the teacher. On the other hand, if the teacher generalizes significantly better than an independently trained student, we would expect the benefits of fidelity to dominate other regularization effects associated with not matching the teacher. This setting reflects the original motivation for knowledge distillation, where we wish to faithfully transfer the representation discovered by a large model or ensemble of models into a more efficient student. ",
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+    {
+        "type": "text",
+        "text": "In Figure $\\mathbb { M } ( { \\mathsf { b } } )$ we see that if we move from self-distillation to the distillation of a 3 ResNet-56 teacher ensemble, fidelity becomes positively correlated with generalization. But there is still a significant gap in fidelity, even after the distillation set is enlarged with $5 0 k$ GAN samples. In practice, the gap remains large enough that higher fidelity students do not always have better generalization, and the regularization effects we see in self-distillation do play a role for more broadly understanding student generalization. We will indeed show in Section $\\boxed { 5 }$ that higher fidelity students do not always generalize better, even if the teacher generalizes much better than the student. ",
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+        "img_path": "images/a6142c0a9a06255373aeb1f38774818c84e68662d3c4d427eee5d2f1eff00a17.jpg",
+        "image_caption": [
+            "Figure 3: Data augmentation and distillation: Test accuracy and teacher-student agreement when distilling a 5-component ResNet-56 teacher ensemble into a ResNet-56 student on CIFAR-100 with varying augmentation policies. The best performing policy is shown in green, results averaged over 3 runs. Additional metrics are reported in Figure $1 \\bar { 1 }$ in Appendix $\\mathbf { C } .$ Mixup and GAN augmentation provide the best generalization, and Mixup $\\tau = 4$ ) provides the best fidelity. The baseline policy (crops and flips) with $\\tau = 4$ is a surprisingly strong baseline. The error bars indicate $\\pm \\sigma$ . "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.3 If distillation already improves generalization, why care about fidelity? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While knowledge distillation does often improve generalization, understanding the relationship between fidelity and generalization, and how to maximize fidelity, is important for several reasons — including better generalization! ",
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+    {
+        "type": "text",
+        "text": "Better generalization in distilling large teacher models and ensembles. Knowledge distillation was initially motivated as a means to deploy powerful models to small devices or low-latency controllers [e.g., 10, 21, 26, 52, 54]. While in self-distillation generalization and fidelity are in tension, there is often a significant disparity in generalization between large teacher models, including ensembles, and smaller students. We have seen this disparity in Figure $\\bar { \\mathbb { M } } ( { \\mathfrak { b } } )$ . We additionally show in Figure 10 in Appendix $\\underline { { \\overline { { \\mathbf { B . l } } } } } ]$ that as we increase the number of ensemble components, the generalization disparity between teacher and distilled student increases. Improving student fidelity is the most obvious way to close the generalization disparity between student and teacher in these settings. Even if one exclusively cares about student accuracy, fidelity is a key consideration outside self-distillation. ",
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+    {
+        "type": "text",
+        "text": "Interpretability and reliability. Knowledge distillation has been identified as a means to transfer representations discovered by large black-box models into simpler more interpretable models, for example to provide insights into medical diagnostics, or discovering rules for understanding sentiment in text [e.g., 23, 24, 6, 33, 8]. The ability to perform this transfer could have extraordinary scientific consequences: large models can often discover structure in data that we would not have anticipated a priori. Moreover, we often want to transfer properties such as well-calibrated uncertainties or robustness, which have been well-established for larger models, so that we can safely deploy more efficient models in their place. In both cases, achieving good distillation fidelity is crucial. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Understanding. The name knowledge distillation implies we are transferring knowledge from the teacher to the student. For this reason, improved student generalization as a consequence of a distillation procedure is sometimes conflated with fidelity. Decoupling fidelity and generalization, and explicitly studying fidelity, is foundational to understanding how knowledge distillation works and how we can make it more useful across a variety of applications. ",
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+    {
+        "type": "text",
+        "text": "4.4 Possible causes of low distillation fidelity ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "If we are able to match the student model to the teacher on a comprehensive distillation dataset, we expect it to match on the test data as well, achieving high distillation fidelity3. Possible causes of the poor distillation fidelity in our CIFAR-100 experiments include: ",
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+    {
+        "type": "image",
+        "img_path": "images/8a2a6126b32fff96f219ed1e9e4115bce416ee0fae8ece2522be6c23d9e96dca.jpg",
+        "image_caption": [
+            "Figure 4: Data recycling and distillation: results on subsampled CIFAR-100. Top: We fix the temperature $( \\tau = 4$ ) and vary the number of ensemble components $( m )$ , comparing students distilled on the same dataset as the teacher $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } )$ , a reserved dataset $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 1 } )$ , or both $( \\mathcal { D } _ { 0 } / \\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 } )$ . Distilling on both produces the best result, while distilling on $\\mathcal { D } _ { 0 }$ increases accuracy and decreases fidelity, relative to $\\mathcal { D } _ { 1 }$ . Bottom: We repeat the experiment, but fix $m = 3$ and vary $\\tau$ . The shaded region corresponds to $\\mu \\pm \\sigma$ , estimated over 3 trials. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Student capacity – We observe low fidelity even in the self-distillation setting, so we can rule out student capacity as a primary cause, but we also confirm in Figure 12 in Appendix $\\mathbb { E . l }$ that increasing the student capacity has very little effect on fidelity in the ensemble-distillation setting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Network architecture – Low fidelity could be specific to ResNet-like architectures, an explanation we rule out by showing similar results with VGG networks [47] in Figure 13 in Appendix C.2. ",
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+    {
+        "type": "text",
+        "text": "Dataset scale and complexity – we provide similar results in Section C.3 for ImageNet, showing that our findings apply to datasets of larger scale and complexity. ",
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+    {
+        "type": "text",
+        "text": "Data domain – Similarly in Section $\\boxed { C . 4 }$ we observe low distillation fidelity in the context of text classification (sentiment analysis on the IMDB dataset), showing our results are relevant beyond image classification. ",
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+    {
+        "type": "text",
+        "text": "Identifiability (Section $5$ ) – the distillation data is insufficient to distinguish high-fidelity and lowfidelity students. In other words, matching the teacher predictions on the distillation dataset does not lead to matching predictions on the test data. ",
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+    {
+        "type": "text",
+        "text": "Optimization (Section 6) – we are unable to solve the distillation optimization problem sufficiently well. The student does not agree with the teacher on test because it does not even agree on train. ",
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+    {
+        "type": "text",
+        "text": "5 Identifiability: Are We Using the Right Distillation Dataset? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We investigate whether it is possible to attain the level of fidelity observed with LeNet-5s on MNIST with ResNets on CIFAR-100 by addressing the identifiability problem — have we shown the student enough of the right input-teacher label pairs to define the solution we want? ",
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+    {
+        "type": "text",
+        "text": "5.1 Should we do more data augmentation? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Data augmentation is a simple and practical method to increase the support of the distillation data distribution. If identifiability is a primary cause of poor distillation fidelity, using a more extensive data augmentation strategy during distillation should improve fidelity. ",
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+    {
+        "type": "text",
+        "text": "To test this hypothesis, we evaluated the effect of several augmentation strategies on student fidelity and generalization. In Figure $^ { 3 , }$ the teacher is a 5-component ensemble of ResNet-56 networks trained on CIFAR-100 with the Baseline augmentation strategy: horizontal flips and random crops. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We report the student accuracy and teacher-student agreement for each augmentation strategy, and also include results for Baseline with $\\tau = 1$ and $\\tau = 4$ to demonstrate the effect of logit tempering. ",
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+    {
+        "type": "text",
+        "text": "We first observe that the best augmentation policies for generalization, $M i x U p$ , and $G A N \\mathbb { H }$ are not the best policies for fidelity. Furthermore, although many augmentation strategies enable slightly higher distillation fidelity compared to Baseline $\\tau = 1 .$ ), even the best augmentation policy, Mixup $\\tau = 4 ,$ ), only achieves a modest $86 \\%$ test agreement. In fact the Baseline $\\tau = 4 ,$ ) policy is quite competitive, achieving $8 4 . 5 \\%$ test agreement. Many of the augmentation strategies also slightly improve teacher-student KL relative to Baseline $\\tau = 4$ ) (see Figure $^ { 1 1 ) }$ ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "In Figure 11 in Appendix ${ \\bf B } . 3$ we report all generalization and fidelity metrics for a range of ensemble sizes, as well as the results for the independent student baseline discussed in Section $\\underline { { \\bar { 3 . 2 } } }$ Often these independent students, taught how to mimic a completely different model, have nearly as good test agreement with the teacher as the student explicitly trained to emulate it. See Appendix $\\mathbf { \\bar { A } } . 1$ for a detailed description of the augmentation procedures. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Should data augmentation be close to the data distribution? In theory, any data augmentation should help with identifiability: if a student matches a teacher on more data, it is more likely to match the teacher elsewhere. However, the Noise and $O O D$ augmentation strategies based on noise and outof-distribution data fail on all metrics, decreasing performance compared to the baseline. In practice, data augmentation has an effect beyond improving identifiability — it has a regularizing effect, making optimization more challenging. We explore this facet of data augmentation in Section 6. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The slight improvements to fidelity with extensive augmentations suggest that increasing the support of the distillation dataset can indeed improve distillation fidelity. However, since the benefit is so small compared to heuristics like logit tempering (which does not modify the support at all), it is very unlikely that an insufficient quantity of teacher labels is the primary obstacle to high fidelity. ",
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+    {
+        "type": "text",
+        "text": "5.2 The data recycling hypothesis ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If simply showing the student more labels does not always significantly improve fidelity, perhaps we are not showing the student the right labels. Additional data augmentation during distillation does give the student more teacher labels to match, but also introduces a distribution shift between the images the teacher was trained on and the images the student is distilling on. Even when the teacher and student have the same augmentation policy, reusing the teacher’s training data for distillation violates the assumptions of empirical risk minimization (ERM) because the distillation data is not an independent draw from the true joint distribution over images and teacher labels. What if there was no augmentation distribution shift, and the student was distilled on a fresh draw from the joint test distribution over images and teacher labels? ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "To investigate the effect of recycling teacher data during distillation we randomly split the CIFAR-100 training dataset $\\mathcal { D }$ into two equal parts, $\\mathcal { D } _ { 0 }$ and $\\mathcal { D } _ { 1 }$ . We train teacher ResNet-56 ensembles on $\\mathcal { D } _ { 0 }$ , and then compare $s _ { 0 }$ , a student distilled on the original $\\mathcal { D } _ { 0 }$ , $s _ { 1 }$ , a student distilled on the unseen $\\mathcal { D } _ { 1 }$ , and $s _ { 0 \\cup 1 }$ , a student distilled on both: $\\mathcal { D } _ { 0 } \\cup \\mathcal { D } _ { 1 }$ . Note that the students cannot access the true labels, only those provided by the teacher. We present the results in Figure 4, varying the ensemble size in the top row and the logit temperature in the bottom row. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surprisingly, $s _ { 0 }$ attains higher test accuracy than $s _ { 1 }$ , while showing worse ECE and lower fidelity (measured by test teacher-student agreement and test teacher-student KL). Therefore, the hypothesis that $s _ { 1 }$ should be a higher fidelity distillation of the teacher than $s _ { 0 }$ does hold, but the gain in fidelity does not result in $s _ { 1 }$ best replicating the teacher’s accuracy. The best attributes of $s _ { 0 }$ and $s _ { 1 }$ are combined by $s _ { 0 \\cup 1 }$ , which coincides with how unlabeled data is typically used in practice $\\pmb { \\left. 2 \\right. }$ . The reason for this puzzling observation is simply that for the larger teachers fidelity has not improved enough to also improve generalization. In fact, the best teacher-student agreement is only around $8 5 \\%$ , no improvement when compared to the results from extensive data augmentation in the last section. We again find that modifying the distillation data can slightly improve fidelity, but the evidence does not support blaming poor distillation fidelity on the wrong choice of distillation data. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac9d8cf1ad2151d45710ce595488770dc75dd7b48df44c2e8e14393f25a3bc6d.jpg",
+        "image_caption": [
+            "Figure 5: The train agreement for teacher ensembles $( m \\in \\{ 1 , 3 , 5 \\} )$ ) and student on the distillation data for a ResNet-56 on CIFAR-100 under different augmentation policies. In all panels, increasing the softness of the teacher labels by adding examples not in the teacher train data makes distillation more difficult. Left: agreement for the synthetic GAN-augmentation policy from Figure 1. Middle: agreement from subsampled CIFAR-100 experiment in Figure $\\sharp$ Right: agreement for some of the augmentation policies in Figure 3. The shaded region is not visible because the variance is very low. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6 Optimization: Does the Student Match the Teacher on Distillation Data? ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "If poor fidelity is not primarily an identifiability problem from the wrong choice of distillation data, perhaps there is a simpler explanation. Up to this point, we have focused on student fidelity on a held-out test set. Now we turn our attention to student behavior on the distillation data itself. Does the student match the teacher on the data it is trained to match it on? ",
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+    {
+        "type": "text",
+        "text": "6.1 More distillation data lowers train agreement ",
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+    {
+        "type": "text",
+        "text": "In Figure 1 we presented an experiment distilling ResNet-56 networks on CIFAR-100 augmented with synthetic GAN-generated images. We saw that enlarging the distillation dataset leads to improved teacher-student agreement on test, but the agreement remains relatively low (below $8 0 \\%$ ) even for the largest distillation dataset that we considered. In Figure $5$ (left panel), we report the teacher-student agreement for the same experiment, but now on the distillation dataset. We now observe the opposite trend: as the distillation dataset becomes larger, it becomes more challenging for the student to match the teacher. Even when the student has identical capacity to the teacher, the student only achieves $9 5 \\%$ agreement with the teacher when we use $5 0 k$ synthetic images for distillation. ",
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+    {
+        "type": "text",
+        "text": "The drop in train agreement is even more pronounced when we use extensive data augmentation. In Figure $\\checkmark$ right panel, we report the teacher-student agreement on the train set with data augmentation for a subset of augmentation strategies presented in Section $\\underline { { \\boldsymbol { \\mathsf { F . 1 } } } } \\big \\| .$ We use the CIFAR-100 dataset and the ResNet-56 model for the teachers and the students (for details, see Section $5 . 1 )$ . In each case, we measure agreement on the augmented training set that was used during distillation. While for the baseline augmentation strategy, we can achieve almost perfect teacher-student agreement, for heavier augmentations the agreement drops dramatically. For the Rotation, Vertical Flip and Color Jitter augmentations, the agreement is between $8 0 \\%$ and $9 0 \\%$ for all the considered teacher sizes. For Combined Augs, the combination of these three augmentation strategies, the agreement drops even further, to just $6 0 \\%$ in self-distillation! ",
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+    {
+        "type": "text",
+        "text": "Our intuition about how knowledge distillation should work largely hinges on the assumption that after distillation the student matches the teacher on the distillation set. However, the results presented in this section suggest that in practice the optimization method is unable to achieve high fidelity even on the distillation dataset when extensive data augmentation or synthetic data is used. The inability to solve the optimization problem undermines distillation: in order to find a student that would match the teacher on all inputs, we need to at least be able to find a student that would match the teacher on all of the distillation data. ",
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+    {
+        "type": "text",
+        "text": "Optimization and the train-test fidelity gap. Notably, despite having the lowest train agreement, the Combined Augs policy results in better test agreement than other polices with better train agreement (Figure $3 )$ ). This result highlights a fundamental trade-off in knowledge distillation: the student needs many teacher labels match the teacher on test, but introducing examples not in the teacher train data makes matching the teacher on the distillation data very difficult. ",
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+        "img_path": "images/1548c429dc0ad80cae5a1fd8ff926d15a7672780d4723132d08c3a676d55517f.jpg",
+        "image_caption": [
+            "Figure 6: Optimization and distillation: self-distillation with ResNet-20s with LayerNorm on CIFAR-100. (a): Final train agreement for SGD and Adam optimizers. Training longer improves agreement, but it remains below $8 5 \\%$ even after $5 k$ epochs. (b): Final train loss and agreement when the initialization is a convex combination of teacher and random weights, $\\theta _ { s } = \\lambda \\theta _ { t } + \\mathbf { \\bar { ( } 1 - } \\lambda ) \\theta _ { r }$ . (c): Projections of the distillation loss surface on the plane intersecting $\\theta _ { t }$ , the initial student weights, and the final student weights for different $\\lambda$ . When $\\lambda$ is small, the student converges to a suboptimal solution with low agreement. The uncertainty regions correspond to $\\mu \\pm \\sigma$ , estimated over 3 trials. "
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+        "type": "text",
+        "text": "6.2 Why is train agreement so low? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A simplified distillation experiment. To simplify our exploration, we focus on self-distillation of a ResNet-20 on CIFAR-100. We use the Baseline data augmentation strategy, as we found that a ResNet-20 student is unable to match the teacher on train even with basic augmentation. We also replace the BatchNorm layers $\\mathbb { \\left. \\overline { { 2 5 } } \\right. }$ in ResNet-20 with LayerNorm $\\pmb { \\left[ \\sqrt { 3 } \\right] }$ , because we found that with BatchNorm layers even when the teacher and the student have identical weights, they can make different predictions due to differences in the activation statistics accumulated by the BatchNorm layers. Layer normalization does not collect any activation statistics, so the student will match the teacher as long as the weights coincide. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Can we solve the optimization problem better? We verify that the distillation fidelity cannot be significantly improved by training longer or with a different optimizer. By default, in our experiments we use stochastic gradient descent (SGD) with momentum, train the student for 300 epochs, and use a weight decay value of $1 0 ^ { - 4 }$ . In Figure $\\boxed { 6 }$ we report the results for the SGD and Adam $\\mathbb { \\ Z } \\mathbb { 1 }$ optimizers run for $1 k$ and $5 k$ epochs without weight decay. Switching from SGD to Adam only reduced fidelity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For both optimizers, training for more epochs does slightly improve train agreement. In particular, with SGD we achieve $8 3 . 3 \\%$ agreement when training for $5 k$ epochs compared to $7 8 . 9 5 \\%$ when training for 300 epochs. It is possible, though unlikely, that if we train for even more epochs the train agreement could reach $1 0 0 \\%$ . However, training for $5 k$ epochs is significantly longer than what is typically done in practice (100 to 500 epochs). Furthermore, the improvement from $1 k$ to $5 k$ epochs is only about $2 \\%$ , suggesting that we would need to train for tens of thousands of epochs, even in the optimistic case that agreement improves linearly, in order to get close to $1 0 0 \\%$ train agreement. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The distillation loss surface hypothesis: If we cannot perfectly distill a ResNet-20 on CIFAR-100 with any of the interventions we have discussed so far, we now ask if there is any modification of the problem that can produce a high-fidelity student. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the self-distillation setting, we do know of at least one set of weights that is optimal w.r.t. the distillation loss — the teacher’s own weights $\\theta _ { t }$ . Letting $\\theta _ { r }$ be a random weight initialization, in Figure $\\boxed { 6 }$ (a) we examine the effect of choosing the student initialization to be a convex combination of the teacher and random weights, $\\theta _ { s } = \\lambda \\bar { \\theta _ { t } } + ( 1 - \\lambda ) \\theta _ { r }$ . After being initialized in this way, the student was trained as before. In other words $\\lambda = 0$ corresponds to a random initialization and $\\lambda = 1$ corresponds to initializing the student weights at the final teacher weights. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We find that if the student is initialized far from the teacher $\\lambda \\leq 0 . 2 5 )$ , the optimizer converges to a sub-optimal value of the distillation loss, producing a student that significantly disagrees with the teacher. However at $\\lambda = 0 . 3 7 5$ there is a sudden change. The final train loss drops to the optimal value and the agreement drastically increases, and the behavior continues for $\\lambda > 0 . 3 7 5$ . To further investigate, in Figure $6 ( \\mathrm { c ) }$ we visualize the distillation loss surface for $\\lambda \\in \\{ 0 , 0 . 2 5 , 0 . 3 7 5 \\}$ projected on the 2D subspace intersecting $\\theta _ { t }$ , the initial student weights, and the final student weights. If the student is initialized far from the teacher $( \\lambda \\in \\{ 0 , 0 . 2 5 \\} )$ , it converges to a distinct, sub-optimal basin of the loss surface. On the other hand, when initialized close to the teacher $\\lambda = 0 . 3 7 5$ ), the student converges to the same basin as the teacher, achieving nearly $100 \\%$ agreement. ",
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+    {
+        "type": "table",
+        "img_path": "images/6261113ac0608e905470acaddd56ab5ee54a3afd4229b650ea76339020420cdb.jpg",
+        "table_caption": [
+            "Table 1: We examine whether fidelity can be improved in the context of ResNet-20 self-distillation on CIFAR-100 if the teacher and student share the same weight initialization. All metrics are computed on the test set. A shared initialization does make the student slightly more similar to the teacher in activation space (measured by CKA), but in function space the results are indistinguishable from randomly initialized students. We report the mean and standard deviation, estimated from 10 trials. The average teacher accuracy was 70.522 (0.412). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\"></td><td colspan=\"3\">CKA (1)</td></tr><tr><td>Init.</td><td>Agree. (↑)</td><td>KL (↓）</td><td>Stage 1</td><td> Stage 2</td><td>Stage 3</td></tr><tr><td>Rand.</td><td>77.174 (0.352)</td><td>0.836 (0.016)</td><td>0.939 (0.017)</td><td>0.925 (0.027)</td><td>0.885 (0.011)</td></tr><tr><td>Teach.</td><td>77.098 (0.238)</td><td>0.838 (0.020)</td><td>0.951 (0.017)</td><td>0.937 (0.020)</td><td>0.890 (0.015)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Is using the initial teacher weights enough for good fidelity? If good fidelity can be obtained by initializing the student near the final teacher weights, it is possible that similar results could be obtained by initializing the student at the initial teacher weights. In Table $^ 1$ we compare students distilled from random initializations with those initialized at the initial teacher weights. In addition to the metrics reported in the rest of the paper, we also include the centered kernel alignment (CKA) $\\pmb { \\Vert 2 8 \\Vert }$ of the preactivations of each of the teacher and student networks. There is a small increase in CKA, indicating that sharing an initialization between teacher and student does increase alignment in activation space, but functionally the students are identical to their randomly initialized counterparts – there is no observable change in accuracy, agreement, or predictive KL when compared to random initialization. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To summarize, we have at last identified a root cause of the ineffectiveness of all our previous interventions on the knowledge distillation procedure. Knowledge distillation is unable to converge to optimal student parameters, even when we know a solution and give the initialization a small head start in the direction of an optimum. Indeed, while identifiability can be an issue, in order to match the teacher on all inputs, the student has to at least match the teacher on the data used for distillation, and achieve a near-optimal value of the distillation loss. Furthermore, the suboptimal convergence of knowledge distillation appears to be a consequence of the optimization dynamics specifically, and not simply initialization bias. In practice, optimization converges to sub-optimal solutions, leading to poor distillation fidelity. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our work provides several new key findings about knowledge distillation: ",
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+    },
+    {
+        "type": "text",
+        "text": "• Good student accuracy does not imply good distillation fidelity: even outside of selfdistillation, the models with the best generalization do not always achieve the best fidelity. • Student fidelity is correlated with calibration when distilling ensembles: although the highest-fidelity student is not always the most accurate, it is always the best calibrated. • Optimization is challenging in knowledge distillation: even in cases when the student has sufficient capacity to match the teacher on the distillation data, it is unable to do so. • There is a trade-off between optimization complexity and distillation data quality: Enlarging the distillation dataset beyond the teacher training data makes it easier for the student to identify the correct solution, but also makes an already difficult optimization problem harder. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In standard deep learning, we are saved by not needing to solve the optimization problem well: while it true that our training loss is highly multimodal, properties such as the flatness of good solutions, the inductive biases of the network, and the implicit biases of SGD, often enable good generalization in practice. In knowledge distillation, however, good fidelity is directly aligned with solving what turns out to be an exceptionally difficult optimization problem. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            338,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Gregory Benton, Marc Finzi, Sanae Lotfi, Nate Gruver, and Ben Poole for helpful feedback. This research is supported by an Amazon Research Award, NSF I-DISRE 193471, NIH R01DA048764-01A1, NSF IIS-1910266, and NSF 1922658NRT-HDR: FUTURE Foundations, Translation, and Responsibility for Data Science. Samuel Stanton is also supported by a United States Department of Defense NDSEG fellowship. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            210,
+            266,
+            227
+        ],
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+    },
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+        "type": "text",
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parse/train/Ad7Gv5NkBPz/Ad7Gv5NkBPz.md

@@ -0,0 +1,438 @@
+# A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We consider the problem of causal structure learning in the setting of heterogeneous   
+2 populations, i.e., populations in which a single causal structure does not adequately   
+3 represent all population members, as is common in biological and social sciences.   
+4 To this end, we introduce a distance covariance-based kernel designed specifically   
+5 to measure the similarity between the underlying nonlinear causal structures of   
+6 different samples. This kernel enables us to perform clustering to identify the   
+7 homogeneous subpopulations. Indeed, we prove the corresponding feature map is   
+8 a statistically consistent estimator of nonlinear independence structure, rendering   
+9 the kernel itself a statistical test for the hypothesis that sets of samples come from   
+10 different generating causal structures. We can then use existing methods to learn   
+11 a causal structure for each of these subpopulations. We demonstrate using our   
+12 kernel for causal clustering with an application in genetics, allowing us to reason   
+13 about the latent transcription factor networks regulating measured gene expression   
+14 levels.
+
+# 15 1 Introduction
+
+16 Learning causal relationships from observational and experimental data is one of the fundamental   
+17 goals of scientific research, and causal inference methods are thus used in a wide variety of fields. The   
+18 resulting variety of applications nevertheless share some common difficulties, such as causal inference   
+19 from complex time-series data (Eichler, 2012) or the underlying causal structure being obscured   
+20 by unmeasured confounders (Greenland et al., 1999). Another common difficulty, especially for   
+21 applications in the biological and social sciences, is causal inference from heterogeneous populations   
+22 (Xie, 2013; Brand and Thomas, 2013)—addressing this difficulty is our main motivation.   
+23 In general terms, we understand a heterogeneous population to be one whose members are not   
+24 adequately described by a single model but rather better described by a collection of models. Within   
+25 our context of causal structure learning, this means a population is heterogeneous if some samples   
+26 are generated by different causal structures—we call this structural heterogeneity. We note that there   
+27 are other kinds of heterogeneity, such as that in samples generated by different joint distributions   
+28 over the same causal structure, which are not the scope of this work.   
+29 A specific example of structural heterogeneity can be found in genetics: causal methods are used to   
+30 learn the structure of gene regulatory networks (Emmert-Streib et al., 2012), and gene expression data   
+31 from a single recording or experiment may include thousands of genes, many of which are involved   
+32 in entirely different networks (Liu, 2015); thus, attempting to learn a single causal structure for all of   
+33 the genes will obscure the fact that different sets of them have different structures.   
+34 The bulk of our work in this paper, and our main contribution, is to introduce the dependence   
+35 contribution kernel, which facilitates a flexible and easily extensible approach to causal clustering:   
+36 first perform clustering to identify structurally homogeneous subsets of samples, and then proceed   
+37 with the actual learning task on each cluster. We prove that our kernel is a statistically consistent   
+38 estimator of the similarity of the causal structures underlying different samples and can thus be used   
+39 to find clusters that minimize structural heterogeneity for causal structure learning tasks. Furthermore,   
+40 the kernel is derived from the distance covariance (Székely et al., 2007), imbuing it with the ability   
+41 to detect nonlinear dependence. It can easily be used in a wide array of clustering algorithms, such   
+42 as $k$ -means, DBSCAN, spectral clustering, or any other method that analogously makes use of a   
+43 similarity (or distance) measure between samples (Filippone et al., 2008).   
+44 The rest of the paper is organized as follows: We finish this section by discussing some of the most   
+45 relevant related work from the causal inference and statistics literature. All of Section 2 is devoted to   
+46 the theory underlying our dependence contribution kernel, including a comparison of the familiar   
+47 product-moment covariance with the distance covariance (Section 2.1), defining an equivalence   
+48 class of causal models with a convenient representation in the kernel space (Section 2.2), and the   
+49 actual definition of our kernel and proofs of its relevant properties (Section 2.3). Next, in Section   
+50 3, we demonstrate causal clustering with the kernel on a heterogeneous gene expression data set,   
+51 finding structurally homogeneous clusters for which we then learn latent causal measurement models,   
+52 allowing us to reason about the different transcription factor networks responsible for regulating the   
+53 measured gene expression levels. Finally, we conclude in Section 4 mentioning possible future work.
+
+# 54 1.1 Related Work
+
+55 Causal inference in heterogeneous populations sometimes refers to data-fusion (Bareinboim and   
+56 Pearl, 2016), i.e., combining known homogeneous subpopulations and performing causal inference   
+57 on the resulting heterogeneous population, or similarly, it can refer to meta-learning using known   
+58 subpopulations (Sharma et al., 2019). Other times, it refers to estimating heterogeneous treatment   
+59 effects (Xie et al., 2012; Athey and Imbens, 2015). However, in our case, the subpopulations are not   
+60 known and we rather consider the problem of learning which samples come from which subpopulation,   
+61 and these are differentiated according to structure instead of treatment effect.   
+62 Previous work on causal clustering has focused more on the causal modeling aspect, using stronger   
+63 assumptions about the underlying structures to learn more detailed models. For example, Kummerfeld   
+64 et al. (2014); Kummerfeld and Ramsey (2016) focus on causal clustering in measurement models,   
+65 with the goal of clustering different features together to study their latent causal structure, based on   
+66 tetrad constraints within the linear product-moment covariance matrix. Huang and Zhang (2019)   
+67 define a class of causal models facilitating mechanism-based clustering, learning causal models both   
+68 for clusters of samples as well as a shared one for all samples, assuming the underlying structures   
+69 are linear non-Gaussian. Saeed et al. (2020) characterize distributions arising from mixtures of   
+70 directed acyclic graph (DAG) causal models (i.e., causal models without latent or selection variables),   
+71 trying to learn both the component DAGs and a representation of how they are mixed. All of these   
+72 approaches, like most causal inference methods, make specific (and for some applications, restrictive)   
+73 assumptions about the underlying distributions or causal structures.   
+74 In contrast, our method is not tied to specific distributional assumptions such as linearity or   
+75 (non)Gaussianity—we assume there are enough samples for statistical inference, as well as the   
+76 usual causal Markov and faithfulness assumptions. For the first step, we cluster samples together if   
+77 they (implicitly, in the kernel space) have similar nonlinear independence structures. For the second   
+78 step, causal structure learning, any existing method (along with its corresponding assumptions) can in   
+79 principle be used. In our gene expression data application (Section 3), the measurement dependence   
+80 inducing latent (MeDIL) causal model framework (Markham and Grosse-Wentrup, 2020), which   
+81 assumes the data consists of measurement variables that are causally connected only through latent   
+82 variables, seems appropriate, however other applications can easily use other methods. For example,   
+83 component and mixture DAGs (Saeed et al., 2020) can be better learned when one first knows which   
+84 samples come from which component—clustering with our kernel ensures samples in different   
+85 clusters come from different DAGs, and so using their method instead of the MeDIL framework   
+86 would be a natural choice for applications in which a DAG (without any latents) is more appropriate.
+
+# 2 Theory
+
+# 2.1 Product-moment Covariance, Distance Covariance, and Dependence Contribution
+
+Though there is more to causal relationships than probabilistic dependence, causal inference methods based on graphical models ultimately rely on at least implicitly learning conditional independence (CI) relations. CI relations can be estimated in many ways, with different dependence measures and tests each having their own theoretical guarantees and being better suited for distributions of various different kinds of data (e.g., categorical, discrete, or continuous) and with various kinds of relationships (e.g., linear, monotonic nonlinear, arbitrary nonlinear) and with different testing assumptions (see Tjøstheim et al., 2018, for a comprehensive overview).
+
+A widely used measure of dependence is the product-moment covariance, often just called covariance, which is defined for two zero-mean random variables $X _ { 1 }$ and $X _ { 2 }$ as the scalar value $\operatorname { c o v } ( X _ { 1 } , X _ { 2 } ) =$ $\operatorname { E } [ X _ { 1 } X _ { 2 } ]$ . This can be extended from a pair of random variables to every pair of variables in a random vector, thus returning a matrix instead of a scalar. The covariance matrix for a vector of zero-mean random variables $\mathbf { X } = ( X _ { 1 } , \ldots , X _ { m } )$ can be estimated from a set $S \in \mathbb { R } ^ { n , m }$ of $n$ samples as $\textstyle { \hat { \Sigma } } _ { \mathbf { X } } = { \frac { 1 } { n } } S ^ { \top } S$ , and the ${ j , j ^ { \prime } }$ -th value of $\hat { \Sigma } _ { \mathbf { X } }$ is thus the estimate $\operatorname { c o v } ( X _ { j } , X _ { j } ^ { \prime } )$ .
+
+102 Two random variables being probabilistically independent (denoted $\perp \perp$ ) implies that their product  
+103 moment covariance is zero, i.e., $X _ { j } \perp \perp X _ { j ^ { \prime } } \implies \operatorname { c o v } ( X _ { j } , X _ { j ^ { \prime } } ) = 0$ (importantly, the inverse of this   
+104 does not hold). Thus, the estimated product-moment covariance can be used in statistical hypothesis   
+105 testing for probabilistic independence (Wasserman, 2013, Ch. 10): $X _ { j }$ and $X _ { j ^ { \prime } }$ are assumed to   
+106 be independent if and only if $\operatorname { c o v } ( X _ { j } , X _ { j ^ { \prime } } )$ is sufficiently close to 0. However, this method has   
+107 an important problem: the product-moment covariance is only a valid test statistic against linear   
+108 dependence.   
+109 Székely et al. (2007) introduce the distance covariance to remedy this problem: random variables are   
+110 probabilistically independent if and only if their distance covariance is zero, i.e., $X _ { j } \perp \perp X _ { j ^ { \prime } } \iff$   
+111 $\operatorname { d C o v } ( X _ { j } , X _ { j ^ { \prime } } ) = 0$ , resulting in the estimated distance covariance being a valid test statistic against   
+112 all types of dependence. The distance covariance is related to the product-moment covariance by   
+113 $\mathrm { d } \mathrm { C o v } ^ { 2 } ( X _ { j } , X _ { j ^ { \prime } } ) = \mathrm { c o v } ( | X _ { j } - X _ { j } ^ { \prime } | , | X _ { j ^ { \prime } } - X _ { j ^ { \prime } } ^ { \prime } | ) - 2 \mathrm { c o v } ( | X _ { j } - X _ { j } ^ { \prime } | , | X _ { j ^ { \prime } } - X _ { j ^ { \prime } } ^ { \prime \prime } | ) .$ , where $( X _ { j } ^ { \prime } , X _ { j ^ { \prime } } ^ { \prime } )$   
+114 and $( X _ { j } ^ { \prime \prime } , X _ { j ^ { \prime } } ^ { \prime \prime } )$ are independent and identically distributed (iid) copies of $( X _ { j } , X _ { j ^ { \prime } } )$ (Székely and   
+115 Rizzo, 2014). The key intuition here is that the distances (e.g., $| X _ { j } - X _ { j } ^ { \prime } | )$ constitute a nonlinear   
+116 projection, so that using the linear product-moment covariance in this projected space allows for the   
+117 detection of nonlinear dependence in the original space.   
+118 Note that dCov is typically defined to be a scalar value when taken between two arbitrary-dimensional   
+119 random vectors, but our restricted presentation of it above in terms of random variables is to make   
+120 it more obviously analogous to the product-moment covariance between random variables. Thus,   
+121 corresponding to $\hat { \Sigma } _ { \mathbf { X } }$ for random vectors, we define the following:   
+22 Definition 1 Let $S \in \mathbb { R } ^ { n , m }$ be a set of $n$ samples from the vector of random variables ${ \textbf { X } } =$   
+23 $( X _ { 1 } , \ldots , X _ { m } )$ . For each $j \in \{ 1 , \dots , m \}$ and $i , i ^ { \prime } \in \{ 1 , \ldots , n \}$ , define the pairwise distance matrix   
+24 $D ^ { j }$ , with values given by $D _ { i , i ^ { \prime } } ^ { \jmath } : = | S _ { i , j } - S _ { i ^ { \prime } , j } |$ . Now define the corresponding doubly-centered   
+25 matrices $C _ { i , i ^ { \prime } } ^ { j } : = { D } _ { i , i ^ { \prime } } ^ { j } - \bar { D ^ { j } } _ { i , \cdot } - \bar { D ^ { j } } _ { \cdot , i ^ { \prime } } + \bar { D ^ { j } } _ { \cdot , \cdot }$ , where putting a bar over the matrix and replacing   
+26 an index $i$ or $i ^ { \prime }$ with $\cdot$ denotes taking the mean over that index. Define the matrix $L \in \mathbb { R } ^ { n ^ { 2 } , m }$ so   
+27 that each column is a flattened doubly-centered distance matrix, $L : = ( \operatorname { v e c } ( C ^ { 1 } ) , \dots , \operatorname { v e c } ( C ^ { m } ) )$ ,   
+28 where $\mathrm { v e c } ( C ^ { j } )$ denotes “flattening” matrix $C ^ { j }$ into a column vector. Finally, the estimated distance   
+29 covariance matrix over sample $S$ is defined as $\begin{array} { r } { \hat { \Delta } \mathbf { x } : = \frac { 1 } { n ^ { 2 } } L ^ { \top } L } \end{array}$ .
+
+Analogous to 130 $\hat { \Sigma } _ { \mathbf { X } }$ , the ${ j , j ^ { \prime } }$ -th entry of $\hat { \Delta } _ { \mathbf { X } }$ corresponds to $\mathrm { d } \hat { \mathrm { C o v } } ^ { 2 } ( X _ { j } , X _ { j ^ { \prime } } ) .$ —indeed it is mathemati131 cally equivalent to computing each pairwise distance covariance value and then manually filling in
+
+132 the matrix. The novelty of our Definition 1 is in finding a matrix of pairwise values instead of a single   
+133 value for the distance covariance between random vectors, which helps provide an intuition for our   
+134 next definition:
+
+Definition 2 Let $S \in \mathbb { R } ^ { n , m }$ be a set of $n$ samples from the vector of random variables ${ \textbf { X } } =$ $( X _ { 1 } , \ldots , X _ { m } )$ ; note that we consistently use indices $i , i ^ { \prime } \in \{ 1 , \ldots , n \}$ and $j , j ^ { \prime } \in \{ 1 , \ldots , m \}$ . Let $D \in \mathbb { R } ^ { n , n , m }$ denote the 3-dimensional array of stacked pairwise distance matrices defined by $D _ { i , i ^ { \prime } , j } : = | S _ { i , j } - S _ { i ^ { \prime } , j } |$ , and use $C \in \mathbb { R } ^ { n , n , m }$ to denote these same distance matrices after being doubly-centered, i.e., $\tilde { C } _ { i , i ^ { \prime } , j } : = D _ { i , i ^ { \prime } , j } - \bar { D } _ { i , \cdot , j } - \bar { D } _ { \cdot , i ^ { \prime } , j } + \bar { D } _ { \cdot , \cdot , j }$ , where replacing an index $i$ or $i ^ { \prime }$ with · denotes the entire (lower-dimensional) subarray over that index, and writing a bar, $\bar { D }$ , denotes taking the mean over that subarray. Then standardize the doubly-centered distances to get $\begin{array} { r } { Z _ { i , i ^ { \prime } , j } : = \frac { C _ { i , i ^ { \prime } , j } } { \bar { D } \cdot \underline { { \mathbf { \Pi } } } _ { \cdot , j } } } \end{array}$ Finally, the dependence contribution map, $\varphi : \mathbb { R } ^ { m }  \mathbb { R } ^ { m , m }$ , is defined as
+
+$$
+\varphi ( S _ { i , \cdot } ) : = Z _ { i , \cdot , \cdot } ^ { \top } Z _ { i , \cdot , \cdot } - \mathcal { T } ( \alpha ) ,
+$$
+
+where 135 $\mathcal { T } ( \alpha ) \in \mathbb { R } ^ { m , m }$ is a matrix of scaled critical values corresponding to a given significance level 136 $\alpha$ with zeros along the diagonal, i.e., $\begin{array} { r } { \mathcal { T } ( \alpha ) _ { j , j ^ { \prime } } = \left\{ \begin{array} { l l } { 0 , } \\ { \frac { 1 } { n } \chi _ { 1 - \alpha } ^ { 2 } ( 1 ) } \end{array} \right. } \end{array}$ if , ot $j = j ^ { \prime }$ e , with $\chi _ { 1 - \alpha } ^ { 2 } ( 1 )$ being the 137 $1 - \alpha$ quantile of the chi-square distribution with 1 degree of freedom.
+
+138 Notice the similarity bestandardization (i.e., use $C$ een Defininstead of $Z$ ons 2 a), then $\begin{array} { r } { \frac { 1 } { n ^ { 2 } } \sum _ { i = 1 } ^ { n } \varphi ( S _ { i , \cdot } ) = \hat { \Delta } _ { \mathbf { X } } } \end{array}$ $\mathcal { T } ( \alpha )$ be a matrix of 0s and. Now, the differences: $\hat { \Delta } _ { \mathbf { X } } \hat { }$ gois   
+140 a single matrix computed over an entire set of samples, whereas $\varphi$ is a map that projects each given   
+141 sample to a new feature space; each entry of $\hat { \Delta } _ { \mathbf { X } }$ is simply a distance covariance value, whereas each   
+142 entry of the sum of $\varphi ( S _ { i , \cdot } )$ over $i$ , by using standardization (using $Z$ instead of $C$ ) and subtracting a   
+143 critical value, corresponds to the result of using a distance covariance value in a statistical hypothesis   
+144 test for independence—indeed:
+
+Lemma 3 Let $S \in \mathbb { R } ^ { n , m }$ be a set of $n$ iid samples from random variables $X _ { 1 } , \ldots , X _ { m }$ with finite first moments. For a given significance level $\alpha$ , under the null hypothesis of $X _ { j } \perp \perp X _ { j ^ { \prime } }$ , the test
+
+$$
+\mathrm { r e j e c t } h _ { \varnothing } \mathrm { i f } \quad \big ( \sum _ { i = 1 } ^ { n } \varphi ( S _ { i , \cdot } ) \big ) _ { j , j ^ { \prime } } > 0
+$$
+
+145 is statistically consistent against all types of dependence.
+
+146 Proof. This follows from (Székely and Rizzo, 2009, Theorem 5 and Corollary 2) and how $\varphi$ is   
+147 defined to correspond to the difference between distance covariance and critical values.    
+148 These differences between $\hat { \Delta } _ { \mathbf { X } }$ and $\varphi$ serve two important purposes: first, they ensure $\varphi$ maps to a   
+149 Hilbert space so that our Definition 9 is a corresponding kernel function (Schölkopf et al., 2001); and   
+150 second, as the name “dependence contribution map” suggests, they ensure $\varphi ( S _ { i , \cdot } )$ is informative not   
+151 just about distance covariance but about nonlinear dependence and about how the inclusion of sample   
+152 $S _ { i , \astrosun }$ ,· in a set of samples $S$ contributes to the dependence patterns estimated from $S _ { \ l }$ — this is the key   
+153 intuition behind how our kernel function is used to learn structurally homogeneous sample subsets,   
+154 as explicated in the following sections.
+
+# 2.2 Causal Graphs in Kernel Space
+
+In general, a full causal structure can only be learned with sufficient data about the effects of interventions, and thus causal structure learning from purely observational data is usually possible only up to an equivalence class of causal graphs (Spirtes et al., 2000; Pearl, 2009). For example, the classic PC and IC algorithms, under the assumptions of no selection bias and no confounding by latent variables, do not necessarily return a fully-specified DAG but instead return a mixed graph, containing possibly directed and undirected edges, representing the Markov equivalence class (Spirtes and Glymour, 1991; Pearl and Verma, 1995).
+
+163 We now define a set of equivalence classes for ancestral graphs (AGs), which—unlike causal DAGs—   
+164 do not assume the absence of selection bias and latent confounders (Richardson et al., 2002):
+
+Definition 4 Consider an arbitrary ancestral graph $\mathcal { A }$ with the set of vertices $V ^ { A }$ and edge function $E ^ { A }$ , and denote the set of unconditional $m$ -connection statements entailed by their corresponding unique maximal ancestral graph as $M ^ { A } = \{ ( j , j ^ { \prime } ) : j \mathcal { L } _ { m } j ^ { \prime } | \emptyset \} \subseteq V ^ { A } \times \dot { V } ^ { A }$ . For any ancestral graph $\mathcal { A } ^ { \prime }$ such that $V ^ { \mathcal { A ^ { \prime } } } = \bar { V } ^ { \mathcal { A } }$ , define the unconditional equivalence relation denoted by $^ \bullet \sim _ { \mathrm { U } }$ ’ as
+
+$$
+\mathcal { A } \sim _ { \mathrm { U } } \mathcal { A } ^ { \prime } \quad \mathrm { i f ~ a n d ~ o n l y ~ i f } \quad M ^ { A } = M ^ { A ^ { \prime } } .
+$$
+
+165 Lemma 5 This lemma has two parts: (i) the relation ${ \sim } _ { \mathrm { U } }$ is an equivalence relation over the set of   
+166 ancestral graphs A; (ii) for an arbitrary ancestral graph $\mathcal A \in \mathbb A$ , the bidirected graph $\mathcal { U } ^ { A } = ( V ^ { A } , E ^ { \mathcal { U } } )$ ,   
+167 where $E ^ { \mathcal { U } }$ maps all pairs $( j , j ^ { \prime } ) \in M ^ { A }$ to the bidirected edge symbol $\cdot $ , is a unique representative   
+168 of the equivalence class $[ A ]$ .   
+169 Proof. For (i), recall that an equivalence relation is any relation satisfying reflexivity, symmetry,   
+170 and transitivity (Devlin, 2003), all of which are satisfied by ${ \sim } _ { \mathrm { U } }$ because of its correspondence to   
+171 the relation $" = "$ between sets. Thus, to prove (ii), it suffices to show that the map $s : \mathbb { A } / \sim _ { \mathrm { U } } \to$   
+172 A, $[ \mathcal { A } ] \mapsto \mathcal { U } ^ { A }$ is injective (i.e, that it is a section) and that $[ s ( [ A ] ) ] = [ A ]$ (Mac Lane, 2013). The   
+173 key to the proof is the observation that $\mathcal { U } ^ { A }$ , because it contains only bidirected edges, is maximal   
+174 and therefore entails exactly the unconditional $m$ -separation statements $M ^ { A }$ , thus by (i) we have   
+175 ${ \mathcal { U } } ^ { A } \sim _ { \mathrm { U } } { \mathcal { A } }$ or equivalently $\mathcal { U } ^ { A } \in [ A ]$ or equivalently $[ \mathcal { U } ^ { A } ] = [ \mathcal { A } ]$ . Let $\mathcal { A } , \mathcal { A } ^ { \prime }$ be arbitrary AGs, and   
+176 assume $s ( [ \mathcal { A } ] ) = s ( [ \mathcal { A } ^ { \prime } ] )$ . Then by definition of $s$ we have $\bar { \mathcal { U } } ^ { A } \bar { = } \bar { \mathcal { U } } ^ { A ^ { \prime } }$ , and by the observation above,   
+177 $\mathcal { U } ^ { A } \in [ \mathcal { A } ^ { \prime } ]$ and thus $[ A ] = [ A ^ { \prime } ]$ , making $s$ injective. And finally, by the definition of $s$ and also by   
+178 the observation above, $[ s ( [ A ] ) ] = [ \mathcal { U } ^ { A } ] = [ A ]$ , completing the proof.    
+179 This equivalence relation and its representatives has some important but perhaps subtle properties.   
+180 First, it is different from Markov equivalence over AGs (which is characterized by partial ancestral   
+181 graphs, PAGs) (Zhang, 2007)—it uses only unconditional $m$ -separation while PAGs are learned from   
+182 conditional $m$ -separation statements. Second, because all DAGs are AGs, ${ \sim } _ { \mathrm { U } }$ is also an equivalence   
+183 relation over DAGs. Third, being a representative means that every equivalence class includes exactly   
+184 one fully bidirected graph (along with other equivalent AGs). Fourth, because each representative is   
+185 formed by considering $m$ -connected paths, $\mathcal { U } ^ { A }$ is not equivalent to what would be generated by some   
+186 “edge-wise” procedure, such as simply replacing every edge in a PAG/AG/DAG/Markov random   
+187 field/moralized DAG with bidirected edges.Finally, its most important property is that it facilitates   
+188 Theorem 8, for which we first need a few more definitions.
+
+Definition 6 Given arbitrary ancestral graphs $\mathcal { A } , \mathcal { A } ^ { \prime } \in \mathbb { A }$ over the same set of vertices, define the Hamming similarity product, denoted $\bullet _ { \bullet } ,$ as
+
+$$
+\bullet : \mathbb { A } \times \mathbb { A } \to \mathbb { A } \quad { \mathrm { a n d } } \quad \mathcal { A } \bullet \mathcal { A } ^ { \prime } \mapsto \mathcal { H } ,
+$$
+
+where 189 $\mathcal { H } = ( V ^ { A } , E ^ { \mathcal { H } } )$ and the function $E ^ { \mathcal { H } } ( j , j ^ { \prime } ) = \ '  ^ { \prime }$ if and only if $E ^ { A } ( j , j ^ { \prime } ) = E ^ { A ^ { \prime } } ( j , j ^ { \prime } )$ .
+
+190 In words, the Hamming similarity product between two ancestral graphs returns a fully bidirected   
+191 graph, with edges only where the two graphs have the same edge type. Now, shifting from ancestral   
+192 graphs to real-valued square matrices:
+
+Definition 7 Let $\cdot _ { \sim _ { \mathrm { O } } }$ ’ denote the orthant equivalence relation (‘orthant’ is the generalization of ‘quadrant’ from $\mathbb { R } ^ { 2 }$ to arbitrarily higher dimensions) in square real matrices, i.e., for matrices Y, Y 0 ∈ Rm,m and with the element-wise function sign(Y )j,j0 = $\mathrm { s i g n } ( Y ) _ { j , j ^ { \prime } } = \left\{ { \begin{array} { l l } { 1 , } \\ { - \ } \end{array} } \right.$ 1, otherwise if $Y _ { j , j ^ { \prime } } > 0$ or $j = j ^ { \prime }$
+
+$$
+Y \sim _ { \mathrm { { o } } } Y ^ { \prime } \quad { \mathrm { i f ~ a n d ~ o n l y ~ i f } } \quad \mathrm { s i g n } ( Y ) _ { j , j ^ { \prime } } = \mathrm { s i g n } ( Y ^ { \prime } ) _ { j , j ^ { \prime } }
+$$
+
+193 Theorem 8 Let $a$ be the map from the set of unconditional equivalence classes over ancestral graphs   
+194 with $m$ vertices, $\mathbb { A } ^ { m } / \sim _ { \mathrm { U } } = \mathbb { U } ^ { m }$ , to the set of orthant equivalence classes over the image of $\varphi$   
+195 i.e., $m \times m$ symmetric real matrices with positive diagonal entries, $\varphi ( \mathbb { R } ^ { m } ) / { \sim } _ { 0 } = \mathbb { O } ^ { m }$ , defined by   
+196 $a : \mathcal { U } \mapsto O$ , where $O _ { j , j ^ { \prime } } = \left\{ { 1 , \atop - 1 } \right.$ if , ot EU (j, j0) = ‘↔’ or j = j0 . Then a is a group isomorphism   
+between 197 $( \mathbb { U } ^ { m } , \bullet )$ and $( \mathbb { O } ^ { m } , \odot )$ , where $\ast _ { \odot } \cdot$ ’ denotes the element-wise product.   
+198 Proof. First, note that $( \mathbb { U } ^ { m } , \bullet )$ is indeed a group, satisfying the three group axioms (Artin, 2011):   
+199 the representative of its identity element is the fully connected bidirected graph over $m$ vertices, $\mathcal { U } ^ { \mathbb { 1 } }$ ;   
+200 each element is its own inverse; and $\bullet$ is associative. Likewise, $( \mathbb { O } ^ { m } , \odot )$ is a group with identity   
+201 element $\left[ \mathbb { 1 } ^ { m , m } \right]$ , each element its own inverse, and the associative element-wise product operator.
+
+Now, to show the two groups are isomorphic, it suffices to show (i) that $a$ is bijective and (ii) that for arbitrary $\mathcal { U } , \mathcal { U } ^ { \prime } \in \mathbb { U } ^ { m }$ $\mathbb { U } ^ { m } , a ( \mathcal { U } ) \odot a ( \mathcal { U } ^ { \prime } ) = a ( \mathcal { U } \bullet \mathcal { U } ^ { \prime } )$ . For (i) notice that if $U \neq U ^ { \prime }$ , then there must be at least one pair of vertices ${ j , j ^ { \prime } }$ such that $\dot { E } ^ { \mathcal { U } } ( j , \dot { j } ^ { \prime } ) \neq E ^ { \mathcal { U } ^ { \prime } } ( j , \dot { j } ^ { \prime } )$ and thus clearly $O _ { j , j ^ { \prime } } \neq O _ { j , j ^ { \prime } } ^ { \prime }$ so $a$ in injective. Furthermore, notice that every distinct $O \in \mathbb { O } ^ { m }$ is the image of some graph $\mathcal { U }$ , so $a$ is also surjective. For (ii), for every $j , j ^ { \prime } \in \{ 1 , \ldots , m \}$ , the definitions of $a , \odot$ , and $\bullet$ ensure $a ( \mathcal { U } ) _ { j , j ^ { \prime } } \odot a ( \mathcal { U } ^ { \prime } ) _ { j , j ^ { \prime } } = 1 \iff E ^ { \mathcal { U } } ( j , j ^ { \prime } ) = E ^ { \mathcal { U } ^ { \prime } } ( j , j ^ { \prime } ) \iff 1 = a ( \mathcal { U } \bullet \mathcal { U } ^ { \prime } )$ , completing the proof. 
+
+209 For causal inference, which (often, but not necessarily) amounts to taking several samples in real   
+210 space and inferring a single corresponding member in the space of ancestral graphs (or, more often,   
+211 its quotient set by some equivalence relation), Theorem 8 means we can compare the different graphs   
+212 of different sample sets without having to first move to the ancestral graph space.
+
+Finally, notice the space of real square matrices is not a typical sample space but rather precisely (a superspace of) the space that our dependence contribution map $\varphi$ (Definition 2) maps samples to—this means that mapping samples with $\varphi$ allows us to make use of the group isomorphism. Though this already provides an intuition for why using $\varphi$ would help with causal clustering, explicitly mapping each sample with it would be unnecessarily computationally expensive, and we are ultimately interested in morphisms between metric spaces (not just groups) of samples and graphs. To address this, we thus now move on to defining a kernel for $\varphi$ .
+
+# 220 2.3 The Dependence Contribution Kernel
+
+Definition 9 Let $S , Z , \mathcal { T }$ , and $\varphi$ be as in Definition 2. We define the dependence contribution kernel using the Frobenius (denoted by the subscript $\mathrm { F }$ ) inner product and norm:
+
+$$
+\kappa ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ) = \frac { \langle \varphi ( S _ { i , \cdot } ) , \varphi ( S _ { i ^ { \prime } , \cdot } ) \rangle _ { \mathrm { F } } } { \| \varphi ( S _ { i , \cdot } ) \\| _ { \mathrm { F } } \| \varphi ( S _ { i ^ { \prime } , \cdot } ) \| _ { \mathrm { F } } }
+$$
+
+221 A more convenient expression for applying the kernel to a data set is obtained by first defining a   
+222 helper kernel, $\gamma$ along with vec from Definition 1:
+
+$$
+\begin{array} { r l } & { \gamma ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ) = \langle \varphi ( S _ { i , \cdot } ) , \varphi ( S _ { i ^ { \prime } , \cdot } ) \rangle _ { \mathrm { F } } } \\ & { \qquad = \left( ( \mathrm { v e c } ( Z _ { i , \cdot } ) ^ { \top } \mathrm { v e c } ( Z _ { i ^ { \prime } , \cdot } ) \right) ^ { 2 } - Z _ { i , \cdot } { \mathcal { T } } Z _ { i , \cdot } ^ { \top } - Z _ { i ^ { \prime } , \cdot } { \mathcal { T } } Z _ { i ^ { \prime } , \cdot } ^ { \top } + \| { \mathcal { T } } \| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+This allows us to write
+
+$$
+\kappa ( s , s ^ { \prime } ) = \frac { \gamma ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ) } { \gamma ( S _ { i , \cdot } , S _ { i , \cdot } ) ^ { \frac { 1 } { 2 } } \gamma ( S _ { i ^ { \prime } , \cdot } , S _ { i ^ { \prime } , \cdot } ) ^ { \frac { 1 } { 2 } } }
+$$
+
+223 Finally, note that $\kappa$ can be readily implemented on an entire set of samples, returning an entire   
+224 Gram (kernel) matrix instead of a scalar value, by replacing the matrix operations above with tensor   
+225 operations and specifying the correct axes along which summation occurs—an implementation can   
+226 be found in our open source Python package at https://non-anonymous-link.after-review.   
+227 A proper distance metric can also be obtained from this kernel through function composition:   
+228 arccos $_ { \mathrm { ~ O ~ } \kappa }$ . The key idea behind the kernel is that it is the cosine similarity in the space that $\varphi$ maps   
+229 to, meaning for arbitrary sample points $x , x ^ { \prime }$ it evaluates to $\cos ( \theta )$ , where $\theta$ is the angle between   
+230 $\varphi ( x )$ and $\varphi ( x ^ { \prime } )$ . In this space, $\theta$ represents the dissimilarity of the dependence patterns underlying   
+231 $x$ and $x ^ { \prime }$ , without being biased by the possibly different magnitudes of $\varphi ( x )$ and $\varphi ( x ^ { \prime } )$ due to   
+232 differing variances. Indeed, it can be used as a statistical test of whether samples come from different   
+233 dependence structures and therefore causal models:
+
+Theorem 10 Let $S \in \mathbb { R } ^ { n , m }$ , $S ^ { \prime } \in \mathbb { R } ^ { n ^ { \prime } , m }$ be sets of $n , n ^ { \prime }$ iid samples drawn respectively from the random variables $X = ( X _ { 1 } , \ldots , X _ { m } )$ and $X ^ { \prime } = ( X _ { 1 } ^ { \prime } , \ldots , X _ { m } ^ { \prime } )$ with finite first moments. Then,
+
+$$
+\sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { n ^ { \prime } } \kappa ( S _ { i \cdot } , S _ { i ^ { \prime } \cdot } ^ { \prime } ) < 0 \implies \exists j , j ^ { \prime } \in \{ 1 , \dots , m \} \mathrm { ~ s u c h ~ t h a t ~ } \mathcal { Z } ( X _ { j } , X _ { j ^ { \prime } } , \emptyset ) \neq \mathcal { Z } ( X _ { j } ^ { \prime } , X _ { j ^ { \prime } } ^ { \prime } , \emptyset ) .
+$$
+
+Proof. Through Slutsky’s Theorem (see Takeshi, 1985, Theorem 3.2.7) and the continuous mapping theorem (see Van der Vaart, 2000, Theorem 2.3), the consistency of $\varphi$ (Lemma 3) guarantees the consistency of $\kappa$ . Because the numerator of $\kappa$ is a Frobenius inner product of $\varphi$ ,
+
+$$
+\sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { n ^ { \prime } } \kappa ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ^ { \prime } ) \propto \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { n ^ { \prime } } \sum _ { j = 1 } ^ { m } \sum _ { j ^ { \prime } = 1 } ^ { m } \varphi ( S _ { i , \cdot } ) _ { j , j ^ { \prime } } \varphi ( S _ { i ^ { \prime } , \cdot } ^ { \prime } ) _ { j , j ^ { \prime } } .
+$$
+
+Thus, in order for $\begin{array} { r } { \sum _ { i , i ^ { \prime } } \kappa ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ^ { \prime } ) < 0 } \end{array}$ , there must be a $j$ and $j ^ { \prime }$ for which $\varphi ( S _ { i , \cdot } ) _ { j , j ^ { \prime } } > 0$ but $\varphi ( S _ { i ^ { \prime } , \cdot } ^ { \prime } ) _ { j , j ^ { \prime } } \ : < \ : 0$ (or vice versa), and thus the hypothesis test in Lemma 3 would reject the null hypothesis that $X _ { j } \perp \perp X _ { j \prime }$ but fail to reject that $X _ { j } ^ { \prime } \perp \perp X _ { j ^ { \prime } } ^ { \prime }$ . 
+
+Corollary 11 Due to the relationship between independence structure and causal structure, an immediate of result of Theorem 10 is that $\begin{array} { r } { \sum _ { i , i } \kappa ( S _ { i , \cdot } , S _ { i ^ { \prime } , \cdot } ^ { \prime } ) < 0 } \end{array}$ implies $X$ and $X ^ { \prime }$ have different causal structures.
+
+Theorem 12 Let $d$ be the distance measure between unconditional equivalence classes of ancestral graphs over $m$ vertices, $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = m ^ { 2 } - | \{ ( j , j ^ { \prime } ) : E ^ { \mathcal { U } \bullet \mathcal { U } ^ { \prime } } ( j , j ^ { \prime } ) = { ^ { * }  } \} | - m$ . For given sample sets $S , S ^ { \prime }$ (i.e., real $n \times m$ matrices), use $\bar { \varphi } ( S )$ to denote the mean of the sample in kernel space, $\textstyle \sum _ { i } \varphi ( S _ { i , . } )$ , and say $S \sim _ { \mathrm { K } } \ S ^ { \prime }$ if and only if $\bar { \varphi } ( S ) \sim _ { 0 } \bar { \varphi } ( S ^ { \prime } )$ ; denote the corresponding quotient set by this equivalence class as $\mathbb { R } ^ { n , m } / \sim _ { \mathrm { K } } = \mathbb { K } ^ { n , m }$ and a representative from each equivalence class as $Q \in [ S ]$ . Let $\delta$ be the distance between sets of samples in $\mathbb { K }$ defined as $\begin{array} { r } { \delta ( Q , Q ^ { \prime } ) = m ^ { 2 } - \frac { 1 } { 2 n ^ { 2 } } \sum _ { i , i ^ { \prime } } \gamma ( Q _ { i , \cdot } , Q _ { i , \cdot } ^ { \ j } ) } \end{array}$ Let $b : \mathbb { U } ^ { m }  \mathbb { K } ^ { n , m } , b : \mathcal { U } \mapsto \Omega$ , where $\Omega$ is the unique element in $\mathbb { K }$ such that $\mathrm { s i g n } ( \bar { \varphi } ( \Omega ) ) = a ( \mathcal { U } )$ . Then $b$ is a distance-preserving map (i.e., an isometry) from the metric space $( \mathbb { U } ^ { m } , d )$ to $( \mathbb { K } ^ { n , m } , \delta )$ .
+
+Proof. Notice that $( \mathbb { U } ^ { m } , d )$ is indeed a metric space (Choudhary, 1993, Ch. 2): $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = 0$ iff $\varkappa ^ { - 1 } \bullet \mathcal { U } ^ { \prime }$ is the empty graph, which happens iff $\mathcal { U } = \mathcal { U } ^ { \prime }$ ; the symmetry of $d$ follows from the symmetry •; and for subadditivity of $d$ , observe that for vertices ${ j , j ^ { \prime } }$ in arbitrary 2-vertex graphs $\boldsymbol { { u } } , \boldsymbol { { u } } ^ { \prime } , \boldsymbol { { u } } ^ { \ast }$ we have either $d ( \mathcal { U } , \mathcal { U } ^ { \ast } ) = 2$ , in which case $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) + d ( \mathcal { U } ^ { \prime } , \mathcal { U } ^ { \prime \prime } ) = 4$ , or we have $d ( \mathcal { U } , \mathcal { U } ^ { \mathfrak { V } } ) = 0$ , in which case $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) + d ( \mathcal { U } ^ { \prime } , \mathcal { U } ^ { \prime \prime } )$ is either 0 or 4—in both cases $d ( \mathcal { U } , \mathcal { U } ^ { \prime \prime } ) \leq d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) + d ( \mathcal { U } ^ { \prime } , \mathcal { U } ^ { \prime \prime } )$ ; this easily extends to graphs of arbitrary numbers of vertices. Likewise, $( \mathbb { K } ^ { n , m } , \delta )$ is a metric space: $\delta ( Q , Q ^ { \prime } ) = 0 \longleftrightarrow \frac { 1 } { 2 n ^ { 2 } } \sum _ { i , i ^ { \prime } } \gamma ( Q _ { i , \cdot } , Q _ { i , \cdot } ^ { \prime } ) = m ^ { 2 } \Longleftrightarrow \bar { \varphi } ( Q ) _ { j , j ^ { \prime } } = \bar { \varphi } ( Q ) _ { j , j ^ { \prime } } ,$ , for all ${ j , j ^ { \prime } }$ , so iff $Q = Q ^ { \prime }$ ; symmetry and subadditivity of $\delta$ follow from the symmetry and subadditivity of $\gamma$ .
+
+Finally, to show $b$ is an isometry, we must show (i) that it is bijective and (ii) that for all $\boldsymbol { \mathcal { U } } , \boldsymbol { \mathcal { U } } ^ { \prime } \in \mathbf { U } ^ { m }$ , $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = \delta ( b ( \mathcal { U } ) , b ( \mathcal { U } ^ { \prime } ) )$ . For (i), observe that by the group isomorphism $a$ and definition of $b$ , we have ${ \mathcal { U } } \neq { \mathcal { U } } ^ { \prime } \implies a ( { \mathcal { U } } ) \neq a ( { \mathcal { U } } ^ { \prime } ) \implies Q \neq Q ^ { \prime } \implies b ( { \mathcal { U } } ) \neq b ( { \mathcal { U } } ^ { \prime } )$ and so $b$ is injective. Also observe that because $\mathbb { K }$ is exactly the set of representatives of orthant equivalence classes of sample sets in kernel space, then for every $Q \in \mathbb { K }$ , there exists a $\mathcal { U }$ such that $b ( \mathcal { U } ) = Q$ , and so $b$ is surjective.
+
+For (ii), isomorphism $a$ and the relation between element-wise product and Frobenius inner product allow us to write $\begin{array} { r } { d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = m ^ { 2 } - \sum _ { j , j ^ { \prime } } ( O \odot O ^ { \prime } ) _ { j , j ^ { \prime } } = m ^ { 2 } - \langle O , O ^ { \prime } \rangle _ { \mathrm { F } } } \end{array}$ . Substituting $O , O ^ { \prime }$ with their corresponding $\Omega , \Omega ^ { \prime }$ , and because the Frobenius inner product is a sesquilinear form, we can write $\begin{array} { r } { d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = m ^ { 2 } - \frac { 1 } { n ^ { 2 } } \sum _ { i , i ^ { \prime } } \langle \varphi ( \Omega _ { i , \cdot } ) , \varphi ( \Omega _ { i , \cdot } ^ { \prime } ) \rangle _ { \mathrm { F } } . } \end{array}$ , which by Definition 10 finally gives us that $d ( \mathcal { U } , \mathcal { U } ^ { \prime } ) = \delta ( \Omega , \Omega ^ { \prime } )$ , completing the proof. 
+
+In less formal terms, Theorem 12 shows how the space of unconditional equivalence classes of ancestral graph corresponds to the space of real matrices, which is a common space for samples to lie in. More specifically, it shows how the structure defined by distances between graphs is the same as the structure defined by distances between sets of samples and how this sample distance is related to our kernel $\kappa$ . Note that this is much stronger than Theorem 10: not only can $\kappa$ tell us that two sets of samples come from different causal models, it gives a measure of just how different the causal models are, in terms of their differing unconditional nonlinear independencies/m-separation statements.
+
+273 To summarize, we began by defining $\varphi$ (Definition 2), which maps a given data set into a new   
+274 higher-dimensional feature space. This feature space corresponds to a space of causal graphical   
+275 models, such that samples which are similar in the new feature space must come from similar causal   
+276 models (Theorem 8). Our main contribution then is to propose the dependence contribution kernel   
+277 $\kappa$ (Definition 9).This kernel $\kappa$ is guaranteed not only to tell us that two sets of samples come from   
+278 different causal models (Theorem 10 and Corollary 11) but furthermore exactly how different the   
+279 causal models are (Theorem 12), all without the computational expense of explicitly projecting   
+280 samples or learning causal models. Thus, $\kappa$ is well-suited for addressing the causal clustering   
+problem and ensures that resulting clusters will be structurally homogeneous so that subsequent   
+causal structure learning will be more informative.
+
+# 283 3 Application
+
+We use kernel $k$ -means with our dependence contribution kernel to cluster a gene expression data set and then use the measurement dependence inducing latent (MeDIL) causal model framework for structure learning within each cluster (Markham and Grosse-Wentrup, 2020). The goal of causal clustering here is to reason about the different latent transcription factor (TF) networks governing gene expression (see Verny et al., 2017; Hackett et al., 2020, for other latent causal model approaches to learning TF networks). The original data set comes from Iyer (1999) and can be found at genome-www.stanford.edu/serum/data/fig2clusterdata.txt, with subsequent analysis by Dhillon et al. (2003, 2004). All of the code for our analysis is open source and available at https://non-anonymous-link.after-review.
+
+The data consists of the measured gene expression levels of 517 different genes from human fibroblast cells in response to serum exposure, measured at 11 different time points, i.e., there are 517 samples and 11 different features. In genetics applications, it is not unusual to consider genes to be samples and expression (over time) to be features—indeed the three previous analyses of this data all have this approach—and the intuition is simply that we wish to cluster genes based on patterns in their expression levels over time, in order to identify subsets of genes that are controlled by the same gene regulatory network. Also notice that such data exemplifies the structurally heterogeneous populations discussed in Section 1: different genes can of course be regulated by different TFs, and so we can better represent the data by first clustering it into subpopulations that are more homogeneous and then performing causal structure learning on each subpopulation.
+
+For clustering, we used $k = 6$ , which we found by looking at both the Variance Ratio Criterion (Calinski and Harabasz, 1974) and the Silhouette Coefficients (Rousseeuw, 1987), computed with the ´ scikit-learn machine learning toolbox (Pedregosa et al., 2011). We implemented (unweighted) kernel $k$ -means ourselves, using the pseudocode given by Dhillon et al. (2004), with initial mean points drawn uniformly at random from the sample set, and with significance level $\alpha = 0 . 1$ for the kernel parameter $\tau ( \alpha )$ . We then used the MeDIL (Markham et al., 2020) package to learn the dependence structure and latent causal models for each cluster.
+
+Figure 1 shows an example of our results for three of the six gene clusters: Figure 1a shows their distance covariance heatmaps and estimated nonlinear dependence structure with significance level $\alpha = 0 . 1$ (so the axes are the 11 different features, i.e. the time, in hours, at which gene expression level was measured), while Figure 1b shows their corresponding causal structures, with measurement variables $M _ { 0 } { - } M _ { 1 0 }$ for each of the features and learned latent variables $L$ for different posited TFs.
+
+The results show a clear difference in causal structure for the different clusters and allow us to reason about the latent TFs regulating genes in different clusters: notice that the latents in cluster K1 each cause only two or three measurement variables that tend to be close together—e.g., $L _ { 1 }$ causes $M _ { 1 }$ and $M _ { 2 }$ , indicating the TF corresponding to $L _ { 1 }$ is “short-acting”, only affecting gene expression from 30 minutes $( M _ { 1 } )$ to 1 hour $( M _ { 2 } )$ after serum exposure; in contrast, the latents in cluster K3 each cause between two and seven measurement variables that tend to be more spread out—e.g., $L _ { 1 }$ causes $M _ { 1 }$ and $M _ { 7 }$ , indicating the corresponding TF is more complicated, “long-acting” but not
+
+![](images/1980f63ea3df4f8129dd14dbc52df12d1755607e16d39aa799cd48672046d8a4.jpg)  
+Figure 1: Results of dependence contribution kernel clustering with significance level $\alpha = 0 . 1$ .
+
+322 continuously so, affecting gene expression 30 minutes $( M _ { 1 } )$ and 12 hours $( M _ { 7 } )$ after serum exposure,   
+323 but independently of gene expression in the time between.
+
+Our results are especially noteworthy compared what happens if one ignores the heterogeneity of the data and learns a causal structure for the entire data set without first clustering with our kernel: in that case, all of the measurement variables are dependent, with a single latent causing all of them, and no meaningful conclusions can be drawn about how unmeasured transcription factors regulate measured gene expression, i.e., the heterogeneity obscures the underlying causal structures.
+
+# 4 Discussion
+
+We address the problem of causal clustering—that is, finding the different causal structures underlying a structurally heterogeneous data set. Our main contribution is to develop the dependence contribution kernel and prove its suitability for the causal clustering task. This allows us to first use the kernel with existing clustering methods, such as kernel $k$ -means or DBSCAN, to identify homogeneous subpopulations. Then we use existing causal structure learning methods on each subpopulation. The kernel guarantees that each subpopulation is more structurally homogeneous and therefore the resulting causal structures better capture the causal structures within the data than if a single model were learned for the entire heterogeneous population.
+
+338 Furthermore, we prove several interesting theoretical properties of our kernel, including (i) that   
+339 it can be used as a statistical test for the hypothesis that two sets of samples come from different   
+340 causal structures, as well as (ii) how it induces a metric space that is isometric to the one defined   
+341 by Hamming distance between ancestral graphs, i.e., comparing sets of samples with our kernel is   
+342 equivalent to first estimating the causal graphs of the different sets and then comparing those graphs.   
+343 Beyond the practical applications of our kernel, as shown by our application in reasoning about latent   
+344 transcription factor networks that regulate gene expression, this work also draws from and suggests   
+345 further fruitful connections between a variety of fields, including causal inference, kernel methods,   
+346 and algebraic statistics.
+
+#
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The first, fifth, and sixth sentences are covered thoroughly in Sections 1, 3, and 4, while the rest are covered thoroughly in Section 2.   
+(b) Did you describe the limitations of your work? [Yes] In Section 1.1 and throughout Section 2   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work has no direct potential negative societal impact—just the same indirect potential most theoretical work has   
+(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] Yes, general assumptions in Section 1.1 as well as more specific assumptions within the statement of each relevant theorem/lemma/etc.   
+(b) Did you include complete proofs of all theoretical results? [Yes] Proofs follow each Theorem and Lemma in the text
+
+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] Though the linked pages contain identifying information, so we’ve included only placeholder “https://non-anonymous-link.after-review” links   
+(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)? [N/A] We didn’t run an experiment multiple times but rather analyzed a real data set
+
+(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)? [No] It runs in just a few seconds, even on an old, underpowered laptop
+
+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] We mentioned that it’s all open source; details can be found in their respective repos/documentation   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] The data is publicly available   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] It’s gene expression data, so neither of these are an issue
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/Ad7Gv5NkBPz/Ad7Gv5NkBPz_content_list.json

@@ -0,0 +1,1160 @@
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+    {
+        "type": "text",
+        "text": "A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 We consider the problem of causal structure learning in the setting of heterogeneous   \n2 populations, i.e., populations in which a single causal structure does not adequately   \n3 represent all population members, as is common in biological and social sciences.   \n4 To this end, we introduce a distance covariance-based kernel designed specifically   \n5 to measure the similarity between the underlying nonlinear causal structures of   \n6 different samples. This kernel enables us to perform clustering to identify the   \n7 homogeneous subpopulations. Indeed, we prove the corresponding feature map is   \n8 a statistically consistent estimator of nonlinear independence structure, rendering   \n9 the kernel itself a statistical test for the hypothesis that sets of samples come from   \n10 different generating causal structures. We can then use existing methods to learn   \n11 a causal structure for each of these subpopulations. We demonstrate using our   \n12 kernel for causal clustering with an application in genetics, allowing us to reason   \n13 about the latent transcription factor networks regulating measured gene expression   \n14 levels. ",
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+    {
+        "type": "text",
+        "text": "15 1 Introduction ",
+        "text_level": 1,
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+    },
+    {
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+        "text": "16 Learning causal relationships from observational and experimental data is one of the fundamental   \n17 goals of scientific research, and causal inference methods are thus used in a wide variety of fields. The   \n18 resulting variety of applications nevertheless share some common difficulties, such as causal inference   \n19 from complex time-series data (Eichler, 2012) or the underlying causal structure being obscured   \n20 by unmeasured confounders (Greenland et al., 1999). Another common difficulty, especially for   \n21 applications in the biological and social sciences, is causal inference from heterogeneous populations   \n22 (Xie, 2013; Brand and Thomas, 2013)—addressing this difficulty is our main motivation.   \n23 In general terms, we understand a heterogeneous population to be one whose members are not   \n24 adequately described by a single model but rather better described by a collection of models. Within   \n25 our context of causal structure learning, this means a population is heterogeneous if some samples   \n26 are generated by different causal structures—we call this structural heterogeneity. We note that there   \n27 are other kinds of heterogeneity, such as that in samples generated by different joint distributions   \n28 over the same causal structure, which are not the scope of this work.   \n29 A specific example of structural heterogeneity can be found in genetics: causal methods are used to   \n30 learn the structure of gene regulatory networks (Emmert-Streib et al., 2012), and gene expression data   \n31 from a single recording or experiment may include thousands of genes, many of which are involved   \n32 in entirely different networks (Liu, 2015); thus, attempting to learn a single causal structure for all of   \n33 the genes will obscure the fact that different sets of them have different structures.   \n34 The bulk of our work in this paper, and our main contribution, is to introduce the dependence   \n35 contribution kernel, which facilitates a flexible and easily extensible approach to causal clustering:   \n36 first perform clustering to identify structurally homogeneous subsets of samples, and then proceed   \n37 with the actual learning task on each cluster. We prove that our kernel is a statistically consistent   \n38 estimator of the similarity of the causal structures underlying different samples and can thus be used   \n39 to find clusters that minimize structural heterogeneity for causal structure learning tasks. Furthermore,   \n40 the kernel is derived from the distance covariance (Székely et al., 2007), imbuing it with the ability   \n41 to detect nonlinear dependence. It can easily be used in a wide array of clustering algorithms, such   \n42 as $k$ -means, DBSCAN, spectral clustering, or any other method that analogously makes use of a   \n43 similarity (or distance) measure between samples (Filippone et al., 2008).   \n44 The rest of the paper is organized as follows: We finish this section by discussing some of the most   \n45 relevant related work from the causal inference and statistics literature. All of Section 2 is devoted to   \n46 the theory underlying our dependence contribution kernel, including a comparison of the familiar   \n47 product-moment covariance with the distance covariance (Section 2.1), defining an equivalence   \n48 class of causal models with a convenient representation in the kernel space (Section 2.2), and the   \n49 actual definition of our kernel and proofs of its relevant properties (Section 2.3). Next, in Section   \n50 3, we demonstrate causal clustering with the kernel on a heterogeneous gene expression data set,   \n51 finding structurally homogeneous clusters for which we then learn latent causal measurement models,   \n52 allowing us to reason about the different transcription factor networks responsible for regulating the   \n53 measured gene expression levels. Finally, we conclude in Section 4 mentioning possible future work. ",
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+        "type": "text",
+        "text": "54 1.1 Related Work ",
+        "text_level": 1,
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+        "text": "55 Causal inference in heterogeneous populations sometimes refers to data-fusion (Bareinboim and   \n56 Pearl, 2016), i.e., combining known homogeneous subpopulations and performing causal inference   \n57 on the resulting heterogeneous population, or similarly, it can refer to meta-learning using known   \n58 subpopulations (Sharma et al., 2019). Other times, it refers to estimating heterogeneous treatment   \n59 effects (Xie et al., 2012; Athey and Imbens, 2015). However, in our case, the subpopulations are not   \n60 known and we rather consider the problem of learning which samples come from which subpopulation,   \n61 and these are differentiated according to structure instead of treatment effect.   \n62 Previous work on causal clustering has focused more on the causal modeling aspect, using stronger   \n63 assumptions about the underlying structures to learn more detailed models. For example, Kummerfeld   \n64 et al. (2014); Kummerfeld and Ramsey (2016) focus on causal clustering in measurement models,   \n65 with the goal of clustering different features together to study their latent causal structure, based on   \n66 tetrad constraints within the linear product-moment covariance matrix. Huang and Zhang (2019)   \n67 define a class of causal models facilitating mechanism-based clustering, learning causal models both   \n68 for clusters of samples as well as a shared one for all samples, assuming the underlying structures   \n69 are linear non-Gaussian. Saeed et al. (2020) characterize distributions arising from mixtures of   \n70 directed acyclic graph (DAG) causal models (i.e., causal models without latent or selection variables),   \n71 trying to learn both the component DAGs and a representation of how they are mixed. All of these   \n72 approaches, like most causal inference methods, make specific (and for some applications, restrictive)   \n73 assumptions about the underlying distributions or causal structures.   \n74 In contrast, our method is not tied to specific distributional assumptions such as linearity or   \n75 (non)Gaussianity—we assume there are enough samples for statistical inference, as well as the   \n76 usual causal Markov and faithfulness assumptions. For the first step, we cluster samples together if   \n77 they (implicitly, in the kernel space) have similar nonlinear independence structures. For the second   \n78 step, causal structure learning, any existing method (along with its corresponding assumptions) can in   \n79 principle be used. In our gene expression data application (Section 3), the measurement dependence   \n80 inducing latent (MeDIL) causal model framework (Markham and Grosse-Wentrup, 2020), which   \n81 assumes the data consists of measurement variables that are causally connected only through latent   \n82 variables, seems appropriate, however other applications can easily use other methods. For example,   \n83 component and mixture DAGs (Saeed et al., 2020) can be better learned when one first knows which   \n84 samples come from which component—clustering with our kernel ensures samples in different   \n85 clusters come from different DAGs, and so using their method instead of the MeDIL framework   \n86 would be a natural choice for applications in which a DAG (without any latents) is more appropriate. ",
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+        "type": "text",
+        "text": "2 Theory ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "2.1 Product-moment Covariance, Distance Covariance, and Dependence Contribution ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Though there is more to causal relationships than probabilistic dependence, causal inference methods based on graphical models ultimately rely on at least implicitly learning conditional independence (CI) relations. CI relations can be estimated in many ways, with different dependence measures and tests each having their own theoretical guarantees and being better suited for distributions of various different kinds of data (e.g., categorical, discrete, or continuous) and with various kinds of relationships (e.g., linear, monotonic nonlinear, arbitrary nonlinear) and with different testing assumptions (see Tjøstheim et al., 2018, for a comprehensive overview). ",
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+        "text": "A widely used measure of dependence is the product-moment covariance, often just called covariance, which is defined for two zero-mean random variables $X _ { 1 }$ and $X _ { 2 }$ as the scalar value $\\operatorname { c o v } ( X _ { 1 } , X _ { 2 } ) =$ $\\operatorname { E } [ X _ { 1 } X _ { 2 } ]$ . This can be extended from a pair of random variables to every pair of variables in a random vector, thus returning a matrix instead of a scalar. The covariance matrix for a vector of zero-mean random variables $\\mathbf { X } = ( X _ { 1 } , \\ldots , X _ { m } )$ can be estimated from a set $S \\in \\mathbb { R } ^ { n , m }$ of $n$ samples as $\\textstyle { \\hat { \\Sigma } } _ { \\mathbf { X } } = { \\frac { 1 } { n } } S ^ { \\top } S$ , and the ${ j , j ^ { \\prime } }$ -th value of $\\hat { \\Sigma } _ { \\mathbf { X } }$ is thus the estimate $\\operatorname { c o v } ( X _ { j } , X _ { j } ^ { \\prime } )$ . ",
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+        "text": "102 Two random variables being probabilistically independent (denoted $\\perp \\perp$ ) implies that their product  \n103 moment covariance is zero, i.e., $X _ { j } \\perp \\perp X _ { j ^ { \\prime } } \\implies \\operatorname { c o v } ( X _ { j } , X _ { j ^ { \\prime } } ) = 0$ (importantly, the inverse of this   \n104 does not hold). Thus, the estimated product-moment covariance can be used in statistical hypothesis   \n105 testing for probabilistic independence (Wasserman, 2013, Ch. 10): $X _ { j }$ and $X _ { j ^ { \\prime } }$ are assumed to   \n106 be independent if and only if $\\operatorname { c o v } ( X _ { j } , X _ { j ^ { \\prime } } )$ is sufficiently close to 0. However, this method has   \n107 an important problem: the product-moment covariance is only a valid test statistic against linear   \n108 dependence.   \n109 Székely et al. (2007) introduce the distance covariance to remedy this problem: random variables are   \n110 probabilistically independent if and only if their distance covariance is zero, i.e., $X _ { j } \\perp \\perp X _ { j ^ { \\prime } } \\iff$   \n111 $\\operatorname { d C o v } ( X _ { j } , X _ { j ^ { \\prime } } ) = 0$ , resulting in the estimated distance covariance being a valid test statistic against   \n112 all types of dependence. The distance covariance is related to the product-moment covariance by   \n113 $\\mathrm { d } \\mathrm { C o v } ^ { 2 } ( X _ { j } , X _ { j ^ { \\prime } } ) = \\mathrm { c o v } ( | X _ { j } - X _ { j } ^ { \\prime } | , | X _ { j ^ { \\prime } } - X _ { j ^ { \\prime } } ^ { \\prime } | ) - 2 \\mathrm { c o v } ( | X _ { j } - X _ { j } ^ { \\prime } | , | X _ { j ^ { \\prime } } - X _ { j ^ { \\prime } } ^ { \\prime \\prime } | ) .$ , where $( X _ { j } ^ { \\prime } , X _ { j ^ { \\prime } } ^ { \\prime } )$   \n114 and $( X _ { j } ^ { \\prime \\prime } , X _ { j ^ { \\prime } } ^ { \\prime \\prime } )$ are independent and identically distributed (iid) copies of $( X _ { j } , X _ { j ^ { \\prime } } )$ (Székely and   \n115 Rizzo, 2014). The key intuition here is that the distances (e.g., $| X _ { j } - X _ { j } ^ { \\prime } | )$ constitute a nonlinear   \n116 projection, so that using the linear product-moment covariance in this projected space allows for the   \n117 detection of nonlinear dependence in the original space.   \n118 Note that dCov is typically defined to be a scalar value when taken between two arbitrary-dimensional   \n119 random vectors, but our restricted presentation of it above in terms of random variables is to make   \n120 it more obviously analogous to the product-moment covariance between random variables. Thus,   \n121 corresponding to $\\hat { \\Sigma } _ { \\mathbf { X } }$ for random vectors, we define the following:   \n22 Definition 1 Let $S \\in \\mathbb { R } ^ { n , m }$ be a set of $n$ samples from the vector of random variables ${ \\textbf { X } } =$   \n23 $( X _ { 1 } , \\ldots , X _ { m } )$ . For each $j \\in \\{ 1 , \\dots , m \\}$ and $i , i ^ { \\prime } \\in \\{ 1 , \\ldots , n \\}$ , define the pairwise distance matrix   \n24 $D ^ { j }$ , with values given by $D _ { i , i ^ { \\prime } } ^ { \\jmath } : = | S _ { i , j } - S _ { i ^ { \\prime } , j } |$ . Now define the corresponding doubly-centered   \n25 matrices $C _ { i , i ^ { \\prime } } ^ { j } : = { D } _ { i , i ^ { \\prime } } ^ { j } - \\bar { D ^ { j } } _ { i , \\cdot } - \\bar { D ^ { j } } _ { \\cdot , i ^ { \\prime } } + \\bar { D ^ { j } } _ { \\cdot , \\cdot }$ , where putting a bar over the matrix and replacing   \n26 an index $i$ or $i ^ { \\prime }$ with $\\cdot$ denotes taking the mean over that index. Define the matrix $L \\in \\mathbb { R } ^ { n ^ { 2 } , m }$ so   \n27 that each column is a flattened doubly-centered distance matrix, $L : = ( \\operatorname { v e c } ( C ^ { 1 } ) , \\dots , \\operatorname { v e c } ( C ^ { m } ) )$ ,   \n28 where $\\mathrm { v e c } ( C ^ { j } )$ denotes “flattening” matrix $C ^ { j }$ into a column vector. Finally, the estimated distance   \n29 covariance matrix over sample $S$ is defined as $\\begin{array} { r } { \\hat { \\Delta } \\mathbf { x } : = \\frac { 1 } { n ^ { 2 } } L ^ { \\top } L } \\end{array}$ . ",
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+        "text": "Analogous to 130 $\\hat { \\Sigma } _ { \\mathbf { X } }$ , the ${ j , j ^ { \\prime } }$ -th entry of $\\hat { \\Delta } _ { \\mathbf { X } }$ corresponds to $\\mathrm { d } \\hat { \\mathrm { C o v } } ^ { 2 } ( X _ { j } , X _ { j ^ { \\prime } } ) .$ —indeed it is mathemati131 cally equivalent to computing each pairwise distance covariance value and then manually filling in ",
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+        "type": "text",
+        "text": "132 the matrix. The novelty of our Definition 1 is in finding a matrix of pairwise values instead of a single   \n133 value for the distance covariance between random vectors, which helps provide an intuition for our   \n134 next definition: ",
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+    {
+        "type": "text",
+        "text": "Definition 2 Let $S \\in \\mathbb { R } ^ { n , m }$ be a set of $n$ samples from the vector of random variables ${ \\textbf { X } } =$ $( X _ { 1 } , \\ldots , X _ { m } )$ ; note that we consistently use indices $i , i ^ { \\prime } \\in \\{ 1 , \\ldots , n \\}$ and $j , j ^ { \\prime } \\in \\{ 1 , \\ldots , m \\}$ . Let $D \\in \\mathbb { R } ^ { n , n , m }$ denote the 3-dimensional array of stacked pairwise distance matrices defined by $D _ { i , i ^ { \\prime } , j } : = | S _ { i , j } - S _ { i ^ { \\prime } , j } |$ , and use $C \\in \\mathbb { R } ^ { n , n , m }$ to denote these same distance matrices after being doubly-centered, i.e., $\\tilde { C } _ { i , i ^ { \\prime } , j } : = D _ { i , i ^ { \\prime } , j } - \\bar { D } _ { i , \\cdot , j } - \\bar { D } _ { \\cdot , i ^ { \\prime } , j } + \\bar { D } _ { \\cdot , \\cdot , j }$ , where replacing an index $i$ or $i ^ { \\prime }$ with · denotes the entire (lower-dimensional) subarray over that index, and writing a bar, $\\bar { D }$ , denotes taking the mean over that subarray. Then standardize the doubly-centered distances to get $\\begin{array} { r } { Z _ { i , i ^ { \\prime } , j } : = \\frac { C _ { i , i ^ { \\prime } , j } } { \\bar { D } \\cdot \\underline { { \\mathbf { \\Pi } } } _ { \\cdot , j } } } \\end{array}$ Finally, the dependence contribution map, $\\varphi : \\mathbb { R } ^ { m }  \\mathbb { R } ^ { m , m }$ , is defined as ",
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+        "type": "equation",
+        "img_path": "images/dfebc778fa5a00282eabb7a6129aa641d36f2b2c61210738a7c2c7a0cea8298f.jpg",
+        "text": "$$\n\\varphi ( S _ { i , \\cdot } ) : = Z _ { i , \\cdot , \\cdot } ^ { \\top } Z _ { i , \\cdot , \\cdot } - \\mathcal { T } ( \\alpha ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where 135 $\\mathcal { T } ( \\alpha ) \\in \\mathbb { R } ^ { m , m }$ is a matrix of scaled critical values corresponding to a given significance level 136 $\\alpha$ with zeros along the diagonal, i.e., $\\begin{array} { r } { \\mathcal { T } ( \\alpha ) _ { j , j ^ { \\prime } } = \\left\\{ \\begin{array} { l l } { 0 , } \\\\ { \\frac { 1 } { n } \\chi _ { 1 - \\alpha } ^ { 2 } ( 1 ) } \\end{array} \\right. } \\end{array}$ if , ot $j = j ^ { \\prime }$ e , with $\\chi _ { 1 - \\alpha } ^ { 2 } ( 1 )$ being the 137 $1 - \\alpha$ quantile of the chi-square distribution with 1 degree of freedom. ",
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+        "text": "138 Notice the similarity bestandardization (i.e., use $C$ een Defininstead of $Z$ ons 2 a), then $\\begin{array} { r } { \\frac { 1 } { n ^ { 2 } } \\sum _ { i = 1 } ^ { n } \\varphi ( S _ { i , \\cdot } ) = \\hat { \\Delta } _ { \\mathbf { X } } } \\end{array}$ $\\mathcal { T } ( \\alpha )$ be a matrix of 0s and. Now, the differences: $\\hat { \\Delta } _ { \\mathbf { X } } \\hat { }$ gois   \n140 a single matrix computed over an entire set of samples, whereas $\\varphi$ is a map that projects each given   \n141 sample to a new feature space; each entry of $\\hat { \\Delta } _ { \\mathbf { X } }$ is simply a distance covariance value, whereas each   \n142 entry of the sum of $\\varphi ( S _ { i , \\cdot } )$ over $i$ , by using standardization (using $Z$ instead of $C$ ) and subtracting a   \n143 critical value, corresponds to the result of using a distance covariance value in a statistical hypothesis   \n144 test for independence—indeed: ",
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+        "text": "Lemma 3 Let $S \\in \\mathbb { R } ^ { n , m }$ be a set of $n$ iid samples from random variables $X _ { 1 } , \\ldots , X _ { m }$ with finite first moments. For a given significance level $\\alpha$ , under the null hypothesis of $X _ { j } \\perp \\perp X _ { j ^ { \\prime } }$ , the test ",
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+        "text": "$$\n\\mathrm { r e j e c t } h _ { \\varnothing } \\mathrm { i f } \\quad \\big ( \\sum _ { i = 1 } ^ { n } \\varphi ( S _ { i , \\cdot } ) \\big ) _ { j , j ^ { \\prime } } > 0\n$$",
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+        "text": "145 is statistically consistent against all types of dependence. ",
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+        "text": "146 Proof. This follows from (Székely and Rizzo, 2009, Theorem 5 and Corollary 2) and how $\\varphi$ is   \n147 defined to correspond to the difference between distance covariance and critical values. \u0003   \n148 These differences between $\\hat { \\Delta } _ { \\mathbf { X } }$ and $\\varphi$ serve two important purposes: first, they ensure $\\varphi$ maps to a   \n149 Hilbert space so that our Definition 9 is a corresponding kernel function (Schölkopf et al., 2001); and   \n150 second, as the name “dependence contribution map” suggests, they ensure $\\varphi ( S _ { i , \\cdot } )$ is informative not   \n151 just about distance covariance but about nonlinear dependence and about how the inclusion of sample   \n152 $S _ { i , \\astrosun }$ ,· in a set of samples $S$ contributes to the dependence patterns estimated from $S _ { \\ l }$ — this is the key   \n153 intuition behind how our kernel function is used to learn structurally homogeneous sample subsets,   \n154 as explicated in the following sections. ",
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+        "type": "text",
+        "text": "2.2 Causal Graphs in Kernel Space ",
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+        "text": "In general, a full causal structure can only be learned with sufficient data about the effects of interventions, and thus causal structure learning from purely observational data is usually possible only up to an equivalence class of causal graphs (Spirtes et al., 2000; Pearl, 2009). For example, the classic PC and IC algorithms, under the assumptions of no selection bias and no confounding by latent variables, do not necessarily return a fully-specified DAG but instead return a mixed graph, containing possibly directed and undirected edges, representing the Markov equivalence class (Spirtes and Glymour, 1991; Pearl and Verma, 1995). ",
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+        "text": "163 We now define a set of equivalence classes for ancestral graphs (AGs), which—unlike causal DAGs—   \n164 do not assume the absence of selection bias and latent confounders (Richardson et al., 2002): ",
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+        "text": "Definition 4 Consider an arbitrary ancestral graph $\\mathcal { A }$ with the set of vertices $V ^ { A }$ and edge function $E ^ { A }$ , and denote the set of unconditional $m$ -connection statements entailed by their corresponding unique maximal ancestral graph as $M ^ { A } = \\{ ( j , j ^ { \\prime } ) : j \\mathcal { L } _ { m } j ^ { \\prime } | \\emptyset \\} \\subseteq V ^ { A } \\times \\dot { V } ^ { A }$ . For any ancestral graph $\\mathcal { A } ^ { \\prime }$ such that $V ^ { \\mathcal { A ^ { \\prime } } } = \\bar { V } ^ { \\mathcal { A } }$ , define the unconditional equivalence relation denoted by $^ \\bullet \\sim _ { \\mathrm { U } }$ ’ as ",
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+        "img_path": "images/3985bfe6ec76a0490110a51a782f1e52fd1eb1225a17c3baf92f862708a31b76.jpg",
+        "text": "$$\n\\mathcal { A } \\sim _ { \\mathrm { U } } \\mathcal { A } ^ { \\prime } \\quad \\mathrm { i f ~ a n d ~ o n l y ~ i f } \\quad M ^ { A } = M ^ { A ^ { \\prime } } .\n$$",
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+        "text": "165 Lemma 5 This lemma has two parts: (i) the relation ${ \\sim } _ { \\mathrm { U } }$ is an equivalence relation over the set of   \n166 ancestral graphs A; (ii) for an arbitrary ancestral graph $\\mathcal A \\in \\mathbb A$ , the bidirected graph $\\mathcal { U } ^ { A } = ( V ^ { A } , E ^ { \\mathcal { U } } )$ ,   \n167 where $E ^ { \\mathcal { U } }$ maps all pairs $( j , j ^ { \\prime } ) \\in M ^ { A }$ to the bidirected edge symbol $\\cdot $ , is a unique representative   \n168 of the equivalence class $[ A ]$ .   \n169 Proof. For (i), recall that an equivalence relation is any relation satisfying reflexivity, symmetry,   \n170 and transitivity (Devlin, 2003), all of which are satisfied by ${ \\sim } _ { \\mathrm { U } }$ because of its correspondence to   \n171 the relation $\" = \"$ between sets. Thus, to prove (ii), it suffices to show that the map $s : \\mathbb { A } / \\sim _ { \\mathrm { U } } \\to$   \n172 A, $[ \\mathcal { A } ] \\mapsto \\mathcal { U } ^ { A }$ is injective (i.e, that it is a section) and that $[ s ( [ A ] ) ] = [ A ]$ (Mac Lane, 2013). The   \n173 key to the proof is the observation that $\\mathcal { U } ^ { A }$ , because it contains only bidirected edges, is maximal   \n174 and therefore entails exactly the unconditional $m$ -separation statements $M ^ { A }$ , thus by (i) we have   \n175 ${ \\mathcal { U } } ^ { A } \\sim _ { \\mathrm { U } } { \\mathcal { A } }$ or equivalently $\\mathcal { U } ^ { A } \\in [ A ]$ or equivalently $[ \\mathcal { U } ^ { A } ] = [ \\mathcal { A } ]$ . Let $\\mathcal { A } , \\mathcal { A } ^ { \\prime }$ be arbitrary AGs, and   \n176 assume $s ( [ \\mathcal { A } ] ) = s ( [ \\mathcal { A } ^ { \\prime } ] )$ . Then by definition of $s$ we have $\\bar { \\mathcal { U } } ^ { A } \\bar { = } \\bar { \\mathcal { U } } ^ { A ^ { \\prime } }$ , and by the observation above,   \n177 $\\mathcal { U } ^ { A } \\in [ \\mathcal { A } ^ { \\prime } ]$ and thus $[ A ] = [ A ^ { \\prime } ]$ , making $s$ injective. And finally, by the definition of $s$ and also by   \n178 the observation above, $[ s ( [ A ] ) ] = [ \\mathcal { U } ^ { A } ] = [ A ]$ , completing the proof. \u0003   \n179 This equivalence relation and its representatives has some important but perhaps subtle properties.   \n180 First, it is different from Markov equivalence over AGs (which is characterized by partial ancestral   \n181 graphs, PAGs) (Zhang, 2007)—it uses only unconditional $m$ -separation while PAGs are learned from   \n182 conditional $m$ -separation statements. Second, because all DAGs are AGs, ${ \\sim } _ { \\mathrm { U } }$ is also an equivalence   \n183 relation over DAGs. Third, being a representative means that every equivalence class includes exactly   \n184 one fully bidirected graph (along with other equivalent AGs). Fourth, because each representative is   \n185 formed by considering $m$ -connected paths, $\\mathcal { U } ^ { A }$ is not equivalent to what would be generated by some   \n186 “edge-wise” procedure, such as simply replacing every edge in a PAG/AG/DAG/Markov random   \n187 field/moralized DAG with bidirected edges.Finally, its most important property is that it facilitates   \n188 Theorem 8, for which we first need a few more definitions. ",
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+        "type": "text",
+        "text": "Definition 6 Given arbitrary ancestral graphs $\\mathcal { A } , \\mathcal { A } ^ { \\prime } \\in \\mathbb { A }$ over the same set of vertices, define the Hamming similarity product, denoted $\\bullet _ { \\bullet } ,$ as ",
+        "bbox": [
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+        "img_path": "images/10aea1b9982dcaab6561bad8759a4e7dc9bd63653db68621e9496cb120b1b9d7.jpg",
+        "text": "$$\n\\bullet : \\mathbb { A } \\times \\mathbb { A } \\to \\mathbb { A } \\quad { \\mathrm { a n d } } \\quad \\mathcal { A } \\bullet \\mathcal { A } ^ { \\prime } \\mapsto \\mathcal { H } ,\n$$",
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+        "type": "text",
+        "text": "where 189 $\\mathcal { H } = ( V ^ { A } , E ^ { \\mathcal { H } } )$ and the function $E ^ { \\mathcal { H } } ( j , j ^ { \\prime } ) = \\ '  ^ { \\prime }$ if and only if $E ^ { A } ( j , j ^ { \\prime } ) = E ^ { A ^ { \\prime } } ( j , j ^ { \\prime } )$ . ",
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+        "type": "text",
+        "text": "190 In words, the Hamming similarity product between two ancestral graphs returns a fully bidirected   \n191 graph, with edges only where the two graphs have the same edge type. Now, shifting from ancestral   \n192 graphs to real-valued square matrices: ",
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+        "type": "text",
+        "text": "Definition 7 Let $\\cdot _ { \\sim _ { \\mathrm { O } } }$ ’ denote the orthant equivalence relation (‘orthant’ is the generalization of ‘quadrant’ from $\\mathbb { R } ^ { 2 }$ to arbitrarily higher dimensions) in square real matrices, i.e., for matrices Y, Y 0 ∈ Rm,m and with the element-wise function sign(Y )j,j0 = $\\mathrm { s i g n } ( Y ) _ { j , j ^ { \\prime } } = \\left\\{ { \\begin{array} { l l } { 1 , } \\\\ { - \\ } \\end{array} } \\right.$ 1, otherwise if $Y _ { j , j ^ { \\prime } } > 0$ or $j = j ^ { \\prime }$ ",
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+        "text": "$$\nY \\sim _ { \\mathrm { { o } } } Y ^ { \\prime } \\quad { \\mathrm { i f ~ a n d ~ o n l y ~ i f } } \\quad \\mathrm { s i g n } ( Y ) _ { j , j ^ { \\prime } } = \\mathrm { s i g n } ( Y ^ { \\prime } ) _ { j , j ^ { \\prime } }\n$$",
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+        "type": "text",
+        "text": "193 Theorem 8 Let $a$ be the map from the set of unconditional equivalence classes over ancestral graphs   \n194 with $m$ vertices, $\\mathbb { A } ^ { m } / \\sim _ { \\mathrm { U } } = \\mathbb { U } ^ { m }$ , to the set of orthant equivalence classes over the image of $\\varphi$   \n195 i.e., $m \\times m$ symmetric real matrices with positive diagonal entries, $\\varphi ( \\mathbb { R } ^ { m } ) / { \\sim } _ { 0 } = \\mathbb { O } ^ { m }$ , defined by   \n196 $a : \\mathcal { U } \\mapsto O$ , where $O _ { j , j ^ { \\prime } } = \\left\\{ { 1 , \\atop - 1 } \\right.$ if , ot EU (j, j0) = ‘↔’ or j = j0 . Then a is a group isomorphism   \nbetween 197 $( \\mathbb { U } ^ { m } , \\bullet )$ and $( \\mathbb { O } ^ { m } , \\odot )$ , where $\\ast _ { \\odot } \\cdot$ ’ denotes the element-wise product.   \n198 Proof. First, note that $( \\mathbb { U } ^ { m } , \\bullet )$ is indeed a group, satisfying the three group axioms (Artin, 2011):   \n199 the representative of its identity element is the fully connected bidirected graph over $m$ vertices, $\\mathcal { U } ^ { \\mathbb { 1 } }$ ;   \n200 each element is its own inverse; and $\\bullet$ is associative. Likewise, $( \\mathbb { O } ^ { m } , \\odot )$ is a group with identity   \n201 element $\\left[ \\mathbb { 1 } ^ { m , m } \\right]$ , each element its own inverse, and the associative element-wise product operator. ",
+        "bbox": [
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+        "text": "",
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+        "type": "text",
+        "text": "Now, to show the two groups are isomorphic, it suffices to show (i) that $a$ is bijective and (ii) that for arbitrary $\\mathcal { U } , \\mathcal { U } ^ { \\prime } \\in \\mathbb { U } ^ { m }$ $\\mathbb { U } ^ { m } , a ( \\mathcal { U } ) \\odot a ( \\mathcal { U } ^ { \\prime } ) = a ( \\mathcal { U } \\bullet \\mathcal { U } ^ { \\prime } )$ . For (i) notice that if $U \\neq U ^ { \\prime }$ , then there must be at least one pair of vertices ${ j , j ^ { \\prime } }$ such that $\\dot { E } ^ { \\mathcal { U } } ( j , \\dot { j } ^ { \\prime } ) \\neq E ^ { \\mathcal { U } ^ { \\prime } } ( j , \\dot { j } ^ { \\prime } )$ and thus clearly $O _ { j , j ^ { \\prime } } \\neq O _ { j , j ^ { \\prime } } ^ { \\prime }$ so $a$ in injective. Furthermore, notice that every distinct $O \\in \\mathbb { O } ^ { m }$ is the image of some graph $\\mathcal { U }$ , so $a$ is also surjective. For (ii), for every $j , j ^ { \\prime } \\in \\{ 1 , \\ldots , m \\}$ , the definitions of $a , \\odot$ , and $\\bullet$ ensure $a ( \\mathcal { U } ) _ { j , j ^ { \\prime } } \\odot a ( \\mathcal { U } ^ { \\prime } ) _ { j , j ^ { \\prime } } = 1 \\iff E ^ { \\mathcal { U } } ( j , j ^ { \\prime } ) = E ^ { \\mathcal { U } ^ { \\prime } } ( j , j ^ { \\prime } ) \\iff 1 = a ( \\mathcal { U } \\bullet \\mathcal { U } ^ { \\prime } )$ , completing the proof. \u0003 ",
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+        "type": "text",
+        "text": "209 For causal inference, which (often, but not necessarily) amounts to taking several samples in real   \n210 space and inferring a single corresponding member in the space of ancestral graphs (or, more often,   \n211 its quotient set by some equivalence relation), Theorem 8 means we can compare the different graphs   \n212 of different sample sets without having to first move to the ancestral graph space. ",
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+    {
+        "type": "text",
+        "text": "Finally, notice the space of real square matrices is not a typical sample space but rather precisely (a superspace of) the space that our dependence contribution map $\\varphi$ (Definition 2) maps samples to—this means that mapping samples with $\\varphi$ allows us to make use of the group isomorphism. Though this already provides an intuition for why using $\\varphi$ would help with causal clustering, explicitly mapping each sample with it would be unnecessarily computationally expensive, and we are ultimately interested in morphisms between metric spaces (not just groups) of samples and graphs. To address this, we thus now move on to defining a kernel for $\\varphi$ . ",
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+        "type": "text",
+        "text": "220 2.3 The Dependence Contribution Kernel ",
+        "text_level": 1,
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+        "text": "Definition 9 Let $S , Z , \\mathcal { T }$ , and $\\varphi$ be as in Definition 2. We define the dependence contribution kernel using the Frobenius (denoted by the subscript $\\mathrm { F }$ ) inner product and norm: ",
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+        "img_path": "images/e08ab9a003245991c22ac50362ba77b77ec7e8993f8171a65a8ff03efdcefc77.jpg",
+        "text": "$$\n\\kappa ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ) = \\frac { \\langle \\varphi ( S _ { i , \\cdot } ) , \\varphi ( S _ { i ^ { \\prime } , \\cdot } ) \\rangle _ { \\mathrm { F } } } { \\| \\varphi ( S _ { i , \\cdot } ) \\\\| _ { \\mathrm { F } } \\| \\varphi ( S _ { i ^ { \\prime } , \\cdot } ) \\| _ { \\mathrm { F } } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "221 A more convenient expression for applying the kernel to a data set is obtained by first defining a   \n222 helper kernel, $\\gamma$ along with vec from Definition 1: ",
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+        "img_path": "images/c406b8f4099fa02802cc96896fb51d1a3919865c83fd08cc5e64cb2181d2b895.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\gamma ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ) = \\langle \\varphi ( S _ { i , \\cdot } ) , \\varphi ( S _ { i ^ { \\prime } , \\cdot } ) \\rangle _ { \\mathrm { F } } } \\\\ & { \\qquad = \\left( ( \\mathrm { v e c } ( Z _ { i , \\cdot } ) ^ { \\top } \\mathrm { v e c } ( Z _ { i ^ { \\prime } , \\cdot } ) \\right) ^ { 2 } - Z _ { i , \\cdot } { \\mathcal { T } } Z _ { i , \\cdot } ^ { \\top } - Z _ { i ^ { \\prime } , \\cdot } { \\mathcal { T } } Z _ { i ^ { \\prime } , \\cdot } ^ { \\top } + \\| { \\mathcal { T } } \\| _ { 2 } ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This allows us to write ",
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+        "type": "equation",
+        "img_path": "images/e9e246ca4381d271e24a239464fbf02a0d4dfcc4e16560af5db152cb90d2683d.jpg",
+        "text": "$$\n\\kappa ( s , s ^ { \\prime } ) = \\frac { \\gamma ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ) } { \\gamma ( S _ { i , \\cdot } , S _ { i , \\cdot } ) ^ { \\frac { 1 } { 2 } } \\gamma ( S _ { i ^ { \\prime } , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ) ^ { \\frac { 1 } { 2 } } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "223 Finally, note that $\\kappa$ can be readily implemented on an entire set of samples, returning an entire   \n224 Gram (kernel) matrix instead of a scalar value, by replacing the matrix operations above with tensor   \n225 operations and specifying the correct axes along which summation occurs—an implementation can   \n226 be found in our open source Python package at https://non-anonymous-link.after-review.   \n227 A proper distance metric can also be obtained from this kernel through function composition:   \n228 arccos $_ { \\mathrm { ~ O ~ } \\kappa }$ . The key idea behind the kernel is that it is the cosine similarity in the space that $\\varphi$ maps   \n229 to, meaning for arbitrary sample points $x , x ^ { \\prime }$ it evaluates to $\\cos ( \\theta )$ , where $\\theta$ is the angle between   \n230 $\\varphi ( x )$ and $\\varphi ( x ^ { \\prime } )$ . In this space, $\\theta$ represents the dissimilarity of the dependence patterns underlying   \n231 $x$ and $x ^ { \\prime }$ , without being biased by the possibly different magnitudes of $\\varphi ( x )$ and $\\varphi ( x ^ { \\prime } )$ due to   \n232 differing variances. Indeed, it can be used as a statistical test of whether samples come from different   \n233 dependence structures and therefore causal models: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 10 Let $S \\in \\mathbb { R } ^ { n , m }$ , $S ^ { \\prime } \\in \\mathbb { R } ^ { n ^ { \\prime } , m }$ be sets of $n , n ^ { \\prime }$ iid samples drawn respectively from the random variables $X = ( X _ { 1 } , \\ldots , X _ { m } )$ and $X ^ { \\prime } = ( X _ { 1 } ^ { \\prime } , \\ldots , X _ { m } ^ { \\prime } )$ with finite first moments. Then, ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9baee6286c27e2be8c77b431b9e1e760c66c99d7f0dd704441aa88f3747d8afa.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { n } \\sum _ { i ^ { \\prime } = 1 } ^ { n ^ { \\prime } } \\kappa ( S _ { i \\cdot } , S _ { i ^ { \\prime } \\cdot } ^ { \\prime } ) < 0 \\implies \\exists j , j ^ { \\prime } \\in \\{ 1 , \\dots , m \\} \\mathrm { ~ s u c h ~ t h a t ~ } \\mathcal { Z } ( X _ { j } , X _ { j ^ { \\prime } } , \\emptyset ) \\neq \\mathcal { Z } ( X _ { j } ^ { \\prime } , X _ { j ^ { \\prime } } ^ { \\prime } , \\emptyset ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Proof. Through Slutsky’s Theorem (see Takeshi, 1985, Theorem 3.2.7) and the continuous mapping theorem (see Van der Vaart, 2000, Theorem 2.3), the consistency of $\\varphi$ (Lemma 3) guarantees the consistency of $\\kappa$ . Because the numerator of $\\kappa$ is a Frobenius inner product of $\\varphi$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ffd00a06fd1589bbd97805a431f96ed8c1cbb4400a2c1a5b780849879d00c771.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { n } \\sum _ { i ^ { \\prime } = 1 } ^ { n ^ { \\prime } } \\kappa ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ^ { \\prime } ) \\propto \\sum _ { i = 1 } ^ { n } \\sum _ { i ^ { \\prime } = 1 } ^ { n ^ { \\prime } } \\sum _ { j = 1 } ^ { m } \\sum _ { j ^ { \\prime } = 1 } ^ { m } \\varphi ( S _ { i , \\cdot } ) _ { j , j ^ { \\prime } } \\varphi ( S _ { i ^ { \\prime } , \\cdot } ^ { \\prime } ) _ { j , j ^ { \\prime } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Thus, in order for $\\begin{array} { r } { \\sum _ { i , i ^ { \\prime } } \\kappa ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ^ { \\prime } ) < 0 } \\end{array}$ , there must be a $j$ and $j ^ { \\prime }$ for which $\\varphi ( S _ { i , \\cdot } ) _ { j , j ^ { \\prime } } > 0$ but $\\varphi ( S _ { i ^ { \\prime } , \\cdot } ^ { \\prime } ) _ { j , j ^ { \\prime } } \\ : < \\ : 0$ (or vice versa), and thus the hypothesis test in Lemma 3 would reject the null hypothesis that $X _ { j } \\perp \\perp X _ { j \\prime }$ but fail to reject that $X _ { j } ^ { \\prime } \\perp \\perp X _ { j ^ { \\prime } } ^ { \\prime }$ . \u0003 ",
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+        "text": "Corollary 11 Due to the relationship between independence structure and causal structure, an immediate of result of Theorem 10 is that $\\begin{array} { r } { \\sum _ { i , i } \\kappa ( S _ { i , \\cdot } , S _ { i ^ { \\prime } , \\cdot } ^ { \\prime } ) < 0 } \\end{array}$ implies $X$ and $X ^ { \\prime }$ have different causal structures. ",
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+        "text": "Theorem 12 Let $d$ be the distance measure between unconditional equivalence classes of ancestral graphs over $m$ vertices, $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = m ^ { 2 } - | \\{ ( j , j ^ { \\prime } ) : E ^ { \\mathcal { U } \\bullet \\mathcal { U } ^ { \\prime } } ( j , j ^ { \\prime } ) = { ^ { * }  } \\} | - m$ . For given sample sets $S , S ^ { \\prime }$ (i.e., real $n \\times m$ matrices), use $\\bar { \\varphi } ( S )$ to denote the mean of the sample in kernel space, $\\textstyle \\sum _ { i } \\varphi ( S _ { i , . } )$ , and say $S \\sim _ { \\mathrm { K } } \\ S ^ { \\prime }$ if and only if $\\bar { \\varphi } ( S ) \\sim _ { 0 } \\bar { \\varphi } ( S ^ { \\prime } )$ ; denote the corresponding quotient set by this equivalence class as $\\mathbb { R } ^ { n , m } / \\sim _ { \\mathrm { K } } = \\mathbb { K } ^ { n , m }$ and a representative from each equivalence class as $Q \\in [ S ]$ . Let $\\delta$ be the distance between sets of samples in $\\mathbb { K }$ defined as $\\begin{array} { r } { \\delta ( Q , Q ^ { \\prime } ) = m ^ { 2 } - \\frac { 1 } { 2 n ^ { 2 } } \\sum _ { i , i ^ { \\prime } } \\gamma ( Q _ { i , \\cdot } , Q _ { i , \\cdot } ^ { \\ j } ) } \\end{array}$ Let $b : \\mathbb { U } ^ { m }  \\mathbb { K } ^ { n , m } , b : \\mathcal { U } \\mapsto \\Omega$ , where $\\Omega$ is the unique element in $\\mathbb { K }$ such that $\\mathrm { s i g n } ( \\bar { \\varphi } ( \\Omega ) ) = a ( \\mathcal { U } )$ . Then $b$ is a distance-preserving map (i.e., an isometry) from the metric space $( \\mathbb { U } ^ { m } , d )$ to $( \\mathbb { K } ^ { n , m } , \\delta )$ . ",
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+        "text": "Proof. Notice that $( \\mathbb { U } ^ { m } , d )$ is indeed a metric space (Choudhary, 1993, Ch. 2): $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = 0$ iff $\\varkappa ^ { - 1 } \\bullet \\mathcal { U } ^ { \\prime }$ is the empty graph, which happens iff $\\mathcal { U } = \\mathcal { U } ^ { \\prime }$ ; the symmetry of $d$ follows from the symmetry •; and for subadditivity of $d$ , observe that for vertices ${ j , j ^ { \\prime } }$ in arbitrary 2-vertex graphs $\\boldsymbol { { u } } , \\boldsymbol { { u } } ^ { \\prime } , \\boldsymbol { { u } } ^ { \\ast }$ we have either $d ( \\mathcal { U } , \\mathcal { U } ^ { \\ast } ) = 2$ , in which case $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) + d ( \\mathcal { U } ^ { \\prime } , \\mathcal { U } ^ { \\prime \\prime } ) = 4$ , or we have $d ( \\mathcal { U } , \\mathcal { U } ^ { \\mathfrak { V } } ) = 0$ , in which case $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) + d ( \\mathcal { U } ^ { \\prime } , \\mathcal { U } ^ { \\prime \\prime } )$ is either 0 or 4—in both cases $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime \\prime } ) \\leq d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) + d ( \\mathcal { U } ^ { \\prime } , \\mathcal { U } ^ { \\prime \\prime } )$ ; this easily extends to graphs of arbitrary numbers of vertices. Likewise, $( \\mathbb { K } ^ { n , m } , \\delta )$ is a metric space: $\\delta ( Q , Q ^ { \\prime } ) = 0 \\longleftrightarrow \\frac { 1 } { 2 n ^ { 2 } } \\sum _ { i , i ^ { \\prime } } \\gamma ( Q _ { i , \\cdot } , Q _ { i , \\cdot } ^ { \\prime } ) = m ^ { 2 } \\Longleftrightarrow \\bar { \\varphi } ( Q ) _ { j , j ^ { \\prime } } = \\bar { \\varphi } ( Q ) _ { j , j ^ { \\prime } } ,$ , for all ${ j , j ^ { \\prime } }$ , so iff $Q = Q ^ { \\prime }$ ; symmetry and subadditivity of $\\delta$ follow from the symmetry and subadditivity of $\\gamma$ . ",
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+        "text": "Finally, to show $b$ is an isometry, we must show (i) that it is bijective and (ii) that for all $\\boldsymbol { \\mathcal { U } } , \\boldsymbol { \\mathcal { U } } ^ { \\prime } \\in \\mathbf { U } ^ { m }$ , $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = \\delta ( b ( \\mathcal { U } ) , b ( \\mathcal { U } ^ { \\prime } ) )$ . For (i), observe that by the group isomorphism $a$ and definition of $b$ , we have ${ \\mathcal { U } } \\neq { \\mathcal { U } } ^ { \\prime } \\implies a ( { \\mathcal { U } } ) \\neq a ( { \\mathcal { U } } ^ { \\prime } ) \\implies Q \\neq Q ^ { \\prime } \\implies b ( { \\mathcal { U } } ) \\neq b ( { \\mathcal { U } } ^ { \\prime } )$ and so $b$ is injective. Also observe that because $\\mathbb { K }$ is exactly the set of representatives of orthant equivalence classes of sample sets in kernel space, then for every $Q \\in \\mathbb { K }$ , there exists a $\\mathcal { U }$ such that $b ( \\mathcal { U } ) = Q$ , and so $b$ is surjective. ",
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+        "type": "text",
+        "text": "For (ii), isomorphism $a$ and the relation between element-wise product and Frobenius inner product allow us to write $\\begin{array} { r } { d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = m ^ { 2 } - \\sum _ { j , j ^ { \\prime } } ( O \\odot O ^ { \\prime } ) _ { j , j ^ { \\prime } } = m ^ { 2 } - \\langle O , O ^ { \\prime } \\rangle _ { \\mathrm { F } } } \\end{array}$ . Substituting $O , O ^ { \\prime }$ with their corresponding $\\Omega , \\Omega ^ { \\prime }$ , and because the Frobenius inner product is a sesquilinear form, we can write $\\begin{array} { r } { d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = m ^ { 2 } - \\frac { 1 } { n ^ { 2 } } \\sum _ { i , i ^ { \\prime } } \\langle \\varphi ( \\Omega _ { i , \\cdot } ) , \\varphi ( \\Omega _ { i , \\cdot } ^ { \\prime } ) \\rangle _ { \\mathrm { F } } . } \\end{array}$ , which by Definition 10 finally gives us that $d ( \\mathcal { U } , \\mathcal { U } ^ { \\prime } ) = \\delta ( \\Omega , \\Omega ^ { \\prime } )$ , completing the proof. \u0003 ",
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+        "type": "text",
+        "text": "In less formal terms, Theorem 12 shows how the space of unconditional equivalence classes of ancestral graph corresponds to the space of real matrices, which is a common space for samples to lie in. More specifically, it shows how the structure defined by distances between graphs is the same as the structure defined by distances between sets of samples and how this sample distance is related to our kernel $\\kappa$ . Note that this is much stronger than Theorem 10: not only can $\\kappa$ tell us that two sets of samples come from different causal models, it gives a measure of just how different the causal models are, in terms of their differing unconditional nonlinear independencies/m-separation statements. ",
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+        "text": "273 To summarize, we began by defining $\\varphi$ (Definition 2), which maps a given data set into a new   \n274 higher-dimensional feature space. This feature space corresponds to a space of causal graphical   \n275 models, such that samples which are similar in the new feature space must come from similar causal   \n276 models (Theorem 8). Our main contribution then is to propose the dependence contribution kernel   \n277 $\\kappa$ (Definition 9).This kernel $\\kappa$ is guaranteed not only to tell us that two sets of samples come from   \n278 different causal models (Theorem 10 and Corollary 11) but furthermore exactly how different the   \n279 causal models are (Theorem 12), all without the computational expense of explicitly projecting   \n280 samples or learning causal models. Thus, $\\kappa$ is well-suited for addressing the causal clustering   \nproblem and ensures that resulting clusters will be structurally homogeneous so that subsequent   \ncausal structure learning will be more informative. ",
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+        "type": "text",
+        "text": "283 3 Application ",
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+        "text": "We use kernel $k$ -means with our dependence contribution kernel to cluster a gene expression data set and then use the measurement dependence inducing latent (MeDIL) causal model framework for structure learning within each cluster (Markham and Grosse-Wentrup, 2020). The goal of causal clustering here is to reason about the different latent transcription factor (TF) networks governing gene expression (see Verny et al., 2017; Hackett et al., 2020, for other latent causal model approaches to learning TF networks). The original data set comes from Iyer (1999) and can be found at genome-www.stanford.edu/serum/data/fig2clusterdata.txt, with subsequent analysis by Dhillon et al. (2003, 2004). All of the code for our analysis is open source and available at https://non-anonymous-link.after-review. ",
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+        "text": "The data consists of the measured gene expression levels of 517 different genes from human fibroblast cells in response to serum exposure, measured at 11 different time points, i.e., there are 517 samples and 11 different features. In genetics applications, it is not unusual to consider genes to be samples and expression (over time) to be features—indeed the three previous analyses of this data all have this approach—and the intuition is simply that we wish to cluster genes based on patterns in their expression levels over time, in order to identify subsets of genes that are controlled by the same gene regulatory network. Also notice that such data exemplifies the structurally heterogeneous populations discussed in Section 1: different genes can of course be regulated by different TFs, and so we can better represent the data by first clustering it into subpopulations that are more homogeneous and then performing causal structure learning on each subpopulation. ",
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+        "type": "text",
+        "text": "For clustering, we used $k = 6$ , which we found by looking at both the Variance Ratio Criterion (Calinski and Harabasz, 1974) and the Silhouette Coefficients (Rousseeuw, 1987), computed with the ´ scikit-learn machine learning toolbox (Pedregosa et al., 2011). We implemented (unweighted) kernel $k$ -means ourselves, using the pseudocode given by Dhillon et al. (2004), with initial mean points drawn uniformly at random from the sample set, and with significance level $\\alpha = 0 . 1$ for the kernel parameter $\\tau ( \\alpha )$ . We then used the MeDIL (Markham et al., 2020) package to learn the dependence structure and latent causal models for each cluster. ",
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+        "type": "text",
+        "text": "Figure 1 shows an example of our results for three of the six gene clusters: Figure 1a shows their distance covariance heatmaps and estimated nonlinear dependence structure with significance level $\\alpha = 0 . 1$ (so the axes are the 11 different features, i.e. the time, in hours, at which gene expression level was measured), while Figure 1b shows their corresponding causal structures, with measurement variables $M _ { 0 } { - } M _ { 1 0 }$ for each of the features and learned latent variables $L$ for different posited TFs. ",
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+        "type": "text",
+        "text": "The results show a clear difference in causal structure for the different clusters and allow us to reason about the latent TFs regulating genes in different clusters: notice that the latents in cluster K1 each cause only two or three measurement variables that tend to be close together—e.g., $L _ { 1 }$ causes $M _ { 1 }$ and $M _ { 2 }$ , indicating the TF corresponding to $L _ { 1 }$ is “short-acting”, only affecting gene expression from 30 minutes $( M _ { 1 } )$ to 1 hour $( M _ { 2 } )$ after serum exposure; in contrast, the latents in cluster K3 each cause between two and seven measurement variables that tend to be more spread out—e.g., $L _ { 1 }$ causes $M _ { 1 }$ and $M _ { 7 }$ , indicating the corresponding TF is more complicated, “long-acting” but not ",
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+    {
+        "type": "image",
+        "img_path": "images/1980f63ea3df4f8129dd14dbc52df12d1755607e16d39aa799cd48672046d8a4.jpg",
+        "image_caption": [
+            "Figure 1: Results of dependence contribution kernel clustering with significance level $\\alpha = 0 . 1$ . "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "322 continuously so, affecting gene expression 30 minutes $( M _ { 1 } )$ and 12 hours $( M _ { 7 } )$ after serum exposure,   \n323 but independently of gene expression in the time between. ",
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+        "type": "text",
+        "text": "Our results are especially noteworthy compared what happens if one ignores the heterogeneity of the data and learns a causal structure for the entire data set without first clustering with our kernel: in that case, all of the measurement variables are dependent, with a single latent causing all of them, and no meaningful conclusions can be drawn about how unmeasured transcription factors regulate measured gene expression, i.e., the heterogeneity obscures the underlying causal structures. ",
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+    {
+        "type": "text",
+        "text": "4 Discussion ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We address the problem of causal clustering—that is, finding the different causal structures underlying a structurally heterogeneous data set. Our main contribution is to develop the dependence contribution kernel and prove its suitability for the causal clustering task. This allows us to first use the kernel with existing clustering methods, such as kernel $k$ -means or DBSCAN, to identify homogeneous subpopulations. Then we use existing causal structure learning methods on each subpopulation. The kernel guarantees that each subpopulation is more structurally homogeneous and therefore the resulting causal structures better capture the causal structures within the data than if a single model were learned for the entire heterogeneous population. ",
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+        "text": "338 Furthermore, we prove several interesting theoretical properties of our kernel, including (i) that   \n339 it can be used as a statistical test for the hypothesis that two sets of samples come from different   \n340 causal structures, as well as (ii) how it induces a metric space that is isometric to the one defined   \n341 by Hamming distance between ancestral graphs, i.e., comparing sets of samples with our kernel is   \n342 equivalent to first estimating the causal graphs of the different sets and then comparing those graphs.   \n343 Beyond the practical applications of our kernel, as shown by our application in reasoning about latent   \n344 transcription factor networks that regulate gene expression, this work also draws from and suggests   \n345 further fruitful connections between a variety of fields, including causal inference, kernel methods,   \n346 and algebraic statistics. ",
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+        "text": "References   \nArtin, M. (2011). Algebra. Pearson Prentice Hall.   \nAthey, S. and Imbens, G. W. (2015). Machine learning methods for estimating heterogeneous causal effects. Stat, 1050(5):1–26.   \nBareinboim, E. and Pearl, J. (2016). Causal inference and the data-fusion problem. Proceedings of the National Academy of Sciences, 113(27):7345–7352.   \nBrand, J. E. and Thomas, J. S. (2013). Causal effect heterogeneity. In Handbook of Causal Analysis for Social Research, pages 189–213. Springer.   \nCai, H., Zheng, V. W., and Chang, K. C.-C. (2018). A comprehensive survey of graph embedding: Problems, techniques, and applications. IEEE Transactions on Knowledge and Data Engineering, 30(9):1616–1637.   \nCalinski, T. and Harabasz, J. (1974). A dendrite method for cluster analysis. ´ Communications in Statistics, 3(1):1–27.   \nChoudhary, B. (1993). The Elements of Complex Analysis. New Age International.   \nDevlin, K. (2003). Sets, functions, and logic: An introduction to abstract mathematics. CRC Press.   \nDhillon, I. S., Guan, Y., and Kulis, B. (2004). Kernel k-means, spectral clustering and normalized cuts. Proceedings of the 2004 ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD ’04.   \nDhillon, I. S., Marcotte, E. M., and Roshan, U. (2003). Diametrical clustering for identifying anti-correlated gene clusters. Bioinformatics, 19(13):1612–1619.   \nEichler, M. (2012). Causal inference in time series analysis. Wiley Series in Probability and Statistics, page 327–354.   \nEmmert-Streib, F., Glazko, G., Gökmen, A., and De Matos Simoes, R. (2012). Statistical inference and reverse engineering of gene regulatory networks from observational expression data. Frontiers in Genetics, 3:8.   \nFilippone, M., Camastra, F., Masulli, F., and Rovetta, S. (2008). A survey of kernel and spectral methods for clustering. 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Harvard university press.   \nTjøstheim, D., Otneim, H., and Støve, B. (2018). Statistical dependence: Beyond pearson’s $\\rho$ . arXiv preprint arXiv:1809.10455.   \nVan der Vaart, A. W. (2000). Asymptotic statistics, volume 3. Cambridge university press.   \nVerny, L., Sella, N., Affeldt, S., Singh, P. P., and Isambert, H. (2017). Learning causal networks with latent variables from multivariate information in genomic data. PLoS computational biology, 13(10):e1005662.   \nWasserman, L. (2013). All of statistics: a concise course in statistical inference. Springer Science & Business Media.   \nXie, Y. (2013). Population heterogeneity and causal inference. Proceedings of the National Academy of Sciences, 110(16):6262–6268.   \nXie, Y., Brand, J. E., and Jann, B. (2012). Estimating heterogeneous treatment effects with observational data. Sociological Methodology, 42(1):314–347.   \nZhang, J. (2007). A characterization of markov equivalence classes for directed acyclic graphs with latent variables. In Conference on Uncertainty in Artificial Intelligence (UAI). ",
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+        "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] The first, fifth, and sixth sentences are covered thoroughly in Sections 1, 3, and 4, while the rest are covered thoroughly in Section 2.   \n(b) Did you describe the limitations of your work? [Yes] In Section 1.1 and throughout Section 2   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our work has no direct potential negative societal impact—just the same indirect potential most theoretical work has   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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parse/train/BJlxmAKlg/BJlxmAKlg.md

@@ -0,0 +1,236 @@
+# REASONET: LEARNING TO STOP READING IN MACHINE COMPREHENSION
+
+Yelong Shen, Po-Sen Huang, Jianfeng Gao, Weizhu Chen Microsoft Research, Redmond, WA, USA {yeshen,pshuang,jfgao,wzchen}@microsoft.com
+
+# ABSTRACT
+
+Teaching a computer to read a document and answer general questions pertaining to the document is a challenging yet unsolved problem. In this paper, we describe a novel neural network architecture called the Reasoning Network (ReasoNet) for machine comprehension tasks. ReasoNets make use of multiple turns to effectively exploit and then reason over the relation among queries, documents, and answers. Different from previous approaches using a fixed number of turns during inference, ReasoNets introduce a termination state to relax this constraint on the reasoning depth. With the use of reinforcement learning, ReasoNets can dynamically determine whether to continue the comprehension process after digesting intermediate results, or to terminate reading when it concludes that existing information is adequate to produce an answer. ReasoNets have achieved state-of-the-art performance in machine comprehension datasets, including unstructured CNN and Daily Mail datasets, and a structured Graph Reachability dataset.
+
+# 1 INTRODUCTION
+
+Teaching machines to read, process, and comprehend natural language documents is a coveted goal for artificial intelligence (Bottou, 2014; Richardson et al., 2013; Hermann et al., 2015). Genuine reading comprehension is extremely challenging, since effective comprehension involves thorough understanding of documents and performing sophisticated inference. Toward solving this machine reading comprehension problem, in recent years, several work has collected various datasets, in the form of question, passage, and answer, to test machine on answering a question based on the provided passage (Richardson et al., 2013; Hermann et al., 2015; Hill et al., 2016; Rajpurkar et al., 2016). Some large-scale cloze-style datasets (Hermann et al., 2015; Hill et al., 2016) have gained significant attention along with powerful deep learning models.
+
+Recent approaches on cloze-style datasets can be separated into two categories: single-turn and multiturn reasoning. Single turn reasoning models utilize attention mechanisms (Bahdanau et al., 2015) with deep learning models to emphasize specific parts of the document which are relevant to the query. These attention models subsequently calculate the relevance between a query and the corresponding weighted representations of document subunits (e.g. sentences or words) to score target candidates (Hill et al., 2016; Hermann et al., 2015; Kadlec et al., 2016). However, considering the sophistication of the problem, after a single-turn comprehension, readers often revisit some specific passage or the question to grasp a better understanding of the problem. With this motivation, recent advances in reading comprehension have made use of multiple turns to infer the relation between query, document and answer (Hill et al., 2016; Dhingra et al., 2016; Trischler et al., 2016; Sordoni et al., 2016). By repeatedly processing the document and question after digesting intermediate information, multi-turn reasoning can generally produce a better answer and all existing work has demonstrated its superior performance consistently.
+
+Existing multi-turn models have a fixed number of hops or iterations in their inference, i.e., with predetermined reasoning depth, without regard to the complexity of each individual query or document. However, when a human reads a document with a question in mind, we often decide whether we want to stop reading if we believe the observed information is adequate already to answer the question, or continue reading after digesting intermediate information until we can answer the question with confidence. This behavior generally varies from document to document, or question to question because it is related to the sophistication of the document or the difficulty of the question. Meanwhile, the analysis in Chen et al. (2016) also illustrates the huge variations in the difficulty level with respect to questions in the CNN/Daily Mail datasets (Hermann et al., 2015). For a significant part of the datasets, this analysis shows that the problem cannot be solved without appropriate reasoning on both its query and document.
+
+With this motivation, we propose a novel neural network architecture called Reasoning Network (ReasoNet). ReasoNets try to mimic the inference process of human readers. With a question in mind, ReasoNets read a document repeatedly, each time focusing on different parts of the document until a satisfying answer is found or formed. This reminds us of a Chinese proverb: “The meaning of a book will become clear if you read it hundreds of times.”. Moreover, unlike previous approaches using fixed number of hops or iterations, ReasoNets introduce a termination state in the inference. This state can decide whether to continue the inference to next turn after digesting intermediate information, or to terminate the whole inference when it concludes that existing information is sufficient to yield an answer. This number of turns in the inference is dynamically modeled by both the document and the query, and can be learned automatically according to the difficulty of the problem.
+
+One of the significant challenges ReasoNets face is how to design an efficient training method, since the termination state is discrete and not connected to the final output. This prohibits canonical backpropagation method being directly applied to train ReasoNets. Inspired by Williams (1992); Mnih et al. (2014), we tackle this challenge by proposing a novel deep reinforcement learning method called Contrastive Reward (CR) to successfully train ReasoNets. Unlike traditional reinforcement learning optimization methods using a global variable to capture rewards, CR utilizes an instance-based reward baseline assignment. Experiments show the superiority of CR in both training speed and accuracy. Finally, by accounting for a dynamic termination state during inference and applying proposed deep reinforcement learning optimization method, ReasoNets achieve the state-of-the-art results in machine comprehension datasets when the paper is first publicly available in arXiv1, including unstructured CNN and Daily Mail datasets, and a proposed structured Graph Reachability dataset.
+
+This paper is organized as follows. In Section 2, we review and compare recent work on machine reading comprehension tasks. In Section 3, we introduce our proposed ReasoNet model architecture and training objectives. Section 4 presents the experimental setting and results on unstructured and structured machine reading comprehension tasks .
+
+# 2 RELATED WORK
+
+Recently, with large-scale datasets available and the impressive advance of various statistical models, machine reading comprehension tasks have attracted much attention. Here we mainly focus on the related work in cloze-style datasets (Hermann et al., 2015; Hill et al., 2016). Based on how they perform the inference, we can classify their models into two categories: single-turn and multi-turn reasoning.
+
+Single-turn reasoning Single turn reasoning models utilize an attention mechanism to emphasis some sections of a document which are relevant to a query. This can be thought of as treating some parts unimportant while focusing on other important ones to find the most probable answer. Hermann et al. (2015) propose the attentive reader and the impatient reader models using neural networks with an attention over passages to predict candidates. Hill et al. (2016) use attention over window-based memory, which encodes a window of words around entity candidates, by leveraging an end-to-end memory network (Sukhbaatar et al., 2015). Meanwhile, given the same entity candidate can appear multiple times in a passage, Kadlec et al. (2016) propose the attention-sum reader to sum up all the attention scores for the same entity. This score captures the relevance between a query and a candidate. Chen et al. (2016) propose using a bilinear term similarity function to calculate attention scores with pretrained word embedding. Trischler et al. (2016) propose the EpiReader which uses two neural network structures: one extracts candidates using the attention-sum reader; the other reranks candidates based on a bilinear term similarity score calculated from query and passage representations.
+
+Multi-turn reasoning For complex passages and complex queries, human readers often revisit the given document in order to perform deeper inference after reading a document. Several recent studies try to simulate this revisit by combining the information in the query with the new information digested from previous iterations (Hill et al., 2016; Dhingra et al., 2016; Sordoni et al., 2016; Weissenborn, 2016; Kumar et al., 2016). Hill et al. (2016) use multiple hops memory network to augment the query with new information from the previous hop. Gated Attention reader (Dhingra et al., 2016) is an extension of the attention-sum reader with multiple iterations by pushing the query encoding into an attention-based gate in each iteration. Iterative Alternative (IA) reader (Sordoni et al., 2016) produces a new query glimpse and document glimpse in each iteration and utilizes them alternatively in the next iteration. Cui et al. (2016) further propose to extend the query-specific attention to both query-to-document attention and document-to-query attention, which is built from the intermediate results in the query-specific attention. By reading documents and enriching the query in an iterative fashion, multi-turn reasoning has demonstrated their superior performance consistently.
+
+Algorithm 1: Stochastic Inference in a ReasoNet   
+
+<table><tr><td></td><td>Input:Memory M; Initial state s1; Step t =1; Maximum Step Tmax Output:Termination Step T,Answer aT Sample t from the distribution p(*Iftg(St; 0tg));</td></tr></table>
+
+Our proposed approach explores the idea of using both attention-sum to aggregate candidate attention scores and multiple turns to attain a better reasoning capability. Unlike previous approaches using fixed number of hops or iterations, motivated by Nogueira & Cho (2016); Mnih et al. (2014), we propose a termination module in the inference. The termination module can decide whether to continue to infer the next turn after digesting intermediate information, or to terminate the whole inference process when it concludes existing information is sufficient to yield an answer. The number of turns in the inference is dynamically modeled by both a document and a query, and is generally related to the complexity of the document and the query.
+
+# 3 REASONING NETWORKS
+
+ReasoNets are devised to mimic the inference process of human readers. ReasoNets read a document repeatedly, with attention on different parts each time until a satisfying answer is found. As shown in Figure 1, a ReasoNet is composed of the following components:
+
+Internal State: The internal state is denoted as $S$ which is a vector representation of the question state. Typically, the initial state $s _ { 1 }$ is the last-word vector representation of query by an RNN. The $t$ -th time step of the internal state is represented by $s _ { t }$ . The sequence of internal state is modeled by an RNN: $s _ { t + 1 } = \mathrm { R N N } ( s _ { t } , x _ { t } ; \theta _ { s } )$ ;
+
+Memory: The external memory is denoted as $M$ . It is a list of word vectors, $M = \{ m _ { i } \} _ { i = 1 \dots D }$ , where $m _ { i }$ is a fixed dimensional vector. In machine comprehensive tasks, $m _ { i }$ is the vector representation of each word in the doc by a bidirectional-RNN.
+
+Attention: Attention vector $x _ { t }$ is generated based on the current internal state $s _ { t }$ and the external memory $M \colon x _ { t } = f _ { a t t } \bigl ( s _ { t } , M ; \theta _ { x } \bigr )$ ;
+
+Termination Gate: Termination gate generates a stochastic random variable according to the current internal state; $t _ { t } \sim p ( \cdot | f _ { t g } ( s _ { t } ; \theta _ { t g } \bar { ) } ) )$ . $t _ { t }$ is a binary random variable. If $t _ { t }$ is true, the ReasoNet stops, and the answer module executes at time step $t$ ; otherwise the ReasoNet generates an attention vector $x _ { t + 1 }$ , and feed into the state network to update the next internal state $s _ { t + 1 }$ .
+
+Answer: The action of answer module is triggered when the termination gate variable is true: $a _ { t } \sim p ( \cdot | f _ { a } ( s _ { t } ; \theta _ { a } ) )$ .
+
+![](images/aef314f8cbc4976d7bc6a8375884e86178fcdf98f061da33c1227bf253c967d5.jpg)  
+Figure 1: A ReasoNet Architecture.
+
+In Algorithm 1, we describe the stochastic inference process of a ReasoNet. The process can be considered as a Partially Observable Markov Decision Process (POMDP) (Kaelbling et al., 1998) in the reinforcement learning (RL) literature. The state sequence $s _ { 1 : T }$ is hidden and dynamic, controlled by an RNN sequence model. The ReasoNet performs an answer action $a _ { T }$ at the $T$ -th step, which implies that the termination gate variables $t _ { 1 : T } = ( t _ { 1 } = 0 , t _ { 2 } = 0 , . . . , t _ { T - 1 } = 0 , t _ { T } = 1 )$ . The ReasoNet learns a stochastic policy $\pi ( ( t _ { t } , a _ { t } ) | s _ { t } ; \theta )$ with parameters $\theta$ to get a distribution over termination actions, to continue reading or to stop, and over answer actions if the model decides to stop at the current step. The termination step $T$ varies from instance to instance.
+
+The parameters $\theta$ of the ReasoNet are given by the parameters of the embedding matrices $W$ , attention network $\theta _ { x }$ , the state RNN network $\theta _ { s }$ , the answer action network $\theta _ { a }$ , and the termination gate network $\theta _ { t g }$ . The parameters $\theta = \{ W , \theta _ { x } , \theta _ { s } , \theta _ { a } , \theta _ { t g } \}$ are trained by maximizing the total expect reward. The expected reward for an instance is defined as:
+
+$$
+J ( \theta ) = \mathbb { E } _ { \pi ( t _ { 1 : T } , a _ { T } ; \theta ) } \left[ \sum _ { t = 1 } ^ { T } r _ { t } \right]
+$$
+
+The reward can only be received at the final termination step when an answer action $a _ { T }$ is performed. We define $r _ { T } = 1$ if $t _ { T } = 1$ and the answer is correct, and $r _ { T } = 0$ otherwise. The rewards on intermediate steps are zeros, $\{ r _ { t } = 0 \} _ { t = 1 \dots T - 1 }$ . $J$ can be maximized by directly applying gradient based optimization methods. The gradient of $J$ is given by:
+
+$$
+\nabla _ { \theta } J ( \theta ) = \mathbb { E } _ { \pi ( t _ { 1 : T } , a _ { T } ; \theta ) } \left[ \nabla _ { \theta } \mathrm { l o g } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) r _ { T } \right]
+$$
+
+We apply the REINFORCE algorithm (Williams, 1992) to compute $\nabla _ { \boldsymbol { \theta } } J ( \boldsymbol { \theta } )$ :
+
+$$
+\tilde { \mathbf { L } } _ { \pi ( t _ { 1 : T } , a _ { T } ; \theta ) } \left[ \nabla _ { \theta } \mathrm { l o g } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) r _ { T } \right] = \sum _ { ( t _ { 1 : T } , a _ { T } ) \in \mathbb { A } ^ { \dagger } } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) \left[ \nabla _ { \theta } \mathrm { l o g } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) ( r _ { T } - b _ { T } ) \right]
+$$
+
+where $\mathbb { A } ^ { \dagger }$ is all the possible episodes, $T , t _ { 1 : T } , a _ { T }$ and $r _ { T }$ are the termination step, termination action, answer action, and reward, respectively, for the $( t _ { 1 : T } , a _ { T } )$ episode. $b _ { T }$ is called the reward baseline in the RL literature to lower variance (Sutton, 1984). It is common to select $b _ { T } = \mathbb { E } _ { \pi } \left[ r _ { T } \right]$ (Sutton et al., 1999), and can be updated via an online moving average approach : $b _ { T } = \lambda b _ { T } + ( \bar { 1 } - \lambda ) r _ { T }$ .
+
+However, we empirically find that above approach leads to slow convergence in training ReasoNets. Intuitively, the average baselines $\{ b _ { T } ; T = 1 . . T _ { \mathrm { m a x } } \}$ are global variables independent of instances. It is hard for these baselines to capture the dynamic termination behavior of ReasoNets. In other words, ReasoNets may stop at different time steps for different instances. The adoption of a global variable without considering the dynamic variance in each instance is inappropriate. To resolve this weakness in traditional methods and account for the dynamic characteristic of ReasoNets, we propose an instance-based baseline method called “Contrastive Reward” (CR) to calculate $\nabla _ { \boldsymbol { \theta } } J ( \boldsymbol { \theta } )$ . The basic idea of CR is to utilize an instance-based baseline assignment. We will elaborate its implementation details in Section 3.1. Empirical results show that the proposed reward schema has produced better results compared to the baseline approach.
+
+# 3.1 TRAINING DETAILS
+
+In the machine reading comprehension tasks, a training dataset can be simplified as a collection of triplets of query q, passage $\mathbf { p }$ , and answer a. Say $\left. q _ { n } , p _ { n } , a _ { n } \right.$ is the $n$ -th training instance.
+
+The first step is to extract memory $M$ from $p _ { n }$ by mapping each symbolic in the passage to a contextual representation given by the concatenation of forward and backward RNN hidden states, i.e., $m _ { k } = [ \bar { \vec { p _ { n } } } ^ { k } , \overleftarrow { \vec { p _ { n } } } ^ { | p _ { n } | - k + 1 } ]$ , and extract initial state $s _ { 1 }$ from $q _ { n }$ by assigning $s _ { 1 } = [ \overrightarrow { q _ { n } } ^ { | q _ { n } | } , \overleftarrow { q _ { n } } ^ { 1 } ]$ Given $M$ and $s _ { 1 }$ for the $n$ -th training instance, a ReasoNet executes $| \dot { \mathbb { A } } ^ { \dagger } |$ episodes, where all possible episodes $\mathbb { A } ^ { \dagger }$ can be enumerated by setting a maximum step. Each episode generates actions and a reward from the last step: h(t1:T , aT ), rT i(t1:T ,aT )∈A† .
+
+Therefore, the gradient of $J$ can be rewritten as:
+
+$$
+\nabla _ { \theta } J ( \theta ) = \sum _ { ( t _ { 1 : T } , a _ { T } ) \in \mathbb { A } ^ { \dagger } } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) \left[ \nabla _ { \theta } \mathrm { l o g } \pi ( t _ { 1 : T } , a _ { T } ; \theta ) ( r _ { T } - b ) \right]
+$$
+
+where the baseline $\begin{array} { r } { b = \sum _ { ( t _ { 1 : T } , a _ { T } ) \in \mathbb { A } ^ { \dagger } } \pi \big ( t _ { 1 : T } , a _ { T } ; \theta \big ) r _ { T } } \end{array}$ is the average reward on the $| \mathbb { A } ^ { \dagger } |$ episodes for the $n$ -th training instance. It allows different baselines for different training instances. This can be beneficial since the complexity of training instances varies significantly. Since the sum of the proposed rewards over $\left| \mathbb { A } ^ { \dagger } \right|$ episodes is zero, $\sum _ { ( t _ { 1 : T } , a _ { T } ) \in \mathbb { A } ^ { \dagger } } \pi \big ( \hat { t _ { 1 : T } } , a _ { T } ; \theta \big ) \big ( r _ { T } - b \big ) = 0$ , we call it Contrastive Reward in this work. In experiments, we empirically find using $\left( \frac { r _ { T } } { b } - 1 \right)$ in replace of $\left( r _ { T } - b \right)$ can lead to a faster convergence. Therefore, we adopt this approach to train ReasoNets in the experiments.
+
+# 4 EXPERIMENTS
+
+# 4.1 CNN AND DAILY MAIL DATASETS
+
+We evaluate the performance of ReasoNets on CNN and Daily Mail datasets.2 The detailed settings of the ReasoNet model are as follows.
+
+Vocab Size: For training our ReasoNet, we keep the most frequent $| V | = 1 0 1 k$ words (not including 584 entities and 1 placeholder marker) in the CNN dataset, and $| V | = 1 5 1 k$ words (not including 530 entities and 1 placeholder marker) in the Daily Mail dataset.
+
+Embedding Layer: We choose word embedding size $d \ : = \ : 3 0 0$ , and use the 300 dimensional pretrained Glove word embeddings (Pennington et al., 2014) for initialization. We also apply dropout with probability 0.2 to the embedding layer.
+
+Bi-GRU Encoder: We apply bi-directional GRU for encoding query and passage into vector representations. We set the number of hidden units to be 256 and 384 for the CNN and Daily Mail datasets, respectively. The recurrent weights of GRUs are initialized with random orthogonal matrices. The other weights in GRU cell are initialized from a uniform distribution between $- 0 . 0 1$ and 0.01. We use a shared GRU model for both query and passage.
+
+Memory and Attention: The memory of the ReasoNet on CNN and Daily Mail dataset is composed of query memory and passage memory. $M = ( M ^ { q u e r y } , M ^ { d o c } )$ , where $M ^ { q u e r y }$ and $M ^ { d \bar { o c } }$ are extracted from query bidirectional-GRU encoder and passage bidirectional-GRU encoder respectively. We choose projected cosine similarity function as the attention module.
+
+# Query: passenger @placeholder , 36 , died at the scene 1 1
+
+Passage: ( $@$ entity0 ) what was supposed to be a fantasy sports car ride at $@$ entity3 turned deadly when a @entity4 crashed into a guardrail . the crash took place sunday at the $@$ entity8 , which bills itself as a chance to drive your dream car on a racetrack . the @entity4 's passenger , 36 - year - old @entity14 of @entity15 , @entity16 , died at the scene ,3 $@$ entity13 said . the driver of the $@$ entity4 , 24 - year - old @entity18 of @entity19 , $@$ entity16 , lost control of the vehicle , the $@$ entity13 said . he was hospitalized with minor injuries . @entity24 , which operates the $@$ entity8 at @entity3 , released a statement sunday night about the crash . " on behalf of everyone in the organization , it is with a very heavy heart that we extend our deepest sympathies to those involved in today 's tragic accident in @entity36 , " the company said . @entity24 also operates the @entity3 -- a chance to drive or ride in $@$ entity39 race cars named for the winningest driver in the sport 's history . @entity0 's @entity43 and $@$ entity44 contributed to this report .
+
+<table><tr><td rowspan=1 colspan=1>Step</td><td rowspan=1 colspan=1>TerminationProbability</td><td rowspan=1 colspan=1>AttentionSum</td></tr><tr><td rowspan=2 colspan=1>12</td><td rowspan=1 colspan=1>0.0011</td><td rowspan=1 colspan=1>0.4916</td></tr><tr><td rowspan=1 colspan=1>0.5747</td><td rowspan=1 colspan=1>0.5486</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>0.9178</td><td rowspan=1 colspan=1>0.5577</td></tr></table>
+
+Answer: @entity14
+
+![](images/59b38a0bbfad8e61b714d9d5b8ab2f98923ebd04389cfbc586f0afb98c0ab8da.jpg)  
+Figure 2: Results of a test example 69e1f777e41bf67d5a22b7c69ae76f0ae873cf43.story from the CNN dataset. The numbers next to the underline bars indicate the rank of the attention scores. The corresponding termination probability and the sum of attention scores for the answer entity are shown in the table on the right.
+
+The attention score adoct,i on memory $m _ { i } ^ { d o c }$ given the state $s _ { t }$ is computed as follows: $a _ { t , i } ^ { d o c } ~ =$ sof $\mathrm { t m a x } _ { i = 1 , \dots , | M ^ { d o c } | } \gamma \cos ( W _ { 1 } ^ { d o c } m _ { i } ^ { d o c } , W _ { 2 } ^ { d o c } s _ { t } )$ , where $\gamma$ is set to 10. $W _ { 1 } ^ { d o c }$ and $W _ { 2 } ^ { d o c }$ are weight vectors associated with $m _ { i } ^ { d o c }$ and $s _ { t }$ , respectively, and are joint trained in the ReasoNet. Thus, attention vector on passage is given by $\begin{array} { r } { x _ { t } ^ { d o c } = \sum _ { i } ^ { | M | } a _ { t , i } m _ { i } ^ { d o c } } \end{array}$ . The final attention vector is the concatenation of the query attention vector and the passage attention vector $x _ { t } = ( x _ { t } ^ { q u e r y } , x _ { t } ^ { d o c } )$ = (xquet ry , xdoct ). The attention module is parameterized by $\theta _ { x } = ( W _ { 1 } ^ { q u e \hat { r } y } , \bar { W _ { 2 } ^ { q u e r y } } , W _ { 1 } ^ { d o c } , W _ { 2 } ^ { d o c } )$ ;
+
+Internal State Controller: We choose GRU model as the internal state controller. The number of hidden units in the GRU state controller is 256 for CNN and 384 for Daily Mail. The initial state of the GRU controller is set to be the last-word of the query representation by a bidirectional-GRU encoder.
+
+Termination Module: We adopt a logistical regression to model the termination variable at each time step : $f _ { t g } ( s _ { t } ; \theta _ { t g } ) = \mathrm { s i g m o i d } ( \bar { W _ { t g } } s _ { t } + b _ { t g } ) \dot { ; } \theta _ { t g } = ( \bar { W _ { t g } } , b _ { t g } )$
+
+Answer Module: We apply a linear projection from GRU outputs and make predictions on the entity candidates. Following the settings in AS Reader (Kadlec et al., 2016), we sum up scores from the same candidate and make a prediction. Thus, AS Reader can be viewed as a special case of ReasoNets with $T _ { \mathrm { m a x } } = 1$ .
+
+Other Details: The maximum reasoning step, $T _ { \mathrm { m a x } }$ is set to 5 in experiments on both CNN and Daily Mail datasets. We use ADAM optimizer (Kingma & Ba, 2015) for parameter optimization with an initial learning rate of 0.0005, $\beta _ { 1 } = 0 . 9$ and $\beta _ { 2 } = 0 . 9 9 9$ ; The absolute value of gradient on each parameter is clipped within 0.001. The batch size is 64 for both CNN and Daily Mail datasets. For each batch of the CNN and Daily Mail datasets we randomly reshuffle the assignment of named entities (Hermann et al., 2015). This forces the model to treat the named entities as semantically meaningless labels. In the prediction of test cases, we randomly reshuffle named entities up to 4 times, and report the averaged answer. Models are trained on GTX TitanX 12GB. It takes 7 hours per epoch to train on the Daily Mail dataset and 3 hours per epoch to train on the CNN dataset. The models are usually converged within 6 epochs on both CNN and Daily Mail datasets.
+
+Table 1 shows the performance of all the existing single model baselines and our proposed ReasoNet. By capturing multi-turn reasoning and learning to stop reading a paragraph, we have achieved the state-of-the-art results in both CNN and Daily Mail datasets. To further understand the inference process of the ReasoNet, Figure 2 shows a test example of the CNN dataset. The model initially focuses on wrong entities with low termination probability. In the second and third steps, the model focuses on the right clue with higher termination probability. Interestingly, we also find that query attention focuses on the placeholder token throughout all the steps.
+
+Table 1: The performance of Reasoning Network on CNN and Daily Mail dataset.   
+
+<table><tr><td></td><td colspan="2">CNN</td><td colspan="2">Daily Mail</td></tr><tr><td></td><td>valid</td><td>test</td><td>valid</td><td>test</td></tr><tr><td>Deep LSTMReader (Hermann etal.,2015)</td><td>55.0</td><td>57.0</td><td>63.3</td><td>62.2</td></tr><tr><td>Attentive Reader (Hermann et al., 2015)</td><td>61.6</td><td>63.0</td><td>70.5</td><td>69.0</td></tr><tr><td>MemNets (Hill et al.,2016)</td><td>63.4</td><td>66.8</td><td>-</td><td>=</td></tr><tr><td>ASReader (Kadlec et al., 2016)</td><td>68.6</td><td>69.5</td><td>75.0</td><td>73.9</td></tr><tr><td>Stanford AR (Chen et al.,2016)</td><td>72.2</td><td>72.4</td><td>76.9</td><td>75.8</td></tr><tr><td>DER Network (Kobayashi et al., 2016)</td><td>71.3</td><td>72.9</td><td>1</td><td>-</td></tr><tr><td>Iterative Attention Reader (Sordoni et al.,2016)</td><td>72.6</td><td>73.3</td><td>=</td><td>=</td></tr><tr><td>EpiReader (Trischler et al.,2016)</td><td>73.4</td><td>74.0</td><td>=</td><td>=</td></tr><tr><td>GA Reader (Dhingra et al.,2016)</td><td>73.0</td><td>73.8</td><td>76.7</td><td>75.7</td></tr><tr><td>AoA Reader (Cui et al., 2016)</td><td>73.1</td><td>74.4</td><td>=</td><td></td></tr><tr><td>ReasoNet</td><td>72.9</td><td>74.7</td><td>77.6</td><td>76.6</td></tr></table>
+
+Table 2: Reachability statistics of the Graph Reachability dataset.   
+
+<table><tr><td></td><td colspan="4">Small Graph</td><td colspan="4">Large Graph</td></tr><tr><td>Reachable Step</td><td>No Reach</td><td>1-3</td><td>4-6</td><td>7-9</td><td>No Reach</td><td>1-3</td><td>4-6</td><td>7-13</td></tr><tr><td>Train (%)</td><td>44.16</td><td>42.06</td><td>13.51</td><td>0.27</td><td>49.02</td><td>25.57</td><td>21.92</td><td>3.49</td></tr><tr><td>Test (%)</td><td>45.00</td><td>41.35</td><td>13.44</td><td>0.21</td><td>49.27</td><td>25.46</td><td>21.74</td><td>3.53</td></tr></table>
+
+# 4.2 GRAPH REACHABILITY TASK
+
+Recent analysis and results (Chen et al., 2016) on the cloze-style machine comprehension tasks have suggested some simple models without multi-turn reasoning can achieve reasonable performance. Based on these results, we construct a synthetic structured Graph Reachability dataset3 to evaluate longer range machine inference and reasoning capability, since we expect ReasoNets have the capability to handle long range relationships.
+
+We generate two synthetic datasets: a small graph dataset and a large graph dataset. In the small graph dataset, it contains $5 0 0 K$ small graphs, where each graph contains 9 nodes, and 16 direct edges to randomly connect pairs of nodes. The large graph dataset contains $5 0 0 K$ graphs, where each graph contains 18 nodes, and 32 random direct edges. Duplicated edges are removed. Table 2 shows the graph reachability statistics on the two datasets.
+
+In Table 3, we show examples of a small graph and a large graph in the synthetic dataset. Both graph and query are represented by a sequence of symbols. In the experiment, we use a 100-dimensional embedding vector for each symbol, and bidirectional-LSTM with 128 and 256 cells for query and graph embedding in the small and the large graph datasets, respectively. The last states of bidirectionalLSTM on query are concatenated to be the initial internal state $s _ { 1 } \overline { { \mathop { = } } } \left[ \overrightarrow { q } ^ { | q | } , \overleftarrow { q } ^ { 1 } \right]$ in the ReasoNet. Another bidirectional-LSTM on graph description maps each symbol $g ^ { i }$ to a contextual representation given by the concatenation of forward and backward LSTM hidden states $m _ { i } = [ \vec { g } ^ { i } , \overleftarrow { g } | \dot { g } | - i + 1 ]$ . The final answer is either “Yes” or “No” and hence logistical regression is used as the answer module: $a _ { t } = \sigma ( W _ { a } s _ { t } + b _ { a } )$ ; $\theta _ { a } = ( W _ { a } , b _ { a } )$ . We apply another logistical regression as the termination gate module: $t _ { t } = \sigma ( W _ { t g } s _ { t } + b _ { t g } )$ . The maximum reasoning step $T _ { \mathrm { m a x } }$ is set to 15 and 25 for the small graph and large graph dataset, respectively.
+
+We denote “ReasoNet” as the standard ReasoNet with termination gate, as described in Section 3.1. To study the effectiveness of the termination gate in ReasoNets, we remove the termination gate and use the prediction from the last state, $\hat { a } = a _ { T _ { \mathrm { m a x } } }$ $\mathcal { T } _ { \mathrm { m a x } }$ is the maximum reasoning step), denoted as “ReasoNet-Last”. To study the effectiveness of multi-turn reasoning, we choose “ReasoNet $- T _ { \mathrm { m a x } } = 2 ^ { , , }$ , which only has single-turn reasoning. We compare ReasoNets with a two layer deep LSTM model (Hermann et al., 2015) with 128 hidden units, denoted as “Deep LSTM Reader”, as a baseline. Table 4 shows the performance of these models on the graph reachability dataset. Deep LSTM Reader achieves $9 0 . 9 2 \%$ and $7 1 . 5 5 \%$ accuracy in the small and large graph dataset, respectively, which indicates the graph reachibility task is not trivial. The results of ReasoNet $- T _ { \mathrm { m a x } } = 2$ are comparable with the results of Deep LSTM Reader, since both Deep LSTM Reader and ReasoNet $- T _ { \mathrm { m a x } } = 2$ perform single-turn reasoning. The ReasoNet-Last model achieves $1 0 0 \%$ accuracy on the small graph dataset, while the ReasoNet-Last model achieves only $7 8 . 9 5 \%$ accuracy on the large graph dataset, as the task becomes more challenging. Meanwhile, the ReasoNet model converges faster than the ReasoNet-Last model. The ReasoNet model converges in 20 epochs in the small graph dataset, and 40 epochs in the large graph dataset, while the ReasoNet-Last model converges around 40 epochs in the small graph dataset, and 70 epochs in the large graph dataset. The results suggest that the termination gate variable in the ReasoNet is helpful when training with sophisticated examples, and makes models converge faster. Both the ReasoNet and ReasoNet-Last models perform better than the ReasoNet $- T _ { \mathrm { m a x } } = 2$ model, which demonstrates the importance of multi-turn reasoning.
+
+Table 3: Small and large random graph in the Graph Reachability dataset. Note that “ $A  B ^ { \prime }$ represents an edge connected from $A$ to $B$ and the $\#$ symbol is used as a delimiter between different edges.   
+
+<table><tr><td></td><td>Small Graph</td><td>Large Graph</td></tr><tr><td>Graph Description</td><td>0→0#0→2#1→→2#2→1# 3→2#3→3#3-6#37# 4→0#4→1#44#57# 6→0#6→1#7→0#</td><td>0→17#1→3#1→→14#1→6# 2→11#2→13#2→15#3→7# 5→0#5→7#6→10#6→5# 7→15#7-→7#8-→11#87# 10 →9#10→6#10→7#12→1# 12 →12#12→6#13-11#14→17# 14→14#15→10#16→2#17→4#</td></tr><tr><td>Query</td><td>7→4</td><td>17→7# 10→17</td></tr><tr><td>Answer</td><td>No</td><td>Yes</td></tr></table>
+
+Table 4: The performance of Reasoning Network on the Graph Reachability dataset.   
+
+<table><tr><td></td><td colspan="3">Small Graph</td><td colspan="3">Large Graph</td></tr><tr><td></td><td>ROC-AUC</td><td>PR-AUC</td><td>Accuracy</td><td>ROC-AUC</td><td>PR-AUC</td><td>Accuracy</td></tr><tr><td>Deep LSTM Reader</td><td>0.9619</td><td>0.9565</td><td>0.9092</td><td>0.7988</td><td>0.7887</td><td>0.7155</td></tr><tr><td>ReasoNet-Tmax = 2</td><td>0.9638</td><td>0.9677</td><td>0.8961</td><td>0.8477</td><td>0.8388</td><td>0.7607</td></tr><tr><td>ReasoNet-Last</td><td>1</td><td>1</td><td>1</td><td>0.8836</td><td>0.8742</td><td>0.7895</td></tr><tr><td>ReasoNet</td><td>1</td><td>1</td><td>1</td><td>0.9988</td><td>0.9989</td><td>0.9821</td></tr></table>
+
+To further understand the inference process in ReasoNets, Figures 3 and 4 show test examples of the large graph dataset. In Figure 3, we can observe that the model does not make a firm prediction till step 9. The highest attention word at each step shows the reasoning process of the model. Interestingly, the model starts from the end node (17), traverses backward till finding the starting node (10) in step 9, and makes a firm termination prediction. On the other hand, in Figure 4, the model learns to stop in step 2. In step 1, the model looks for neighbor nodes (12, 6, 16) to 4 and 9. Then, the model gives up in step 2 and predict “No". All of these demonstrate the dynamic termination characteristic and potential reasoning capability of ReasoNets.
+
+We show the distribution of termination steps in ReasoNets on the test set in Appendix A. The termination step is chosen with the maximum termination probability $\begin{array} { r } { p ( k ) = t _ { k } \prod _ { i = 1 } ^ { k - 1 } \left( 1 - t _ { i } \right) } \end{array}$ , where $t _ { i }$ is the termination probability at step $i$ .
+
+# 5 CONCLUSION
+
+In this paper, we propose ReasoNets that dynamically decide whether to continue or to terminate the inference process in machine comprehension tasks. Using reinforcement learning with the proposed contractive reward, our proposed model achieves the start-of-the-art results in machine comprehension datasets, including unstructured CNN and Daily Mail datasets, and a proposed structured Graph
+
+![](images/828b0ab37652d632533e42c570b2fc01a99bda3e77251fac95aac51b54196208.jpg)  
+Figure 3: An example of graph reachability result, given a query $^ { \cdot } 1 0  1 7 ^ { \cdot \cdot }$ (Answer: Yes). The red circles highlight the nodes/edges which have the highest attention in each step. The corresponding termination probability and prediction results are shown in the table. The model terminates at step 10.
+
+![](images/1b009fa598e45632dfae6caf3412153f62b648f633352ded1b11e14f26e05cef.jpg)  
+Figure 4: An example of graph reachability result, given a query $" 4  9 "$ (Answer: No). The numbers next to the underline bars indicate the rank of the attention scores. The corresponding termination probability and prediction results are shown in the table.
+
+Reachability dataset. For future work, ReasoNets can be generalized to other tasks that requires reasoning capability, such as question answering and knowledge graph inference.
+
+# ACKNOWLEDGMENTS
+
+We thank Ming-Wei Chang, Li Deng, Lihong Li, and Xiaodong Liu for their thoughtful feedback and discussions.
+
+# REFERENCES
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of the International Conference on Learning Representations, 2015.
+
+Léon Bottou. From machine learning to machine reasoning. Machine Learning, 94(2):133–149, 2014.
+
+Danqi Chen, Jason Bolton, and Christopher D Manning. A thorough examination of the CNN / Daily Mail reading comprehension task. In ACL, 2016.
+
+Yiming Cui, Zhipeng Chen, Si Wei, Shijin Wang, Ting Liu, and Guoping Hu. Attention-over-attention neural networks for reading comprehension. CoRR, abs/1607.04423, 2016.
+
+Bhuwan Dhingra, Hanxiao Liu, William W. Cohen, and Ruslan Salakhutdinov. Gated-attention readers for text comprehension. CoRR, abs/1606.01549, 2016.
+
+Karm Moritz Hermann, Tomáš Kociský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and ˇ Phil Blunsom. Teaching machines to read and comprehend. In Advances in Neural Information Processing Systems, pp. 1693–1701, 2015.
+
+Felix Hill, Antoine Bordes, Sumit Chopra, and Jason Weston. The Goldilocks principle: Reading children’s books with explicit memory representations. In Proceedings of the International Conference on Learning Representations, 2016.
+
+Rudolf Kadlec, Martin Schmid, Ondrej Bajgar, and Jan Kleindienst. Text understanding with the attention sum reader network. arXiv:1603.01547v1 [cs.CL], 2016.
+
+Leslie Pack Kaelbling, Michael L. Littman, and Anthony R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101:99–134, 1998.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of the International Conference on Learning Representations, 2015.
+
+Sosuke Kobayashi, Ran Tian, Naoaki Okazaki, and Kentaro Inui. Dynamic entity representation with maxpooling improves machine reading. In Proceedings of the North American Chapter of the Association for Computational Linguistics and Human Language Technologies (NAACL-HLT), 2016.
+
+Ankit Kumar, Ozan Irsoy, Peter Ondruska, Mohit Iyyer, James Bradbury, Ishaan Gulrajani, Victor Zhong, Romain Paulus, and Richard Socher. Ask me anything: Dynamic memory networks for natural language processing. In Proceedings of the International Conference on Machine Learning, 2016.
+
+Volodymyr Mnih, Nicolas Heess, Alex Graves, et al. Recurrent models of visual attention. In Advances in Neural Information Processing Systems, pp. 2204–2212, 2014.
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+Rodrigo Nogueira and Kyunghyun Cho. Webnav: A new large-scale task for natural language based sequential decision making. In Advances in Neural Information Processing Systems, 2016.
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+Jeffrey Pennington, Richard Socher, and Christopher D. Manning. Glove: Global vectors for word representation. In EMNLP, 2014.
+
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: 100, $0 0 0 +$ questions for machine comprehension of text. In EMNLP, 2016.
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+Matthew Richardson, Christopher JC Burges, and Erin Renshaw. MCTest: A challenge dataset for the opendomain machine comprehension of text. In EMNLP, 2013.
+
+Alessandro Sordoni, Phillip Bachman, and Yoshua Bengio. Iterative alternating neural attention for machine reading. CoRR, abs/1606.02245, 2016.
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+Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015.
+
+Richard S. Sutton, David McAllester, Satinder Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. In Advances in Neural Information Processing Systems, 1999.
+
+Richard Stuart Sutton. Temporal Credit Assignment in Reinforcement Learning. PhD thesis, 1984.
+
+Adam Trischler, Zheng Ye, Xingdi Yuan, and Kaheer Suleman. Natural language comprehension with the EpiReader. In EMNLP, 2016.
+
+Dirk Weissenborn. Separating answers from queries for neural reading comprehension. CoRR, abs/1607.03316, 2016.
+
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8(3-4):229–256, 1992.
+
+# A THE TERMINATION STEP DISTRIBUTION IN REASONETS
+
+In this section, we present the termination step distribution of ReasoNets. Figure 5 and Figure 6 show the termination step distribution of ReasoNets in the CNN dataset and the graph reachability dataset, respectively. The distributions spread out across different steps and there are a large number of instances that terminate in the last step. We study the correlation between the termination steps and the complexity of test instances in Figure 7. We use Breadth-First Search (BFS) algorithm over the target graph given the query to analyze the complexity of test instances. For example, BFS-Step $= 2$ indicates that there are two intermediate nodes in the shortest reachability path. Test instances with larger BFS-Steps are more challenging. We denote ${ \bf B } \mathrm { F S - S t e p = - 1 }$ as there is no reachable path for the given query. Figure 7 shows that test instances with larger BFS-Steps require more reasoning steps.
+
+![](images/adcde0f9d5715d34527b5d7e8969cdd91ba5497d289bbec43ef5d78dfa4d27b9.jpg)  
+Figure 5: The termination step distribution of a ReasoNet $T _ { m a x } = 5$ ) in the CNN dataset.
+
+![](images/5878c3f9906a18e1bb1f43a9acaf057c6babbb5ec0bf2c90c555091dc480302a.jpg)  
+Figure 6: Termination step distribution of ReasoNets in the graph reachability dataset, where $T _ { m a x }$ is set to 15 and 25 in the small graph and large graph dataset, respectively.
+
+![](images/f59afdcdb6641db05b3f411dae89cf7fd4794c3030b12d3625992d0dfd6211b7.jpg)  
+Figure 7: The correlation between BFS steps and ReasoNet termination steps in the graph reachability dataset, where $T _ { m a x }$ is set to 15 and 25 in the small graph and large graph dataset, respectively, and BFS-Step $\ c = - 1$ denotes unreachable cases. The value indicates the number of instances in each case.

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+    {
+        "type": "text",
+        "text": "REASONET: LEARNING TO STOP READING IN MACHINE COMPREHENSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yelong Shen, Po-Sen Huang, Jianfeng Gao, Weizhu Chen Microsoft Research, Redmond, WA, USA {yeshen,pshuang,jfgao,wzchen}@microsoft.com ",
+        "bbox": [
+            183,
+            169,
+            602,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Teaching a computer to read a document and answer general questions pertaining to the document is a challenging yet unsolved problem. In this paper, we describe a novel neural network architecture called the Reasoning Network (ReasoNet) for machine comprehension tasks. ReasoNets make use of multiple turns to effectively exploit and then reason over the relation among queries, documents, and answers. Different from previous approaches using a fixed number of turns during inference, ReasoNets introduce a termination state to relax this constraint on the reasoning depth. With the use of reinforcement learning, ReasoNets can dynamically determine whether to continue the comprehension process after digesting intermediate results, or to terminate reading when it concludes that existing information is adequate to produce an answer. ReasoNets have achieved state-of-the-art performance in machine comprehension datasets, including unstructured CNN and Daily Mail datasets, and a structured Graph Reachability dataset. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            489,
+            336,
+            505
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Teaching machines to read, process, and comprehend natural language documents is a coveted goal for artificial intelligence (Bottou, 2014; Richardson et al., 2013; Hermann et al., 2015). Genuine reading comprehension is extremely challenging, since effective comprehension involves thorough understanding of documents and performing sophisticated inference. Toward solving this machine reading comprehension problem, in recent years, several work has collected various datasets, in the form of question, passage, and answer, to test machine on answering a question based on the provided passage (Richardson et al., 2013; Hermann et al., 2015; Hill et al., 2016; Rajpurkar et al., 2016). Some large-scale cloze-style datasets (Hermann et al., 2015; Hill et al., 2016) have gained significant attention along with powerful deep learning models. ",
+        "bbox": [
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+            520,
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+            646
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent approaches on cloze-style datasets can be separated into two categories: single-turn and multiturn reasoning. Single turn reasoning models utilize attention mechanisms (Bahdanau et al., 2015) with deep learning models to emphasize specific parts of the document which are relevant to the query. These attention models subsequently calculate the relevance between a query and the corresponding weighted representations of document subunits (e.g. sentences or words) to score target candidates (Hill et al., 2016; Hermann et al., 2015; Kadlec et al., 2016). However, considering the sophistication of the problem, after a single-turn comprehension, readers often revisit some specific passage or the question to grasp a better understanding of the problem. With this motivation, recent advances in reading comprehension have made use of multiple turns to infer the relation between query, document and answer (Hill et al., 2016; Dhingra et al., 2016; Trischler et al., 2016; Sordoni et al., 2016). By repeatedly processing the document and question after digesting intermediate information, multi-turn reasoning can generally produce a better answer and all existing work has demonstrated its superior performance consistently. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing multi-turn models have a fixed number of hops or iterations in their inference, i.e., with predetermined reasoning depth, without regard to the complexity of each individual query or document. However, when a human reads a document with a question in mind, we often decide whether we want to stop reading if we believe the observed information is adequate already to answer the question, or continue reading after digesting intermediate information until we can answer the question with confidence. This behavior generally varies from document to document, or question to question because it is related to the sophistication of the document or the difficulty of the question. Meanwhile, the analysis in Chen et al. (2016) also illustrates the huge variations in the difficulty level with respect to questions in the CNN/Daily Mail datasets (Hermann et al., 2015). For a significant part of the datasets, this analysis shows that the problem cannot be solved without appropriate reasoning on both its query and document. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With this motivation, we propose a novel neural network architecture called Reasoning Network (ReasoNet). ReasoNets try to mimic the inference process of human readers. With a question in mind, ReasoNets read a document repeatedly, each time focusing on different parts of the document until a satisfying answer is found or formed. This reminds us of a Chinese proverb: “The meaning of a book will become clear if you read it hundreds of times.”. Moreover, unlike previous approaches using fixed number of hops or iterations, ReasoNets introduce a termination state in the inference. This state can decide whether to continue the inference to next turn after digesting intermediate information, or to terminate the whole inference when it concludes that existing information is sufficient to yield an answer. This number of turns in the inference is dynamically modeled by both the document and the query, and can be learned automatically according to the difficulty of the problem. ",
+        "bbox": [
+            174,
+            181,
+            825,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One of the significant challenges ReasoNets face is how to design an efficient training method, since the termination state is discrete and not connected to the final output. This prohibits canonical backpropagation method being directly applied to train ReasoNets. Inspired by Williams (1992); Mnih et al. (2014), we tackle this challenge by proposing a novel deep reinforcement learning method called Contrastive Reward (CR) to successfully train ReasoNets. Unlike traditional reinforcement learning optimization methods using a global variable to capture rewards, CR utilizes an instance-based reward baseline assignment. Experiments show the superiority of CR in both training speed and accuracy. Finally, by accounting for a dynamic termination state during inference and applying proposed deep reinforcement learning optimization method, ReasoNets achieve the state-of-the-art results in machine comprehension datasets when the paper is first publicly available in arXiv1, including unstructured CNN and Daily Mail datasets, and a proposed structured Graph Reachability dataset. ",
+        "bbox": [
+            174,
+            325,
+            825,
+            478
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper is organized as follows. In Section 2, we review and compare recent work on machine reading comprehension tasks. In Section 3, we introduce our proposed ReasoNet model architecture and training objectives. Section 4 presents the experimental setting and results on unstructured and structured machine reading comprehension tasks . ",
+        "bbox": [
+            176,
+            486,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            563,
+            344,
+            579
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, with large-scale datasets available and the impressive advance of various statistical models, machine reading comprehension tasks have attracted much attention. Here we mainly focus on the related work in cloze-style datasets (Hermann et al., 2015; Hill et al., 2016). Based on how they perform the inference, we can classify their models into two categories: single-turn and multi-turn reasoning. ",
+        "bbox": [
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+            593,
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+            662
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Single-turn reasoning Single turn reasoning models utilize an attention mechanism to emphasis some sections of a document which are relevant to a query. This can be thought of as treating some parts unimportant while focusing on other important ones to find the most probable answer. Hermann et al. (2015) propose the attentive reader and the impatient reader models using neural networks with an attention over passages to predict candidates. Hill et al. (2016) use attention over window-based memory, which encodes a window of words around entity candidates, by leveraging an end-to-end memory network (Sukhbaatar et al., 2015). Meanwhile, given the same entity candidate can appear multiple times in a passage, Kadlec et al. (2016) propose the attention-sum reader to sum up all the attention scores for the same entity. This score captures the relevance between a query and a candidate. Chen et al. (2016) propose using a bilinear term similarity function to calculate attention scores with pretrained word embedding. Trischler et al. (2016) propose the EpiReader which uses two neural network structures: one extracts candidates using the attention-sum reader; the other reranks candidates based on a bilinear term similarity score calculated from query and passage representations. ",
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+    {
+        "type": "text",
+        "text": "Multi-turn reasoning For complex passages and complex queries, human readers often revisit the given document in order to perform deeper inference after reading a document. Several recent studies try to simulate this revisit by combining the information in the query with the new information digested from previous iterations (Hill et al., 2016; Dhingra et al., 2016; Sordoni et al., 2016; Weissenborn, 2016; Kumar et al., 2016). Hill et al. (2016) use multiple hops memory network to augment the query with new information from the previous hop. Gated Attention reader (Dhingra et al., 2016) is an extension of the attention-sum reader with multiple iterations by pushing the query encoding into an attention-based gate in each iteration. Iterative Alternative (IA) reader (Sordoni et al., 2016) produces a new query glimpse and document glimpse in each iteration and utilizes them alternatively in the next iteration. Cui et al. (2016) further propose to extend the query-specific attention to both query-to-document attention and document-to-query attention, which is built from the intermediate results in the query-specific attention. By reading documents and enriching the query in an iterative fashion, multi-turn reasoning has demonstrated their superior performance consistently. ",
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+    {
+        "type": "table",
+        "img_path": "images/ab4be3e769795a60eafc33d71dc657eaf4d922b66a42e38f97f4c5051694fb14.jpg",
+        "table_caption": [
+            "Algorithm 1: Stochastic Inference in a ReasoNet "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Input:Memory M; Initial state s1; Step t =1; Maximum Step Tmax Output:Termination Step T,Answer aT Sample t from the distribution p(*Iftg(St; 0tg));</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our proposed approach explores the idea of using both attention-sum to aggregate candidate attention scores and multiple turns to attain a better reasoning capability. Unlike previous approaches using fixed number of hops or iterations, motivated by Nogueira & Cho (2016); Mnih et al. (2014), we propose a termination module in the inference. The termination module can decide whether to continue to infer the next turn after digesting intermediate information, or to terminate the whole inference process when it concludes existing information is sufficient to yield an answer. The number of turns in the inference is dynamically modeled by both a document and a query, and is generally related to the complexity of the document and the query. ",
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+    {
+        "type": "text",
+        "text": "3 REASONING NETWORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "ReasoNets are devised to mimic the inference process of human readers. ReasoNets read a document repeatedly, with attention on different parts each time until a satisfying answer is found. As shown in Figure 1, a ReasoNet is composed of the following components: ",
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+        "type": "text",
+        "text": "Internal State: The internal state is denoted as $S$ which is a vector representation of the question state. Typically, the initial state $s _ { 1 }$ is the last-word vector representation of query by an RNN. The $t$ -th time step of the internal state is represented by $s _ { t }$ . The sequence of internal state is modeled by an RNN: $s _ { t + 1 } = \\mathrm { R N N } ( s _ { t } , x _ { t } ; \\theta _ { s } )$ ; ",
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+    {
+        "type": "text",
+        "text": "Memory: The external memory is denoted as $M$ . It is a list of word vectors, $M = \\{ m _ { i } \\} _ { i = 1 \\dots D }$ , where $m _ { i }$ is a fixed dimensional vector. In machine comprehensive tasks, $m _ { i }$ is the vector representation of each word in the doc by a bidirectional-RNN. ",
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+    {
+        "type": "text",
+        "text": "Attention: Attention vector $x _ { t }$ is generated based on the current internal state $s _ { t }$ and the external memory $M \\colon x _ { t } = f _ { a t t } \\bigl ( s _ { t } , M ; \\theta _ { x } \\bigr )$ ; ",
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+    {
+        "type": "text",
+        "text": "Termination Gate: Termination gate generates a stochastic random variable according to the current internal state; $t _ { t } \\sim p ( \\cdot | f _ { t g } ( s _ { t } ; \\theta _ { t g } \\bar { ) } ) )$ . $t _ { t }$ is a binary random variable. If $t _ { t }$ is true, the ReasoNet stops, and the answer module executes at time step $t$ ; otherwise the ReasoNet generates an attention vector $x _ { t + 1 }$ , and feed into the state network to update the next internal state $s _ { t + 1 }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Answer: The action of answer module is triggered when the termination gate variable is true: $a _ { t } \\sim p ( \\cdot | f _ { a } ( s _ { t } ; \\theta _ { a } ) )$ . ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/aef314f8cbc4976d7bc6a8375884e86178fcdf98f061da33c1227bf253c967d5.jpg",
+        "image_caption": [
+            "Figure 1: A ReasoNet Architecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Algorithm 1, we describe the stochastic inference process of a ReasoNet. The process can be considered as a Partially Observable Markov Decision Process (POMDP) (Kaelbling et al., 1998) in the reinforcement learning (RL) literature. The state sequence $s _ { 1 : T }$ is hidden and dynamic, controlled by an RNN sequence model. The ReasoNet performs an answer action $a _ { T }$ at the $T$ -th step, which implies that the termination gate variables $t _ { 1 : T } = ( t _ { 1 } = 0 , t _ { 2 } = 0 , . . . , t _ { T - 1 } = 0 , t _ { T } = 1 )$ . The ReasoNet learns a stochastic policy $\\pi ( ( t _ { t } , a _ { t } ) | s _ { t } ; \\theta )$ with parameters $\\theta$ to get a distribution over termination actions, to continue reading or to stop, and over answer actions if the model decides to stop at the current step. The termination step $T$ varies from instance to instance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The parameters $\\theta$ of the ReasoNet are given by the parameters of the embedding matrices $W$ , attention network $\\theta _ { x }$ , the state RNN network $\\theta _ { s }$ , the answer action network $\\theta _ { a }$ , and the termination gate network $\\theta _ { t g }$ . The parameters $\\theta = \\{ W , \\theta _ { x } , \\theta _ { s } , \\theta _ { a } , \\theta _ { t g } \\}$ are trained by maximizing the total expect reward. The expected reward for an instance is defined as: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/30f328f770fbf94c5ddb11730eb36d8b0e45083ace2e771da8eff9ae5bc7963a.jpg",
+        "text": "$$\nJ ( \\theta ) = \\mathbb { E } _ { \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) } \\left[ \\sum _ { t = 1 } ^ { T } r _ { t } \\right]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The reward can only be received at the final termination step when an answer action $a _ { T }$ is performed. We define $r _ { T } = 1$ if $t _ { T } = 1$ and the answer is correct, and $r _ { T } = 0$ otherwise. The rewards on intermediate steps are zeros, $\\{ r _ { t } = 0 \\} _ { t = 1 \\dots T - 1 }$ . $J$ can be maximized by directly applying gradient based optimization methods. The gradient of $J$ is given by: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df6f74dc64393ac22e2eac0a133c8d648502a62b32bb179f4c5972f4b17b9841.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J ( \\theta ) = \\mathbb { E } _ { \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) } \\left[ \\nabla _ { \\theta } \\mathrm { l o g } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) r _ { T } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply the REINFORCE algorithm (Williams, 1992) to compute $\\nabla _ { \\boldsymbol { \\theta } } J ( \\boldsymbol { \\theta } )$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/623f414be214a74574e542bdc1b4528fba4063b66efc679c8a6cf17ab1d2b5ad.jpg",
+        "text": "$$\n\\tilde { \\mathbf { L } } _ { \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) } \\left[ \\nabla _ { \\theta } \\mathrm { l o g } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) r _ { T } \\right] = \\sum _ { ( t _ { 1 : T } , a _ { T } ) \\in \\mathbb { A } ^ { \\dagger } } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) \\left[ \\nabla _ { \\theta } \\mathrm { l o g } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) ( r _ { T } - b _ { T } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { A } ^ { \\dagger }$ is all the possible episodes, $T , t _ { 1 : T } , a _ { T }$ and $r _ { T }$ are the termination step, termination action, answer action, and reward, respectively, for the $( t _ { 1 : T } , a _ { T } )$ episode. $b _ { T }$ is called the reward baseline in the RL literature to lower variance (Sutton, 1984). It is common to select $b _ { T } = \\mathbb { E } _ { \\pi } \\left[ r _ { T } \\right]$ (Sutton et al., 1999), and can be updated via an online moving average approach : $b _ { T } = \\lambda b _ { T } + ( \\bar { 1 } - \\lambda ) r _ { T }$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "However, we empirically find that above approach leads to slow convergence in training ReasoNets. Intuitively, the average baselines $\\{ b _ { T } ; T = 1 . . T _ { \\mathrm { m a x } } \\}$ are global variables independent of instances. It is hard for these baselines to capture the dynamic termination behavior of ReasoNets. In other words, ReasoNets may stop at different time steps for different instances. The adoption of a global variable without considering the dynamic variance in each instance is inappropriate. To resolve this weakness in traditional methods and account for the dynamic characteristic of ReasoNets, we propose an instance-based baseline method called “Contrastive Reward” (CR) to calculate $\\nabla _ { \\boldsymbol { \\theta } } J ( \\boldsymbol { \\theta } )$ . The basic idea of CR is to utilize an instance-based baseline assignment. We will elaborate its implementation details in Section 3.1. Empirical results show that the proposed reward schema has produced better results compared to the baseline approach. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.1 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the machine reading comprehension tasks, a training dataset can be simplified as a collection of triplets of query q, passage $\\mathbf { p }$ , and answer a. Say $\\left. q _ { n } , p _ { n } , a _ { n } \\right.$ is the $n$ -th training instance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The first step is to extract memory $M$ from $p _ { n }$ by mapping each symbolic in the passage to a contextual representation given by the concatenation of forward and backward RNN hidden states, i.e., $m _ { k } = [ \\bar { \\vec { p _ { n } } } ^ { k } , \\overleftarrow { \\vec { p _ { n } } } ^ { | p _ { n } | - k + 1 } ]$ , and extract initial state $s _ { 1 }$ from $q _ { n }$ by assigning $s _ { 1 } = [ \\overrightarrow { q _ { n } } ^ { | q _ { n } | } , \\overleftarrow { q _ { n } } ^ { 1 } ]$ Given $M$ and $s _ { 1 }$ for the $n$ -th training instance, a ReasoNet executes $| \\dot { \\mathbb { A } } ^ { \\dagger } |$ episodes, where all possible episodes $\\mathbb { A } ^ { \\dagger }$ can be enumerated by setting a maximum step. Each episode generates actions and a reward from the last step: h(t1:T , aT ), rT i(t1:T ,aT )∈A† . ",
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+    {
+        "type": "text",
+        "text": "Therefore, the gradient of $J$ can be rewritten as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5db686893161c65d946bf1b31b4c9f5855955cdb91d2c0b9563bc2289fcfe47a.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J ( \\theta ) = \\sum _ { ( t _ { 1 : T } , a _ { T } ) \\in \\mathbb { A } ^ { \\dagger } } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) \\left[ \\nabla _ { \\theta } \\mathrm { l o g } \\pi ( t _ { 1 : T } , a _ { T } ; \\theta ) ( r _ { T } - b ) \\right]\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where the baseline $\\begin{array} { r } { b = \\sum _ { ( t _ { 1 : T } , a _ { T } ) \\in \\mathbb { A } ^ { \\dagger } } \\pi \\big ( t _ { 1 : T } , a _ { T } ; \\theta \\big ) r _ { T } } \\end{array}$ is the average reward on the $| \\mathbb { A } ^ { \\dagger } |$ episodes for the $n$ -th training instance. It allows different baselines for different training instances. This can be beneficial since the complexity of training instances varies significantly. Since the sum of the proposed rewards over $\\left| \\mathbb { A } ^ { \\dagger } \\right|$ episodes is zero, $\\sum _ { ( t _ { 1 : T } , a _ { T } ) \\in \\mathbb { A } ^ { \\dagger } } \\pi \\big ( \\hat { t _ { 1 : T } } , a _ { T } ; \\theta \\big ) \\big ( r _ { T } - b \\big ) = 0$ , we call it Contrastive Reward in this work. In experiments, we empirically find using $\\left( \\frac { r _ { T } } { b } - 1 \\right)$ in replace of $\\left( r _ { T } - b \\right)$ can lead to a faster convergence. Therefore, we adopt this approach to train ReasoNets in the experiments. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 CNN AND DAILY MAIL DATASETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the performance of ReasoNets on CNN and Daily Mail datasets.2 The detailed settings of the ReasoNet model are as follows. ",
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+        "type": "text",
+        "text": "Vocab Size: For training our ReasoNet, we keep the most frequent $| V | = 1 0 1 k$ words (not including 584 entities and 1 placeholder marker) in the CNN dataset, and $| V | = 1 5 1 k$ words (not including 530 entities and 1 placeholder marker) in the Daily Mail dataset. ",
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+        "type": "text",
+        "text": "Embedding Layer: We choose word embedding size $d \\ : = \\ : 3 0 0$ , and use the 300 dimensional pretrained Glove word embeddings (Pennington et al., 2014) for initialization. We also apply dropout with probability 0.2 to the embedding layer. ",
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+        "type": "text",
+        "text": "Bi-GRU Encoder: We apply bi-directional GRU for encoding query and passage into vector representations. We set the number of hidden units to be 256 and 384 for the CNN and Daily Mail datasets, respectively. The recurrent weights of GRUs are initialized with random orthogonal matrices. The other weights in GRU cell are initialized from a uniform distribution between $- 0 . 0 1$ and 0.01. We use a shared GRU model for both query and passage. ",
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+    {
+        "type": "text",
+        "text": "Memory and Attention: The memory of the ReasoNet on CNN and Daily Mail dataset is composed of query memory and passage memory. $M = ( M ^ { q u e r y } , M ^ { d o c } )$ , where $M ^ { q u e r y }$ and $M ^ { d \\bar { o c } }$ are extracted from query bidirectional-GRU encoder and passage bidirectional-GRU encoder respectively. We choose projected cosine similarity function as the attention module. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Query: passenger @placeholder , 36 , died at the scene 1 1 ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Passage: ( $@$ entity0 ) what was supposed to be a fantasy sports car ride at $@$ entity3 turned deadly when a @entity4 crashed into a guardrail . the crash took place sunday at the $@$ entity8 , which bills itself as a chance to drive your dream car on a racetrack . the @entity4 's passenger , 36 - year - old @entity14 of @entity15 , @entity16 , died at the scene ,3 $@$ entity13 said . the driver of the $@$ entity4 , 24 - year - old @entity18 of @entity19 , $@$ entity16 , lost control of the vehicle , the $@$ entity13 said . he was hospitalized with minor injuries . @entity24 , which operates the $@$ entity8 at @entity3 , released a statement sunday night about the crash . \" on behalf of everyone in the organization , it is with a very heavy heart that we extend our deepest sympathies to those involved in today 's tragic accident in @entity36 , \" the company said . @entity24 also operates the @entity3 -- a chance to drive or ride in $@$ entity39 race cars named for the winningest driver in the sport 's history . @entity0 's @entity43 and $@$ entity44 contributed to this report . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/8c567ad13c16f62297f13251531bd2acfc9d6af35871b8227db656e2dea3a854.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Step</td><td rowspan=1 colspan=1>TerminationProbability</td><td rowspan=1 colspan=1>AttentionSum</td></tr><tr><td rowspan=2 colspan=1>12</td><td rowspan=1 colspan=1>0.0011</td><td rowspan=1 colspan=1>0.4916</td></tr><tr><td rowspan=1 colspan=1>0.5747</td><td rowspan=1 colspan=1>0.5486</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>0.9178</td><td rowspan=1 colspan=1>0.5577</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Answer: @entity14 ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/59b38a0bbfad8e61b714d9d5b8ab2f98923ebd04389cfbc586f0afb98c0ab8da.jpg",
+        "image_caption": [
+            "Figure 2: Results of a test example 69e1f777e41bf67d5a22b7c69ae76f0ae873cf43.story from the CNN dataset. The numbers next to the underline bars indicate the rank of the attention scores. The corresponding termination probability and the sum of attention scores for the answer entity are shown in the table on the right. "
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+        "text": "The attention score adoct,i on memory $m _ { i } ^ { d o c }$ given the state $s _ { t }$ is computed as follows: $a _ { t , i } ^ { d o c } ~ =$ sof $\\mathrm { t m a x } _ { i = 1 , \\dots , | M ^ { d o c } | } \\gamma \\cos ( W _ { 1 } ^ { d o c } m _ { i } ^ { d o c } , W _ { 2 } ^ { d o c } s _ { t } )$ , where $\\gamma$ is set to 10. $W _ { 1 } ^ { d o c }$ and $W _ { 2 } ^ { d o c }$ are weight vectors associated with $m _ { i } ^ { d o c }$ and $s _ { t }$ , respectively, and are joint trained in the ReasoNet. Thus, attention vector on passage is given by $\\begin{array} { r } { x _ { t } ^ { d o c } = \\sum _ { i } ^ { | M | } a _ { t , i } m _ { i } ^ { d o c } } \\end{array}$ . The final attention vector is the concatenation of the query attention vector and the passage attention vector $x _ { t } = ( x _ { t } ^ { q u e r y } , x _ { t } ^ { d o c } )$ = (xquet ry , xdoct ). The attention module is parameterized by $\\theta _ { x } = ( W _ { 1 } ^ { q u e \\hat { r } y } , \\bar { W _ { 2 } ^ { q u e r y } } , W _ { 1 } ^ { d o c } , W _ { 2 } ^ { d o c } )$ ; ",
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+    {
+        "type": "text",
+        "text": "Internal State Controller: We choose GRU model as the internal state controller. The number of hidden units in the GRU state controller is 256 for CNN and 384 for Daily Mail. The initial state of the GRU controller is set to be the last-word of the query representation by a bidirectional-GRU encoder. ",
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+        "text": "Termination Module: We adopt a logistical regression to model the termination variable at each time step : $f _ { t g } ( s _ { t } ; \\theta _ { t g } ) = \\mathrm { s i g m o i d } ( \\bar { W _ { t g } } s _ { t } + b _ { t g } ) \\dot { ; } \\theta _ { t g } = ( \\bar { W _ { t g } } , b _ { t g } )$ ",
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+        "type": "text",
+        "text": "Answer Module: We apply a linear projection from GRU outputs and make predictions on the entity candidates. Following the settings in AS Reader (Kadlec et al., 2016), we sum up scores from the same candidate and make a prediction. Thus, AS Reader can be viewed as a special case of ReasoNets with $T _ { \\mathrm { m a x } } = 1$ . ",
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+        "type": "text",
+        "text": "Other Details: The maximum reasoning step, $T _ { \\mathrm { m a x } }$ is set to 5 in experiments on both CNN and Daily Mail datasets. We use ADAM optimizer (Kingma & Ba, 2015) for parameter optimization with an initial learning rate of 0.0005, $\\beta _ { 1 } = 0 . 9$ and $\\beta _ { 2 } = 0 . 9 9 9$ ; The absolute value of gradient on each parameter is clipped within 0.001. The batch size is 64 for both CNN and Daily Mail datasets. For each batch of the CNN and Daily Mail datasets we randomly reshuffle the assignment of named entities (Hermann et al., 2015). This forces the model to treat the named entities as semantically meaningless labels. In the prediction of test cases, we randomly reshuffle named entities up to 4 times, and report the averaged answer. Models are trained on GTX TitanX 12GB. It takes 7 hours per epoch to train on the Daily Mail dataset and 3 hours per epoch to train on the CNN dataset. The models are usually converged within 6 epochs on both CNN and Daily Mail datasets. ",
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+        "type": "text",
+        "text": "Table 1 shows the performance of all the existing single model baselines and our proposed ReasoNet. By capturing multi-turn reasoning and learning to stop reading a paragraph, we have achieved the state-of-the-art results in both CNN and Daily Mail datasets. To further understand the inference process of the ReasoNet, Figure 2 shows a test example of the CNN dataset. The model initially focuses on wrong entities with low termination probability. In the second and third steps, the model focuses on the right clue with higher termination probability. Interestingly, we also find that query attention focuses on the placeholder token throughout all the steps. ",
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+        "type": "table",
+        "img_path": "images/67b3c6f3e0912a8573a6e0ed240bb05f9e460af2ba06df484e4a4cd394a764a0.jpg",
+        "table_caption": [
+            "Table 1: The performance of Reasoning Network on CNN and Daily Mail dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">CNN</td><td colspan=\"2\">Daily Mail</td></tr><tr><td></td><td>valid</td><td>test</td><td>valid</td><td>test</td></tr><tr><td>Deep LSTMReader (Hermann etal.,2015)</td><td>55.0</td><td>57.0</td><td>63.3</td><td>62.2</td></tr><tr><td>Attentive Reader (Hermann et al., 2015)</td><td>61.6</td><td>63.0</td><td>70.5</td><td>69.0</td></tr><tr><td>MemNets (Hill et al.,2016)</td><td>63.4</td><td>66.8</td><td>-</td><td>=</td></tr><tr><td>ASReader (Kadlec et al., 2016)</td><td>68.6</td><td>69.5</td><td>75.0</td><td>73.9</td></tr><tr><td>Stanford AR (Chen et al.,2016)</td><td>72.2</td><td>72.4</td><td>76.9</td><td>75.8</td></tr><tr><td>DER Network (Kobayashi et al., 2016)</td><td>71.3</td><td>72.9</td><td>1</td><td>-</td></tr><tr><td>Iterative Attention Reader (Sordoni et al.,2016)</td><td>72.6</td><td>73.3</td><td>=</td><td>=</td></tr><tr><td>EpiReader (Trischler et al.,2016)</td><td>73.4</td><td>74.0</td><td>=</td><td>=</td></tr><tr><td>GA Reader (Dhingra et al.,2016)</td><td>73.0</td><td>73.8</td><td>76.7</td><td>75.7</td></tr><tr><td>AoA Reader (Cui et al., 2016)</td><td>73.1</td><td>74.4</td><td>=</td><td></td></tr><tr><td>ReasoNet</td><td>72.9</td><td>74.7</td><td>77.6</td><td>76.6</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/18dc6d01109907830d4cf572e73eb7f2c77179c41fd867a0f60dee9232800357.jpg",
+        "table_caption": [
+            "Table 2: Reachability statistics of the Graph Reachability dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Small Graph</td><td colspan=\"4\">Large Graph</td></tr><tr><td>Reachable Step</td><td>No Reach</td><td>1-3</td><td>4-6</td><td>7-9</td><td>No Reach</td><td>1-3</td><td>4-6</td><td>7-13</td></tr><tr><td>Train (%)</td><td>44.16</td><td>42.06</td><td>13.51</td><td>0.27</td><td>49.02</td><td>25.57</td><td>21.92</td><td>3.49</td></tr><tr><td>Test (%)</td><td>45.00</td><td>41.35</td><td>13.44</td><td>0.21</td><td>49.27</td><td>25.46</td><td>21.74</td><td>3.53</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 GRAPH REACHABILITY TASK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Recent analysis and results (Chen et al., 2016) on the cloze-style machine comprehension tasks have suggested some simple models without multi-turn reasoning can achieve reasonable performance. Based on these results, we construct a synthetic structured Graph Reachability dataset3 to evaluate longer range machine inference and reasoning capability, since we expect ReasoNets have the capability to handle long range relationships. ",
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+        "type": "text",
+        "text": "We generate two synthetic datasets: a small graph dataset and a large graph dataset. In the small graph dataset, it contains $5 0 0 K$ small graphs, where each graph contains 9 nodes, and 16 direct edges to randomly connect pairs of nodes. The large graph dataset contains $5 0 0 K$ graphs, where each graph contains 18 nodes, and 32 random direct edges. Duplicated edges are removed. Table 2 shows the graph reachability statistics on the two datasets. ",
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+    {
+        "type": "text",
+        "text": "In Table 3, we show examples of a small graph and a large graph in the synthetic dataset. Both graph and query are represented by a sequence of symbols. In the experiment, we use a 100-dimensional embedding vector for each symbol, and bidirectional-LSTM with 128 and 256 cells for query and graph embedding in the small and the large graph datasets, respectively. The last states of bidirectionalLSTM on query are concatenated to be the initial internal state $s _ { 1 } \\overline { { \\mathop { = } } } \\left[ \\overrightarrow { q } ^ { | q | } , \\overleftarrow { q } ^ { 1 } \\right]$ in the ReasoNet. Another bidirectional-LSTM on graph description maps each symbol $g ^ { i }$ to a contextual representation given by the concatenation of forward and backward LSTM hidden states $m _ { i } = [ \\vec { g } ^ { i } , \\overleftarrow { g } | \\dot { g } | - i + 1 ]$ . The final answer is either “Yes” or “No” and hence logistical regression is used as the answer module: $a _ { t } = \\sigma ( W _ { a } s _ { t } + b _ { a } )$ ; $\\theta _ { a } = ( W _ { a } , b _ { a } )$ . We apply another logistical regression as the termination gate module: $t _ { t } = \\sigma ( W _ { t g } s _ { t } + b _ { t g } )$ . The maximum reasoning step $T _ { \\mathrm { m a x } }$ is set to 15 and 25 for the small graph and large graph dataset, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We denote “ReasoNet” as the standard ReasoNet with termination gate, as described in Section 3.1. To study the effectiveness of the termination gate in ReasoNets, we remove the termination gate and use the prediction from the last state, $\\hat { a } = a _ { T _ { \\mathrm { m a x } } }$ $\\mathcal { T } _ { \\mathrm { m a x } }$ is the maximum reasoning step), denoted as “ReasoNet-Last”. To study the effectiveness of multi-turn reasoning, we choose “ReasoNet $- T _ { \\mathrm { m a x } } = 2 ^ { , , }$ , which only has single-turn reasoning. We compare ReasoNets with a two layer deep LSTM model (Hermann et al., 2015) with 128 hidden units, denoted as “Deep LSTM Reader”, as a baseline. Table 4 shows the performance of these models on the graph reachability dataset. Deep LSTM Reader achieves $9 0 . 9 2 \\%$ and $7 1 . 5 5 \\%$ accuracy in the small and large graph dataset, respectively, which indicates the graph reachibility task is not trivial. The results of ReasoNet $- T _ { \\mathrm { m a x } } = 2$ are comparable with the results of Deep LSTM Reader, since both Deep LSTM Reader and ReasoNet $- T _ { \\mathrm { m a x } } = 2$ perform single-turn reasoning. The ReasoNet-Last model achieves $1 0 0 \\%$ accuracy on the small graph dataset, while the ReasoNet-Last model achieves only $7 8 . 9 5 \\%$ accuracy on the large graph dataset, as the task becomes more challenging. Meanwhile, the ReasoNet model converges faster than the ReasoNet-Last model. The ReasoNet model converges in 20 epochs in the small graph dataset, and 40 epochs in the large graph dataset, while the ReasoNet-Last model converges around 40 epochs in the small graph dataset, and 70 epochs in the large graph dataset. The results suggest that the termination gate variable in the ReasoNet is helpful when training with sophisticated examples, and makes models converge faster. Both the ReasoNet and ReasoNet-Last models perform better than the ReasoNet $- T _ { \\mathrm { m a x } } = 2$ model, which demonstrates the importance of multi-turn reasoning. ",
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+    {
+        "type": "table",
+        "img_path": "images/52e283742603be905aebaf4f3d3dad432a1023357f6ff1d4a4433b66944727f7.jpg",
+        "table_caption": [
+            "Table 3: Small and large random graph in the Graph Reachability dataset. Note that “ $A  B ^ { \\prime }$ represents an edge connected from $A$ to $B$ and the $\\#$ symbol is used as a delimiter between different edges. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Small Graph</td><td>Large Graph</td></tr><tr><td>Graph Description</td><td>0→0#0→2#1→→2#2→1# 3→2#3→3#3-6#37# 4→0#4→1#44#57# 6→0#6→1#7→0#</td><td>0→17#1→3#1→→14#1→6# 2→11#2→13#2→15#3→7# 5→0#5→7#6→10#6→5# 7→15#7-→7#8-→11#87# 10 →9#10→6#10→7#12→1# 12 →12#12→6#13-11#14→17# 14→14#15→10#16→2#17→4#</td></tr><tr><td>Query</td><td>7→4</td><td>17→7# 10→17</td></tr><tr><td>Answer</td><td>No</td><td>Yes</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b48c3d747d0b78802ab00733a261621b3ba656ce41893ec1f1d2b615b32aa5f6.jpg",
+        "table_caption": [
+            "Table 4: The performance of Reasoning Network on the Graph Reachability dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Small Graph</td><td colspan=\"3\">Large Graph</td></tr><tr><td></td><td>ROC-AUC</td><td>PR-AUC</td><td>Accuracy</td><td>ROC-AUC</td><td>PR-AUC</td><td>Accuracy</td></tr><tr><td>Deep LSTM Reader</td><td>0.9619</td><td>0.9565</td><td>0.9092</td><td>0.7988</td><td>0.7887</td><td>0.7155</td></tr><tr><td>ReasoNet-Tmax = 2</td><td>0.9638</td><td>0.9677</td><td>0.8961</td><td>0.8477</td><td>0.8388</td><td>0.7607</td></tr><tr><td>ReasoNet-Last</td><td>1</td><td>1</td><td>1</td><td>0.8836</td><td>0.8742</td><td>0.7895</td></tr><tr><td>ReasoNet</td><td>1</td><td>1</td><td>1</td><td>0.9988</td><td>0.9989</td><td>0.9821</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "To further understand the inference process in ReasoNets, Figures 3 and 4 show test examples of the large graph dataset. In Figure 3, we can observe that the model does not make a firm prediction till step 9. The highest attention word at each step shows the reasoning process of the model. Interestingly, the model starts from the end node (17), traverses backward till finding the starting node (10) in step 9, and makes a firm termination prediction. On the other hand, in Figure 4, the model learns to stop in step 2. In step 1, the model looks for neighbor nodes (12, 6, 16) to 4 and 9. Then, the model gives up in step 2 and predict “No\". All of these demonstrate the dynamic termination characteristic and potential reasoning capability of ReasoNets. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We show the distribution of termination steps in ReasoNets on the test set in Appendix A. The termination step is chosen with the maximum termination probability $\\begin{array} { r } { p ( k ) = t _ { k } \\prod _ { i = 1 } ^ { k - 1 } \\left( 1 - t _ { i } \\right) } \\end{array}$ , where $t _ { i }$ is the termination probability at step $i$ . ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose ReasoNets that dynamically decide whether to continue or to terminate the inference process in machine comprehension tasks. Using reinforcement learning with the proposed contractive reward, our proposed model achieves the start-of-the-art results in machine comprehension datasets, including unstructured CNN and Daily Mail datasets, and a proposed structured Graph ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/828b0ab37652d632533e42c570b2fc01a99bda3e77251fac95aac51b54196208.jpg",
+        "image_caption": [
+            "Figure 3: An example of graph reachability result, given a query $^ { \\cdot } 1 0  1 7 ^ { \\cdot \\cdot }$ (Answer: Yes). The red circles highlight the nodes/edges which have the highest attention in each step. The corresponding termination probability and prediction results are shown in the table. The model terminates at step 10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b009fa598e45632dfae6caf3412153f62b648f633352ded1b11e14f26e05cef.jpg",
+        "image_caption": [
+            "Figure 4: An example of graph reachability result, given a query $\" 4  9 \"$ (Answer: No). The numbers next to the underline bars indicate the rank of the attention scores. The corresponding termination probability and prediction results are shown in the table. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Reachability dataset. For future work, ReasoNets can be generalized to other tasks that requires reasoning capability, such as question answering and knowledge graph inference. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            866,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Ming-Wei Chang, Li Deng, Lihong Li, and Xiaodong Liu for their thoughtful feedback and discussions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
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+        "text": "A THE TERMINATION STEP DISTRIBUTION IN REASONETS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we present the termination step distribution of ReasoNets. Figure 5 and Figure 6 show the termination step distribution of ReasoNets in the CNN dataset and the graph reachability dataset, respectively. The distributions spread out across different steps and there are a large number of instances that terminate in the last step. We study the correlation between the termination steps and the complexity of test instances in Figure 7. We use Breadth-First Search (BFS) algorithm over the target graph given the query to analyze the complexity of test instances. For example, BFS-Step $= 2$ indicates that there are two intermediate nodes in the shortest reachability path. Test instances with larger BFS-Steps are more challenging. We denote ${ \\bf B } \\mathrm { F S - S t e p = - 1 }$ as there is no reachable path for the given query. Figure 7 shows that test instances with larger BFS-Steps require more reasoning steps. ",
+        "bbox": [
+            173,
+            223,
+            825,
+            363
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/adcde0f9d5715d34527b5d7e8969cdd91ba5497d289bbec43ef5d78dfa4d27b9.jpg",
+        "image_caption": [
+            "Figure 5: The termination step distribution of a ReasoNet $T _ { m a x } = 5$ ) in the CNN dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            310,
+            382,
+            686,
+            573
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/5878c3f9906a18e1bb1f43a9acaf057c6babbb5ec0bf2c90c555091dc480302a.jpg",
+        "image_caption": [
+            "Figure 6: Termination step distribution of ReasoNets in the graph reachability dataset, where $T _ { m a x }$ is set to 15 and 25 in the small graph and large graph dataset, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            640,
+            813,
+            881
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/f59afdcdb6641db05b3f411dae89cf7fd4794c3030b12d3625992d0dfd6211b7.jpg",
+        "image_caption": [
+            "Figure 7: The correlation between BFS steps and ReasoNet termination steps in the graph reachability dataset, where $T _ { m a x }$ is set to 15 and 25 in the small graph and large graph dataset, respectively, and BFS-Step $\\ c = - 1$ denotes unreachable cases. The value indicates the number of instances in each case. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            396,
+            805,
+            575
+        ],
+        "page_idx": 11
+    }
+]

parse/train/BkfbpsAcF7/BkfbpsAcF7.md

@@ -0,0 +1,426 @@
+# EXCESSIVE INVARIANCE CAUSES ADVERSARIAL VULNERABILITY
+
+Jorn-Henrik Jacobsen ¨ 1∗, Jens Behrmann1,2, Richard Zemel1, Matthias Bethge3
+
+1Vector Institute and University of Toronto 2University of Bremen, Center for Industrial Mathematics 3University of Tubingen ¨ ∗j.jacobsen@vectorinstitute.ai
+
+# ABSTRACT
+
+Despite their impressive performance, deep neural networks exhibit striking failures on out-of-distribution inputs. One core idea of adversarial example research is to reveal neural network errors under such distribution shifts. We decompose these errors into two complementary sources: sensitivity and invariance. We show deep networks are not only too sensitive to task-irrelevant changes of their input, as is well-known from $\epsilon$ -adversarial examples, but are also too invariant to a wide range of task-relevant changes, thus making vast regions in input space vulnerable to adversarial attacks. We show such excessive invariance occurs across various tasks and architecture types. On MNIST and ImageNet one can manipulate the class-specific content of almost any image without changing the hidden activations. We identify an insufficiency of the standard cross-entropy loss as a reason for these failures. Further, we extend this objective based on an informationtheoretic analysis so it encourages the model to consider all task-dependent features in its decision. This provides the first approach tailored explicitly to overcome excessive invariance and resulting vulnerabilities.
+
+# 1 INTRODUCTION
+
+![](images/0c5aefdbacb589aa565e0fca4bb21d81f39d66085e00e2ad22a8271ebac3ee23.jpg)  
+Top-1: Bullfrog Top-2:Acorn Top-3:Garter snake   
+Figure 1: All images shown cause a competitive ImageNet-trained network to output the exact same probabilities over all 1000 classes (logits shown above each image). The leftmost image is from the ImageNet validation set; all other images are constructed such that they match the non-class related information of images taken from other classes (for details see section 2.1). The excessive invariance revealed by this set of adversarial examples demonstrates that the logits contain only a small fraction of the information perceptually relevant to humans for discrimination between the classes.
+
+Adversarial vulnerability is one of the most iconic failure cases of modern machine learning models (Szegedy et al., 2013) and a prime example of their weakness in out-of-distribution generalization. It is particularly striking that under i.i.d. settings deep networks show superhuman performance on many tasks (LeCun et al., 2015), while tiny targeted shifts of the input distribution can cause them to make unintuitive mistakes. The reason for these failures and how they may be avoided or at least mitigated is an active research area (Schmidt et al., 2018; Gilmer et al., 2018b; Bubeck et al., 2018).
+
+So far, the study of adversarial examples has mostly been concerned with the setting of small perturbation, or $\epsilon$ -adversaries (Goodfellow et al., 2015; Madry et al., 2017; Raghunathan et al., 2018).
+
+Perturbation-based adversarial examples are appealing because they allow to quantitatively measure notions of adversarial robustness (Brendel et al., 2018). However, recent work argued that the perturbation-based approach is unrealistically restrictive and called for the need of generalizing the concept of adversarial examples to the unrestricted case, including any input crafted to be misinterpreted by the learned model (Song et al., 2018; Brown et al., 2018). Yet, settings beyond $\epsilon$ -robustness are hard to formalize (Gilmer et al., 2018a).
+
+We argue here for an alternative, complementary viewpoint on the problem of adversarial examples. Instead of focusing on transformations erroneously crossing the decision-boundary of classifiers, we focus on excessive invariance as a major cause for adversarial vulnerability. To this end, we introduce the concept of invariance-based adversarial examples and show that class-specific content of almost any input can be changed arbitrarily without changing activations of the network, as illustrated in figure 1 for ImageNet. This viewpoint opens up new directions to analyze and control crucial aspects underlying vulnerability to unrestricted adversarial examples.
+
+The invariance perspective suggests that adversarial vulnerability is a consequence of narrow learning, yielding classifiers that rely only on few highly predictive features in their decisions. This has also been supported by the observation that deep networks strongly rely on spectral statistical regularities (Jo & Bengio, 2017), or stationary statistics (Gatys et al., 2017) to make their decisions, rather than more abstract features like shape and appearance. We hypothesize that a major reason for this excessive invariance can be understood from an information-theoretic viewpoint of crossentropy, which maximizes a bound on the mutual information between labels and representation, giving no incentive to explain all class-dependent aspects of the input. This may be desirable in some cases, but to achieve truly general understanding of a scene or an object, machine learning models have to learn to successfully separate essence from nuisance and subsequently generalize even under shifted input distributions.
+
+Our contributions:
+
+• We identify excessive invariance underlying striking failures in deep networks and formalize the connection to adversarial examples. We show invariance-based adversarial examples can be observed across various tasks and types of deep network architectures.   
+• We propose an invertible network architecture that gives explicit access to its decision space, enabling class-specific manipulations to images while leaving all dimensions of the representation seen by the final classifier invariant.   
+• From an information-theoretic viewpoint, we identify the cross-entropy objective as a major reason for the observed failures. Leveraging invertible networks, we propose an alternative objective that provably reduces excessive invariance and works well in practice.
+
+# 2 TWO COMPLEMENTARY APPROACHES TO ADVERSARIAL EXAMPLES
+
+In this section, we define pre-images and establish a link to adversarial examples.
+
+Definition 1 (Pre-images / Invariance). Let $F : \mathbb { R } ^ { d }  \mathbb { R } ^ { C }$ be a neural network, $F = f _ { L } \circ \cdot \cdot \cdot \circ f _ { 1 }$ with layers $f _ { i }$ and let $F _ { i }$ denote the network up to layer i. Further, let $D : \mathbb { R } ^ { d }  \{ 1 , \dots , C \}$ be a classifier with $D = \arg \operatorname* { m a x } _ { k = 1 , \ldots , C } s o f t m a x ( F ( x ) ) _ { k }$ . Then, for input $\boldsymbol { x } \in \mathbb { R } ^ { d }$ , we define the following pre-images
+
+(i) i-th Layer pre-image: $\{ x ^ { * } \in \mathbb { R } ^ { d } \mid F _ { i } ( x ^ { * } ) = F _ { i } ( x ) \}$ (ii) Logit pre-image: $\{ x ^ { * } \in \mathbb { R } ^ { d } \mid F ( x ^ { * } ) = F ( x ) \}$ (iii) Argmax pre-image: $\{ x ^ { * } \in \mathbb { R } ^ { d } \mid D ( x ^ { * } ) = D ( x ) \}$ , where $( i ) \subset ( i i ) \subset ( i i i )$ by the compositional nature of $D$ . Moreover, the (sub-)network is invariant to perturbations $\Delta x$ which satisfy $x ^ { * } = x + \Delta x$
+
+![](images/c0f0bdabfb8999576b5eacd6bad061d4e004e90bb9a9f87cefb6c26365da9a34.jpg)  
+Figure 2: Connection between (1) invariance-based (long pink arrow) and (2) perturbation-based adversarial examples (short orange arrow). Class distributions are shown in green and blue; dashed line is the decision-boundary of a classifier. All adversarial examples can be reached either by crossing the decision-boundary of the classifier via perturbations, or by moving within the pre-image of the classifier to mis-classified regions. The two viewpoints are complementary to one another and highlight that adversarial vulnerability is not only caused by excessive sensitivity to semantically meaningless perturbations, but also by excessive insensitivity to semantically meaningful transformations.
+
+Non-trivial pre-images (pre-images containing more elements than input $x$ ) after the $i$ -th layer occur if the chain $f _ { i } \circ \cdots \circ f _ { 1 }$ is not injective, for instance due to subsampling or non-injective activation functions like ReLU (Behrmann et al., 2018a). This accumulated invariance can become problematic if not controlled properly, as we will show in the following.
+
+We define perturbation-based adversarial examples by introducing the notion of an oracle (e.g., a human decision-maker or the unknown input-output function considered in learning theory):
+
+Definition 2 (Perturbation-based Adversarial Examples). A Perturbation-based adversarial example $x ^ { * } \in \mathbb { R } ^ { d }$ of $x \in \mathbb { R } ^ { d }$ fulfills:
+
+(i) Perturbation of decision: $D ( x ^ { * } ) \neq o ( x ^ { * } )$ and $D ( x ) \neq D ( x ^ { * } )$ , where $D : \mathbb { R } ^ { d }  \{ 1 , \ldots , C \}$ is the classifier and $o : \mathbb { R } ^ { d }  \{ 1 , . . . , C \}$ is the oracle.
+
+(ii) Created by adversary: $x ^ { * } \in \mathbb { R } ^ { d }$ is created by an algorithm $\mathcal { A } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ with $x \mapsto x ^ { * }$
+
+Further, -bounded adversarial ex. $x ^ { * }$ of $x$ fulfill $\| x - x ^ { * } \| < \epsilon , \| \cdot \|$ a norm on $\mathbb { R } ^ { d }$ and $\epsilon > 0$ .
+
+Usually, such examples are constructed as $\epsilon$ -bounded adversarial examples (Goodfellow et al., 2015). However, as our goal is to characterize general invariances of the network, we do not restrict ourselves to bounded perturbations.
+
+Definition 3 (Invariance-based Adversarial Examples). Let $G$ denote the i-th layer, logits or the classifier (Definition 1) and let $x ^ { * } \neq x$ be in the $G$ pre-image of $x$ and and o an oracle (Definition 2). Then, an invariance-based adversarial example fulfills $o ( x ) \neq o ( x ^ { * } )$ , while $G ( x ) = G ( x ^ { * } )$ (and hence $D ( x ) = D ( x ^ { * } ) ,$ .
+
+Intuitively, adversarial perturbations cause the output of the classifier to change while the oracle would still consider the new input $x ^ { * }$ as being from the original class. Hence in the context of $\epsilon$ - bounded perturbations, the classifier is too sensitive to task-irrelevant changes. On the other hand, movements in the pre-image leave the classifier invariant. If those movements induce a change in class as judged by the oracle, we call these invariance-based adversarial examples. In this case, however, the classifier is too insensitive to task-relevant changes. In conclusion, these two modes are complementary to each other, whereas both constitute failure modes of the learned classifier.
+
+When not restricting to $\epsilon$ -perturbations, perturbation-based and invariance-based adversarial examples yield the same input $x ^ { * }$ via
+
+$$
+\begin{array} { r l } & { x ^ { * } = x _ { 1 } + \Delta x _ { 1 } , \quad D ( x ^ { * } ) \neq D ( x _ { 1 } ) , \quad o ( x ^ { * } ) = o ( x _ { 1 } ) } \\ & { x ^ { * } = x _ { 2 } + \Delta x _ { 2 } , \quad D ( x ^ { * } ) = D ( x _ { 2 } ) , \quad o ( x ^ { * } ) \neq o ( x _ { 2 } ) , } \end{array}
+$$
+
+with different reference points $x _ { 1 }$ and $x _ { 2 }$ , see Figure 2. Hence, the key difference is the change of reference, which allows us to approach these failure modes from different directions. To connect these failure modes with an intuitive understanding of variations in the data, we now introduce the notion of invariance to nuisance and semantic variations, see also (Achille & Soatto, 2018).
+
+Definition 4 (Semantic/ Nuisance perturbation of an input). Let o be an oracle (Definition 2) and $\boldsymbol { x } \in \mathbb { R } ^ { d }$ . Then, a perturbation $\Delta x$ of an input $\boldsymbol { x } \in \mathbb { R } ^ { d }$ is called semantic, $i f o ( x ) \neq o ( x + \Delta x )$ and nuisance if $o ( x ) = o ( x + \Delta x )$ .
+
+For example, such a nuisance perturbation could be a translation or occlusion in image classification. Further in Appendix A, we discuss the synthetic example called Adversarial Spheres from (Gilmer et al., 2018b), where nuisance and semantics can be explicitly formalized as rotation and norm scaling.
+
+# 2.1 USING BIJECTIVE NETWORKS TO ANALYZE EXCESSIVE INVARIANCE
+
+As invariance-based adversarial examples manifest themselves in changes which do not affect the output of the network $F$ , we need a generic approach that gives us access to the discarded nuisance variability. While feature nuisances are intractable to access for general architectures (see comment after Definition 1), invertible classifiers only remove nuisance variability in their final projection (Jacobsen et al., 2018). For $C < d$ , we denote the classifier as $D : \mathbb { R } ^ { d }  \{ 1 , . . . , C \}$ . Our contributions in this section are: (1) Introduce an invertible architecture with a simplified readout structure, allowing to exactly visualize manipulations in the hidden-space, (2) Propose an analytic attack based on this architecture allowing to analyze its decision-making, (3) Reveal striking invariance-based vulnerability in competitive classifiers.
+
+Bijective classifiers with simplified readout. We build deep networks that give access to their decision space by removing the final linear mapping onto the class probes in invertible RevNet-classifiers and call these networks fully invertible RevNets. The fully invertible RevNet classifier can be written as $\begin{array} { r l } { D _ { \theta } } & { { } = } \end{array}$ arg $\mathrm { m a x } _ { k = 1 , \ldots , C }$ sof tmax $( F _ { \theta } ( x ) _ { k } )$ , where $F _ { \theta }$ represents the bijective network. We denote $z = F _ { \theta } ( x )$ , $z _ { s } = z _ { 1 , . . . , C }$ as the logits (semantic variables) and $z _ { n } = z _ { C + 1 , \dots , d }$ as the nuisance variables ( $z _ { n }$ is not used for classification). In practice we choose the first C indices of the final $z$ tensor or apply a more sophiscticated DCT scheme (see appendix D) to set the subspace $z _ { s }$ , but other choices work as well. The architecture of the network is similar to iRevNets (Jacobsen et al., 2018) with some additional Glow components like actnorm (Kingma & Dhariwal, 2018), squeezing, dimension splitting and affine block structure (Dinh et al., 2017), see Figure 3 for a graphical description. As all components are common in the bijective network literature, we refer the reader to Appendix D for exact training and architecture details. Due to its simple readout structure, the resulting invertible network allows to qualitatively and quantitatively investigate the task-specific content in nuisance and logit variables. Despite this restriction, we achieve performance on par with commonly-used baselines on MNIST and ImageNet, see Table 1 and Appendix D.
+
+![](images/f78c381fb42ee49eac102ae6637b3dd6749fe9bab74d5ab5f291c5398e0a2c9d.jpg)  
+Figure 3: The fully invertible RevNet, a hybrid of Glow and iRevNet with simple readout structure. $z _ { s }$ represents the logits and $z _ { n }$ the nuisance.
+
+Table 1: The table shows error rates on the ILSVRC-2012 validation set of our proposed fully invertible RevNet compared to a VGG (Simonyan & Zisserman, 2014) and two ResNet (He et al., 2016) variants, as well as an iRevNet (Jacobsen et al., 2018) with a non-invertible final projection onto the logits. Our proposed fully invertible RevNet performs roughly on par with others.   
+
+<table><tr><td>% Error</td><td>fi-RevNet48(Ours)</td><td>VGG19</td><td>ResNet18</td><td>ResNet50</td><td>iRevNet300</td></tr><tr><td>ILSVRC2012 Val Top1</td><td>29.50</td><td>28.70</td><td>30.43</td><td>24.70</td><td>26.70</td></tr><tr><td>ILSVRC2012 Val Top5</td><td>11.30</td><td>9.90</td><td>10.80</td><td>7.89</td><td>1</td></tr></table>
+
+Analytic attack. To analyze the trained models, we can sample elements from the logit pre-image by computing $x _ { m e t } = \dot { F } ^ { - 1 } ( z _ { s } , \tilde { z } _ { n } )$ , where $z _ { s }$ and $\tilde { z } _ { n }$ are taken from two different inputs. We term this heuristic metameric sampling. The samples would be from the true data distribution if the subspaces would be factorized as $P ( z _ { s } , z _ { n } ) = P ( z _ { s } ) P ( z _ { n } )$ . Experimentally we find that logit metamers are revealing adversarial subspaces and are visually close to natural images on ImageNet.
+
+![](images/312912477085d17d938475783720b40414291d57f774eebbb9a82a2e2680bf15.jpg)  
+Figure 4: Left: Decision-boundaries in 2D subspace spanned by two random data points $x _ { 1 } , x _ { 2 }$ . Right: Decision-boundaries in 2D subspace spanned by random datapoint $x$ and metamer $x _ { m e t }$ .
+
+Thus, metameric sampling gives us an analytic tool to inspect dependencies between semantic and nuisance variables without the need for expensive and approximate optimization procedures.
+
+Attack on adversarial spheres. First, we evaluate our analytic attack on the synthetic spheres dataset, where the task is to classify samples as belonging to one out of two spheres with different radii. We choose the sphere dimensionality to be $d = 1 0 0$ and the radii: $R _ { 1 } = 1$ , $R _ { 2 } = 1 0$ . By training a fully-connected fully invertible RevNet, we obtain $100 \%$ accuracy. After training we visualize the decision-boundaries of the original classifier $D$ and a posthoc trained classifier on $z _ { n }$ (nuisance classifier), see Figure 4. We densely sample points in a 2D subspace, following Gilmer et al. (2018b), to visualize two cases: 1) the decision-boundary on a 2D plane spanned by two randomly chosen data points, 2) the decision-boundary spanned by metameric sample $x _ { m e t }$ and reference point $x$ . In the metameric sample subspace we identify excessive invariance of the classifier. Here, it is possible to move any point from the inner sphere to the outer sphere without changing the classifiers predictions. However, this is not possible for the classifier trained on $z _ { n }$ . Most notably, the visualized failure is not due to a lack of data seen during training, but rather due to excessive invariance of the original classifier $D$ on $z _ { s }$ . Thus, the nuisance classifier on $z _ { n }$ does not exhibit the same adversarial vulnerability in its subspace.
+
+![](images/f5d93067addd2b7eec8ae5acd4eb220e3b2efaa5b441ac7a436dc0c871109964.jpg)  
+Figure 5: Each column shows three images belonging together. Top row are source images from which we sample the logits, middle row are logit metamers and bottom row images from which we sample the nuisances. Top row and middle row have the same (approximately for ResNets, exactly for fully invertible RevNets) logit activations. Thus, it is possible to change the image content completely without changing the 10- and 1000-dimensional logit vectors respectively. This highlights a striking failure of classifiers to capture all task-dependent variability.
+
+Attack on MNIST and ImageNet. After validating its potential to uncover adversarial subspaces, we apply metameric sampling to fully invertible RevNets trained on MNIST and Imagenet, see Figure 5. The result is striking, as the nuisance variables $z _ { n }$ are dominating the visual appearance of the logit metamers, making it possible to attach any semantic content to any logit activation pattern. Note that the entire 1000-dimensional feature vector containing probabilities over all ImageNet classses remains unchanged by any of the transformations we apply. To show our findings are not a particular property of bijective networks, we attack an ImageNet trained ResNet152 with a gradientbased version of our metameric attack, also known as feature adversaries (Sabour et al., 2016). The attack minimizes the mean squared error between a given set of logits from one image to another image (see appendix B for details). The attack shows the same failures for non-bijective models. This result highlights the general relevance of our finding and poses the question of the origin of this excessive invariance, which we will analyze in the following section.
+
+# 3 OVERCOMING INSUFFICIENCY OF CROSSENTROPY-BASED INFORMATION-MAXIMIZATION
+
+In this section we identify why the cross-entropy objective does not necessarily encourage to explain all task-dependent variations of the data and propose a way to fix this. As shown in figure 4, the nuisance classifier on $z _ { n }$ uses task-relevant information not captured by the logit classifier $D _ { \theta }$ on $z _ { s }$ (evident by its superior performance in the adversarial subspace).
+
+We leverage the simple readout-structure of our invertible network and turn this observation into a formal explanation framework using information theory: Let $( x , y ) \sim \mathcal { D }$ with labels $y \in \{ 0 , 1 \} ^ { C }$ . Then the goal of a classifier can be stated as maximizing the mutual information (Cover & Thomas, 2006) between semantic features $z _ { s }$ (logits) extracted by network $F _ { \theta }$ and labels $y$ , denoted by $I ( y ; z _ { s } )$ .
+
+Adversarial distribution shift. As the previously discussed failures required to modify input data from distribution $\mathcal { D }$ , we introduce the concept of an adversarial distribution shift $\mathcal { D } _ { A d v } \neq \mathcal { D }$ to formalize these modifications. Our first assumptions for $\mathcal { D } _ { A d v }$ is $I _ { \mathcal { D } _ { A d v } } ( z _ { n } ; y ) \ \le \ I _ { \mathcal { D } } ( z _ { n } ; y )$ . Intuitively, the nuisance variables $z _ { n }$ of our network do not become more informative about $y$ . Thus, the distribution shift may reduce the predictiveness of features encoded in $z _ { s }$ , but does not introduce or increase the predictive value of variations captured in $z _ { n }$ . Second, we assume $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } | z _ { n } ) \leq$ $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } )$ , which corresponds to positive or zero interaction information, see e.g. (Ghassami & Kiyavash, 2017). While the information in $z _ { s }$ and $z _ { n }$ can be redundant in this assumption, synergetic effects where conditioning on $z _ { n }$ increase the mutual information between $y$ and $z _ { s }$ are excluded.
+
+Bijective networks $F _ { \theta }$ capture all variations by design which translates to information preservation $I ( y ; x ) = I ( y ; F _ { \theta } ( x ) )$ , see (Kraskov et al., 2004). Consider the reformulation
+
+$$
+I ( y ; x ) = I ( y ; F _ { \theta } ( x ) ) = I ( y ; z _ { s } , z _ { n } ) = I ( y ; z _ { s } ) + I ( y ; z _ { n } | z _ { s } ) = I ( y ; z _ { n } ) + I ( y ; z _ { s } | z _ { n } )
+$$
+
+by the chain rule of mutual information (Cover & Thomas, 2006), where $I ( y ; z _ { n } | z _ { s } )$ denotes the conditional mutual information. Most strikingly, equation 5 offers two ways forward:
+
+1. Direct increase of $I ( y ; z _ { s } )$   
+2. Indirect increase of $I ( y ; z _ { s } | z _ { n } )$ via decreasing $I ( y ; z _ { n } )$ .
+
+Usually in a classification task, only $I ( y ; z _ { s } )$ is increased actively via training a classifier. While this approach is sufficient in most cases, expressed via high accuracies on training and test data, it may fail under $\mathcal { D } _ { A d v }$ . This highlights why cross-entropy training may not be sufficient to overcome excessive semantic invariance. However, by leveraging the bijection $F _ { \theta }$ we can minimize the unused information $I ( y ; z _ { n } )$ using the intuition of a nuisance classifier.
+
+Definition 5 (Independence cross-entropy loss). Let $F _ { \theta } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ a bijective network with parameters $\theta \in \mathbb { R } ^ { p _ { 1 } }$ and $\tilde { F } _ { \theta } ( x ) = s o f t m a x ( F _ { \theta } ( x ) _ { 1 , . . . , C } )$ . Furthermore, let $D _ { \theta _ { n c } } : \mathbb { R } ^ { d - C }  [ 0 , 1 ] ^ { C }$ be the nuisance classifier with $\theta _ { n c } \in \mathbb { R } ^ { p _ { 2 } }$ . Then, the independence cross-entropy loss is defined as:
+
+$$
+\operatorname* { m i n } _ { \theta } \operatorname* { m m a x } _ { \theta _ { n c } } \mathcal { L } _ { i C E } ( \theta , \theta _ { n c } ) = \underbrace { \sum _ { i = 1 } ^ { C } - y _ { i } \log \tilde { F } _ { \theta } ^ { z _ { s } } ( x ) } _ { = : \mathcal { L } _ { s C E } ( \theta ) } + \underbrace { \sum _ { i = 1 } ^ { C } y _ { i } \log D _ { \theta _ { n c } } ( F _ { \theta } ^ { z _ { n } } ( x ) ) _ { i } } _ { = : \mathcal { L } _ { n C E } ( \theta , \theta _ { n c } ) } .
+$$
+
+The underlying principles of the nuisance classification loss $\mathcal { L } _ { n C E }$ can be understood using a variational lower bound on mutual information from Barber $\&$ Agakov (2003). In summary, the minimization is with respect to a lower bound on $I _ { \mathcal { D } } ( y ; z _ { n } )$ , while the maximization aims to tighten the bound (see Lemma 10 in Appendix C). By using these results, we now state the main result under the assumed distribution shift and successful minimization (proof in Appendix C.1):
+
+Theorem 6 (Information $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } )$ maximal after distribution shift). Let $\mathcal { D } _ { A d v }$ denote the adversarial distribution and $\mathcal { D }$ the training distribution. Assume $I _ { \mathcal { D } } ( y ; z _ { n } ) = 0$ by minimizing $\mathcal { L } _ { i C E }$ and the distribution shift satisfies $I _ { \mathcal { D } _ { A d v } } ( z _ { n } ; y ) \le I _ { \mathcal { D } } ( z _ { n } ; y )$ and $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } | z _ { n } ) \le I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } )$ . Then,
+
+$$
+I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } ) = I _ { \mathcal { D } } ( y ; x ) .
+$$
+
+![](images/73d5fe6f47b49141239978bf86ee7d741b2c00e7f149f70f085c270429adb6af.jpg)  
+Figure 6: Left: Mutual information under distribution $\mathcal { D } _ { t r a i n }$ , Right: Effect of distributional shift to $\mathcal { D } _ { A d v }$ . Each case under training with cross-entropy (CE) and independence cross-entropy (iCE). Under distribution $\mathcal { D }$ , the iCE-loss minimizes $I ( y ; z _ { n } )$ (Lemma 10, Appendix C), but has no effect as the CE-loss already maximizes $I ( y ; z _ { s } )$ . However under the shift to $\mathcal { D } _ { A d v }$ , the information $I ( y ; z _ { s } )$ decreases when training only under the CE-loss (orange arrow), while the iCE-loss induces $I ( y ; z _ { n } ) = 0$ and thus leaves $I ( y ; z _ { s } )$ unchanged (Theorem 6).
+
+Thus, incorporating the nuisance classifier allows for the discussed indirect increase of $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } )$ under an adversarial distribution shift, visualized in Figure 6.
+
+To aid stability and further encourage factorization of $z _ { s }$ and $z _ { n }$ in practice, we add a maximum likelihood term to our independence cross-entropy objective as
+
+$$
+\operatorname* { m i n } _ { \theta } \operatorname* { m a x } _ { \theta _ { n c } } \mathcal { L } ( \theta , \theta _ { n c } ) = \mathcal { L } _ { i C E } ( \theta , \theta _ { n c } ) - \underbrace { \sum _ { k = 1 } ^ { d - C } \log \big ( p _ { k } ( F _ { \theta } ^ { z _ { n } } ( x ) _ { k } ) | \mathsf { d e t } ( J _ { \theta } ^ { x } ) | \big ) } _ { = : \mathcal { L } _ { M L E _ { n } } ( \theta ) } ,
+$$
+
+where $\operatorname* { d e t } ( J _ { \theta } ^ { x } )$ denotes the determinant of the Jacobian of $F _ { \theta } ( x )$ and $p _ { k } \sim \mathcal N ( \beta _ { k } , \gamma _ { k } )$ with $\beta _ { k } , \gamma _ { k }$ learned parameter. The log-determinant can be computed exactly in our model with negligible additional cost. Note, that optimizing $\mathcal { L } _ { M L E _ { n } }$ on the nuisance variables together with $\mathcal { L } _ { s C E }$ amounts to maximum-likelihood under a factorial prior (see Lemma 11 in Appendix C).
+
+Just as in GANs the quality of the result relies on a tight bound provided by the nuisance classifier and convergence of the MLE term. Thus, it is important to analyze the success of the objective after training. We do this by applying our metameric sampling attack, but there are also other ways like evaluating a more powerful nuisance classifier after training.
+
+# 4 APPLYING INDEPENDENCE CROSS-ENTROPY
+
+In this section, we show that our proposed independence cross-entropy loss is effective in reducing invariance-based vulnerability in practice by comparing it to vanilla cross-entropy training in four aspects: (1) error on train and test set, (2) effect under distribution shift, perturbing nuisances via metameric sampling, (3) evaluate accuracy of a classifier on the nuisance variables to quantify the class-specific information in them and (4) on our newly introduced shiftMNIST, an augmented version of MNIST to benchmark adversarial distribution shifts according to Theorem 6.
+
+For all experiments we use the same network architecture and settings, the only difference being the two additional loss terms as explained in Definition 5 and equation 6. In terms of test error of the logit classifier, both losses perform approximately on par, whereas the gap between train and test error vanishes for our proposed loss function, indicating less overfitting. For classification errors see Table 2 in appendix D.
+
+Robustness under metameric sampling attack. To analyze if our proposed loss indeed leads to independence between $z _ { n }$ and labels $y$ , we attack it with our metameric sampling procedure. As we are only looking on data samples and not on samples from the model (factorized gaussian on nuisances), this attack should reveal if the network learned to trick the objective. In Figure 7 we show interpolations between original images and logit metamers in CE- and iCE-trained fully invertible RevNets. In particular, we are holding the activations $z _ { s }$ constant, while linearly interpolating nuisances $z _ { n }$ down the column. The CE-trained network allows us to transform any image into any class without changing the logits. However, when training with our proposed iCE, the picture changes fundamentally and interpolations in the pre-image only change the style of a digit, but not its semantic content. This shows our loss has the ability to overcome excessive task-related invariance and encourages the model to explain and separate all task-related variability of the input from the nuisances of the task.
+
+![](images/2878ac37ae1f52e0860ae2bbe9863a4363816bf0b1fe4fc1fe72b942ed272dab.jpg)  
+Figure 7: Samples $\tilde { x } = F ^ { - 1 } ( z _ { s } , \tilde { z } _ { n } )$ with logit activations $z _ { s }$ taken from original image and $\tilde { z } _ { n }$ obtained by linearly interpolating from the original nuisance $z _ { n }$ (first row) to the nuisance of a target example $z _ { n } ^ { * }$ (last row upper block). The used target example is shown at the bottom. When training with cross-entropy, virtually any image can be turned into any class without changing the logits $z _ { s }$ , illustrating strong vulnerability to invariance-based adversaries. Yet, training with independence cross-entropy solves the problem and interpolations between nuisances $z _ { n }$ and $z _ { n } ^ { * }$ preserve the semantic content of the image.
+
+A classifier trained on the nuisance variables of the cross-entropy trained model performs even better than the logit classifier. Yet, a classifier on the nuisances of the independence cross-entropy trained model is performing poorly (Table 2 in appendix D). This indicates little class-specific information in the nuisances $z _ { n }$ , as intended by our objective function. Note also that this inability of the nuisance classifier to decode class-specific information is not due to it being hard to read out from $z _ { n }$ , as this would be revealed by the metameric sampling attack (see Figure 7).
+
+![](images/9878f27952dcfb71dbf48a4f121956a443404c4ecba376079d04801f339ac19e.jpg)  
+Figure 8: shiftMNIST experiments. (a): Binary shiftMNIST, where the class is additionally encoded with a location-based binary code on the left border of the image (highlighted with red circles). The shifted adversarial test distribution does not have the binary class encoding. (b): Texture shiftMNIST, where the class is additionally encoded in background texture type. The texture-class coupling is randomized in the shifted adversarial test distribution. Right: Results of CE-trained ResNet, fully invertible RevNet and iCE-trained fully invertible RevNet. The CE-based models build excessive invariance with respect to the digit identity on $\mathcal { D } _ { t r a i n }$ and fail on $\mathcal { D } _ { A d v }$ . Difference denotes the largest improvement between CE-trained and iCE-trained model. The iCE model is more resilient to removing informative features, and reduces the error on $\mathcal { D } _ { A d v }$ up to $38 \%$ .
+
+shiftMNIST: Benchmarking adversarial distribution shift. To further test the efficacy of our proposed independence cross-entropy, we introduce a simple, but challenging new dataset termed shiftMNIST to test classifiers under adversarial distribution shifts $\mathcal { D } _ { A d v }$ . The dataset is based on vanilla MNIST, augmented by introducing additional, highly predictive features at train time that are randomized or removed at test time. Randomization or removal ensures that there are no synergy effects between digits and planted features under $\mathcal { D } _ { A d v }$ . This setup allows us to reduce mutual information between category and the newly introduced feature in a targeted manner. (a) Binary shiftMNIST is vanilla MNIST augmented by coding the category for each digit into a single binary pixel scheme. The location of the binary pixel reveals the category of each image unambigiously, while only minimally altering the image’s appearance.
+
+At test time, the binary code is not present and the network can not rely on it anymore. (b) Textured shiftMNIST introduces textured backgrounds for each digit category which are patches sampled from the describable texture dataset (Cimpoi et al., 2014).
+
+At train time the same type of texture is underlayed each digit of the same category, while texture types across categories differ. At test time, the relationship is broken and texture backgrounds are paired with digits randomly, again minimizing the mutual information between background and label in a targeted manner. See Figure 8 for examples1.
+
+It turns out that this task is indeed very hard for standard classifiers and their tendency to become excessively invariant to semantically meaningful features, as predicted by our theoretical analysis. When trained with cross-entropy, ResNets and fi-RevNets make zero errors on the train set, while having error rates of up to $87 \%$ on the shifted test set. This is striking, given that e.g. in binary shiftMNIST, only one single pixel is removed under $\mathcal { D } _ { A d v }$ , leaving the whole image almost unchanged. When applying our independence cross-entropy, the picture changes again. The errors made by the network improve by up to almost $38 \%$ on binary shiftMNIST and around $28 \%$ on textured shiftMNIST. This highlights the effectiveness of our proposed loss function and its ability to minimize catastrophic failure under severe distribution shifts exploiting excessive invariance.
+
+# 5 RELATED WORK
+
+Adversarial examples. Adversarial examples often include $\epsilon$ -norm restrictions (Szegedy et al., 2013), while (Gilmer et al., 2018a) argue for a broader definition to fully capture the implications for security. The $\epsilon$ -adversarial examples have also been extended to $\epsilon$ -feature adversaries (Sabour et al., 2016), which are equivalent to our approximate metameric sampling attack. Some works (Song et al., 2018; Fawzi et al., 2018) consider unrestricted adversarial examples, which are closely related to invariance-based adversarial vulnerability. The difference to human perception revealed by adversarial examples fundamentally questions which statistics deep networks use to base their decisions (Jo & Bengio, 2017; Tsipras et al., 2019).
+
+Relationship between standard and bijective networks. We leverage recent advances in reversible (Gomez et al., 2017) and bijective networks (Jacobsen et al., 2018; Ardizzone et al., 2019; Kingma & Dhariwal, 2018) for our analysis. It has been shown that ResNets and iRevNets behave similarly on various levels of their representation on challenging tasks (Jacobsen et al., 2018) and that iRevNets as well as Glow-type networks are related to ResNets by the choice of dimension splitting applied in their residual blocks (Grathwohl et al., 2019). Perhaps unsurprisingly, given so many similarities, ResNets themselves have been shown to be provably bijective under mild conditions (Behrmann et al., 2018b). Further, excessive invariance of the type we discuss here has been shown to occur in non residual-type architectures as well (Gilmer et al., 2018b; Behrmann et al., 2018a). For instance, it has been observed that up to $60 \%$ of semantically meaningful input dimensions on the adversarial spheres problem are learned to be ignored, while retaining virtually perfect performance (Gilmer et al., 2018b). In summary, there is ample evidence that RevNet-type networks are closely related to ResNets, while providing a principled framework to study widely observed issues related to excessive invariance in deep learning in general and adversarial robustness in particular.
+
+Information theory. The information-theoretic view has gained recent interest in machine learning due to the information bottleneck (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017; Alemi et al., 2017) and usage in generative modelling (Chen et al., 2016; Hjelm et al., 2019). As a consequence, the estimation of mutual information (Barber & Agakov, 2003; Alemi et al., 2018; Achille & Soatto, 2018; Belghazi et al., 2018) has attracted growing attention. The concept of group-wise independence between latent variables goes back to classical independent subspace analysis (Hyvarinen ¨ & Hoyer, 2000) and received attention in learning unbiased representations, e.g. see the Fair Variational Autoencoder (Louizos et al., 2015). Furthermore, extended cross-entropy losses via entropy terms (Pereyra et al., 2017) or minimizing predictability of variables (Schmidhuber, 1991) has been introduced for other applications. Our proposed loss also shows similarity to the GAN loss (Goodfellow et al., 2014). However, in our case there is no notion of real or fake samples, but exploring similarities in the optimization are a promising avenue for future work.
+
+# 6 CONCLUSION
+
+Failures of deep networks under distribution shift and their difficulty in out-of-distribution generalization are prime examples of the limitations in current machine learning models. The field of adversarial example research aims to close this gap from a robustness point of view. While a lot of work has studied $\epsilon$ -adversarial examples, recent trends extend the efforts towards the unrestricted case. However, adversarial examples with no restriction are hard to formalize beyond testing error. We introduce a reverse view on the problem to: (1) show that a major cause for adversarial vulnerability is excessive invariance to semantically meaningful variations, (2) demonstrate that this issue persists across tasks and architectures; and (3) make the control of invariance tractable via fully-invertible networks.
+
+In summary, we demonstrated how a bijective network architecture enables us to identify large adversarial subspaces on multiple datasets like the adversarial spheres, MNIST and ImageNet. Afterwards, we formalized the distribution shifts causing such undesirable behavior via information theory. Using this framework, we find one of the major reasons is the insufficiency of the vanilla cross-entropy loss to learn semantic representations that capture all task-dependent variations in the input. We extend the loss function by components that explicitly encourage a split between semantically meaningful and nuisance features. Finally, we empirically show that this split can remove unwanted invariances by performing a set of targeted invariance-based distribution shift experiments.
+
+# 7 ACKNOWLEDGEMENTS
+
+We thank Ryota Tomioka for spotting a mistake in the proof for Theorem 6. We thank thank the anonymous reviewers, Ricky Chen, Will Grathwohl and Jesse Bettencourt for helpful comments on the manuscript. We gratefully acknowledge the financial support from the German Science Foundation for the CRC 1233 on ”Robust Vision” and RTG $2 2 2 4 \cdots 3$ : Parameter Identification - Analysis, Algorithms, Applications”
+
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+# A SEMANTIC AND NUISANCE VARIATION ON ADVERSARIAL SPHERES
+
+Example 7 (Semantic and nuisance on Adversarial Spheres (Gilmer et al., 2018b)). Consider classifying inputs $x$ from two classes given by radii $R _ { 1 }$ or $R _ { 2 }$ . Further, let $( r , \phi )$ denote the spherical coordinates of $x$ . Then, any perturbation $\Delta x _ { \mathrm { { \ell } } }$ , $x ^ { * } = x + \Delta x$ with $r ^ { * } \neq r$ is semantic. On the other hand, $i f r ^ { * } = r$ the perturbation is a nuisance with respect to the task of discriminating two spheres.
+
+In this example, the max-margin classifier $\begin{array} { r } { D ( x ) = s i g n \left( \| x \| - \frac { R _ { 1 } + R _ { 2 } } { 2 } \right) } \end{array}$ is invariant to any nuisance perturbation, while being only sensitive to semantic perturbations. In summary, the transform to spherical coordinates allows to linearize semantic and nuisance perturbations. Using this notion, invariance-based adversarial examples can be attributed to perturbations of $x ^ { * } = x + \Delta x$ with following two properties
+
+1. Perturbed sample $x ^ { * }$ stays in the pre-image $\{ x ^ { * } \in \mathbb { R } ^ { d } \mid D ( x ^ { * } ) = D ( x ) \}$ of the classifier
+
+2. Perturbation $\Delta x$ is semantic, as $o ( x ) \neq o ( x + \Delta x )$ .
+
+Thus, the failure of the classifier $D$ can be thought of a mis-alignment between its invariance (expressed through the pre-image) and the semantics of the data and task (expressed by the oracle).
+
+Example 8 (Mis-aligned classifier on Adversarial Spheres). Consider the classifier
+
+$$
+D ( x ) = s i g n \left( \left\| x _ { 1 , \ldots , d - 1 } \right\| - { \frac { R _ { 1 } + R _ { 2 } } { 2 } } \right) ,
+$$
+
+which computes the norm of $x$ from its first $d - 1$ cartesian-coordinates. Then, $D$ is invariant to a semantic perturbation with $\Delta r = R _ { 2 } - R _ { 1 }$ if only changes in the last coordinate $x _ { d }$ are made.
+
+We empirically evaluate the classifier in equation 7 on the spheres problem (10M/2M samples setting (Gilmer et al., 2018b)) and validate that it can reach perfect classification accuracy. However, by construction, perturbing the invariant dimension $x _ { d } ^ { * } = x _ { d } + \Delta x _ { d }$ allows us to move all samples from the inner sphere to the outer sphere. Thus, the accuracy of the classifier drops to chance level when evaluating its performance under such a distributional shift.
+
+To conclude, this underlines how classifiers with optimal performance on finite samples can exhibit non-intuitive failure modes due to excessive invariance with respect to semantic variations.
+
+# B APPROXIMATE GRADIENT-BASED METAMERIC SAMPLES
+
+We use a standard Imagenet pre-trained Resnet-154 as provided by the torchvision package (Paszke et al., 2017) and choose a logit percept $\mathbf { y } = G ( \mathbf { x } )$ that can be based on any seed image. Then we optimize various images $\tilde { x }$ to be metameric to $\mathbf { x }$ by simply minimizing a mean squared error loss of the form:
+
+$$
+\mathcal { L } _ { \mathrm { M S E } } ( G ( x ) , G ( \tilde { x } ) ) = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } ( G ( x ) _ { k } - G ( \tilde { x } ) _ { k } ) ^ { 2 }
+$$
+
+in the 1000-dimensional semantic logit space via stochastic gradient descent. We optimize with Adam in Pytorch default settings and a learning rate of 0.01 for 3000 iterations. The optimization thus takes the form of an adversarial attack targeting all logit entries and with no norm restriction on the input distance. Note that our metameric sampling attack in bijective networks is the analytic reverse equivalent of this attack. It leads to the exact solution at the cost of one inverse pass instead of an approximate solution here at the cost of thousands of gradient steps.
+
+![](images/12e7f16bcbe83dd180d562b2e7c487f38ad425681ed46752814169a18d94d312.jpg)  
+Figure 9: Here we show a batch of randomly sampled metamers from our ImageNet-trained fully invertible RevNet-48. The quality is generally similar, sometimes colored artifacts appear.
+
+# C INFORMATION THEORY
+
+Computing mutual information is often intractable as it requires the joint probability $p ( x , y )$ , see (Cover & Thomas, 2006) for an extensive treatment of information theory. However, following variational lower bound can be used for approximation, see (Barber & Agakov, 2003).
+
+Lemma 9 (Variational lower bound on mutual information). Let $X , Y$ be random variables with conditional density $p ( y | x )$ . Further, let $q _ { \theta } ( y | x )$ be a variational density depending on parameter $\theta$ . Then, the lower bound
+
+$$
+\begin{array} { r l } & { I ( Y ; X ) = h ( Y ) - h ( Y \vert X ) = h ( Y ) + \mathbb { E } _ { X } \mathbb { E } _ { Y \vert X } \log q _ { \theta } ( y \vert x ) + \mathbb { E } _ { X } ( p ( y \vert x ) \parallel q _ { \theta } ( y \vert x ) ) } \\ & { \qquad \ge h ( Y ) + \mathbb { E } _ { X } \mathbb { E } _ { Y \vert X } \log q _ { \theta } ( y \vert x ) } \end{array}
+$$
+
+holds with equality if $p ( y | x ) = q _ { \theta } ( y | x )$ .
+
+While above lower bound removes the need for the computation of $p ( y | x )$ , estimating the expectation $\mathbb { E } _ { Y \mid X }$ still requires sampling from it. Using this bound, we can now state the effect of the nuisance classifiation loss.
+
+Lemma 10 (Effect of nuisance classifier). Define semantics as $z _ { s } = F _ { \theta } ( x ) _ { 1 , . . . , C }$ and nuisances as $z _ { n } = F _ { \theta } ( x ) _ { C + 1 , \dots , d } ,$ , where $( x , y ) \sim \mathcal { D }$ . Then, the nuisance classification loss yields
+
+(i) Minimization of lower bound on $I _ { \mathcal { D } } ( y ; z _ { n } )$ : $\theta ^ { * } ~ = ~ \arg \operatorname* { m i n } _ { \theta } \mathcal { L } _ { n C E } ( \theta , \theta _ { n c } ^ { * } )$ minimizes $I _ { \theta _ { n c } ^ { * } } ( y ; z _ { n } )$ , where $I _ { \theta _ { n c } ^ { * } } ( y ; z _ { n } ) \le I _ { \mathcal { D } } ( y ; z _ { n } )$ and $\begin{array} { r } { \theta _ { n c } ^ { * } = \arg \operatorname* { m a x } _ { \theta _ { 2 } } \mathcal { L } _ { n C E } ( \theta , \theta _ { n c } ) } \end{array}$ .
+
+(ii) Maximization to tighten bound on $I _ { \mathcal { D } } ( y ; z _ { n } )$ : Under a perfect model of the conditional density, $D _ { \theta _ { n c } ^ { * } } ( z _ { n } ) \stackrel {  } { = } p ( y | z _ { n } ) .$ , it holds $I _ { \theta _ { n c } ^ { * } } ( y ; z _ { n } ) = I _ { \mathcal { D } } ( y ; z _ { n } )$ .
+
+Proof. To proof above result, we need to draw the connection to the variational lower bound on mutual information from Lemma 9. Let the nuisance classifier $D _ { \theta _ { n c } } ( z _ { n } )$ model the variational posterior $q _ { \theta _ { n c } } ( y | z _ { n } )$ . Then we have the lower bound
+
+$$
+I ( y ; z _ { n } ) \geq h ( y ) + \mathbb { E } _ { z _ { n } } \mathbb { E } _ { y | z _ { n } } \log D _ { \theta _ { n c } } ( z _ { n } ) = : I _ { \theta _ { n c } } ( y ; z _ { n } ) .
+$$
+
+From Lemma 9 follows, that if $D _ { \theta _ { n c } } ( z _ { n } ) = p ( y | z _ { n } )$ , it holds $I ( y ; z _ { n } ) = I _ { \theta _ { n c } } ( y ; z _ { n } )$ . Hence, the nuisance classifier needs to model the conditional density perfectly.
+
+Estimating this bound via Monte Carlo simulation requires sampling from the conditional density $p ( y | z _ { n } )$ . Following (Alemi et al., 2017), we have the Markov property $y  x  z _ { n }$ as labels $y$ interact with inputs $x$ and representation $z _ { n }$ interacts with inputs $x$ . Hence,
+
+$$
+\begin{array} { r l } {  { p ( y | z _ { n } ) p ( z _ { n } ) = p ( y , z _ { n } ) } } \\ & { = \int _ { \mathcal X } p ( x , y , z _ { n } ) d x } \\ & { = \int _ { \mathcal X } p ( z _ { n } | x , y ) p ( y | x ) p ( x ) d x } \\ & { = \displaystyle \int _ { \mathcal X } p ( z _ { n } | x ) p ( y | x ) p ( x ) d x } \\ & { = \mathbb E _ { \alpha } [ p ( z _ { n } | x ) p ( y | x ) ] . } \end{array}
+$$
+
+Including above and assuming $F _ { \theta } ( x ) = z _ { n }$ to be a deterministic function, we have
+
+$$
+\begin{array} { r } { \mathbb { E } _ { z _ { n } } \mathbb { E } _ { y | z _ { n } } \log D _ { \theta _ { n c } } ( z _ { n } ) = \mathbb { E } _ { x } \mathbb { E } _ { y | x } \mathbb { E } _ { z _ { n } | x } \log D _ { \theta _ { n c } } ( z _ { n } ) = \mathbb { E } _ { x } \mathbb { E } _ { y | x } \log D _ { \theta _ { n c } } ( z _ { n } ) . } \end{array}
+$$
+
+Lemma 11 (Effect of MLE-term). Define semantics as $z _ { s } = F _ { \theta } ( x ) _ { 1 , . . . , C }$ and nuisances as $z _ { n } =$ $F _ { \theta } ( x ) _ { C + 1 , . . . , d }$ , where $( x , y ) \sim \mathcal { D }$ . Then, the MLE-term in equation $6$ together with cross-entropy on the semantics
+
+$$
+\theta ^ { * } = \arg \operatorname* { m i n } _ { \theta } \mathcal { L } _ { s C E } ( \theta ) + \mathcal { L } _ { M L E _ { n } } ( \theta )
+$$
+
+minimizes the mutual information $I ( z _ { s } ; z _ { n } )$
+
+Proof. Let $\tilde { z } _ { s } = s o f t m a x ( z _ { s } )$ . Then minimizing the loss terms $\mathcal { L } _ { s C E }$ and $ { \mathcal { L } } _ { M L E _ { n } }$ is a maximum likelihood estimation under the factorial prior
+
+$$
+\begin{array} { c } { { p ( \tilde { z } _ { s } , z _ { n } ) = p ( \tilde { z } _ { s } ) p ( z _ { n } ) } } \\ { { \ } } \\ { { = C a t ( ( \tilde { z } _ { s } ) _ { 1 } , \ldots , ( \tilde { z } _ { s } ) _ { C } ) \displaystyle \prod _ { k = 1 } ^ { d - C } p _ { k } ( z _ { n } ) _ { k } , } } \end{array}
+$$
+
+where $C a t$ is a categorical distribution. As sof tmax is shift-invariant, $s o f t m a x ( x + c ) \ =$ $s o f t m a x ( x )$ , above factorial prior for $\tilde { z } _ { s }$ and $z _ { n }$ yields independence between logits $z _ { s }$ and $z _ { n }$ up to a constant $c$ . Finally note, the log term and summation in $\mathcal { L } _ { M L E _ { n } }$ and $\mathcal { L } _ { C E }$ is re-formulation for computational ease but does not change its minimizer as the logarithm is monotone. □
+
+# C.1 PROOF OF THEOREM 6
+
+From the assumptions follows $I _ { \mathcal { D } _ { A d v } } ( y ; z _ { n } ) = 0$ . Furthermore, we have the assumption
+
+$$
+I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } | z _ { n } ) \leq I _ { \mathcal { D } _ { A d v } } ( z _ { s } ; y ) ,
+$$
+
+excluding synergetic effects in the interaction information (Ghassami & Kiyavash, 2017). By information preservation under homeomorphisms (Kraskov et al., 2004) and the chain rule of mutual information (Cover & Thomas, 2006), we have
+
+$$
+\begin{array} { r l } & { I _ { \mathcal { D } _ { A d v } } ( y ; x ) = I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } , z _ { n } ) } \\ & { \qquad = I _ { \mathcal { D } _ { A d v } } ( y ; z _ { n } ) + I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } | z _ { n } ) } \\ & { \qquad \leq I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } ) . } \end{array}
+$$
+
+As $z _ { s } = F ( x ) _ { 1 , . . . , C }$ is obtained by the deterministic transform $F$ , by the data processing inequality (Cover & Thomas, 2006) we have the inequality $I _ { \mathcal { D } _ { A d v } } ( y ; x ) \ge I _ { \mathcal { D } _ { A d v } } ( y ; z _ { s } )$ . Thus, the claimed equality must hold.
+
+# C.2 MUTUAL INFORMATION BOUNDED
+
+Remark 12. Since our goal is to maximize the mutual information $I ( y ; z _ { s } )$ while minimizing $I ( y ; z _ { n } )$ , we need to ensure that this objective is well defined as mutual information can be unbounded from above for continuous random variables. However, due to the data processing inequality (Cover & Thomas, 2006) we have $I ( y ; z _ { n } ) = I ( y ; F _ { \theta } ( x ) ) \le I ( y ; x )$ . Hence, we have a fixed upper bound given by our data $( x , y )$ . Compared to (Belghazi et al., 2018) there is thus no need for gradient clipping or a switch to the bounded Jensen-Shannon divergence as in (Hjelm et al., 2019) is not necessary.
+
+# D TRAINING AND ARCHITECTURAL DETAILS
+
+All experiments were based on a fully invertible RevNet model with different hyperparameters for each dataset. For the spheres experiment we used Pytorch (Paszke et al., 2017) and for MNIST, as well as Imagenet Tensorflow (Abadi et al., 2016).
+
+# D.1 SPHERES EXPERIMENTS
+
+The network is a fully connected fully invertible RevNet. It has 4 RevNet-type ReLU bottleneck blocks with additive couplings and uses no batchnorm. We train it via cross-entropy and use the Adam optimizer (Kingma & Ba, 2014) with a learning rate of 0.0001 and otherwise default Pytorch settings. The nuisance classifier is a 3 layer ReLU network with 1000 hidden units per layer.
+
+We choose the spheres to be 100-dimensional, with $R _ { 1 } = 1$ and $R _ { 2 } = 1 0$ , train on $5 0 0 \mathrm { k }$ samples for 10 epochs and then validate on another $1 0 0 \mathrm { k }$ holdout set. We achieve $100 \%$ train and validation accuracy for logit and nuisance classifier.
+
+# D.2 MNIST EXPERIMENTS
+
+We use a convolutional fully invertible RevNet with additional actnorm and invertible 1x1 convolutions between each layer as introduced in Kingma & Dhariwal (2018). The network has 3 stages, after which half of the variables are factored out and an invertible downsampling, or squeezing (Dinh et al., 2017; Jacobsen et al., 2018) is applied. The network has 16 RevNet blocks with batch norm per stage and 128 filters per layer. We also dequantize the inputs as is typically done in flow-based generative models.
+
+The network is trained via Adamax (Kingma & Ba, 2014) with a base learning rate of 0.001 for 100 epochs and we multiply the it with a factor of 0.2 every 30 epochs and use a batch size of 64 and l2 weight decay of 1e-4. For training we compare vanilla cross-entropy training with our proposed independence cross-entropy loss. To have a more balanced loss signal, we normalize $\mathcal { L } _ { n C E }$ by the number of input dimensions it receives for the maximization step. The nuisance classifier is a fullyconnected 3 layer ReLU network with 512 units. As data-augmentation we use random shifts of 3 pixels. For classification errors of the different architectures we compare, see Table 2.
+
+# D.3 IMAGENET EXPERIMENTS
+
+We use a convolutional fully invertible RevNet with 4 stages, 4 RevNet blocks per stage and invertible downsampling after each stage, as well as two invertible downsamplings on the input of the network. The first three stages consist of additive and the last of affine coupling layers. After the final layer we apply an orthogonal 2D DCT type-II to all feature maps and read out the classes in the low-pass components of the transformation. This effectively gives us an invertible global average pooling and makes our network even more similar to ResNets, that always apply global average pooling on their final feature maps. We train the network with momentum SGD for 128 epochs, a batch size of 480 (distributed to 6 GPUs), a base learning rate of 0.1, which is reduced by a factor of 0.1 every 32 epochs. We apply momentum of 0.9 and l2 weight decay of 1e-4.
+
+Table 2: Results comparing cross-entropy training (CE) with independence cross-entropy training (iCE) from Definition 5 and two architectures from the literature. The accuracy of the logit classifiers is on par for the CE and iCE networks, but the train error is higher for CE compared to test error, indicating less overfitting for iCE. Further, a classifier independently trained on the nuisance variables is able to reach even smaller error than on the logits for CE, but just $2 7 . 7 0 \%$ error for iCE, indicating that we have successfully removed most of the information of the label from the nuisance variables and fixed the problem of excessive invariance to semantically meaningful variability with no cost in test error.   
+
+<table><tr><td>MNIST</td><td>SOTA</td><td>LeNet</td><td>CE</td><td>iCE (ours)</td><td>CE</td><td>iCE (ours)</td></tr><tr><td>Readout</td><td>Logit</td><td>Logit</td><td>Logit</td><td>Logit</td><td>Nuisance</td><td>Nuisance</td></tr><tr><td>% Test Error</td><td>0.21</td><td>1.70</td><td>0.39</td><td>0.38</td><td>0.34</td><td>27.70</td></tr><tr><td>% Train Error</td><td>1</td><td>1</td><td>0.00</td><td>0.37</td><td>0.00</td><td>40.21</td></tr></table>

parse/train/DAaaaqPv9-q/DAaaaqPv9-q.md

@@ -0,0 +1,484 @@
+# SELF-SUPERVISED GRAPH-LEVEL REPRESENTATION LEARNING WITH LOCAL AND GLOBAL STRUCTURE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper focuses on unsupervised/self-supervised whole-graph representation learning, which is critical in many tasks including drug and material discovery. Current methods can effectively model the local structure between different graph instances, but they fail to discover the global semantic structure of the entire dataset. In this work, we propose a unified framework called Local-instance and Global-semantic Learning (GraphLoG) for self-supervised whole-graph representation learning. Specifically, besides preserving the local instance-level structure, GraphLoG leverages a nonparametric strategy to learn hierarchical prototypes of the data. These prototypes capture the semantic clusters in the latent space, and the number of prototypes can automatically adapt to different feature distributions. We evaluate GraphLoG by pre-training it on massive unlabeled graphs followed by fine-tuning on downstream tasks. Extensive experiments on both chemical and biological benchmark datasets demonstrate the effectiveness of our approach.
+
+# 1 INTRODUCTION
+
+Learning informative representations of whole graphs is a fundamental problem in a variety of domains and tasks, such as molecule properties prediction in drug and material discovery (Gilmer et al., 2017; Wu et al., 2018), protein function forecast in biological networks (Alvarez & Yan, 2012; Jiang et al., 2017), and predicting the properties of circuits in circuit design (Zhang et al., 2019). Recently, Graph Neural Networks (GNNs) have attracted a surge of interest and showed the effectiveness in learning graph representations. These methods are usually trained in a supervised fashion, which requires a large number of labeled data. Nevertheless, in many scientific domains, labeled data are very limited and expensive to obtain. Therefore, it is becoming increasingly important to learn the representations of graphs in an unsupervised or self-supervised fashion.
+
+Self-supervised learning has recently achieved profound success for both natural language processing, e.g. GPT (Radford et al., 2018) and BERT (Devlin et al., 2019), and image understanding, e.g. MoCo (He et al., 2019) and SimCLR (Chen et al., 2020). However, how to effectively learn the representations of graphs in a self-supervised way is still an open problem. Intuitively, a desirable graph representation should be able to preserve the local-instance structure, so that similar graphs are embedded close to each other and dissimilar ones stay far apart. In addition, the representations of a set of graphs should also reflect the global-semantic structure of the data, so that the graphs with similar semantic properties are compactly embedded, which benefits various downstream tasks, e.g. graph classification or regression. Such structure can be sufficiently captured by semantic clusters (Caron et al., 2018; Ji et al., 2019), especially in a hierarchical fashion (Li et al., 2020).
+
+There are some recent works that learn graph representation in a self-supervised manner, such as local-global mutual information maximization (Velickovic et al., 2019; Sun et al., 2019), structuralsimilarity/context prediction (Navarin et al., 2018; Hu et al., 2019; You et al., 2020) and contrastive multi-view learning (Hassani & Ahmadi, 2020). However, all these methods are capable of modeling only the local structure between different graph instances but fail to discover the global-semantic structure. To address this shortcoming, we are seeking for an approach that is sufficient to model both the local and global structure of a given set of graphs.
+
+To attain this goal, we propose a Local-instance and Global-semantic Learning (GraphLoG) framework for self-supervised graph representation learning. In specific, for preserving the local similarity between various graph instances, we first align the embeddings of correlated graphs/subgraphs1 by maximizing their mutual information. In this locally smooth embedding space, we further represent the distribution of different graph embeddings with hierarchical prototypes2 whose number is adaptively determined by the data in a nonparametric fashion. During training, these prototypes guide each graph to map to the semantically-similar feature cluster, and, simultaneously, the prototypes are maintained by online-updated graph embeddings. In this process, the global-semantic structure of the data is gradually discovered and refined. The whole model is pre-trained with a large number of unlabeled graphs, and then fine-tuned and evaluated on some downstream tasks.
+
+![](images/ae4c7de28b439b080f04652edbc11df9c28d3e311375d558cf580690687cd5fd.jpg)  
+Figure 1: Illustration of GraphLoG. (a) Correlated graphs are constrained to be adjacently embedded to pursue the local-instance structure of the data. (b) Hierarchical prototypes are employed to discover and refine the global-semantic structure of the data.
+
+We summarize our contributions as follows:
+
+• We contribute a unified framework called Local-instance and Global-semantic Learning (GraphLoG) for self-supervised graph representation learning, which is able to model the structure of a set of graphs both locally and globally.   
+We novelly propose to infer the global-semantic structure underlying the unlabeled graphs by learning hierarchical prototypes via a nonparametric strategy.   
+We empirically verify our framework’s superior performance on different GNN architectures through pre-training on a large-scale unlabeled dataset and fine-tuning on benchmark tasks in both the chemistry and biology domains.
+
+# 2 PROBLEM DEFINITION AND PRELIMINARIES
+
+# 2.1 PROBLEM DEFINITION
+
+An ideal representation should preserve the local structure among the data instances. More specifically, we define it as follows:
+
+Definition 1 (Local-instance Structure). The local-instance structure refers to the local pairwise similarity between different instances (Roweis & Saul, 2000; Belkin & Niyogi, 2002). To preserve the local-instance structure of graph-structured data, a pair of similar graphs/subgraphs, $\mathcal { G }$ and $\mathcal { G } ^ { \prime }$ , are expected to be mapped to the nearby positions of embedding space, as illustrated in Fig. 1(a), while the dissimilar pairs should be mapped to far apart.
+
+The pursuit of local-instance structure is usually insufficient to capture the semantics underlying the entire dataset. It is therefore important to discover the global-semantic structure of the data, which is concretely defined as follows:
+
+Definition 2 (Global-semantic Structure). A real-world dataset is usually distributed as different semantic clusters (Furnas et al., 2017; Ji et al., 2019). Therefore, we define the global-semantic structure of a dataset as the distribution of its semantic clusters, and each cluster is represented by a prototype (i.e. a representative cluster embedding). Since the semantics of a set of graphs can be structured in a hierarchical way (Ashburner et al., 2000; Chen et al., 2012), we represent the whole dataset with hierarchical prototypes. A detailed example can be seen in Fig. 1(b).
+
+Problem Definition. For self-supervised graph representation learning, a set of unlabeled graphs $\mathbb { G } = \{ \mathcal { G } _ { 1 } , \mathcal { G } _ { 2 } , \cdot \cdot \cdot , \mathcal { G } _ { M } \}$ are given, and we aim to learn a low-dimensional vector $h _ { \mathcal { G } _ { i } } \ \in \ \bar { \mathbb { R } } ^ { \delta }$ for each graph $\mathcal { G } _ { i } \in \mathbb { G }$ under the guidance of the data itself. In specific, we expect the derived graph embeddings $\dot { \mathbf { H } } \in \mathbb { R } ^ { M \times \delta }$ follow both the local-instance and global-semantic structure.
+
+# 2.2 PRELIMINARIES
+
+Graph Neural Networks (GNNs). Given a graph $\mathcal { G } = ( \nu , \mathcal { E } )$ with node attributes $X _ { \mathcal { V } } = \{ X _ { v } | v \in$ $\nu \}$ and edge attributes $X _ { \mathcal { E } } = \{ X _ { u v } | ( u , v ) \in \mathcal { E } \}$ , a GNN aims to learn an embedding vector $h _ { v }$ for each node $v \in \nu$ and also a vector $h _ { \mathcal { G } }$ for the entire graph $\mathcal { G }$ . For an $L$ -layer GNN, a neighborhood aggregation scheme is performed to capture the $L$ -hop information surrounding each node. The $l$ -th layer of a GNN can be formalized as follows:
+
+$$
+h _ { v } ^ { ( l ) } = \mathbf { C O M B I N E } ^ { ( l ) } \Big ( h _ { v } ^ { ( l - 1 ) } , \mathbf { A G G R E G A T E } ^ { ( l ) } \big ( \Big \{ \Big ( h _ { v } ^ { ( l - 1 ) } , h _ { u } ^ { ( l - 1 ) } , X _ { u v } \Big ) : u \in \mathcal { N } ( v ) \Big \} \Big ) \Big ) ,
+$$
+
+where $\mathcal { N } ( v )$ is the neighborhood set of $v$ , $h _ { v } ^ { ( l ) }$ denotes the representation of node $v$ at the $l$ -th layer, and h(0)v i s initialized as the node attribute $X _ { v }$ . Since $h _ { v }$ summarizes the information of a patch centered around node $v$ , we will refer to $h _ { v }$ as patch embedding to underscore this point. The entire graph’s embedding can be derived by a permutation-invariant readout function:
+
+$$
+h _ { \mathcal { G } } = \mathrm { R E A D O U T } \big ( \big \{ h _ { v } | v \in \mathcal { V } \big \} \big ) .
+$$
+
+Mutual Information Estimation. Mutual information (MI) can measure both the linear and nonlinear dependency between two random variables. Some recent works (Belghazi et al., 2018; Hjelm et al., 2019) employed neural networks to estimate the lower bound of MI. Among which, InfoNCE loss (van den Oord et al., 2018) has been introduced to maximize a lower bound of MI by minimizing itself, and we also adopt it in this work for its simplicity and effectiveness. In practice, given a query $q$ , InfoNCE loss is optimized to score the positive sample $z _ { + }$ higher than a set of distractors $\{ z _ { i } \} _ { i = 1 } ^ { K }$
+
+$$
+\mathcal { L } _ { \mathrm { N C E } } \big ( q , z _ { + } , \{ z _ { i } \} _ { i = 1 } ^ { K } \big ) = - \log \frac { \exp \big ( T ( q , z _ { + } ) \big ) } { \exp \big ( T ( q , z _ { + } ) \big ) + \sum _ { i = 1 } ^ { K } \exp \big ( T ( q , z _ { i } ) \big ) } ,
+$$
+
+where $T ( \cdot , \cdot )$ is a parameterized discriminator function which maps two representation vectors to a scalar value, whose architecture is detailed in Sec. 6.1.
+
+Rival Penalized Competitive Learning (RPCL). The RPCL method (Xu et al., 1993) is a variant of classical competitive learning approaches, e.g. K-means clustering. Concretely, given a sample for update, RPCL-based clustering not only pulls the winning cluster center (i.e. the closest one) towards the sample, but also pushes the rival cluster center (i.e. the second closest one) away from the sample. We adopt this clustering algorithm for its strong capability of discovering feature clusters without specifying the number of clusters beforehand (i.e. in a nonparametric fashion), which is critical in the self-supervised learning scenarios where the number of semantic categories is not given.
+
+# 3 LOCAL-INSTANCE AND GLOBAL-SEMANTIC LEARNING
+
+# 3.1 LEARNING LOCAL-INSTANCE STRUCTURE OF GRAPH REPRESENTATIONS
+
+We first define the correlated graphs that are expected to be embedded close to each other in the embedding space. Since the graphs from a dataset lie in a highly discrete space, it is hard to seek out the correlated counterpart of each graph from the dataset. To tackle this limitation, we propose to construct pairs of correlated graphs via the attribute masking strategy (Hu et al., 2019) which randomly masks a part of node/edge attributes in a graph (theoretical analysis is stated in Sec. A).
+
+Through applying this technique to a randomly sampled mini-batch $B _ { \mathcal { G } } = \{ \mathcal { G } _ { j } = ( \nu _ { j } , \mathcal { E } _ { j } ) \} _ { j = 1 } ^ { N }$ with graphs, the correlated counterpart of each graph can be obtained, which forms another minibatch $\bar { B _ { \mathcal { G } } ^ { \prime } } = \{ \mathcal { G } _ { j } ^ { \prime } = ( \mathcal { V } _ { j } ^ { \prime } , \mathcal { E } _ { j } ^ { \prime } ) \} _ { j = 1 } ^ { N }$ $\mathcal { G } _ { j }$ and $\mathcal { G } _ { j } ^ { \prime }$ are deemed as a pair of correlated graphs). Taking both mini-batches as input, the corresponding patch and graph embeddings are derived as follows:
+
+$$
+\begin{array} { r l } { h _ { \mathcal { V } _ { j } } = \{ h _ { v } | v \in \mathcal { V } _ { j } \} = \mathbf { G N N } ( X _ { \mathcal { V } _ { j } } , X _ { \mathcal { E } _ { j } } ) , \quad h _ { \mathcal { V } _ { j } ^ { \prime } } = \{ h _ { v } | v \in \mathcal { V } _ { j } ^ { \prime } \} = \mathbf { G N N } ( X _ { \mathcal { V } _ { j } ^ { \prime } } , X _ { \mathcal { E } _ { j } ^ { \prime } } ) , } & { } \\ { h _ { \mathcal { G } _ { j } } = \mathbf { R E A D O U T } ( h _ { \mathcal { V } _ { j } } ) , \quad h _ { \mathcal { G } _ { j } ^ { \prime } } = \mathbf { R E A D O U T } ( h _ { \mathcal { V } _ { j } ^ { \prime } } ) , } & { } \end{array}
+$$
+
+where $h _ { \mathcal { V } _ { j } } \ ( h _ { \mathcal { V } _ { j } ^ { \prime } } )$ is the set of patch embeddings for graph $\mathcal { G } _ { j }$ $( \mathcal { G } _ { j } ^ { \prime } )$ , and $h _ { \mathcal { G } _ { j } } \ ( h _ { \mathcal { G } _ { j } ^ { \prime } } )$ denotes the embedding of entire graph. With these ingredients, we design the learning objective for local-instance structure based on two desiderata: (1) similar subgraphs (i.e. patches) have similar feature representations; (2) graphs with a set of similar patches are embedded close to each other. To attain these goals, we propose to maximize the mutual information (i.e. minimize the InfoNCE loss) between correlated patches/graphs, which derives two constraints for the local-instance structure:
+
+$$
+\mathcal { L } _ { \mathrm { p a t c h } } = \frac { 1 } { \sum _ { j = 1 } ^ { N } | \mathcal { V } _ { j } ^ { \prime } | } \sum _ { j = 1 } ^ { N } \sum _ { v ^ { \prime } \in \mathcal { V } _ { j } ^ { \prime } } \sum _ { v \in \mathcal { V } _ { j } } \mathbb { 1 } _ { v  v ^ { \prime } } \cdot \mathcal { L } _ { \mathrm { N C E } } \big ( h _ { v ^ { \prime } } , h _ { v } , \{ h _ { \tilde { v } } | \tilde { v } \in \mathcal { V } _ { j } , \tilde { v } \neq v \} \big ) ,
+$$
+
+$$
+\mathcal { L } _ { \mathrm { g r a p h } } = \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \mathcal { L } _ { \mathrm { N C E } } \big ( h _ { \mathcal { G } _ { j } ^ { \prime } } , h _ { \mathcal { G } _ { j } } , \{ h _ { \mathcal { G } _ { k } } | 1 \leqslant k \leqslant N , k \neq j \} \big ) ,
+$$
+
+$$
+\mathcal { L } _ { \mathrm { l o c a l } } = \mathcal { L } _ { \mathrm { p a t c h } } + \mathcal { L } _ { \mathrm { g r a p h } } ,
+$$
+
+where $\mathcal { L } _ { \mathrm { N C E } } ( \cdot , \cdot , \cdot )$ is the InfoNCE loss function defined in Eq. 3, and $\mathbb { 1 } _ { v  v ^ { \prime } }$ denotes the indicator function judging whether $v$ and $v ^ { \prime }$ are the corresponding nodes in a pair of correlated graphs. Note that, masking node/edge attributes doesn’t change the topology of a graph, which makes it easy to determine these corresponding nodes in our method.
+
+# 3.2 LEARNING GLOBAL-SEMANTIC STRUCTURE OF GRAPH REPRESENTATIONS
+
+It is worth noticing that the graphs in a dataset may possess hierarchical semantic information. For example, drugs (i.e. molecular graphs) are represented by a five-level hierarchy in the Anatomical Therapeutic Chemical (ATC) classification system (Chen et al., 2012). Moreover, the biological functions of proteins (i.e. graphs of amino acid residues) can be organized in a hierarchical structure (e.g. Gene Ontology (Ashburner et al., 2000) and FunCat (Ruepp et al., 2004) protein functionaldefinition schemes).
+
+Motivated by this fact, we propose the notion of hierarchical prototypes to describe the distributions of graph embeddings. These prototypes are structured as a set of trees (Fig. 1(b)), in which each node denotes a prototype (i.e. a representative embedding of feature cluster) and corresponds to an unique parent node unless it is at the top layer. Formally, the hierarchical prototypes can be represented as $\{ c _ { i } ^ { l } \} _ { i = 1 } ^ { M _ { l } } ~ ( l = 1 , 2 , \cdot \cdot \cdot , L _ { p } )$ , where $L _ { p }$ denotes the depth of hierarchical prototypes, and $M _ { l }$ is the number of prototypes at the -th layer. Except for the leaf nodes, each prototype possesses a set of child nodes, denoted as $\mathbb { C } ( c _ { i } ^ { l } )$ $\ 1 \leqslant i \leqslant M _ { l }$ , $l = 1 , 2 , \cdots , L _ { p } - 1 )$ . During training, the derivation of these variables can be divided into two stages, i.e. initialization and maintenance.
+
+Initialization of hierarchical prototypes. In order to establish appropriate priors of graph embeddings, we first pre-train the GNN by minimizing $\mathcal { L } _ { \mathrm { l o c a l } }$ for one epoch and utilize it to extract the embeddings of all graphs in the training set, denoted as $\{ h g _ { i } \} _ { i = 1 } ^ { N _ { D } }$ ( $N _ { D }$ is the size of training i i=1  set). These embeddings are used to initialize the bottom layer prototypes (i.e. $\{ c _ { i } ^ { L _ { p } } \} _ { i = 1 } ^ { M _ { L _ { p } } } )$ via the
+
+$$
+\{ c _ { i } ^ { L _ { p } } \} _ { i = 1 } ^ { M _ { L _ { p } } } = \mathrm { R P C L } \big ( \{ h _ { \mathcal { G } _ { i } } \} _ { i = 1 } ^ { N _ { D } } \big ) ,
+$$
+
+where $\operatorname { R P C L } ( \cdot )$ outputs the cluster centers that are assigned with at least one sample. After that, the prototypes of upper layers are initialized by iteratively applying RPCL-based clustering to the prototypes of the layer below:
+
+$$
+\{ c _ { i } ^ { l } \} _ { i = 1 } ^ { M _ { l } } = \mathrm { R P C L } \big ( \{ c _ { i } ^ { l + 1 } \} _ { i = 1 } ^ { M _ { l + 1 } } \big ) , \quad l = 1 , 2 , \cdots , L _ { p } - 1 .
+$$
+
+It is noteworthy that, in the initialization scheme, the number of prototypes is automatically adapted to the distribution of graph embeddings. As a result, this scheme is nonparametric and can adapt to different datasets without the prior knowledges about them.
+
+Maintenance of hierarchical prototypes. In the training process, since the embedding of each graph is dynamically changing, we propose a strategy to maintain hierarchical prototypes with online updated graph embeddings. Concretely, under the guidance of a similarity measurement (e.g. cosine similarity in our implementation), the graph embeddings extracted from mini-batch $B _ { \mathcal { G } }$ are
+
+Algorithm 1 Training procedure of Local-instance and Global-semantic Learning (GraphLoG).
+
+Input: Training set $D = \{ \mathcal { G } _ { j } \} _ { j = 1 } ^ { N _ { D } }$ , the number of training iterations $N _ { T }$ , hierarchical prototypes’ depth $L _ { p }$ and exponential decay rate $\beta$ .
+
+Output: The pre-trained GNN.
+
+Initialize hierarchical prototypes $\{ c _ { i } ^ { l } \} _ { i = 1 } ^ { M _ { l } } ( l = 1 , 2 , \cdot \cdot \cdot , L _ { p } )$
+
+$$
+\begin{array} { r l } & { B _ { \mathcal { G } } ^ { \prime } \gets \mathrm { A t t r M a s k i n g } ( B _ { \mathcal { G } } ) } \\ & { h _ { \mathcal { V } _ { j } } , h _ { \mathcal { V } _ { j } ^ { \prime } } , h _ { \mathcal { G } _ { j } } , h _ { \mathcal { G } _ { j } ^ { \prime } } \gets \mathrm { E q s . } ( 4 , 5 ) ( j = 1 , 2 , \cdots , N ) } \\ & { \mathcal { L } _ { \mathrm { l o c a l } } , \mathcal { L } _ { \mathrm { g l o b a l } } \gets \mathrm { E q s . } ( 8 , 1 3 ) } \\ & { \theta _ { \mathrm { G N N } } \stackrel {  } { \ } - \nabla _ { \theta _ { \mathrm { G N N } } } ( \mathcal { L } _ { \mathrm { l o c a l } } + \mathcal { L } _ { \mathrm { g l o b a l } } ) } \\ & { \theta _ { T } \stackrel {  } { \cup } - \nabla _ { \theta _ { T } } ( \mathcal { L } _ { \mathrm { l o c a l } } + \mathcal { L } _ { \mathrm { g l o b a l } } ) } \\ & { \{ c _ { i } ^ { l } \} _ { i = 1 } ^ { M }  \mathrm { E q s . } ( 1 1 , 1 2 ) ( l = 1 , 2 , \cdots , L _ { p } ) } \end{array}
+$$
+
+divided into $M _ { L _ { p } }$ groups according to their most similar bottom layer prototype, and the mean graph embeddings are computed within each group, denoted as $\{ \widehat { c } _ { i } ^ { L _ { p } } \} _ { i = 1 } ^ { M _ { L _ { p } } }$ . These mean embeddings are bemployed to update bottom layer prototypes via an exponential moving average scheme:
+
+$$
+c _ { i } ^ { L _ { p } } \gets \beta c _ { i } ^ { L _ { p } } + ( 1 - \beta ) \widehat { c } _ { i } ^ { L _ { p } } , \quad 1 \leqslant i \leqslant M _ { L _ { p } } ,
+$$
+
+where $\beta$ is the exponential decay rate. For the prototypes of upper layers, they are updated with the mean of their child prototypes in the corresponding tree:
+
+$$
+c _ { i } ^ { l } \gets \frac { 1 } { \left| \mathbb { C } ( c _ { i } ^ { l } ) \right| } \sum _ { \substack { c _ { k } ^ { l + 1 } \in \mathbb { C } ( c _ { i } ^ { l } ) } } c _ { k } ^ { l + 1 } , \quad 1 \leqslant i \leqslant M _ { l } , ~ l = 1 , 2 , \cdots , L _ { p } - 1 .
+$$
+
+Constraint for global-semantic structure. Now that the latent semantic structure of the data has been represented by hierarchical prototypes, we seek to constrain the distributions of graph embeddings with these prototypes. The major goal here is to map correlated graphs to the same set of feature clusters. In practice, according to cosine similarity, we first search for the prototypes most similar to the embedding of graph $\mathcal { G } _ { j }$ in each layer, denoted as $s ( \mathcal { G } _ { j } ) = \{ s _ { 1 } ( \mathcal { G } _ { j } ) , s _ { 2 } ( \mathcal { G } _ { j } ) , \cdots , s _ { L _ { p } } ( \mathcal { G } _ { j } ) \}$ . Note that, this search process follows the topology of hierarchical prototypes, which means that: $s _ { l + 1 } ( \mathcal { G } _ { j } ) \in \mathbb { C } ( s _ { l } ( \mathcal { G } _ { j } ) )$ (l = 1, 2, · · · , Lp − 1). Correspondingly, when using the embedding of the correlated graph $\mathcal { G } _ { j } ^ { \prime }$ for search, we expect an identical searching path, and such objective is pursued by maximizing the mutual information (i.e. minimizing the InfoNCE loss) between graph embedding $h _ { \mathcal { G } _ { j } ^ { \prime } }$ and the prototypes in $s ( \mathcal { G } _ { j } )$ :
+
+$$
+\mathcal { L } _ { \mathrm { g l o b a l } } = \frac { 1 } { N \cdot L _ { p } } \sum _ { j = 1 } ^ { N } \sum _ { l = 1 } ^ { L _ { p } } \mathcal { L } _ { \mathrm { N C E } } \big ( h _ { \mathcal { G } _ { j } ^ { \prime } } , s _ { l } ( \mathcal { G } _ { j } ) , \{ c _ { i } ^ { l } | 1 \leqslant i \leqslant M _ { l } , c _ { i } ^ { l } \neq s _ { l } ( \mathcal { G } _ { j } ) \} \big ) .
+$$
+
+Discussion. A recent work (Li et al., 2020) employed hierarchical prototypes for visual representation learning. The semantic hierarchy established in that work is derived from multiple times of clustering with different numbers of clusters, which relies on heuristically selected cluster numbers and fails to model the relations between the prototypes from different hierarchies. In contrast, our method is free from pre-defined cluster numbers, and a set of relational trees are constructed to embody the hierarchical relations between different prototypes.
+
+# 3.3 MODEL OPTIMIZATION
+
+The training procedure of Local-instance and Global-semantic Learning (GraphLoG) is summarized in Algorithm 1. Along training, using the online-updated graph embeddings, the constraints for local-instance and global-semantic structure are derived, and the hierarchical prototypes are maintained. In each iteration, the parameters of GNN and discriminator are optimized with gradient descent using the following objective:
+
+$$
+\operatorname* { m i n } _ { \mathrm { G N N } , T } \mathcal { L } _ { \mathrm { l o c a l } } + \mathcal { L } _ { \mathrm { g l o b a l } } .
+$$
+
+# 4 SUP-GRAPHLOG: A SUPERVISED BASELINE FOR LOCAL-INSTANCE ANDGLOBAL-SEMANTIC LEARNING
+
+In order to verify the effectiveness of local-instance and global-semantic learning when it is directly applied to supervised downstream tasks, we propose a baseline model, named as sup-GraphLoG, which combines a plain GNN and the proposed hierarchical prototypes.
+
+In the training phase, in order to establish appropriate local-instance structure of graph embeddings, the GNN is first pre-trained along with a linear classifier to perform graph classification on the training set. For the initialization of hierarchical prototypes, the number of bottom layer prototypes is set as the class number of the supervised task (e.g. $2 K _ { T }$ bottom layer prototypes for a task with $K _ { T }$ binary classification problems), and each bottom layer prototype is the mean embedding of all the training graphs belonging to the corresponding class. The upper layer prototypes are initialized as in Eq. 10. For maintenance, given a mini-batch of labeled graphs, each bottom layer prototype is updated by the mean embedding of all the graphs belonging to the corresponding class using an exponential moving average scheme as in Eq. 11, and the prototypes of upper layers are maintained following Eq. 12.
+
+For constraining the global-semantic structure, compared with the top-down search in the selfsupervised model (Sec. 3.2), in this supervised setting, we first use the label of graph $\mathcal { G } _ { j }$ to randomly select a matched bottom layer prototype $s _ { L _ { p } } ( \mathcal { G } _ { j } )$ and then obtain the whole searching path $\dot { s } ( \mathcal { G } _ { j } ) = \{ s _ { 1 } ( \mathcal { G } _ { j } ) , s _ { 2 } ( \mathcal { G } _ { j } ) , \cdots , \overline { { s _ { L _ { p } } ( \mathcal { G } _ { j } ) } } \}$ from bottom to up. Based on this positive searching path, we randomly sample a negative path $s ^ { n } ( { \mathcal { G } } _ { j } ) = \{ s _ { 1 } ^ { n } ( { \bar { \mathcal { G } } } _ { j } ) , s _ { 2 } ^ { n } ( { \mathcal { G } } _ { j } ) , \cdot \cdot \cdot , s _ { L _ { p } } ^ { \bar { n } } ( { \mathcal { G } } _ { j } ) \}$ satisfying that graph $\mathcal { G } _ { j }$ does not belong to the corresponding class of $s _ { L _ { p } } ^ { n } ( \mathcal { G } _ { j } )$ , and $s _ { l } ^ { n } ( \mathcal { G } _ { j } ) \neq s _ { l } ( \mathcal { G } _ { j } )$ , $s _ { l + 1 } ^ { n } ( \mathcal { G } _ { j } ) \in \mathbb { C } ( s _ { l } ^ { n } ( \mathcal { G } _ { j } ) ) ( l = 1 , 2 , \cdot \cdot \cdot , L _ { p } - 1 )$ . It is expected that the embedding of graph $\mathcal { G } _ { j }$ is more similar with the prototypes on path $s ( \mathcal { G } _ { j } )$ than the ones on path $s ^ { n } ( { \mathcal { G } } _ { j } )$ , which defines the loss constraint on a mini-batch as follows:
+
+$$
+\mathcal { L } _ { \mathrm { g l o b a l } } ^ { \mathrm { s u p } } = \frac { 1 } { N \cdot L _ { p } } \sum _ { j = 1 } ^ { N } \sum _ { l = 1 } ^ { L _ { p } } \mathcal { L } _ { \mathrm { N C E } } \big ( h _ { \mathcal { G } _ { j } } , s _ { l } ( \mathcal { G } _ { j } ) , s _ { l } ^ { n } ( \mathcal { G } _ { j } ) \big ) .
+$$
+
+We further optimize the GNN by minimizing this loss, which refines the global-semantic structure in the embedding space.
+
+In the inference phase, given an unlabeled graph, we first compute the similarity between its embedding and all the bottom layer prototypes via the cosine similarity function. After that, the taskspecific prediction is derived by comparing the similarity scores of the classes corresponding to these prototypes, in which the classes with larger scores serve as the prediction result.
+
+# 5 RELATED WORK
+
+Graph Neural Networks (GNNs). Recently, following the efforts of learning graph representations via optimizing random walk (Perozzi et al., 2014; Tang et al., 2015; Grover & Leskovec, 2016; Narayanan et al., 2017) or matrix factorization (Cao et al., 2015; Wang et al., 2016) objectives, GNNs are proposed to explicitly derive proximity-preserved feature vectors in a neighborhood aggregation manner. As suggested in Gilmer et al. (2017), the forward pass of most GNNs can be depicted in two phases, Message Passing and Readout phase, and various works (Kipf & Welling, 2017; Hamilton et al., 2017; Velickovic et al., 2018; Ying et al., 2018; Zhang et al., 2018; Xu et al., 2019) sought to improve the effectiveness of these two phases. Unlike these methods which are mainly trained in a supervised fashion, our approach aims for unsupervised/self-supervised learning for GNNs.
+
+Self-supervised Learning for GNNs. There are some recent works that explored self-supervised graph representation learning with GNNs. Garc´ıa-Duran & Niepert ´ (2017) learned graph representations by embedding propagation, and Velickovic et al. (2019), Sun et al. (2019) and Hassani & Ahmadi (2020) achieved this goal through mutual information maximization. Also, some selfsupervised tasks, e.g. edge prediction (Kipf & Welling, 2016), context prediction (Hu et al., 2019; Rong et al., 2020a), graph partitioning (You et al., 2020) and edge/attribute generation (Hu et al., 2020), have been designed to acquire knowledges from unlabeled graphs. Nevertheless, all these methods are only able to model the local relations between different graph instances. The proposed framework seeks to discover both the local-instance and global-semantic structure of a set of graphs.
+
+Self-supervised Semantic Learning. Clustering-based methods (Xie et al., 2016; Yang et al., 2016; 2017; Caron et al., 2018; Ji et al., 2019; Li et al., 2020) are commonly used to learn the semantic information of the data in a self-supervised fashion. Among which, DeepCluster (Caron et al., 2018) proved the strong transferability of the visual representations learnt by clustering prediction to various downstream visual tasks. Prototypical Contrastive Learning (Li et al., 2020) set a new stateof-the-art for unsupervised visual representation learning. These methods are mainly developed for images but not for graph-structured data. Furthermore, the hierarchical semantic structure of the data has been less explored in previous works.
+
+# 6 EXPERIMENTS
+
+# 6.1 EXPERIMENTAL SETUP
+
+Pre-training details. Following Hu et al. (2019), we adopt a five-layer Graph Isomorphism Network (GIN) $\mathrm { { X u } }$ et al., 2019) with 300-dimensional hidden units and a mean pooling readout function for performance comparisons (Secs. 6.2 and 6.3). The discriminator for mutual information estimation is formalized as: $T ( x _ { 1 } , x _ { 2 } ) = g ( f ( x _ { 1 } ) , f ( x _ { 2 } ) )$ , where $f ( \cdot )$ is a projection function fitted by two linear layers and a ReLU nonlinearity between them, and $g ( \cdot , \cdot )$ is a similarity function (e.g. cosine similarity in our method). In all experiments, we use an Adam optimizer (Kingma & Ba, 2015) (learning rate: $1 \times 1 0 ^ { - 3 }$ , batch size: 512) to train the model for 20 epochs. Unless otherwise specified, the hierarchical prototypes’ depth $L _ { p }$ is set as 3, and the exponential decay rate $\beta$ is set as 0.95. For attribute masking, $30 \%$ node attributes in molecular graphs are masked, and $30 \%$ edge attributes in Protein-Protein Interaction (PPI) networks are masked. These hyperparameters are selected by the grid search on the validation sets of four downstream molecule datasets (i.e. BBBP, SIDER, ClinTox and BACE), and their sensitivity is analyzed in Secs. 6.4 and F.
+
+Fine-tuning details. For fine-tuning on a downstream task, a linear classifier is appended on the top of pre-trained GNN, and an Adam optimizer (classifier’s learning rate: $1 \times 1 0 ^ { - 3 }$ , GNN’s learning rate: $1 \times 1 0 ^ { - 4 }$ , batch size: 32) is employed to train the model for 100 epochs. For sup-GraphLoG, the GNN is first trained along with a linear classifier for 50 epochs using an Adam optimizer (learning rate: $1 \times 1 0 ^ { - 3 }$ , batch size: 32), and it is then fine-tuned under the guidance of hierarchical prototypes by an Adam optimizer (learning rate: $1 \times 1 0 ^ { - 4 }$ , batch size: 32). All reported results are averaged over five independent runs under the same configuration. Our approach is implemented with PyTorch (Paszke et al., 2017), and the source code will be released for reproducibility.
+
+Performance comparison. We compare the proposed method with existing self-supervised graph representation learning algorithms (i.e. EdgePred (Kipf & Welling, 2016), InfoGraph (Sun et al., 2019), AttrMasking (Hu et al., 2019), ContextPred (Hu et al., 2019) and GraphPartition (You et al., 2020)) to verify its effectiveness. Following the setting in Hu et al. (2019), after pre-training GNN models with self-supervised methods, a graph-level multi-task supervised pre-training is conducted to achieve more transferable graph representations, and the performance on downstream tasks is respectively evaluated before and after this graph-level supervised pre-training.
+
+# 6.2 EXPERIMENTS ON CHEMISTRY DOMAIN
+
+Datasets. For fair comparison, we use the same datasets as in Hu et al. (2019). In specific, a subset of ZINC15 database (Sterling & Irwin, 2015) with 2 million unlabeled molecules is employed for self-supervised pre-training, and a preprocessed ChEMBL dataset (Mayr et al., 2018) with 456K labeled molecules is used for graph-level supervised pre-training. Eight binary classification datasets contained in MoleculeNet (Wu et al., 2018) serve as downstream tasks.
+
+Results. Tab. 1 reports the performance of proposed GraphLoG method compared with other works. Among all self-supervised learning strategies, our approach achieves the best performance on seven of eight tasks, and a $3 \%$ performance gain is obtained in terms of average ROC-AUC. After applying a subsequent graph-level supervised pre-training, our models’ performance is further promoted. In particular, a $2 . 9 \%$ increase is observed on the SIDER dataset. Also, the comparison between two supervised methods without self-supervised pre-training is presented in the table, the proposed supGraphLoG outperforms the vanilla GIN model with random initialization, which demonstrates the benefit of learning global-semantic structure. The training curves of eight downstream tasks are provided in Sec. G.
+
+Table 1: Test ROC-AUC $( \% )$ on molecular property prediction benchmarks.   
+
+<table><tr><td>Methods</td><td>BBBP</td><td>Tox21</td><td>ToxCast</td><td>SIDER</td><td>ClinTox</td><td>MUV</td><td>HIV</td><td>BACE</td><td>Avg</td></tr><tr><td>Random</td><td>65.8±4.5</td><td>74.0±0.8</td><td></td><td></td><td></td><td>63.4±0.6 57.3±1.6 58.0±4.4 71.8± 2.5</td><td>75.3±1.9</td><td>70.1±5.4</td><td>67.0</td></tr><tr><td>sup-GraphLoG (ours)</td><td>71.1 ± 0.3</td><td>72.9±0.2</td><td>63.8 ± 0.1</td><td>61.4 ± 0.6</td><td>64.0 ±0.6</td><td>72.5± 1.0</td><td>76.7 ± 0.5</td><td>76.5 ± 1.1</td><td>69.9</td></tr><tr><td>EdgePred (2016)</td><td>67.3±2.4</td><td>76.0±0.6</td><td>64.1±0.6</td><td>60.4±0.7</td><td>64.1±3.7</td><td>74.1 ± 2.1</td><td>76.3±1.0</td><td>79.9±0.9</td><td>70.3</td></tr><tr><td>InfoGraph (2019)</td><td>68.2±0.7</td><td>75.5±0.6</td><td>63.1±0.3</td><td>59.4 ±1.0</td><td>70.5±1.8</td><td>75.6±1.2</td><td>77.6± 0.4</td><td>：78.9±1.1</td><td>71.1</td></tr><tr><td>AttrMasking (2019)</td><td>64.3± 2.8</td><td>76.7±0.4</td><td>64.2 ± 0.5</td><td>61.0±0.7</td><td>71.8± 4.1</td><td>74.7± 1.4</td><td>77.2 ± 1.1</td><td>79.3 ± 1.6</td><td>71.1</td></tr><tr><td>ContextPred (2019)</td><td>68.0±2.0</td><td>75.7±0.7</td><td>63.9±0.6</td><td>60.9±0.6</td><td>65.9±3.8</td><td>75.8±1.7</td><td>77.3 3±1.0</td><td>79.6 ± 1.2</td><td>70.9</td></tr><tr><td>GraphPartition (2020)</td><td>70.3±0.7</td><td>75.2±0.4</td><td>63.2±0.3</td><td>61.0±0.8</td><td>64.2±0.5</td><td>75.4±1.7</td><td>77.1±0.7</td><td>79.6 ±1.8</td><td>70.8</td></tr><tr><td>GraphLoG (ours)</td><td>73.9 ± 0.7</td><td>76.2 ± 0.2</td><td>64.2 ± 0.5</td><td>61.7 ± 1.2</td><td>78.6 ± 1.5</td><td>76.4± 1.0</td><td>78.2 ± 0.6</td><td>83.3±1.4</td><td>74.1</td></tr><tr><td>Supervised</td><td>68.3±0.7</td><td>77.0±0.3</td><td>64.4±0.4</td><td>62.1±0.5</td><td>57.2 ± 2.5</td><td>79.4 ± 1.3</td><td>74.4 ±1.2</td><td>76.0±1.0</td><td>70.0</td></tr><tr><td>EdgePred* (2016)</td><td>66.6±2.2</td><td>78.3 ±0.3</td><td>66.5± 0.3</td><td>63.3 ±0.9</td><td>70.9±4.6</td><td>78.5±2.4</td><td>77.5±0.8</td><td>79.1 ± 3.7</td><td>72.6</td></tr><tr><td>InfoGraph*(2019)</td><td>68.4±1.0</td><td>77.6±0.7</td><td>65.3±0.4</td><td>62.5±0.7</td><td>73.8±1.9</td><td>79.3 ±1.6</td><td>78.0±1.1</td><td>82.4 ±1.3</td><td>73.4</td></tr><tr><td>AttrMasking*(2019)</td><td>66.5 ± 2.5</td><td>77.9±0.4</td><td>65.1±0.3</td><td>63.9 ± 0.9</td><td>73.7± 2.8</td><td>81.2 ± 1.9</td><td>77.1 ± 1.2</td><td>80.3±0.9</td><td>73.2</td></tr><tr><td>ContextPred*(2019)</td><td>68.7 ±1.3</td><td>78.1±0.6</td><td>65.7±0.6</td><td>62.7±0.8</td><td>72.6 ±1.5</td><td>81.3 ± 2.1</td><td>79.9±0.7</td><td>84.5±0.7</td><td>74.2</td></tr><tr><td>GraphPartition* (2020)</td><td>71.1±0.5</td><td>77.4±0.4</td><td>64.2±0.1</td><td>63.4±0.2</td><td>72.9±0.4</td><td>78.2±0.7</td><td>78.6±0.4</td><td>80.4±0.2</td><td>73.3</td></tr><tr><td>GraphLoG*(ours)</td><td>74.0 ± 0.8</td><td>78.5±0.2</td><td>66.5± 0.5</td><td>64.6± 0.8</td><td>78.6± 0.7</td><td>79.5 ± 1.3</td><td>80.1± 0.7</td><td>85.1±0.9</td><td>75.9</td></tr></table>
+
+“\*” denotes the model composed of a specific self-supervised pre-training and a subsequent graph-level supervised pre-training.
+
+Table 2: Performance comparison and ablation study on biological function prediction benchmark.   
+(a) Test ROC-AUC $( \% )$ of different methods.   
+
+<table><tr><td>Methods</td><td>ROC-AUC (%)</td></tr><tr><td>Random sup-GraphLoG (ours)</td><td>64.8±1.0 67.6± 0.8</td></tr><tr><td>EdgePred (Kipf &amp;Welling,2016)</td><td>70.5± 0.7</td></tr><tr><td>InfoGraph (Sun et al.,2019)</td><td>70.7 ± 0.5</td></tr><tr><td>AttrMasking (Hu et al.,2019)</td><td>70.5 ± 0.5</td></tr><tr><td>ContextPred (Hu et al.,2019)</td><td>69.9 ±0.3</td></tr><tr><td>GraphPartition (You et al.,2020)</td><td>71.0±0.2</td></tr><tr><td>GraphLoG (ours)</td><td>72.8 ±0.4</td></tr></table>
+
+(b) Ablation study for different loss terms.
+
+<table><tr><td>Lpatch</td><td>Lgraph</td><td>Lglobal</td><td>ROC-AUC (%)</td></tr><tr><td>√</td><td></td><td></td><td>70.6 ± 0.5</td></tr><tr><td></td><td>1</td><td>√</td><td>71.1 ± 0.7</td></tr><tr><td></td><td></td><td></td><td>71.4 ± 0.4</td></tr><tr><td>√</td><td>√</td><td></td><td>72.0±0.3</td></tr><tr><td>√</td><td></td><td>√</td><td>72.0±0.6</td></tr><tr><td></td><td>√</td><td>√</td><td>71.8±0.5</td></tr><tr><td>√</td><td>√</td><td>√</td><td>72.8 ±0.4</td></tr></table>
+
+# 6.3 EXPERIMENTS ON BIOLOGY DOMAIN
+
+Datasets. For biology domain, following the settings in Hu et al. (2019), 395K unlabeled protein ego-networks are utilized for self-supervised pre-training, and the prediction of 5000 coarsegrained biological functions on 88K labeled protein ego-networks serves as graph-level supervised pre-training. The downstream task is to predict 40 fine-grained biological functions of 8 species.
+
+Results. In Tab. 2a, we report the test ROC-AUC of various self-supervised learning techniques, and more results on biology domain can be found in Sec. C. It can be observed that the proposed GraphLoG method outperforms existing approaches with a clear margin, i.e. a $1 . 8 \%$ performance gain. This result illustrates that the proposed scheme is beneficial to fine-grained downstream tasks. In addition, sup-GraphLoG is able to promote the performance of a plain GIN model by $2 . 8 \%$ on this biological downstream task.
+
+# 6.4 ANALYSIS
+
+Effect of different loss terms. In Tab. 2b, we analyze the effect of three loss terms on biological function prediction, and we continue using the GIN depicted in Sec. 6.1 in this experiment. When each loss is independently applied (1st, 2nd and 3rd row), the loss for global-semantic structure performs best, which probably benefits from its exploration of data’s semantic information. Through combining these losses, the full model (last row) achieves the best performance, which illustrates that the learning of local-instance and global-semantic structure are complementary to each other. We provide more ablation studies on different model components in Sec. E.
+
+Results on different GNNs. Fig. 2(a) presents the effect of self-supervised pre-training on four kinds of GNNs, GCN (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), GAT (Velickovic et al., 2018) and GIN (Xu et al., 2019). We can observe that the proposed GraphLoG scheme outperforms two existing methods, AttrMasking and ContextPred, on all configurations, and it avoids the performance decrease relative to random initialization baseline on GAT.
+
+![](images/209e82dc00a45c66a111b37383d4bc068c4b59c4e7441dc46ec5d08bde621e79.jpg)  
+Figure 2: (a) Experimental results on different GNNs. (b)&(c) Sensitivity analysis of hierarchical prototypes’ depth $L _ { p }$ and exponential decay rate $\beta$ . (All results are reported on biology domain.)
+
+![](images/15ea04b4935b86d72b3b5e96b14d422300fa74c5fd1e09217456ca3037c9c248.jpg)  
+Figure 3: The t-SNE (Maaten & Hinton, 2008) visualization of graph embeddings and hierarchical prototypes on ZINC15 database (i.e. the pre-training dataset for chemistry domain).
+
+Sensitivity of hierarchical prototypes’ depth $L _ { p }$ . In this part, we discuss the selection of parameter $L _ { p }$ which controls the number of discovered semantic hierarchies. In Fig. 2(b), we plot model’s performance under different $L _ { p }$ values. It can be observed that deeper hierarchical prototypes (i.e. $L _ { p } \geqslant 3 _ { \mathrm { , } }$ ) achieve stable performance gain compared to the shallow ones (i.e. $L _ { p } \leqslant 2$ ).
+
+Sensitivity of exponential decay rate $\beta$ . In this experiment, we evaluate our approach’s sensitivity to the parameter $\beta$ . Fig. 2(c) shows the test ROC-AUC on downstream task using different $\beta$ values. From the line chart, we can observe that the proposed model’s performance is not sensitive to $\beta$ , which makes the maintenance scheme of hierarchical prototypes easy to tune.
+
+Visualization. In Fig. 3, we utilize t-SNE (Maaten & Hinton, 2008) to visualize the distributions of graph embeddings and hierarchical prototypes on ZINC15 dataset. Compared to the model with only ${ \mathcal { L } } _ { \mathrm { p a t c h } }$ constraint, some feature clusters are formed after constraining the relations between correlated graphs’ embeddings by ${ \mathcal { L } } _ { \mathrm { g r a p h } }$ . More obvious feature separation is achieved after applying $\mathcal { L } _ { \mathrm { g l o b a l } }$ , which illustrates its effectiveness on discovering the global-semantic structure of the data.
+
+# 7 CONCLUSIONS AND FUTURE WORK
+
+We devise a unified framework called Local-instance and Global-semantic Learning (GraphLoG) for self-supervised graph representation learning, which models the structure of a set of unlabeled graphs both locally and globally. In this framework, we novelly propose to learn hierarchical prototypes upon graph embeddings to infer the global-semantic structure in graphs. Using the benchmark datasets from both chemistry and biology domains, we empirically verify our method’s superior performance over state-of-the-art approaches on different GNN architectures.
+
+Our future works will include exploring novel ways to construct correlated graphs, improving selfsupervised learning manners, unifying pre-training and fine-tuning, and extending our framework to other domains such as sociology, physics and material science.
+
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+# A THEORETICAL ANALYSIS OF CORRELATED GRAPH CONSTRUCTION
+
+In our method, we choose the attribute masking (Hu et al., 2019) strategy to generate correlated graph pairs, which is widely used in recent self-supervised graph representation learning algorithms (Hu et al., 2019; 2020; Qiu et al., 2020). Since the graph structure has not been changed by the masking operation, the masked node attribute information can be partially recovered by its surrounding neighbors after being fed into a GNN. Therefore, the embeddings of correlated graph pairs can maintain a high degree of consistency in the feature space, which is desirable for the proposed GraphLoG model. We formally elucidate this point as follows.
+
+Given an attributed graph $\mathcal { G } = ( \mathcal { V } , \mathcal { E } , X _ { \mathcal { V } } , X _ { \mathcal { E } } )$ $X \nu$ : node attributes, $X _ { \mathcal { E } }$ : edge attributes), we assume that its correlated graph $\mathcal { G } ^ { \prime } = ( \mathcal { V } , \mathcal { E } , X _ { \mathcal { V } } ^ { \prime } , X _ { \mathcal { E } } )$ is obtained by masking the attribute of a node $v$ , i.e. $X _ { \nu } ^ { \prime } \gets X _ { \nu - \{ v \} } \cup \{ X _ { v } ^ { m } \}$ .
+
+Proposition 1. The $L$ -layer GNN can repair the lost information induced by attribute masking operation by $\mathcal { T } _ { \mathrm { r e p a i r } } \geqslant \dot { \mathcal { T } } \big ( X _ { v } , \{ X _ { \tilde { v } } | \tilde { v } \in \dot { \mathcal { N } } _ { v } ^ { L } \} \big )$ , where $\mathcal { T } ( \cdot , \cdot )$ denotes the mutual information, and $\mathcal { N } _ { v } ^ { L }$ is the $L$ -hop neighborhood set of $v$ .
+
+Proof. Before information propagation by GNN, we define the lost information $\mathcal { T } _ { \mathrm { l o s t } }$ induced by attribute masking as the conditional entropy of graph $\mathcal { G }$ conditioned on its correlated graph $\mathcal { G } ^ { \prime }$ :
+
+$$
+\mathcal { T } _ { \mathrm { l o s t } } = \mathcal { H } ( \mathcal { G } | \mathcal { G } ^ { \prime } ) = \mathcal { H } ( X _ { v } ) .
+$$
+
+After the information propagation by GNN, we have:
+
+$$
+\tilde { \mathcal { T } } _ { \mathrm { l o s t } } = \mathcal { H } ( h _ { \mathcal { G } } | h _ { \mathcal { G } ^ { \prime } } ) = \mathcal { H } ( h _ { \mathcal { G } } ) - \mathcal { T } ( h _ { \mathcal { G } } , h _ { \mathcal { G } ^ { \prime } } ) ,
+$$
+
+where $\tilde { \mathcal { T } } _ { \mathrm { l o s t } }$ is the information lost in the embedding of correlated graph $\mathcal { G } ^ { \prime }$ compared with the embedding of origin graph $\mathcal { G }$ .
+
+According to the neighborhood aggregation scheme in GNN, we can derive:
+
+$$
+\mathcal { T } ( h _ { \mathcal { G } } , h _ { \mathcal { G } ^ { \prime } } ) = \mathcal { H } ( h _ { \mathcal { G } } ) - \mathcal { H } ( h _ { v } ) + \mathcal { T } \big ( X _ { v } , \{ X _ { \widetilde { v } } | \widetilde { v } \in \mathcal { N } _ { v } ^ { L } \} \big ) ,
+$$
+
+where $\mathcal { T } \big ( X _ { v } , \{ X _ { \tilde { v } } | \tilde { v } \in \mathcal { N } _ { v } ^ { L } \} \big )$ 
 denotes the recovered information for the masked node $v$ from its $L$ -hop neighbors. Combining Eqs. 17 and 18 leads to:
+
+$$
+\tilde { \mathcal { T } } _ { \mathrm { l o s t } } = \mathcal { H } ( h _ { v } ) - \mathcal { T } \big ( \boldsymbol { X } _ { v } , \{ \boldsymbol { X } _ { \tilde { v } } | \tilde { v } \in \mathcal { N } _ { v } ^ { L } \} \big ) .
+$$
+
+We can derive the information repaired by GNN, $\mathcal { T } _ { \mathrm { r e p a i r } }$ , as:
+
+$$
+\mathcal { T } _ { \mathrm { r e p a i r } } = \mathcal { Z } _ { \mathrm { l o s t } } - \tilde { \mathcal { Z } } _ { \mathrm { l o s t } } = \mathcal { Z } \big ( X _ { v } , \{ X _ { \tilde { v } } | \tilde { v } \in \mathcal { N } _ { v } ^ { L } \} \big ) + \big ( \mathcal { H } ( X _ { v } ) - \mathcal { H } ( h _ { v } ) \big ) .
+$$
+
+Since $h _ { v }$ is the low-dimensional embedding of node attribute $X _ { v }$ , we can deduce that:
+
+$$
+\mathcal { H } ( X _ { v } ) \geqslant \mathcal { H } ( h _ { v } ) .
+$$
+
+Therefore, combining Eqs. 20 and 21, we can conclude:
+
+$$
+\mathcal { T } _ { \mathrm { r e p a i r } } \geqslant \mathcal { T } \big ( X _ { v } , \{ X _ { \tilde { v } } | \tilde { v } \in \mathcal { N } _ { v } ^ { L } \} \big ) .
+$$
+
+We also evaluate the GraphLoG model with different correlated graph construction strategies, and the experimental results can be found in Sec. E.1. It empirically shows that attribute masking is more reliable to our method.
+
+# B MORE IMPLEMENTATION DETAILS
+
+Attribute masking strategy. We add an extra dimension in the vector of node/edge attribute and set only that dimension as 1 when the corresponding node/edge is masked. Given a mini-batch of graphs, we mask the same proportion of node/edge attributes in each graph, and, for undirected graphs, the attributes on the both directions of an edge are masked/unmasked.
+
+GNN architecture. All the GNNs in our experiments (i.e. GCN (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), GAT (Velickovic et al., 2018) and GIN (Xu et al., 2019)) are with 5 layers, 300-dimensional hidden units and a mean pooling readout function. In addition, two attention heads are employed in each layer of the GAT model.
+
+# C MORE RESULTS ON BIOLOGY DOMAIN
+
+Table 3: Test ROC-AUC $( \% )$ on biological function prediction benchmark.   
+
+<table><tr><td>Methods</td><td>ROC-AUC (%)</td><td>Methods</td><td>ROC-AUC (%)</td></tr><tr><td>Random</td><td>64.8 ±1.0</td><td>Supervised</td><td>72.9 ± 0.5</td></tr><tr><td>EdgePred (Kipf &amp; Welling,2016)</td><td>70.5± 0.7</td><td>EdgePred* (Kipf &amp; Welling,2016)</td><td>73.1 ± 0.5</td></tr><tr><td>InfoGraph (Sun et al., 2019)</td><td>70.7± 0.5</td><td>InfoGraph* (Sun et al.,2019)</td><td>73.7± 0.4</td></tr><tr><td>AttrMasking (Hu et al., 2019)</td><td>70.5± 0.5</td><td>AttrMasking* (Hu et al.,2019)</td><td>74.2 ± 1.5</td></tr><tr><td>ContextPred (Hu et al.,2019)</td><td>69.9 ± 0.3</td><td>ContextPred*(Hu et al.,2019)</td><td>74.3 ± 0.6</td></tr><tr><td>GraphPartition (You et al.,2020)</td><td>71.0 ± 0.2</td><td>GraphPartition*(You et al.,2020)</td><td>73.5 ± 0.1</td></tr><tr><td>GraphLoG (ours)</td><td>72.8 ± 0.4</td><td>GraphLoG* (ours)</td><td>75.7 ± 0.6</td></tr></table>
+
+“\*” denotes the model composed of a specific self-supervised pre-training and a subsequent graph-level supervised pre-training.
+
+In Tab. 3, we report the performance of different approaches on the downstream task of biology domain, and the results before and after applying a subsequent graph-level supervised pre-training are respectively reported. It can be observed that the proposed GraphLoG method outperforms existing approaches with a clear margin under both settings, which illustrates the effectiveness of proposed learning scheme with and without the guidance of graph-level supervisory signal.
+
+# D MORE RESULTS ON GRAPH CLASSIFICATION BENCHMARKS
+
+Table 4: The 10-fold cross validation accuracy ( $\mathrm { m e a n } \pm \mathrm { s t d } \%$ ) of self-supervised methods on graph classification benchmarks.   
+
+<table><tr><td>Methods</td><td>MUTAG</td><td>PTC-MR</td><td>IMDB-Binary</td><td>IMDB-Multi Reddit-Binary</td><td></td></tr><tr><td>random walk (Gärtner et al., 2003)</td><td>83.7± 1.5</td><td>57.9 ± 1.3</td><td>50.7± 0.3</td><td>34.7±0.2</td><td>1</td></tr><tr><td>node2vec (Grover &amp; Leskovec,2016)</td><td>72.6 ± 10.2</td><td>58.6±8.0</td><td></td><td></td><td>1</td></tr><tr><td>graph2vec (Narayanan et al., 2017)</td><td>83.2±9.6</td><td>60.2 ± 6.9</td><td>71.1 ± 0.5</td><td>50.4± 0.9</td><td>75.8± 1.0</td></tr><tr><td>sub2vec (Adhikari et al., 2018)</td><td>61.1 ± 15.8</td><td>60.0±6.4</td><td>55.3 ± 1.5</td><td>36.7±0.8</td><td>71.5 ± 0.4</td></tr><tr><td>InfoGraph (Sun et al.,2019)</td><td>89.0 ± 1.1</td><td>61.7 ± 1.4</td><td>73.0 ± 0.9</td><td>49.7± 0.5</td><td>82.5 ± 1.4</td></tr><tr><td>Contrastive (Hassani &amp; Ahmadi, 2020)</td><td>89.7±1.1</td><td>62.5 ± 1.7</td><td>74.2 ± 0.7</td><td>51.2 ± 0.5</td><td>84.5± 0.6</td></tr><tr><td>GraphLoG (ours)</td><td>89.9 ± 1.5</td><td>63.8 ± 1.6</td><td>76.6 ± 4.2</td><td>53.0 ± 3.5</td><td>85.9 ± 2.9</td></tr></table>
+
+Setups. In this experiment, we compare GraphLoG with six self-supervised graph representation learning methods, i.e. random walk (Gartner et al. ¨ , 2003), node2vec (Grover & Leskovec, 2016), graph2vec (Narayanan et al., 2017), sub2vec (Adhikari et al., 2018), InfoGraph (Sun et al., 2019) and Contrastive (Hassani & Ahmadi, 2020). We strictly follow the linear evaluation protocol in Sun et al. (2019) and report the mean accuracy of 10-fold cross validation. Five conventional graph classification benchmark datasets, i.e. MUTAG (Kriege et al., 2016), PTC (Kriege et al., 2016), IMDB-Binary (Yanardag & Vishwanathan, 2015), IMDB-Multi (Yanardag & Vishwanathan, 2015) and Reddit-Binary (Yanardag & Vishwanathan, 2015), are used for evaluation. The settings of network architecture, optimizer and training parameters follow those in Sec. 6.1.
+
+Results. Tab. 4 presents the comparisons of self-supervised approaches on five graph classification benchmark datasets. The proposed GraphLoG model ranks the first place in every task, and, especially, it outperforms a recent contrastive-learning-based method (Hassani & Ahmadi, 2020), which demonstrates the effectiveness of learning local-instance and global-semantic structure.
+
+# E MORE ABLATION STUDIES
+
+# E.1 ABLATION STUDY ON CONSTRUCTING CORRELATED GRAPHS
+
+In this part, we analyze three ways of constructing correlated graphs, i.e AttrMasking (Hu et al., 2019) (with $30 \%$ attribute masking rate), DropEdge (Rong et al., 2020b) (with $10 \%$ edges dropped) and GraphDiffusion (Klicpera et al., 2019) (with heat kernel to derive a denser adjacency matrix), and evaluate them under the proposed GraphLoG framework. As shown in the first segment of Tab. 5, the AttrMasking strategy outperforms other two techniques with a clear margin, which is mainly ascribed to the fact that, compared with dropping or adding edges, masking node attributes can preserve the matching degree of correlated graph pairs to a greater extent (referring to the theoretical analysis in Sec. A).
+
+Table 5: Ablation study for three model components on biological function prediction benchmark.   
+
+<table><tr><td rowspan=1 colspan=1>Model Components</td><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>ROC-AUC (%)</td></tr><tr><td rowspan=1 colspan=1>Correlated Graph Construction</td><td rowspan=1 colspan=1>AttrMaskingDropEdgeGraphDiffusion</td><td rowspan=1 colspan=1>72.8±0.471.5 ± 0.570.4± 0.7</td></tr><tr><td rowspan=1 colspan=1>Loss Format</td><td rowspan=1 colspan=1>InfoNCEHinge Loss</td><td rowspan=1 colspan=1>72.8 ± 0.472.3± 0.8</td></tr><tr><td rowspan=1 colspan=1>Clustering Algorithm</td><td rowspan=1 colspan=1>K-meansRPCLAdaptive-RPCL</td><td rowspan=1 colspan=1>72.2 ± 0.372.8± 0.472.9 ± 0.3</td></tr></table>
+
+# E.2 ABLATION STUDY ON LOSS FORMAT
+
+We investigate the effect of two loss formats, InfoNCE loss (van den Oord et al., 2018) and hingeloss-based contrastive loss (Hadsell et al., 2006), on our model. The conventional contrastive loss uses one negative sample for each positive pair and is in a hinge loss form, while the InfoNCE loss employs a large number of negative samples and is in the form of softmax. We modify the loss format in Eqs. 6, 7 and 13 to conduct the comparison. According to the second segment of Tab. 5, the InfoNCE loss marginally improve model’s performance, and both losses can achieve superior performance under the GraphLoG framework.
+
+# E.3 ABLATION STUDY ON CLUSTERING ALGORITHM
+
+We employ three clustering algorithms to derive hierarchical prototypes in our method and evaluate their corresponding pre-trained models on downstream task. For $\mathbf { K }$ -means, in the prototype initialization stage, we perform clustering hierarchically as in Eqs. 9 and 10 and obtain three hierarchies of prototypes with fixed number $M _ { 1 } = 1 0$ , $M _ { 2 } = 3 0$ and $M _ { 3 } = 1 0 0$ for each hierarchy, respectively. Also, we design an adaptive variant of RPCL (Xu et al., 1993), named as Adaptive-RPCL, which is able to adjust the number of prototypes during training. Specifically, a counter is additionally maintained for each bottom layer prototype to record the number of iterations from the last time that the prototype is updated by graph embeddings. When a counter reaches threshold $\gamma = 1 0 0$ , the corresponding bottom layer prototype is removed, and an upper layer prototype is removed if all the bottom layer prototypes in its corresponding tree are eliminated.
+
+In the third segment of Tab. 5, the performance on biological downstream task is reported for the pre-trained models using different clustering algorithms. It can be observed that two RPCL-based clustering methods marginally outperform K-means, and their performance is comparable with each other. These results illustrate that the proposed GraphLoG model is not too sensitive to the selection of clustering algorithm.
+
+# E.4 ROBUSTNESS OF CLUSTERING ALGORITHM
+
+In this experiment, we examine the robustness of RPCL clustering algorithm in the proposed GraphLoG model. Specifically, we conduct clustering-based hierarchical prototype initialization for six times and obtain six different self-supervised pre-training models based on distinct initialization. Fig. 4 plots the biological downstream task performance of these six models. We can observe that they perform comparably with each other, which demonstrates that the GraphLoG model is fairly robust to different clustering outputs.
+
+![](images/7328eb37d58f7c4216f2e7a32da3502bc9c80cbe191b012ebb5bb1e086be0f3d.jpg)  
+Figure 4: Test ROC-AUC $( \% )$ of six models derived by different clustering results on biological downstream task.
+
+# F SENSITIVITY ANALYSIS OF ATTRIBUTE MASKING
+
+![](images/877f48ab8db33104bf33f31dcdc051dd7cbb500a5708488c85d816a7f21e4254.jpg)  
+Figure 5: Sensitivity analysis of attribute masking rate on (a) molecular property prediction benchmarks, where the average test ROC-AUC $( \% )$ on eight downstream tasks is reported, and (b) biological function prediction benchmark.
+
+Setups. When varying the attribute masking rate to evaluate its sensitivity, other hyperparameters are fixed as the values depicted in Sec. 6.1. In specific, the hierarchical prototypes’ depth $L _ { p }$ is set as 3, and the exponential decay rate $\beta$ equals to 0.95.
+
+Results. In Fig. 5, we plot model’s performance on the downstream tasks of chemistry and biology domains under different masking rates. The highest test ROC-AUC is achieved when attribute masking rate is around $3 0 \%$ , which means that, under such settings, the constructed correlated graphs benefit the proposed learning scheme most.
+
+# G TRAINING CURVES
+
+In Fig. 6, we plot the training curves of four approaches, i.e. the random initialization baseline, context prediction (Hu et al., 2019), attribute masking (Hu et al., 2019) and the proposed GraphLoG method, on eight molecular property prediction tasks. From these line charts, we can observe that, through pre-training on a large-scale unlabeled dataset by GraphLoG, GNN model is able to converge at a higher ROC-AUC on the training set compared with other three methods.
+
+![](images/b990ff6ce2677b95844d98108563302066351bb6333340af2e9ff93da70005f9.jpg)  
+Figure 6: Training curves of different methods on eight downstream tasks of chemistry domain. The ROC-AUC $( \% )$ on training set is recorded along the training process.

parse/train/HJgExaVtwr/HJgExaVtwr.md

@@ -0,0 +1,313 @@
+# DIVIDEMIX: LEARNING WITH NOISY LABELS AS SEMI-SUPERVISED LEARNING
+
+Junnan Li, Richard Socher, Steven C.H. Hoi Salesforce Research {junnan.li,rsocher,shoi}@salesforce.com
+
+# ABSTRACT
+
+Deep neural networks are known to be annotation-hungry. Numerous efforts have been devoted to reducing the annotation cost when learning with deep networks. Two prominent directions include learning with noisy labels and semi-supervised learning by exploiting unlabeled data. In this work, we propose DivideMix, a novel framework for learning with noisy labels by leveraging semi-supervised learning techniques. In particular, DivideMix models the per-sample loss distribution with a mixture model to dynamically divide the training data into a labeled set with clean samples and an unlabeled set with noisy samples, and trains the model on both the labeled and unlabeled data in a semi-supervised manner. To avoid confirmation bias, we simultaneously train two diverged networks where each network uses the dataset division from the other network. During the semi-supervised training phase, we improve the MixMatch strategy by performing label co-refinement and label co-guessing on labeled and unlabeled samples, respectively. Experiments on multiple benchmark datasets demonstrate substantial improvements over state-of-the-art methods. Code is available at https://github.com/LiJunnan1992/DivideMix.
+
+# 1 INTRODUCTION
+
+The remarkable success in training deep neural networks (DNNs) is largely attributed to the collection of large datasets with human annotated labels. However, it is extremely expensive and time-consuming to label extensive data with high-quality annotations. On the other hand, there exist alternative and inexpensive methods for mining large-scale data with labels, such as querying commercial search engines (Li et al., 2017a), downloading social media images with tags (Mahajan et al., 2018), leveraging machine-generated labels (Kuznetsova et al., 2018), or using a single annotator to label each sample (Tanno et al., 2019). These alternative methods inevitably yield samples with noisy labels. A recent study (Zhang et al., 2017) shows that DNNs can easily overfit to noisy labels and results in poor generalization performance.
+
+Existing methods on learning with noisy labels (LNL) primarily take a loss correction approach. Some methods estimate the noise transition matrix and use it to correct the loss function (Patrini et al., 2017; Goldberger & Ben-Reuven, 2017). However, correctly estimating the noise transition matrix is challenging. Some methods leverage the predictions from DNNs to correct labels and modify the loss accordingly (Reed et al., 2015; Tanaka et al., 2018). These methods do not perform well under high noise ratio as the predictions from DNNs would dominate training and cause overfitting. To overcome this, Arazo et al. (2019) adopt MixUp (Zhang et al., 2018) augmentation. Another approach selects or reweights samples so that noisy samples contribute less to the loss (Jiang et al., 2018; Ren et al., 2018). A challenging issue is to design a reliable criteria to select clean samples. It has been shown that DNNs tend to learn simple patterns first before fitting label noise (Arpit et al., 2017). Therefore, many methods treat samples with small loss as clean ones (Jiang et al., 2018; Arazo et al., 2019). Among those methods, Co-teaching (Han et al., 2018) and Co-teaching $^ +$ (Yu et al., 2019) train two networks where each network selects small-loss samples in a mini-batch to train the other.
+
+Another active area of research that also aims to reduce annotation cost is semi-supervised learning (SSL). In SSL, the training data consists of unlabeled samples in addition to the labeled samples. Significant progress has been made in leveraging unlabeled samples by enforcing the model to produce low entropy predictions on unlabeled data (Grandvalet & Bengio, 2004) or consistent predictions on perturbed input (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2019). Recently, Berthelot et al. (2019) propose MixMatch, which unifies several dominant SSL approaches in one framework and achieves state-of-the-art performance.
+
+Despite the individual advances in LNL and SSL, their connection has been underexplored. In this work, we propose DivideMix, which addresses learning with label noise in a semi-supervised manner. Different from most existing LNL approaches, DivideMix discards the sample labels that are highly likely to be noisy, and leverages the noisy samples as unlabeled data to regularize the model from overfitting and improve generalization performance. The key contributions of this work are:
+
+• We propose co-divide, which trains two networks simultaneously. For each network, we dynamically fit a Gaussian Mixture Model (GMM) on its per-sample loss distribution to divide the training samples into a labeled set and an unlabeled set. The divided data is then used to train the other network. Co-divide keeps the two networks diverged, so that they can filter different types of error and avoid confirmation bias in self-training. During SSL phase, we improve MixMatch with label co-refinement and co-guessing to account for label noise. For labeled samples, we refine their ground-truth labels using the network’s predictions guided by the GMM for the other network. For unlabeled samples, we use the ensemble of both networks to make reliable guesses for their labels. We experimentally show that DivideMix significantly advances state-of-the-art results on multiple benchmarks with different types and levels of label noise. We also provide extensive ablation study and qualitative results to examine the effect of different components.
+
+# 2 RELATED WORK
+
+# 2.1 LEARNING WITH NOISY LABELS
+
+Most existing methods for training DNNs with noisy labels seek to correct the loss function. The correction can be categorized in two types. The first type treats all samples equally and correct loss either explicitly or implicitly through relabeling the noisy samples. For relabeling methods, the noisy samples are modeled with directed graphical models (Xiao et al., 2015), Conditional Random Fields (Vahdat, 2017), knowledge graph (Li et al., 2017b), or DNNs (Veit et al., 2017; Lee et al., 2018). However, they require access to a small set of clean samples. Recently, Tanaka et al. (2018) and Yi & Wu (2019) propose iterative methods which relabel samples using network predictions. For explicit loss correction. Reed et al. (2015) propose a bootstrapping method which modifies the loss with model predictions, and Ma et al. (2018) improve the bootstrapping method by exploiting the dimensionality of feature subspaces. Patrini et al. (2017) estimate the label corruption matrix for loss correction, and Hendrycks et al. (2018) improve the corruption matrix by using a clean set of data. The second type of correction focuses on reweighting training samples or separating clean and noisy samples, which results in correcting the loss function (Thulasidasan et al., 2019; Konstantinov & Lampert, 2019). A common method is to consider samples with smaller loss as clean ones (Shen & Sanghavi, 2019). Jiang et al. (2018) train a mentor network to guide a student network by assigning weights to samples. Ren et al. (2018) reweight samples based on their gradient directions. Chen et al. (2019) apply cross validation to identify clean samples. Arazo et al. (2019) calculate sample weights by modeling per-sample loss with a mixture model. Han et al. (2018) train two networks which select small-loss samples within each mini-batch to train each other, and Yu et al. (2019) improve it by updating the network on disagreement data to keep the two networks diverged.
+
+Contrary to all aforementioned methods, our method discards the labels that are highly likely to be noisy, and utilize the noisy samples as unlabeled data to regularize training in a SSL manner. Ding et al. (2018) and Kong et al. (2019) have shown that SSL method is effective in LNL. However, their methods do not perform well under high levels of noise, whereas our method can better distinguish and utilize noisy samples. Besides leveraging SSL, our method also introduces other advantages. Compared to self-training methods (Jiang et al., 2018; Arazo et al., 2019), our method can avoid the confirmation bias problem (Tarvainen & Valpola, 2017) by training two networks to filter error for each other. Compared to Co-teaching (Han et al., 2018) and Co-teaching $^ +$ (Yu et al., 2019), our method is more robust to noise by enabling the two networks to teach each other implicitly at each epoch (co-divide) and explicitly at each mini-batch (label co-refinement and co-guessing).
+
+![](images/a67a30a4e81a352859b3682687cd3a281448848f5d56340f469951f6cf837e0b.jpg)  
+Figure 1: DivideMix trains two networks (A and B) simultaneously. At each epoch, a network models its per-sample loss distribution with a GMM to divide the dataset into a labeled set (mostly clean) and an unlabeled set (mostly noisy), which is then used as training data for the other network (i.e. co-divide). At each mini-batch, a network performs semi-supervised training using an improved MixMatch method. We perform label co-refinement on the labeled samples and label co-guessing on the unlabeled samples.
+
+# 2.2 SEMI-SUPERVISED LEARNING
+
+SSL methods aim to improve the model’s performance by leveraging unlabeled data. Current state-of-the-art SSL methods mostly involve adding an additional loss term on unlabeled data to regularize training. The regularization falls into two classes: consistency regularization (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2019) enforces the model to produce consistent predictions on augmented input data; entropy minimization (Grandvalet & Bengio, 2004; Lee, 2013) encourages the model to give high-confidence predictions on unlabeled data. Recently, Berthelot et al. (2019) propose MixMatch, which unifies consistency regularization, entropy minimization, and the MixUp (Zhang et al., 2018) regularization into one framework.
+
+# 3 METHOD
+
+In this section, we introduce DivideMix, our proposed method for learning with noisy labels. An overview of the method is shown in Figure 1. To avoid confirmation bias of self-training where the model would accumulate its errors, we simultaneously train two networks to filter errors for each other through epoch-level implicit teaching and batch-level explicit teaching. At each epoch, we perform co-divide, where one network divides the noisy training dataset into a clean labeled set $( \mathcal { X } )$ and a noisy unlabeled set $( \mathcal { U } )$ , which are then used by the other network. At each mini-batch, one network utilizes both labeled and unlabeled samples to perform semi-supervised learning guided by the other network. Algorithm 1 delineates the full algorithm.
+
+# 3.1 CO-DIVIDE BY LOSS MODELING
+
+Deep networks tend to learn clean samples faster than noisy samples (Arpit et al., 2017), leading to lower loss for clean samples (Han et al., 2018; Chen et al., 2019). Following Arazo et al. (2019), we aim to find the probability of a sample being clean by fitting a mixture model to the per-sample loss distribution. Formally, let $\mathcal D = ( \dot { \mathcal X } , \mathcal y ) = \bar { \{ ( x _ { i } , y _ { i } ) \} } _ { i = 1 } ^ { \bar { N } }$ denote the training data, where $x _ { i }$ is an image and $y _ { i } \in \{ 0 , 1 \} ^ { C }$ is the one-hot label over $C$ classes. Given a model with parameters $\theta$ , the cross-entropy loss $\ell ( \theta )$ reflects how well the model fits the training samples:
+
+$$
+\ell ( \theta ) = \left\{ \ell _ { i } \right\} _ { i = 1 } ^ { N } = \big \{ - \sum _ { c = 1 } ^ { C } y _ { i } ^ { c } \log ( \mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } } ( x _ { i } ; \theta ) ) \big \} _ { i = 1 } ^ { N } ,
+$$
+
+where $\mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } }$ is the model’s output softmax probability for class $c$ .
+
+Arazo et al. (2019) fit a two-component Beta Mixture Model (BMM) to the max-normalized loss $\ell$ to model the distribution of clean and noisy samples. However, we find that BMM tends to produce undesirable flat distributions and fails when the label noise is asymmetric. Instead, Gaussian Mixture Model (GMM) (Permuter et al., 2006) can better distinguish clean and noisy samples due to its flexibility in the sharpness of distribution. Therefore, we fit a two-component GMM to $\ell$ using the Expectation-Maximization algorithm. For each sample, its clean probability $w _ { i }$ is the posterior probability $p ( g | \ell _ { i } )$ , where $g$ is the Gaussian component with smaller mean (smaller loss).
+
+We divide the training data into a labeled set and an unlabeled set by setting a threshold $\tau$ on $w _ { i }$ However, training a model using the data divided by itself could lead to confirmation bias (i.e. the model is prone to confirm its mistakes (Tarvainen & Valpola, 2017)), as noisy samples that are wrongly grouped into the labeled set would keep having lower loss due to the model overfitting to their labels. Therefore, we propose co-divide to avoid error accumulation. In co-divide, the GMM for one network is used to divide training data for the other network. The two networks are kept diverged from each other due to different (random) parameter initialization, different training data division, different (random) mini-batch sequence, and different training targets. Being diverged offers the two networks distinct abilities to filter different types of error, making the model more robust to noise.
+
+Algorithm 1: DivideMix. Line 4-8: co-divide; Line 17-18: label co-refinement; Line 20: label co-guessing.   
+
+<table><tr><td rowspan="18">1 Input: 0(1） and θ(2),training dataset (X,V),clean probability threshold T, number of augmentations M, sharpening temperature T,unsupervised loss weight Xu,Beta distribution parameter α for MixMatch. 2 0(1),0(2) =WarmUp(X,V,0(1),0(2)) 3 whilee&lt;MaxEpoch do W(2) = GMM(X,V,0(1)) 4 W(1) = GMM(X,V,0(2)) 5 for k = 1,2 do 6 x(k）={(xi,yi,Wi)|wi ≥ T,∀(xi,yi, Wi)∈(x,V,W(k）)} uck）={xilwi&lt;T,(xi,Wi）∈(x,W()} I/ unlabeled training set for 0(k) 8 for iter = 1 to num-iters do 9 From x(k),draw a mini-batch {(xb, yb,wb);b∈ (1,., B)} 10 FromU(k),draw a mini-batch {ub;b ∈(1,.., B)} 11 12 for b=1to B do 13 for m=1to M do xb,m = Augment(xb) I apply mth round of augmentation to xb 14 Ub,m = Augment(ub) Il apply mth round of augmentation to Ub 15 16 end 17 pb = M∑m Pmodel(@b,m;0(k)) ll average the predictions across augmentations of xb 18 yb = Wbyb+(1-Wb)pb / refine ground-truth label guided by the clean probability produced by the other network 19 yb = Sharpen(yb,T) // apply temperature sharpening to the refined label 20 qb = 2M ∑m (Pmodel(ub,m;0(1)) + Pmodel(ub,m;(2)) Il co-guessing: average the predictions from both networks across augmentations of ub 21 qb = Sharpen(qb,T) // apply temperature sharpening to the guessed label 22 end x={(xb,m, yb);b ∈(1,..,B),m ∈ (1,.,M)} 23 /augmentedlabeledmini-batch U={(ub,m,qb);b ∈(1,.,B),m ∈(1,.,M)} 24 I/augmentedunlabeledmini-batch</td></tr><tr><td>I/ standard training (with confidence penalty) I/ model per-sample loss with θ(1） to obtain clean proabability for θ(2) I// model per-sample loss with 0(2) to obtain clean proabability for 0(1) I/ train the two networks one by one I/ labeled training set for 0(k)</td></tr></table>
+
+Confidence Penalty for Asymmetric Noise. For initial convergence of the algorithm, we need to “warm up” the model for a few epochs by training on all data using the standard cross-entropy loss. The warm up is effective for symmetric (i.e. uniformly random) label noise. However, for asymmetric (i.e. class-conditional) label noise, the network would quickly overfit to noise during warm up and produce over-confident (low entropy) predictions, which leads to most samples having near-zero normalized loss (see Figure 2a). In such cases, the GMM cannot effectively distinguish clean and noisy samples based on the loss distribution. To address this issue, we penalize confident predictions from the network by adding a negative entropy term, $- \mathcal { H }$ (Pereyra et al., 2017), to the cross-entropy loss during warm up. The entropy of a model’s prediction for an input $x$ is defined as:
+
+$$
+\mathcal { H } = - \sum _ { c } \mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } } ( { \boldsymbol x } ; \theta ) \log ( \mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } } ( { \boldsymbol x } ; \theta ) ) ,
+$$
+
+By maximizing the entropy, $\ell$ becomes more evenly distributed (see Figure 2b) and easier to be modeled by the GMM. Furthermore, in Figure $2 \mathrm { c }$ we show $\ell$ when the model is trained with
+
+![](images/a06b0f0d146c0e8fb255b1fbdb2183f97e00c7fdd53ec56c742e21ea9e8896aa.jpg)  
+Figure 2: Training on CIFAR-10 with $40 \%$ asymmetric noise, warm up for 10 epochs. (a) Standard training with cross-entropy loss causes the model to overfit and produce over-confident predictions, making $\ell$ difficult to be modeled by the GMM. (b) Adding a confidence penalty (negative entropy) during warm up leads to more evenly-distributed $\ell$ . (c) Training with DivideMix can effectively reduce the loss for clean samples while keeping the loss larger for most noisy samples.
+
+DivideMix for 10 more epochs after warm up. The proposed method can significantly reduce the loss for clean samples while keeping the loss larger for most noisy samples.
+
+# 3.2 MIXMATCH WITH LABEL CO-REFINEMENT AND CO-GUESSING
+
+At each epoch, having divided the training data, we train the two networks one at a time while keeping the other one fixed. Given a mini-batch of labeled samples with their corresponding one-hot labels and clean probability, $\{ ( x _ { b } , y _ { b } , w _ { b } ) ; b \in ( 1 , . . . , B ) \}$ , and a mini-batch of unlabeled samples $\{ u _ { b } ; b \in ( 1 , . . . , \bar { B } ) \}$ , we exploit MixMatch (Berthelot et al., 2019) for SSL. MixMatch utilizes unlabeled data by merging consistency regularization (i.e. encourage the model to output same predictions on perturbed unlabeled data) and entropy minimization (i.e. encourage the model to output confident predictions on unlabeled data) with the MixUp (Zhang et al., 2018) augmentation (i.e. encourage the model to have linear behaviour between samples).
+
+To account for label noise, we make two improvements to MixMatch which enable the two networks to teach each other. First, we perform label co-refinement for labeled samples by linearly combining the ground-truth label $y _ { b }$ with the network’s prediction $p _ { b }$ (averaged across multiple augmentations of $x _ { b }$ ), guided by the clean probability $w _ { b }$ produced by the other network:
+
+$$
+\bar { y } _ { b } = w _ { b } y _ { b } + ( 1 - w _ { b } ) p _ { b } .
+$$
+
+Then we apply a sharpening function on the refined label to reduce its temperature:
+
+$$
+\hat { y } _ { b } = \mathrm { S h a r p e n } ( \bar { y } _ { b } , T ) = { \bar { y } _ { b } ^ { c } } ^ { \frac { 1 } { T } } \bigg / \sum _ { c = 1 } ^ { C } { \bar { y } _ { b } ^ { c } } ^ { \frac { 1 } { T } } , \mathrm { ~ f o r } c = 1 , 2 , . . . , C .
+$$
+
+Second, we use the ensemble of predictions from both networks to “co-guess” the labels for unlabeled samples (algorithm 1, line 20), which can produce more reliable guessed labels.
+
+Having acquired $\hat { \mathcal X }$ (and $\hat { \mathcal { U } }$ ) which consists of multiple augmentations of labeled (unlabeled) samples and their refined (guessed) labels, we follow MixMatch to “mix” the data, where each sample is interpolated with another sample randomly chosen from the combined mini-batch of $\hat { \mathcal X }$ and $\hat { \mathcal { U } }$ . Specifically, for a pair of samples $( x _ { 1 } , x _ { 2 } )$ and their corresponding labels $( p _ { 1 } , p _ { 2 } )$ , the mixed $( x ^ { \prime } , p ^ { \prime } )$ is computed by:
+
+$$
+\begin{array} { r l } & { \lambda \sim \mathrm { B e t a } ( \alpha , \alpha ) , } \\ & { \lambda ^ { \prime } = \operatorname* { m a x } ( \lambda , 1 - \lambda ) , } \\ & { x ^ { \prime } = \lambda ^ { \prime } x _ { 1 } + ( 1 - \lambda ^ { \prime } ) x _ { 2 } , } \\ & { p ^ { \prime } = \lambda ^ { \prime } p _ { 1 } + ( 1 - \lambda ^ { \prime } ) p _ { 2 } . } \end{array}
+$$
+
+MixMatch transforms $\hat { \mathcal X }$ and $\hat { \mathcal { U } }$ into $\mathcal { X } ^ { \prime }$ and $\mathcal { U } ^ { \prime }$ . Equation 6 ensures that $\mathcal { X } ^ { \prime }$ are “closer” to $\hat { \mathcal X }$ than $\hat { \mathcal { U } }$ The loss on $\mathcal { X } ^ { \prime }$ is the cross-entropy loss and the loss on $\mathcal { U } ^ { \prime }$ is the mean squared error:
+
+$$
+\begin{array} { l } { \mathcal { L } _ { \mathcal { X } } = - \displaystyle \frac { 1 } { | \mathcal { X } ^ { \prime } | } \sum _ { x , p \in \mathcal { X } ^ { \prime } } \sum _ { c } p _ { c } \log ( \mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } } ( x ; \theta ) ) , } \\ { \mathcal { L } _ { \mathcal { U } } = \displaystyle \frac { 1 } { | \mathcal { U } ^ { \prime } | } \sum _ { x , p \in \mathcal { U } ^ { \prime } } \| p - \mathrm { p } _ { \mathrm { m o d e l } } ( x ; \theta ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Under high levels of noise, the network would be encouraged to predict the same class to minimize the loss. To prevent assigning all samples to a single class, we apply the regularization term used by Tanaka et al. (2018) and Arazo et al. (2019), which uses a uniform prior distribution $\pi$ (i.e. $\pi _ { c } = 1 / C )$ ) to regularize the model’s average output across all samples in the mini-batch:
+
+$$
+\mathcal { L } _ { \mathrm { r e g } } = \sum _ { c } \pi _ { c } \log \left( \pi _ { c } \bigg / \frac { 1 } { | \mathcal { X } ^ { \prime } | + | \mathcal { U } ^ { \prime } | } \sum _ { \boldsymbol { x } \in \mathcal { X } ^ { \prime } + \mathcal { U } ^ { \prime } } \mathrm { p } _ { \mathrm { m o d e l } } ^ { \mathrm { c } } ( \boldsymbol { x } ; \boldsymbol { \theta } ) \right) .
+$$
+
+Finally, the total loss is:
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathcal { X } } + \lambda _ { u } \mathcal { L } _ { \mathcal { U } } + \lambda _ { r } \mathcal { L } _ { \mathrm { r e g } } .
+$$
+
+In our experiments, we set $\lambda _ { r }$ as 1 and use $\lambda _ { u }$ to control the strength of the unsupervised loss.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS AND IMPLEMENTATION DETAILS
+
+We extensively validate our method on four benchmark datasets, namely CIFAR-10, CIFAR100 (Krizhevsky & Hinton, 2009), Clothing1M (Xiao et al., 2015), and WebVision (Li et al., 2017a). Both CIFAR-10 and CIFAR-100 contain 50K training images and 10K test images of size $3 2 \times 3 2$ . Following previous works (Tanaka et al., 2018; Li et al., 2019), we experiment with two types of label noise: symmetric and asymmetric. Symmetric noise is generated by randomly replacing the labels for a percentage of the training data with all possible labels. Note that there is another criterion for symmetric label noise injection where the true labels cannot be maintained (Jiang et al., 2018; Wang et al., 2018), for which we also report the results (Table 6 in Appendix). Asymmetric noise is designed to mimic the structure of real-world label noise, where labels are only replaced by similar classes (e.g. deer horse, $\mathrm { 1 o g  c a t }$ ).
+
+We use an 18-layer PreAct Resnet (He et al., 2016) and train it using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 300 epochs. We set the initial learning rate as 0.02, and reduce it by a factor of 10 after 150 epochs. The warm up period is 10 epochs for CIFAR-10 and 30 epochs for CIFAR-100. We find that most hyperparameters introduced by DivideMix do not need to be heavily tuned. For all CIFAR experiments, we use the same hyperparameters $M = 2$ , $T = 0 . 5$ , and $\alpha = 4$ . $\tau$ is set as 0.5 except for $5 0 \%$ noise ratio when it is set as 0.6. We choose $\lambda _ { u }$ from $\{ 0 , 2 5 , 5 0 , 1 5 0 \}$ using a small validation set.
+
+Clothing1M and WebVision 1.0 are two large-scale datasets with real-world noisy labels. Clothing1M consists of 1 million training images collected from online shopping websites with labels generated from surrounding texts. We follow previous work (Li et al., 2019) and use ResNet-50 with ImageNet pretrained weights. WebVision contains 2.4 million images crawled from the web using the 1,000 concepts in ImageNet ILSVRC12. Following previous work (Chen et al., 2019), we compare baseline methods on the first 50 classes of the Google image subset using the inception-resnet v2 (Szegedy et al., 2017). The training details are delineated in Appendix B.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+We compare DivideMix with multiple baselines using the same network architecture. Here we introduce some of the most recent state-of-the-art methods: Meta-Learning (Li et al., 2019) proposes a gradient based method to find model parameters that are more noise-tolerant; Joint-Optim (Tanaka et al., 2018) and P-correction (Yi & Wu, 2019) jointly optimize the sample labels and the network parameters; M-correction (Arazo et al., 2019) models sample loss with BMM and applies MixUp.
+
+Table 1: Comparison with state-of-the-art methods in test accuracy $( \% )$ on CIFAR-10 and CIFAR-100 with symmetric noise. Methods marked by \* denote re-implementations based on public code.   
+
+<table><tr><td colspan="2">Dataset</td><td colspan="4">CIFAR-10 1</td><td colspan="4">CIFAR-100</td></tr><tr><td>Method/Noise ratio</td><td></td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td>Cross-Entropy</td><td>Best Last</td><td>86.8 82.7</td><td>79.4 57.9</td><td>62.9 26.1</td><td>42.7 16.8</td><td>62.0 61.8</td><td>46.7 37.3</td><td>19.9 8.8</td><td>10.1 3.5</td></tr><tr><td>Bootstrap (Reed et al., 2015)</td><td>Best Last</td><td>86.8 82.9</td><td>79.8 58.4</td><td>63.3 26.8</td><td>42.9 17.0</td><td>62.1 62.0</td><td>46.6 37.9</td><td>19.9 8.9</td><td>10.2 3.8</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>Best Last</td><td>86.8 83.1</td><td>79.8 59.4</td><td>63.3 26.2</td><td>42.9 18.8</td><td>61.5 61.4</td><td>46.6 37.3</td><td>19.9 9.0</td><td>10.2 3.4</td></tr><tr><td>Co-teaching+* (Yu et al., 2019)</td><td>Best Last</td><td>89.5 88.2</td><td>85.7 84.1</td><td>67.4 45.5</td><td>47.9 30.1</td><td>65.6 64.1</td><td>51.8 45.3</td><td>27.9 15.5</td><td>13.7 8.8</td></tr><tr><td>Mixup (Zhang et al., 2018)</td><td>Best Last</td><td>95.6 92.3</td><td>87.1 77.6</td><td>71.6 46.7</td><td>52.2 43.9</td><td>67.8 66.0</td><td>57.3 46.6</td><td>30.8 17.6</td><td>14.6 8.1</td></tr><tr><td>P-correction* (Yi &amp; Wu,2019)</td><td>Best Last</td><td>92.4 92.0</td><td>89.1 88.7</td><td>77.5 76.5</td><td>58.9 58.2</td><td>69.4 68.1</td><td>57.5 56.4</td><td>31.1 20.7</td><td>15.3 8.8</td></tr><tr><td>Meta-Learning* (Li et al., 2019)</td><td>Best Last</td><td>92.9 92.0</td><td>89.3 88.8</td><td>77.4 76.1</td><td>58.7 58.3</td><td>68.5 67.7</td><td>59.2 58.0</td><td>42.4 40.1</td><td>19.5 14.3</td></tr><tr><td>M-correction (Arazo et al., 2019)</td><td>Best Last</td><td>94.0 93.8</td><td>92.0 91.9</td><td>86.8 86.6</td><td>69.1 68.7</td><td>73.9 73.4</td><td>66.1 65.4</td><td>48.2 47.6</td><td>24.3 20.5</td></tr><tr><td>DivideMix</td><td>Best Last</td><td>96.1 95.7</td><td>94.6 94.4</td><td>93.2 92.9</td><td>76.0 75.4</td><td>77.3 76.9</td><td>74.6 74.2</td><td>60.2 59.6</td><td>31.5 31.0</td></tr></table>
+
+Note that none of these methods can consistently outperform others across different datasets. Mcorrection excels at symmetric noise, whereas Meta-Learning performs better for asymmetric noise.
+
+Table 1 shows the results on CIFAR-10 and CIFAR-100 with different levels of symmetric label noise ranging from $2 0 \%$ to $9 0 \%$ . We report both the best test accuracy across all epochs and the averaged test accuracy over the last 10 epochs. DivideMix outperforms state-of-the-art methods by a large margin across all noise ratios. The improvement is substantial $\mathord { \sim } 1 0 \%$ in accuracy) for the more challenging CIFAR-100 with high noise ratios. Appendix A shows comparison with more methods in Table 6. The results on CIFAR-10 with asymmetric noise is shown in Table 2. We use $4 0 \%$ because certain classes become theoretically indistinguishable for asymmetric noise larger than $5 0 \%$ .
+
+Table 2: Comparison with state-of-the-art methods in test accuracy $( \% )$ on CIFAR-10 with $40 \%$ asymmetric noise. We re-implement all methods under the same setting.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Best</td><td rowspan=1 colspan=1>Last</td></tr><tr><td rowspan=5 colspan=1>Cross-EntropyF-correction (Patrini et al.,2017)M-correction (Arazo et al., 2019)Iterative-CV (Chen et al.,2019)P-correction (Yi&amp; Wu,2019)Joint-Optim (Tanaka et al., 2018)Meta-Learning (Li et al., 2019)</td><td rowspan=2 colspan=1>85.087.287.4</td><td rowspan=1 colspan=1>72.3</td></tr><tr><td rowspan=2 colspan=1>83.186.388.088.1</td></tr><tr><td rowspan=1 colspan=1>88.688.5</td></tr><tr><td rowspan=1 colspan=1>88.9</td><td rowspan=1 colspan=1>88.4</td></tr><tr><td rowspan=1 colspan=1>89.2</td><td rowspan=1 colspan=1>88.6</td></tr><tr><td rowspan=1 colspan=1>DivideMix</td><td rowspan=1 colspan=1>93.4</td><td rowspan=1 colspan=1>92.1</td></tr></table>
+
+Table 3 and Table 4 show the results on Clothing1M and WebVision, respectively. DivideMix consistently outperforms state-of-the-art methods across all datasets with different types of label noise. For WebVision, we achieve more than $12 \%$ improvement in top-1 accuracy.
+
+Table 3: Comparison with state-of-the-art methods in test accuracy $( \% )$ on Clothing1M. Results for baselines are copied from original papers.   
+
+<table><tr><td>Method</td><td>Test Accuracy</td></tr><tr><td>Cross-Entropy</td><td>69.21</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>69.84</td></tr><tr><td>M-correction (Arazo et al., 2019)</td><td>71.00</td></tr><tr><td>Joint-Optim (Tanaka et al.,2018)</td><td>72.16</td></tr><tr><td>Meta-Cleaner (Zhang et al., 2019)</td><td>72.50</td></tr><tr><td>Meta-Learning (Li et al., 2019)</td><td>73.47</td></tr><tr><td>P-correction (Yi&amp; Wu, 2019)</td><td>73.49</td></tr><tr><td>DivideMix</td><td>74.76</td></tr></table>
+
+Table 4: Comparison with state-of-the-art methods trained on (mini) WebVision dataset. Numbers denote top-1 (top-5) accuracy $( \% )$ on the WebVision validation set and the ImageNet ILSVRC12 validation set. Results for baseline methods are copied from Chen et al. (2019).   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">WebVision</td><td colspan="2">ILSVRC12</td></tr><tr><td>top1</td><td>top5</td><td>top1</td><td>top5</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>61.12</td><td>82.68</td><td>57.36</td><td>82.36</td></tr><tr><td>Decoupling (Malach &amp; Shalev-Shwartz,2017)</td><td>62.54</td><td>84.74</td><td>58.26</td><td>82.26</td></tr><tr><td>D2L (Ma et al., 2018)</td><td>62.68</td><td>84.00</td><td>57.80</td><td>81.36</td></tr><tr><td>MentorNet (Jiang et al., 2018)</td><td>63.00</td><td>81.40</td><td>57.80</td><td>79.92</td></tr><tr><td>Co-teaching (Han et al., 2018)</td><td>63.58</td><td>85.20</td><td>61.48</td><td>84.70</td></tr><tr><td>Iterative-CV (Chen et al.,2019)</td><td>65.24</td><td>85.34</td><td>61.60</td><td>84.98</td></tr><tr><td>DivideMix</td><td>77.32</td><td>91.64</td><td>75.20</td><td>90.84</td></tr></table>
+
+# 4.3 ABLATION STUDY
+
+We study the effect of removing different components to provide insights into what makes DivideMix successful. We analyze the results in Table 5 as follows. Appendix C contains additional explanations.   
+
+<table><tr><td colspan="2">Dataset</td><td colspan="5">CIFAR-10</td><td colspan="4">CIFAR-100</td></tr><tr><td colspan="2">Noise type</td><td colspan="4">Sym.</td><td>I Asym.I</td><td colspan="4">Sym.</td></tr><tr><td colspan="2">Methods/Noise ratio</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>40%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td>DivideMix</td><td>Best Last</td><td>96.1 95.7</td><td>94.6 94.4</td><td>93.2 92.9</td><td>76.0 75.4</td><td>93.4 92.1</td><td>77.3 76.9</td><td>74.6 74.2</td><td>60.2 59.6</td><td>31.5 31.0</td></tr><tr><td>DivideMix with 0(1） test</td><td>Best Last</td><td>95.2 95.0</td><td>94.2 93.7</td><td>93.0 92.4</td><td>75.5 74.2</td><td>92.7 91.4</td><td>75.2 74.8</td><td>72.8 72.1</td><td>58.3 57.6</td><td>29.9 29.2</td></tr><tr><td>DivideMix w/o co-training</td><td>Best Last</td><td>95.0 94.8</td><td>94.0 93.3</td><td>92.6 92.2</td><td>74.3 73.2</td><td>91.9 90.6</td><td>74.8 74.1</td><td>72.3 71.7</td><td>56.7 56.3</td><td>27.7 27.2</td></tr><tr><td>DivideMix w/o label refinement</td><td>Best Last</td><td>96.0 95.5</td><td>94.6 94.2</td><td>93.0 92.7</td><td>73.7 73.0</td><td>87.7 86.3</td><td>76.9 76.4</td><td>74.2 73.9</td><td>58.7 58.2</td><td>26.9 26.3</td></tr><tr><td>DivideMix w/o augmentation</td><td>Best Last</td><td>95.3 94.9</td><td>94.1 93.5</td><td>92.2 91.8</td><td>73.9 73.0</td><td>89.5 88.4</td><td>76.5 76.2</td><td>73.1 72.6</td><td>58.2 58.0</td><td>26.9 26.4</td></tr><tr><td>Divide and MixMatch</td><td>Best Last</td><td>94.1 93.5</td><td>92.8 92.3</td><td>89.7 89.1</td><td>70.1 68.6</td><td>86.5 85.2</td><td>73.7 72.4</td><td>70.5 69.7</td><td>55.3 53.9</td><td>25.0 23.7</td></tr></table>
+
+Table 5: Ablation study results in terms of test accuracy $( \% )$ on CIFAR-10 and CIFAR-100.
+
+• To study the effect of model ensemble during test, we use the prediction from a single model $\theta ^ { ( 1 ) }$ instead of averaging the predictions from both networks as in DivideMix. Note that the training process remains unchanged. The decrease in accuracy suggests that the ensemble of two diverged networks consistently yields better performance during inference.
+
+• To study the effect of co-training, we train a single network using self-divide (i.e. divide the training data based on its own loss). The performance further decreases compared to $\theta ^ { ( 1 ) }$ .
+
+• We find that both label refinement and input augmentation are beneficial for DivideMix.   
+• We combine self-divide with the original MixMatch as a naive baseline for using SLL in LNL.
+
+Appendix A also introduces more in-depth studies in examining the robustness of our method to label noise, including the AUC for clean/noisy sample classification on CIFAR-10 training data, qualitative examples from Clothing1M where our method can effectively identify the noisy samples and leverage them as unlabeled data, and visualization results using t-SNE.
+
+# 5 CONCLUSION
+
+In this paper, we propose DivideMix for learning with noisy labels by leveraging SSL. Our method trains two networks simultaneously and achieves robustness to noise through dataset co-divide, label co-refinement and co-guessing. Through extensive experiments across multiple datasets, we show that DivideMix consistently exhibits substantial performance improvements compared to state-of-the-art methods. For future work, we are interested in incorporating additional ideas from SSL to LNL, and vice versa. Furthermore, we are also interested in adapting DivideMix to other domains such as NLP.
+
+# REFERENCES
+
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+
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+
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+
+# APPENDIX A ADDITIONAL EXPERIMENT RESULTS
+
+In Table 6, we compare DivideMix with previous methods under the symmetric noise setting where true labels cannot be maintained. DivideMix significantly outperforms previous methods which use deeper or wider network architectures.
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Architecture</td><td colspan="4">CIFAR-10</td><td colspan="4">I CIFAR-100</td></tr><tr><td>20%</td><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40%</td><td>60%</td><td>80%</td></tr><tr><td>MentorNet (Jiang et al.,2018)</td><td>WRN-101</td><td>92.0</td><td>89.0</td><td>1</td><td>49.0</td><td>73.0</td><td>68.0</td><td>1</td><td>35.0</td></tr><tr><td>D2L (Ma et al.,2018)</td><td>CNN-12/RN-44</td><td>85.1</td><td>83.4</td><td>72.8</td><td>1</td><td>62.2</td><td>52.0</td><td>42.3</td><td>-</td></tr><tr><td>Reweight (Ren et al.,2018)</td><td>WRN-28</td><td>86.9</td><td>1</td><td>1</td><td>-</td><td>61.3</td><td>1</td><td>1</td><td>-</td></tr><tr><td>Abstention (Thulasidasan et al.,2019)</td><td>WRN-28</td><td>93.4</td><td>90.9</td><td>87.6</td><td>70.8</td><td>75.8</td><td>68.2</td><td>59.4</td><td>34.1</td></tr><tr><td>DivideMix</td><td>PRN-18</td><td>96.2</td><td>94.9</td><td>94.3</td><td>79.8</td><td>77.2</td><td>75.2</td><td>72.0</td><td>60.0</td></tr></table>
+
+Table 6: Comparison with state-of-the-art methods in test accuracy $( \% )$ on CIFAR-10 and CIFAR-100 with symmetric noise. Numbers are copied from original papers. Key: WRN (Wide ResNet), PRN (PreActivation ResNet). DivideMix outperforms previous methods that use deeper/wider networks.
+
+In Figure 3, we show the Area Under a Curve (AUC) for clean/noisy sample classification on CIFAR10 training data from one of the GMMs during the first 100 epochs. Our method can effectively separate clean and noisy samples as training proceeds, even for high noise ratio.
+
+![](images/bcb55372fcaed61efff0227dc6c6de49ae78ac01fb4da87ef1bd0b38a9b1efca.jpg)  
+Figure 3: Area Under a Curve for clean/noisy image classification on CIFAR-10 training samples. Our method can effectively filter out the noisy samples and leverage them as unlabeled data.
+
+In Figure 4, we show example images in Clothing1M identified by our method as noisy samples. Our method achieves noise filtering by discarding the noisy labels (shown in red) and using the co-guessed labels (shown in blue) to regularize training.
+
+![](images/ef853d5909df4b7d178349eb6a60a79f51528ec2d16aa6131d951fb850dda7d2.jpg)  
+Figure 4: Clothing1M images identified as noisy samples by our method. Ground-truth labels are shown above in red and the co-guessed labels are shown below in blue.
+
+In Figure 5, we visualize the features of training images using t-SNE (Maaten & Hinton, 2008). The model is trained using DivideMix for 200 epochs on CIFAR-10 with $80 \%$ label noise. The embeddings form 10 distinct clusters corresponding to the true class labels, not the noisy training labels, which demonstrates our method’s robustness to label noise.
+
+![](images/698cb0a2f655f9563e0202a89cf2bbf53bd5df66057821d0f10b8d528c14484a.jpg)  
+Figure 5: T-SNE of training images after training the model using DivideMix for 200 epochs on CIFAR-10 with $80 \%$ label noise. Different colors indicate (a) noisy training labels or (b) true labels. DivideMix is able to learn the true class distribution of the training data despite the label noise.
+
+# APPENDIX B ADDITIONAL TRAINING DETAILS
+
+For CIFAR experiments, the only hyperparameter that we tune on a per-experiment basis is the unsupervised loss weight $\lambda _ { u }$ . Table 7 shows the value that we use. A larger $\lambda _ { u }$ is required for stronger regularization under high noise ratios or with more classes.
+
+For both Clothing1M and WebVision, we use the same set of hyperparameters $M = 2$ , $T = 0 . 5$ , $\tau = 0 . 5$ , $\lambda _ { u } = 0$ , $\alpha = 0 . 5$ , and train the network using SGD with a momentum of 0.9, a weight decay of 0.001, and a batch size of 32. The warm up period is 1 epoch. For Clothing1M, we train the network for 80 epochs. The initial learning rate is set as 0.002 and reduced by a factor of 10 after 40 epochs. For each epoch, we sample $1 0 0 0 \mathrm { { m i n i } }$ -batches from the training data while ensuring the labels (noisy) are balanced. For WebVision, we train the network for 100 epochs. The initial learning rate is set as 0.01 and reduced by a factor of 10 after 50 epochs.
+
+Table 7: Unsupervised loss weight $\lambda _ { u }$ for CIFAR experiments. Higher noise ratio requires stronger regularization from unlabeled samples.   
+
+<table><tr><td rowspan="2">Hyperparameter</td><td colspan="5">CIFAR-10</td><td colspan="4">CIFAR-100</td></tr><tr><td>Asym. 40%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>I 20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td></td><td>0</td><td>0</td><td>25</td><td>25</td><td>50</td><td>25</td><td>150</td><td>150</td><td>150</td></tr></table>
+
+# APPENDIX C ADDITIONAL EXPLANATIONS FOR ABLATION STUDY
+
+Here we clarify some details for the baseline methods in the ablation study. First, DivideMix w/o co-training still has dataset division, label refinement and label guessing, but performed by the same model. Thus, the performance drop (especially for CIFAR-100 with high noise ratio) suggests the disadvantage of self-training. Second, label refinement is important for high noise ratio because more noisy samples would be mistakenly divided into the labeled set. Third, augmentation improves performance through both producing more reliable predictions and achieving consistency regularization. In addition, same as Berthelot et al. (2019), we also find that temperature sharpening is essential for our method to perform well.
+
+# APPENDIX D TRAINING TIME ANALYSIS
+
+We analyse the training time of DivideMix to understand its efficiency. In Table 8, we compare the total training time of DivideMix on CIFAR-10 with several state-of-the-art methods, using a single Nvidia V100 GPU. DivideMix is slower than Co-teaching $^ +$ (Yu et al., 2019), but faster than P-correction (Yi & Wu, 2019) and Meta-Learning (Li et al., 2019) which involve multiple training iterations. In Table 9, we also break down the computation time for each operation in DivideMix.
+
+<table><tr><td>Co-teaching+*|P-correction丨Meta-Learning</td><td></td><td></td><td>DivideMix</td></tr><tr><td>4.3 h</td><td>6.0h</td><td>8.6h</td><td>5.2h</td></tr></table>
+
+Table 8: Comparison of total training time (hours) on CIFAR-10.   
+
+<table><tr><td></td><td></td><td>Co-Divide (Alg.1,line 4-8)|Data MixMatch (Alg.1,line 12-24)|Forward-Backward (Alg.1,line 25-27)</td></tr><tr><td>17.2 s</td><td>16.0 s</td><td>12.5 s</td></tr></table>
+
+Table 9: Computation time (seconds) per-epoch for each operation in DivideMix during training.

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+    {
+        "type": "text",
+        "text": "DIVIDEMIX: LEARNING WITH NOISY LABELS AS SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Junnan Li, Richard Socher, Steven C.H. Hoi Salesforce Research {junnan.li,rsocher,shoi}@salesforce.com ",
+        "bbox": [
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+            169,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            248,
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+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks are known to be annotation-hungry. Numerous efforts have been devoted to reducing the annotation cost when learning with deep networks. Two prominent directions include learning with noisy labels and semi-supervised learning by exploiting unlabeled data. In this work, we propose DivideMix, a novel framework for learning with noisy labels by leveraging semi-supervised learning techniques. In particular, DivideMix models the per-sample loss distribution with a mixture model to dynamically divide the training data into a labeled set with clean samples and an unlabeled set with noisy samples, and trains the model on both the labeled and unlabeled data in a semi-supervised manner. To avoid confirmation bias, we simultaneously train two diverged networks where each network uses the dataset division from the other network. During the semi-supervised training phase, we improve the MixMatch strategy by performing label co-refinement and label co-guessing on labeled and unlabeled samples, respectively. Experiments on multiple benchmark datasets demonstrate substantial improvements over state-of-the-art methods. Code is available at https://github.com/LiJunnan1992/DivideMix. ",
+        "bbox": [
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+            281,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            336,
+            546
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The remarkable success in training deep neural networks (DNNs) is largely attributed to the collection of large datasets with human annotated labels. However, it is extremely expensive and time-consuming to label extensive data with high-quality annotations. On the other hand, there exist alternative and inexpensive methods for mining large-scale data with labels, such as querying commercial search engines (Li et al., 2017a), downloading social media images with tags (Mahajan et al., 2018), leveraging machine-generated labels (Kuznetsova et al., 2018), or using a single annotator to label each sample (Tanno et al., 2019). These alternative methods inevitably yield samples with noisy labels. A recent study (Zhang et al., 2017) shows that DNNs can easily overfit to noisy labels and results in poor generalization performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing methods on learning with noisy labels (LNL) primarily take a loss correction approach. Some methods estimate the noise transition matrix and use it to correct the loss function (Patrini et al., 2017; Goldberger & Ben-Reuven, 2017). However, correctly estimating the noise transition matrix is challenging. Some methods leverage the predictions from DNNs to correct labels and modify the loss accordingly (Reed et al., 2015; Tanaka et al., 2018). These methods do not perform well under high noise ratio as the predictions from DNNs would dominate training and cause overfitting. To overcome this, Arazo et al. (2019) adopt MixUp (Zhang et al., 2018) augmentation. Another approach selects or reweights samples so that noisy samples contribute less to the loss (Jiang et al., 2018; Ren et al., 2018). A challenging issue is to design a reliable criteria to select clean samples. It has been shown that DNNs tend to learn simple patterns first before fitting label noise (Arpit et al., 2017). Therefore, many methods treat samples with small loss as clean ones (Jiang et al., 2018; Arazo et al., 2019). Among those methods, Co-teaching (Han et al., 2018) and Co-teaching $^ +$ (Yu et al., 2019) train two networks where each network selects small-loss samples in a mini-batch to train the other. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another active area of research that also aims to reduce annotation cost is semi-supervised learning (SSL). In SSL, the training data consists of unlabeled samples in addition to the labeled samples. Significant progress has been made in leveraging unlabeled samples by enforcing the model to produce low entropy predictions on unlabeled data (Grandvalet & Bengio, 2004) or consistent predictions on perturbed input (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2019). Recently, Berthelot et al. (2019) propose MixMatch, which unifies several dominant SSL approaches in one framework and achieves state-of-the-art performance. ",
+        "bbox": [
+            176,
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+        "page_idx": 0
+    },
+    {
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+    },
+    {
+        "type": "text",
+        "text": "Despite the individual advances in LNL and SSL, their connection has been underexplored. In this work, we propose DivideMix, which addresses learning with label noise in a semi-supervised manner. Different from most existing LNL approaches, DivideMix discards the sample labels that are highly likely to be noisy, and leverages the noisy samples as unlabeled data to regularize the model from overfitting and improve generalization performance. The key contributions of this work are: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose co-divide, which trains two networks simultaneously. For each network, we dynamically fit a Gaussian Mixture Model (GMM) on its per-sample loss distribution to divide the training samples into a labeled set and an unlabeled set. The divided data is then used to train the other network. Co-divide keeps the two networks diverged, so that they can filter different types of error and avoid confirmation bias in self-training. During SSL phase, we improve MixMatch with label co-refinement and co-guessing to account for label noise. For labeled samples, we refine their ground-truth labels using the network’s predictions guided by the GMM for the other network. For unlabeled samples, we use the ensemble of both networks to make reliable guesses for their labels. We experimentally show that DivideMix significantly advances state-of-the-art results on multiple benchmarks with different types and levels of label noise. We also provide extensive ablation study and qualitative results to examine the effect of different components. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 LEARNING WITH NOISY LABELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Most existing methods for training DNNs with noisy labels seek to correct the loss function. The correction can be categorized in two types. The first type treats all samples equally and correct loss either explicitly or implicitly through relabeling the noisy samples. For relabeling methods, the noisy samples are modeled with directed graphical models (Xiao et al., 2015), Conditional Random Fields (Vahdat, 2017), knowledge graph (Li et al., 2017b), or DNNs (Veit et al., 2017; Lee et al., 2018). However, they require access to a small set of clean samples. Recently, Tanaka et al. (2018) and Yi & Wu (2019) propose iterative methods which relabel samples using network predictions. For explicit loss correction. Reed et al. (2015) propose a bootstrapping method which modifies the loss with model predictions, and Ma et al. (2018) improve the bootstrapping method by exploiting the dimensionality of feature subspaces. Patrini et al. (2017) estimate the label corruption matrix for loss correction, and Hendrycks et al. (2018) improve the corruption matrix by using a clean set of data. The second type of correction focuses on reweighting training samples or separating clean and noisy samples, which results in correcting the loss function (Thulasidasan et al., 2019; Konstantinov & Lampert, 2019). A common method is to consider samples with smaller loss as clean ones (Shen & Sanghavi, 2019). Jiang et al. (2018) train a mentor network to guide a student network by assigning weights to samples. Ren et al. (2018) reweight samples based on their gradient directions. Chen et al. (2019) apply cross validation to identify clean samples. Arazo et al. (2019) calculate sample weights by modeling per-sample loss with a mixture model. Han et al. (2018) train two networks which select small-loss samples within each mini-batch to train each other, and Yu et al. (2019) improve it by updating the network on disagreement data to keep the two networks diverged. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contrary to all aforementioned methods, our method discards the labels that are highly likely to be noisy, and utilize the noisy samples as unlabeled data to regularize training in a SSL manner. Ding et al. (2018) and Kong et al. (2019) have shown that SSL method is effective in LNL. However, their methods do not perform well under high levels of noise, whereas our method can better distinguish and utilize noisy samples. Besides leveraging SSL, our method also introduces other advantages. Compared to self-training methods (Jiang et al., 2018; Arazo et al., 2019), our method can avoid the confirmation bias problem (Tarvainen & Valpola, 2017) by training two networks to filter error for each other. Compared to Co-teaching (Han et al., 2018) and Co-teaching $^ +$ (Yu et al., 2019), our method is more robust to noise by enabling the two networks to teach each other implicitly at each epoch (co-divide) and explicitly at each mini-batch (label co-refinement and co-guessing). ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/a67a30a4e81a352859b3682687cd3a281448848f5d56340f469951f6cf837e0b.jpg",
+        "image_caption": [
+            "Figure 1: DivideMix trains two networks (A and B) simultaneously. At each epoch, a network models its per-sample loss distribution with a GMM to divide the dataset into a labeled set (mostly clean) and an unlabeled set (mostly noisy), which is then used as training data for the other network (i.e. co-divide). At each mini-batch, a network performs semi-supervised training using an improved MixMatch method. We perform label co-refinement on the labeled samples and label co-guessing on the unlabeled samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 SEMI-SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "SSL methods aim to improve the model’s performance by leveraging unlabeled data. Current state-of-the-art SSL methods mostly involve adding an additional loss term on unlabeled data to regularize training. The regularization falls into two classes: consistency regularization (Laine & Aila, 2017; Tarvainen & Valpola, 2017; Miyato et al., 2019) enforces the model to produce consistent predictions on augmented input data; entropy minimization (Grandvalet & Bengio, 2004; Lee, 2013) encourages the model to give high-confidence predictions on unlabeled data. Recently, Berthelot et al. (2019) propose MixMatch, which unifies consistency regularization, entropy minimization, and the MixUp (Zhang et al., 2018) regularization into one framework. ",
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+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we introduce DivideMix, our proposed method for learning with noisy labels. An overview of the method is shown in Figure 1. To avoid confirmation bias of self-training where the model would accumulate its errors, we simultaneously train two networks to filter errors for each other through epoch-level implicit teaching and batch-level explicit teaching. At each epoch, we perform co-divide, where one network divides the noisy training dataset into a clean labeled set $( \\mathcal { X } )$ and a noisy unlabeled set $( \\mathcal { U } )$ , which are then used by the other network. At each mini-batch, one network utilizes both labeled and unlabeled samples to perform semi-supervised learning guided by the other network. Algorithm 1 delineates the full algorithm. ",
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+    {
+        "type": "text",
+        "text": "3.1 CO-DIVIDE BY LOSS MODELING ",
+        "text_level": 1,
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+        "text": "Deep networks tend to learn clean samples faster than noisy samples (Arpit et al., 2017), leading to lower loss for clean samples (Han et al., 2018; Chen et al., 2019). Following Arazo et al. (2019), we aim to find the probability of a sample being clean by fitting a mixture model to the per-sample loss distribution. Formally, let $\\mathcal D = ( \\dot { \\mathcal X } , \\mathcal y ) = \\bar { \\{ ( x _ { i } , y _ { i } ) \\} } _ { i = 1 } ^ { \\bar { N } }$ denote the training data, where $x _ { i }$ is an image and $y _ { i } \\in \\{ 0 , 1 \\} ^ { C }$ is the one-hot label over $C$ classes. Given a model with parameters $\\theta$ , the cross-entropy loss $\\ell ( \\theta )$ reflects how well the model fits the training samples: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/e5ceb5cbb72e40ae62652296ba0e15bdd57826ba89f029947f1892409e6cabd1.jpg",
+        "text": "$$\n\\ell ( \\theta ) = \\left\\{ \\ell _ { i } \\right\\} _ { i = 1 } ^ { N } = \\big \\{ - \\sum _ { c = 1 } ^ { C } y _ { i } ^ { c } \\log ( \\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } } ( x _ { i } ; \\theta ) ) \\big \\} _ { i = 1 } ^ { N } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } }$ is the model’s output softmax probability for class $c$ . ",
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+        "type": "text",
+        "text": "Arazo et al. (2019) fit a two-component Beta Mixture Model (BMM) to the max-normalized loss $\\ell$ to model the distribution of clean and noisy samples. However, we find that BMM tends to produce undesirable flat distributions and fails when the label noise is asymmetric. Instead, Gaussian Mixture Model (GMM) (Permuter et al., 2006) can better distinguish clean and noisy samples due to its flexibility in the sharpness of distribution. Therefore, we fit a two-component GMM to $\\ell$ using the Expectation-Maximization algorithm. For each sample, its clean probability $w _ { i }$ is the posterior probability $p ( g | \\ell _ { i } )$ , where $g$ is the Gaussian component with smaller mean (smaller loss). ",
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+        "type": "text",
+        "text": "We divide the training data into a labeled set and an unlabeled set by setting a threshold $\\tau$ on $w _ { i }$ However, training a model using the data divided by itself could lead to confirmation bias (i.e. the model is prone to confirm its mistakes (Tarvainen & Valpola, 2017)), as noisy samples that are wrongly grouped into the labeled set would keep having lower loss due to the model overfitting to their labels. Therefore, we propose co-divide to avoid error accumulation. In co-divide, the GMM for one network is used to divide training data for the other network. The two networks are kept diverged from each other due to different (random) parameter initialization, different training data division, different (random) mini-batch sequence, and different training targets. Being diverged offers the two networks distinct abilities to filter different types of error, making the model more robust to noise. ",
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+    {
+        "type": "table",
+        "img_path": "images/4e2ce29a44debd04f96e18d88fc0b23e81023203bf1a7ac45689558fe5fc36ef.jpg",
+        "table_caption": [
+            "Algorithm 1: DivideMix. Line 4-8: co-divide; Line 17-18: label co-refinement; Line 20: label co-guessing. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"18\">1 Input: 0(1） and θ(2),training dataset (X,V),clean probability threshold T, number of augmentations M, sharpening temperature T,unsupervised loss weight Xu,Beta distribution parameter α for MixMatch. 2 0(1),0(2) =WarmUp(X,V,0(1),0(2)) 3 whilee&lt;MaxEpoch do W(2) = GMM(X,V,0(1)) 4 W(1) = GMM(X,V,0(2)) 5 for k = 1,2 do 6 x(k）={(xi,yi,Wi)|wi ≥ T,∀(xi,yi, Wi)∈(x,V,W(k）)} uck）={xilwi&lt;T,(xi,Wi）∈(x,W()} I/ unlabeled training set for 0(k) 8 for iter = 1 to num-iters do 9 From x(k),draw a mini-batch {(xb, yb,wb);b∈ (1,., B)} 10 FromU(k),draw a mini-batch {ub;b ∈(1,.., B)} 11 12 for b=1to B do 13 for m=1to M do xb,m = Augment(xb) I apply mth round of augmentation to xb 14 Ub,m = Augment(ub) Il apply mth round of augmentation to Ub 15 16 end 17 pb = M∑m Pmodel(@b,m;0(k)) ll average the predictions across augmentations of xb 18 yb = Wbyb+(1-Wb)pb / refine ground-truth label guided by the clean probability produced by the other network 19 yb = Sharpen(yb,T) // apply temperature sharpening to the refined label 20 qb = 2M ∑m (Pmodel(ub,m;0(1)) + Pmodel(ub,m;(2)) Il co-guessing: average the predictions from both networks across augmentations of ub 21 qb = Sharpen(qb,T) // apply temperature sharpening to the guessed label 22 end x={(xb,m, yb);b ∈(1,..,B),m ∈ (1,.,M)} 23 /augmentedlabeledmini-batch U={(ub,m,qb);b ∈(1,.,B),m ∈(1,.,M)} 24 I/augmentedunlabeledmini-batch</td></tr><tr><td>I/ standard training (with confidence penalty) I/ model per-sample loss with θ(1） to obtain clean proabability for θ(2) I// model per-sample loss with 0(2) to obtain clean proabability for 0(1) I/ train the two networks one by one I/ labeled training set for 0(k)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Confidence Penalty for Asymmetric Noise. For initial convergence of the algorithm, we need to “warm up” the model for a few epochs by training on all data using the standard cross-entropy loss. The warm up is effective for symmetric (i.e. uniformly random) label noise. However, for asymmetric (i.e. class-conditional) label noise, the network would quickly overfit to noise during warm up and produce over-confident (low entropy) predictions, which leads to most samples having near-zero normalized loss (see Figure 2a). In such cases, the GMM cannot effectively distinguish clean and noisy samples based on the loss distribution. To address this issue, we penalize confident predictions from the network by adding a negative entropy term, $- \\mathcal { H }$ (Pereyra et al., 2017), to the cross-entropy loss during warm up. The entropy of a model’s prediction for an input $x$ is defined as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/833933b04987ab910ce9d352112bab0f64d6b3c56dfc159f2f80b31da2b47083.jpg",
+        "text": "$$\n\\mathcal { H } = - \\sum _ { c } \\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } } ( { \\boldsymbol x } ; \\theta ) \\log ( \\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } } ( { \\boldsymbol x } ; \\theta ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "By maximizing the entropy, $\\ell$ becomes more evenly distributed (see Figure 2b) and easier to be modeled by the GMM. Furthermore, in Figure $2 \\mathrm { c }$ we show $\\ell$ when the model is trained with ",
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+    {
+        "type": "image",
+        "img_path": "images/a06b0f0d146c0e8fb255b1fbdb2183f97e00c7fdd53ec56c742e21ea9e8896aa.jpg",
+        "image_caption": [
+            "Figure 2: Training on CIFAR-10 with $40 \\%$ asymmetric noise, warm up for 10 epochs. (a) Standard training with cross-entropy loss causes the model to overfit and produce over-confident predictions, making $\\ell$ difficult to be modeled by the GMM. (b) Adding a confidence penalty (negative entropy) during warm up leads to more evenly-distributed $\\ell$ . (c) Training with DivideMix can effectively reduce the loss for clean samples while keeping the loss larger for most noisy samples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "DivideMix for 10 more epochs after warm up. The proposed method can significantly reduce the loss for clean samples while keeping the loss larger for most noisy samples. ",
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+    {
+        "type": "text",
+        "text": "3.2 MIXMATCH WITH LABEL CO-REFINEMENT AND CO-GUESSING",
+        "text_level": 1,
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+        "type": "text",
+        "text": "At each epoch, having divided the training data, we train the two networks one at a time while keeping the other one fixed. Given a mini-batch of labeled samples with their corresponding one-hot labels and clean probability, $\\{ ( x _ { b } , y _ { b } , w _ { b } ) ; b \\in ( 1 , . . . , B ) \\}$ , and a mini-batch of unlabeled samples $\\{ u _ { b } ; b \\in ( 1 , . . . , \\bar { B } ) \\}$ , we exploit MixMatch (Berthelot et al., 2019) for SSL. MixMatch utilizes unlabeled data by merging consistency regularization (i.e. encourage the model to output same predictions on perturbed unlabeled data) and entropy minimization (i.e. encourage the model to output confident predictions on unlabeled data) with the MixUp (Zhang et al., 2018) augmentation (i.e. encourage the model to have linear behaviour between samples). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To account for label noise, we make two improvements to MixMatch which enable the two networks to teach each other. First, we perform label co-refinement for labeled samples by linearly combining the ground-truth label $y _ { b }$ with the network’s prediction $p _ { b }$ (averaged across multiple augmentations of $x _ { b }$ ), guided by the clean probability $w _ { b }$ produced by the other network: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7d8beda8a792f12c3a531f3025555af44d89d692d3493eccf43a84e3d8e7a990.jpg",
+        "text": "$$\n\\bar { y } _ { b } = w _ { b } y _ { b } + ( 1 - w _ { b } ) p _ { b } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Then we apply a sharpening function on the refined label to reduce its temperature: ",
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+    {
+        "type": "equation",
+        "img_path": "images/81073b78a9b2fb895b85689405431c27322aa7ebdfb7b5e12a38f1e4ad685cbb.jpg",
+        "text": "$$\n\\hat { y } _ { b } = \\mathrm { S h a r p e n } ( \\bar { y } _ { b } , T ) = { \\bar { y } _ { b } ^ { c } } ^ { \\frac { 1 } { T } } \\bigg / \\sum _ { c = 1 } ^ { C } { \\bar { y } _ { b } ^ { c } } ^ { \\frac { 1 } { T } } , \\mathrm { ~ f o r } c = 1 , 2 , . . . , C .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Second, we use the ensemble of predictions from both networks to “co-guess” the labels for unlabeled samples (algorithm 1, line 20), which can produce more reliable guessed labels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having acquired $\\hat { \\mathcal X }$ (and $\\hat { \\mathcal { U } }$ ) which consists of multiple augmentations of labeled (unlabeled) samples and their refined (guessed) labels, we follow MixMatch to “mix” the data, where each sample is interpolated with another sample randomly chosen from the combined mini-batch of $\\hat { \\mathcal X }$ and $\\hat { \\mathcal { U } }$ . Specifically, for a pair of samples $( x _ { 1 } , x _ { 2 } )$ and their corresponding labels $( p _ { 1 } , p _ { 2 } )$ , the mixed $( x ^ { \\prime } , p ^ { \\prime } )$ is computed by: ",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/3389117c0334befb06dec3ebb0dc695d87ca7007b0078986cef8de6f00d02979.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\lambda \\sim \\mathrm { B e t a } ( \\alpha , \\alpha ) , } \\\\ & { \\lambda ^ { \\prime } = \\operatorname* { m a x } ( \\lambda , 1 - \\lambda ) , } \\\\ & { x ^ { \\prime } = \\lambda ^ { \\prime } x _ { 1 } + ( 1 - \\lambda ^ { \\prime } ) x _ { 2 } , } \\\\ & { p ^ { \\prime } = \\lambda ^ { \\prime } p _ { 1 } + ( 1 - \\lambda ^ { \\prime } ) p _ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MixMatch transforms $\\hat { \\mathcal X }$ and $\\hat { \\mathcal { U } }$ into $\\mathcal { X } ^ { \\prime }$ and $\\mathcal { U } ^ { \\prime }$ . Equation 6 ensures that $\\mathcal { X } ^ { \\prime }$ are “closer” to $\\hat { \\mathcal X }$ than $\\hat { \\mathcal { U } }$ The loss on $\\mathcal { X } ^ { \\prime }$ is the cross-entropy loss and the loss on $\\mathcal { U } ^ { \\prime }$ is the mean squared error: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5347751a765c1637b78edd3f9cacebfa96464ac7f4f84e42a725c7205191699b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathcal { L } _ { \\mathcal { X } } = - \\displaystyle \\frac { 1 } { | \\mathcal { X } ^ { \\prime } | } \\sum _ { x , p \\in \\mathcal { X } ^ { \\prime } } \\sum _ { c } p _ { c } \\log ( \\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } } ( x ; \\theta ) ) , } \\\\ { \\mathcal { L } _ { \\mathcal { U } } = \\displaystyle \\frac { 1 } { | \\mathcal { U } ^ { \\prime } | } \\sum _ { x , p \\in \\mathcal { U } ^ { \\prime } } \\| p - \\mathrm { p } _ { \\mathrm { m o d e l } } ( x ; \\theta ) \\| _ { 2 } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Under high levels of noise, the network would be encouraged to predict the same class to minimize the loss. To prevent assigning all samples to a single class, we apply the regularization term used by Tanaka et al. (2018) and Arazo et al. (2019), which uses a uniform prior distribution $\\pi$ (i.e. $\\pi _ { c } = 1 / C )$ ) to regularize the model’s average output across all samples in the mini-batch: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fc8c285624f0d6d136fc9c54151515d9e9d103651e83757d5a61881fb3b1ac82.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { r e g } } = \\sum _ { c } \\pi _ { c } \\log \\left( \\pi _ { c } \\bigg / \\frac { 1 } { | \\mathcal { X } ^ { \\prime } | + | \\mathcal { U } ^ { \\prime } | } \\sum _ { \\boldsymbol { x } \\in \\mathcal { X } ^ { \\prime } + \\mathcal { U } ^ { \\prime } } \\mathrm { p } _ { \\mathrm { m o d e l } } ^ { \\mathrm { c } } ( \\boldsymbol { x } ; \\boldsymbol { \\theta } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Finally, the total loss is: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/835caf651d5b0f425b86c809ff47a7cbb532816c8d8c1b781c02836fcf1f4d46.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { \\mathcal { X } } + \\lambda _ { u } \\mathcal { L } _ { \\mathcal { U } } + \\lambda _ { r } \\mathcal { L } _ { \\mathrm { r e g } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In our experiments, we set $\\lambda _ { r }$ as 1 and use $\\lambda _ { u }$ to control the strength of the unsupervised loss. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS AND IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We extensively validate our method on four benchmark datasets, namely CIFAR-10, CIFAR100 (Krizhevsky & Hinton, 2009), Clothing1M (Xiao et al., 2015), and WebVision (Li et al., 2017a). Both CIFAR-10 and CIFAR-100 contain 50K training images and 10K test images of size $3 2 \\times 3 2$ . Following previous works (Tanaka et al., 2018; Li et al., 2019), we experiment with two types of label noise: symmetric and asymmetric. Symmetric noise is generated by randomly replacing the labels for a percentage of the training data with all possible labels. Note that there is another criterion for symmetric label noise injection where the true labels cannot be maintained (Jiang et al., 2018; Wang et al., 2018), for which we also report the results (Table 6 in Appendix). Asymmetric noise is designed to mimic the structure of real-world label noise, where labels are only replaced by similar classes (e.g. deer horse, $\\mathrm { 1 o g  c a t }$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use an 18-layer PreAct Resnet (He et al., 2016) and train it using SGD with a momentum of 0.9, a weight decay of 0.0005, and a batch size of 128. The network is trained for 300 epochs. We set the initial learning rate as 0.02, and reduce it by a factor of 10 after 150 epochs. The warm up period is 10 epochs for CIFAR-10 and 30 epochs for CIFAR-100. We find that most hyperparameters introduced by DivideMix do not need to be heavily tuned. For all CIFAR experiments, we use the same hyperparameters $M = 2$ , $T = 0 . 5$ , and $\\alpha = 4$ . $\\tau$ is set as 0.5 except for $5 0 \\%$ noise ratio when it is set as 0.6. We choose $\\lambda _ { u }$ from $\\{ 0 , 2 5 , 5 0 , 1 5 0 \\}$ using a small validation set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Clothing1M and WebVision 1.0 are two large-scale datasets with real-world noisy labels. Clothing1M consists of 1 million training images collected from online shopping websites with labels generated from surrounding texts. We follow previous work (Li et al., 2019) and use ResNet-50 with ImageNet pretrained weights. WebVision contains 2.4 million images crawled from the web using the 1,000 concepts in ImageNet ILSVRC12. Following previous work (Chen et al., 2019), we compare baseline methods on the first 50 classes of the Google image subset using the inception-resnet v2 (Szegedy et al., 2017). The training details are delineated in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPARISON WITH STATE-OF-THE-ART METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare DivideMix with multiple baselines using the same network architecture. Here we introduce some of the most recent state-of-the-art methods: Meta-Learning (Li et al., 2019) proposes a gradient based method to find model parameters that are more noise-tolerant; Joint-Optim (Tanaka et al., 2018) and P-correction (Yi & Wu, 2019) jointly optimize the sample labels and the network parameters; M-correction (Arazo et al., 2019) models sample loss with BMM and applies MixUp. ",
+        "bbox": [
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+            853,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/5242e14238fcd3653947be2deccb5dcdf634cbaa20f6aa7d6b972065b9e2bd60.jpg",
+        "table_caption": [
+            "Table 1: Comparison with state-of-the-art methods in test accuracy $( \\% )$ on CIFAR-10 and CIFAR-100 with symmetric noise. Methods marked by \\* denote re-implementations based on public code. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Dataset</td><td colspan=\"4\">CIFAR-10 1</td><td colspan=\"4\">CIFAR-100</td></tr><tr><td>Method/Noise ratio</td><td></td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td>Cross-Entropy</td><td>Best Last</td><td>86.8 82.7</td><td>79.4 57.9</td><td>62.9 26.1</td><td>42.7 16.8</td><td>62.0 61.8</td><td>46.7 37.3</td><td>19.9 8.8</td><td>10.1 3.5</td></tr><tr><td>Bootstrap (Reed et al., 2015)</td><td>Best Last</td><td>86.8 82.9</td><td>79.8 58.4</td><td>63.3 26.8</td><td>42.9 17.0</td><td>62.1 62.0</td><td>46.6 37.9</td><td>19.9 8.9</td><td>10.2 3.8</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>Best Last</td><td>86.8 83.1</td><td>79.8 59.4</td><td>63.3 26.2</td><td>42.9 18.8</td><td>61.5 61.4</td><td>46.6 37.3</td><td>19.9 9.0</td><td>10.2 3.4</td></tr><tr><td>Co-teaching+* (Yu et al., 2019)</td><td>Best Last</td><td>89.5 88.2</td><td>85.7 84.1</td><td>67.4 45.5</td><td>47.9 30.1</td><td>65.6 64.1</td><td>51.8 45.3</td><td>27.9 15.5</td><td>13.7 8.8</td></tr><tr><td>Mixup (Zhang et al., 2018)</td><td>Best Last</td><td>95.6 92.3</td><td>87.1 77.6</td><td>71.6 46.7</td><td>52.2 43.9</td><td>67.8 66.0</td><td>57.3 46.6</td><td>30.8 17.6</td><td>14.6 8.1</td></tr><tr><td>P-correction* (Yi &amp; Wu,2019)</td><td>Best Last</td><td>92.4 92.0</td><td>89.1 88.7</td><td>77.5 76.5</td><td>58.9 58.2</td><td>69.4 68.1</td><td>57.5 56.4</td><td>31.1 20.7</td><td>15.3 8.8</td></tr><tr><td>Meta-Learning* (Li et al., 2019)</td><td>Best Last</td><td>92.9 92.0</td><td>89.3 88.8</td><td>77.4 76.1</td><td>58.7 58.3</td><td>68.5 67.7</td><td>59.2 58.0</td><td>42.4 40.1</td><td>19.5 14.3</td></tr><tr><td>M-correction (Arazo et al., 2019)</td><td>Best Last</td><td>94.0 93.8</td><td>92.0 91.9</td><td>86.8 86.6</td><td>69.1 68.7</td><td>73.9 73.4</td><td>66.1 65.4</td><td>48.2 47.6</td><td>24.3 20.5</td></tr><tr><td>DivideMix</td><td>Best Last</td><td>96.1 95.7</td><td>94.6 94.4</td><td>93.2 92.9</td><td>76.0 75.4</td><td>77.3 76.9</td><td>74.6 74.2</td><td>60.2 59.6</td><td>31.5 31.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that none of these methods can consistently outperform others across different datasets. Mcorrection excels at symmetric noise, whereas Meta-Learning performs better for asymmetric noise. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the results on CIFAR-10 and CIFAR-100 with different levels of symmetric label noise ranging from $2 0 \\%$ to $9 0 \\%$ . We report both the best test accuracy across all epochs and the averaged test accuracy over the last 10 epochs. DivideMix outperforms state-of-the-art methods by a large margin across all noise ratios. The improvement is substantial $\\mathord { \\sim } 1 0 \\%$ in accuracy) for the more challenging CIFAR-100 with high noise ratios. Appendix A shows comparison with more methods in Table 6. The results on CIFAR-10 with asymmetric noise is shown in Table 2. We use $4 0 \\%$ because certain classes become theoretically indistinguishable for asymmetric noise larger than $5 0 \\%$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/73a65fc92a28d8057383a4de40d710e5bcf8e2ed68747a3a0451d17ba8bef640.jpg",
+        "table_caption": [
+            "Table 2: Comparison with state-of-the-art methods in test accuracy $( \\% )$ on CIFAR-10 with $40 \\%$ asymmetric noise. We re-implement all methods under the same setting. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Best</td><td rowspan=1 colspan=1>Last</td></tr><tr><td rowspan=5 colspan=1>Cross-EntropyF-correction (Patrini et al.,2017)M-correction (Arazo et al., 2019)Iterative-CV (Chen et al.,2019)P-correction (Yi&amp; Wu,2019)Joint-Optim (Tanaka et al., 2018)Meta-Learning (Li et al., 2019)</td><td rowspan=2 colspan=1>85.087.287.4</td><td rowspan=1 colspan=1>72.3</td></tr><tr><td rowspan=2 colspan=1>83.186.388.088.1</td></tr><tr><td rowspan=1 colspan=1>88.688.5</td></tr><tr><td rowspan=1 colspan=1>88.9</td><td rowspan=1 colspan=1>88.4</td></tr><tr><td rowspan=1 colspan=1>89.2</td><td rowspan=1 colspan=1>88.6</td></tr><tr><td rowspan=1 colspan=1>DivideMix</td><td rowspan=1 colspan=1>93.4</td><td rowspan=1 colspan=1>92.1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 3 and Table 4 show the results on Clothing1M and WebVision, respectively. DivideMix consistently outperforms state-of-the-art methods across all datasets with different types of label noise. For WebVision, we achieve more than $12 \\%$ improvement in top-1 accuracy. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5e7af623c6766a1addac489ef4df53b9010c794c0ec209735a71781edde1f71.jpg",
+        "table_caption": [
+            "Table 3: Comparison with state-of-the-art methods in test accuracy $( \\% )$ on Clothing1M. Results for baselines are copied from original papers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Test Accuracy</td></tr><tr><td>Cross-Entropy</td><td>69.21</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>69.84</td></tr><tr><td>M-correction (Arazo et al., 2019)</td><td>71.00</td></tr><tr><td>Joint-Optim (Tanaka et al.,2018)</td><td>72.16</td></tr><tr><td>Meta-Cleaner (Zhang et al., 2019)</td><td>72.50</td></tr><tr><td>Meta-Learning (Li et al., 2019)</td><td>73.47</td></tr><tr><td>P-correction (Yi&amp; Wu, 2019)</td><td>73.49</td></tr><tr><td>DivideMix</td><td>74.76</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/11d20d3a1f46de7c35d9f0dcd31c3acbba863287ab4169d22242ec16e3b987a8.jpg",
+        "table_caption": [
+            "Table 4: Comparison with state-of-the-art methods trained on (mini) WebVision dataset. Numbers denote top-1 (top-5) accuracy $( \\% )$ on the WebVision validation set and the ImageNet ILSVRC12 validation set. Results for baseline methods are copied from Chen et al. (2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">WebVision</td><td colspan=\"2\">ILSVRC12</td></tr><tr><td>top1</td><td>top5</td><td>top1</td><td>top5</td></tr><tr><td>F-correction (Patrini et al., 2017)</td><td>61.12</td><td>82.68</td><td>57.36</td><td>82.36</td></tr><tr><td>Decoupling (Malach &amp; Shalev-Shwartz,2017)</td><td>62.54</td><td>84.74</td><td>58.26</td><td>82.26</td></tr><tr><td>D2L (Ma et al., 2018)</td><td>62.68</td><td>84.00</td><td>57.80</td><td>81.36</td></tr><tr><td>MentorNet (Jiang et al., 2018)</td><td>63.00</td><td>81.40</td><td>57.80</td><td>79.92</td></tr><tr><td>Co-teaching (Han et al., 2018)</td><td>63.58</td><td>85.20</td><td>61.48</td><td>84.70</td></tr><tr><td>Iterative-CV (Chen et al.,2019)</td><td>65.24</td><td>85.34</td><td>61.60</td><td>84.98</td></tr><tr><td>DivideMix</td><td>77.32</td><td>91.64</td><td>75.20</td><td>90.84</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4f0e73f48fc64a2af3e39772c33248cf4e324b889c78e78dcb92408f7d47c1d7.jpg",
+        "table_caption": [
+            "We study the effect of removing different components to provide insights into what makes DivideMix successful. We analyze the results in Table 5 as follows. Appendix C contains additional explanations. "
+        ],
+        "table_footnote": [
+            "Table 5: Ablation study results in terms of test accuracy $( \\% )$ on CIFAR-10 and CIFAR-100. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Dataset</td><td colspan=\"5\">CIFAR-10</td><td colspan=\"4\">CIFAR-100</td></tr><tr><td colspan=\"2\">Noise type</td><td colspan=\"4\">Sym.</td><td>I Asym.I</td><td colspan=\"4\">Sym.</td></tr><tr><td colspan=\"2\">Methods/Noise ratio</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>40%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td>DivideMix</td><td>Best Last</td><td>96.1 95.7</td><td>94.6 94.4</td><td>93.2 92.9</td><td>76.0 75.4</td><td>93.4 92.1</td><td>77.3 76.9</td><td>74.6 74.2</td><td>60.2 59.6</td><td>31.5 31.0</td></tr><tr><td>DivideMix with 0(1） test</td><td>Best Last</td><td>95.2 95.0</td><td>94.2 93.7</td><td>93.0 92.4</td><td>75.5 74.2</td><td>92.7 91.4</td><td>75.2 74.8</td><td>72.8 72.1</td><td>58.3 57.6</td><td>29.9 29.2</td></tr><tr><td>DivideMix w/o co-training</td><td>Best Last</td><td>95.0 94.8</td><td>94.0 93.3</td><td>92.6 92.2</td><td>74.3 73.2</td><td>91.9 90.6</td><td>74.8 74.1</td><td>72.3 71.7</td><td>56.7 56.3</td><td>27.7 27.2</td></tr><tr><td>DivideMix w/o label refinement</td><td>Best Last</td><td>96.0 95.5</td><td>94.6 94.2</td><td>93.0 92.7</td><td>73.7 73.0</td><td>87.7 86.3</td><td>76.9 76.4</td><td>74.2 73.9</td><td>58.7 58.2</td><td>26.9 26.3</td></tr><tr><td>DivideMix w/o augmentation</td><td>Best Last</td><td>95.3 94.9</td><td>94.1 93.5</td><td>92.2 91.8</td><td>73.9 73.0</td><td>89.5 88.4</td><td>76.5 76.2</td><td>73.1 72.6</td><td>58.2 58.0</td><td>26.9 26.4</td></tr><tr><td>Divide and MixMatch</td><td>Best Last</td><td>94.1 93.5</td><td>92.8 92.3</td><td>89.7 89.1</td><td>70.1 68.6</td><td>86.5 85.2</td><td>73.7 72.4</td><td>70.5 69.7</td><td>55.3 53.9</td><td>25.0 23.7</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• To study the effect of model ensemble during test, we use the prediction from a single model $\\theta ^ { ( 1 ) }$ instead of averaging the predictions from both networks as in DivideMix. Note that the training process remains unchanged. The decrease in accuracy suggests that the ensemble of two diverged networks consistently yields better performance during inference. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• To study the effect of co-training, we train a single network using self-divide (i.e. divide the training data based on its own loss). The performance further decreases compared to $\\theta ^ { ( 1 ) }$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• We find that both label refinement and input augmentation are beneficial for DivideMix.   \n• We combine self-divide with the original MixMatch as a naive baseline for using SLL in LNL. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Appendix A also introduces more in-depth studies in examining the robustness of our method to label noise, including the AUC for clean/noisy sample classification on CIFAR-10 training data, qualitative examples from Clothing1M where our method can effectively identify the noisy samples and leverage them as unlabeled data, and visualization results using t-SNE. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            219,
+            318,
+            236
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose DivideMix for learning with noisy labels by leveraging SSL. Our method trains two networks simultaneously and achieves robustness to noise through dataset co-divide, label co-refinement and co-guessing. Through extensive experiments across multiple datasets, we show that DivideMix consistently exhibits substantial performance improvements compared to state-of-the-art methods. For future work, we are interested in incorporating additional ideas from SSL to LNL, and vice versa. Furthermore, we are also interested in adapting DivideMix to other domains such as NLP. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond empirical ´ risk minimization. In ICLR, 2018. ",
+        "bbox": [
+            173,
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "Weihe Zhang, Yali Wang, and Yu Qiao. Metacleaner: Learning to hallucinate clean representations for noisy-labeled visual recognition. In CVPR, 2019. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A ADDITIONAL EXPERIMENT RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            558,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Table 6, we compare DivideMix with previous methods under the symmetric noise setting where true labels cannot be maintained. DivideMix significantly outperforms previous methods which use deeper or wider network architectures. ",
+        "bbox": [
+            173,
+            136,
+            825,
+            178
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5e49ffa584abd15a1b668c6e88091409843b1617bd57185a2d83097793bda8d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Architecture</td><td colspan=\"4\">CIFAR-10</td><td colspan=\"4\">I CIFAR-100</td></tr><tr><td>20%</td><td>40%</td><td>60%</td><td>80%</td><td>20%</td><td>40%</td><td>60%</td><td>80%</td></tr><tr><td>MentorNet (Jiang et al.,2018)</td><td>WRN-101</td><td>92.0</td><td>89.0</td><td>1</td><td>49.0</td><td>73.0</td><td>68.0</td><td>1</td><td>35.0</td></tr><tr><td>D2L (Ma et al.,2018)</td><td>CNN-12/RN-44</td><td>85.1</td><td>83.4</td><td>72.8</td><td>1</td><td>62.2</td><td>52.0</td><td>42.3</td><td>-</td></tr><tr><td>Reweight (Ren et al.,2018)</td><td>WRN-28</td><td>86.9</td><td>1</td><td>1</td><td>-</td><td>61.3</td><td>1</td><td>1</td><td>-</td></tr><tr><td>Abstention (Thulasidasan et al.,2019)</td><td>WRN-28</td><td>93.4</td><td>90.9</td><td>87.6</td><td>70.8</td><td>75.8</td><td>68.2</td><td>59.4</td><td>34.1</td></tr><tr><td>DivideMix</td><td>PRN-18</td><td>96.2</td><td>94.9</td><td>94.3</td><td>79.8</td><td>77.2</td><td>75.2</td><td>72.0</td><td>60.0</td></tr></table>",
+        "bbox": [
+            174,
+            194,
+            821,
+            318
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Comparison with state-of-the-art methods in test accuracy $( \\% )$ on CIFAR-10 and CIFAR-100 with symmetric noise. Numbers are copied from original papers. Key: WRN (Wide ResNet), PRN (PreActivation ResNet). DivideMix outperforms previous methods that use deeper/wider networks. ",
+        "bbox": [
+            176,
+            328,
+            825,
+            371
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 3, we show the Area Under a Curve (AUC) for clean/noisy sample classification on CIFAR10 training data from one of the GMMs during the first 100 epochs. Our method can effectively separate clean and noisy samples as training proceeds, even for high noise ratio. ",
+        "bbox": [
+            173,
+            388,
+            826,
+            433
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/bcb55372fcaed61efff0227dc6c6de49ae78ac01fb4da87ef1bd0b38a9b1efca.jpg",
+        "image_caption": [
+            "Figure 3: Area Under a Curve for clean/noisy image classification on CIFAR-10 training samples. Our method can effectively filter out the noisy samples and leverage them as unlabeled data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            454,
+            700,
+            664
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 4, we show example images in Clothing1M identified by our method as noisy samples. Our method achieves noise filtering by discarding the noisy labels (shown in red) and using the co-guessed labels (shown in blue) to regularize training. ",
+        "bbox": [
+            173,
+            715,
+            823,
+            758
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef853d5909df4b7d178349eb6a60a79f51528ec2d16aa6131d951fb850dda7d2.jpg",
+        "image_caption": [
+            "Figure 4: Clothing1M images identified as noisy samples by our method. Ground-truth labels are shown above in red and the co-guessed labels are shown below in blue. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            775,
+            818,
+            888
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we visualize the features of training images using t-SNE (Maaten & Hinton, 2008). The model is trained using DivideMix for 200 epochs on CIFAR-10 with $80 \\%$ label noise. The embeddings form 10 distinct clusters corresponding to the true class labels, not the noisy training labels, which demonstrates our method’s robustness to label noise. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            159
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/698cb0a2f655f9563e0202a89cf2bbf53bd5df66057821d0f10b8d528c14484a.jpg",
+        "image_caption": [
+            "Figure 5: T-SNE of training images after training the model using DivideMix for 200 epochs on CIFAR-10 with $80 \\%$ label noise. Different colors indicate (a) noisy training labels or (b) true labels. DivideMix is able to learn the true class distribution of the training data despite the label noise. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            183,
+            795,
+            345
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B ADDITIONAL TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            436,
+            534,
+            452
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For CIFAR experiments, the only hyperparameter that we tune on a per-experiment basis is the unsupervised loss weight $\\lambda _ { u }$ . Table 7 shows the value that we use. A larger $\\lambda _ { u }$ is required for stronger regularization under high noise ratios or with more classes. ",
+        "bbox": [
+            174,
+            472,
+            825,
+            513
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For both Clothing1M and WebVision, we use the same set of hyperparameters $M = 2$ , $T = 0 . 5$ , $\\tau = 0 . 5$ , $\\lambda _ { u } = 0$ , $\\alpha = 0 . 5$ , and train the network using SGD with a momentum of 0.9, a weight decay of 0.001, and a batch size of 32. The warm up period is 1 epoch. For Clothing1M, we train the network for 80 epochs. The initial learning rate is set as 0.002 and reduced by a factor of 10 after 40 epochs. For each epoch, we sample $1 0 0 0 \\mathrm { { m i n i } }$ -batches from the training data while ensuring the labels (noisy) are balanced. For WebVision, we train the network for 100 epochs. The initial learning rate is set as 0.01 and reduced by a factor of 10 after 50 epochs. ",
+        "bbox": [
+            174,
+            520,
+            825,
+            619
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/63205127d28a5df62be19d820e005ee318490695cc789fc5cb7af4234ab8990e.jpg",
+        "table_caption": [
+            "Table 7: Unsupervised loss weight $\\lambda _ { u }$ for CIFAR experiments. Higher noise ratio requires stronger regularization from unlabeled samples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Hyperparameter</td><td colspan=\"5\">CIFAR-10</td><td colspan=\"4\">CIFAR-100</td></tr><tr><td>Asym. 40%</td><td>20%</td><td>50%</td><td>80%</td><td>90%</td><td>I 20%</td><td>50%</td><td>80%</td><td>90%</td></tr><tr><td></td><td>0</td><td>0</td><td>25</td><td>25</td><td>50</td><td>25</td><td>150</td><td>150</td><td>150</td></tr></table>",
+        "bbox": [
+            204,
+            638,
+            794,
+            698
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C ADDITIONAL EXPLANATIONS FOR ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            777,
+            666,
+            792
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here we clarify some details for the baseline methods in the ablation study. First, DivideMix w/o co-training still has dataset division, label refinement and label guessing, but performed by the same model. Thus, the performance drop (especially for CIFAR-100 with high noise ratio) suggests the disadvantage of self-training. Second, label refinement is important for high noise ratio because more noisy samples would be mistakenly divided into the labeled set. Third, augmentation improves performance through both producing more reliable predictions and achieving consistency regularization. In addition, same as Berthelot et al. (2019), we also find that temperature sharpening is essential for our method to perform well. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX D TRAINING TIME ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            490,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We analyse the training time of DivideMix to understand its efficiency. In Table 8, we compare the total training time of DivideMix on CIFAR-10 with several state-of-the-art methods, using a single Nvidia V100 GPU. DivideMix is slower than Co-teaching $^ +$ (Yu et al., 2019), but faster than P-correction (Yi & Wu, 2019) and Meta-Learning (Li et al., 2019) which involve multiple training iterations. In Table 9, we also break down the computation time for each operation in DivideMix. ",
+        "bbox": [
+            173,
+            132,
+            825,
+            204
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/c7a71e927b83fc62c14eda5712ef98d0c6f800a187051588b0234f83321755cf.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Co-teaching+*|P-correction丨Meta-Learning</td><td></td><td></td><td>DivideMix</td></tr><tr><td>4.3 h</td><td>6.0h</td><td>8.6h</td><td>5.2h</td></tr></table>",
+        "bbox": [
+            297,
+            217,
+            700,
+            257
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f374c2ef21fb96dbca04d9d72a3e9584ba31f1e364fa96cbb47327d1baaeb4ff.jpg",
+        "table_caption": [
+            "Table 8: Comparison of total training time (hours) on CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>Co-Divide (Alg.1,line 4-8)|Data MixMatch (Alg.1,line 12-24)|Forward-Backward (Alg.1,line 25-27)</td></tr><tr><td>17.2 s</td><td>16.0 s</td><td>12.5 s</td></tr></table>",
+        "bbox": [
+            179,
+            304,
+            816,
+            345
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 9: Computation time (seconds) per-epoch for each operation in DivideMix during training. ",
+        "bbox": [
+            176,
+            354,
+            816,
+            371
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HygrAR4tPS/HygrAR4tPS.md

@@ -0,0 +1,617 @@
+# ON EMPIRICAL COMPARISONS OF OPTIMIZERS FOR DEEP LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Selecting an optimizer is a central step in the contemporary deep learning pipeline. In this paper, we demonstrate the sensitivity of optimizer comparisons to the hyperparameter tuning protocol. Our findings suggest that the hyperparameter search space may be the single most important factor explaining the rankings obtained by recent empirical comparisons in the literature. In fact, we show that these results can be contradicted when hyperparameter search spaces are changed. As tuning effort grows without bound, more general optimizers should never underperform the ones they can approximate (i.e., Adam should never perform worse than momentum), but recent attempts to compare optimizers either assume these inclusion relationships are not practically relevant or restrict the hyperparameters in ways that break the inclusions. In our experiments, we find that inclusion relationships between optimizers matter in practice and always predict optimizer comparisons. In particular, we find that the popular adaptive gradient methods never underperform momentum or gradient descent. We also report practical tips around tuning often ignored hyperparameters of adaptive gradient methods and raise concerns about fairly benchmarking optimizers for neural network training.
+
+# 1 INTRODUCTION
+
+The optimization algorithm chosen by a deep learning practitioner determines the training speed and the final predictive performance of their model. To date, there is no theory that adequately explains how to make this choice. Instead, our community relies on empirical studies (Wilson et al., 2017) and benchmarking (Schneider et al., 2019). Indeed, it is the de facto standard that papers introducing new optimizers report extensive comparisons across a large number of workloads. Therefore, to maximize scientific progress, we must have confidence in our ability to make empirical comparisons between optimization algorithms.
+
+Although there is no theory guiding us when comparing optimizers, the popular first-order optimizers form a natural inclusion hierarchy. For example, ADAM (Kingma and Ba, 2015) and RMSPROP (Tieleman and Hinton, 2012) can approximately simulate MOMENTUM (Polyak, 1964) if the $\epsilon$ term in the denominator of their parameter updates is allowed to grow very large. However, these relationships may not matter in practice. For example, the settings of ADAM’s hyperparameters that allow it to match the performance of MOMENTUM may be too difficult to find (for instance, they may be infinite).
+
+In this paper, we demonstrate two important and interrelated points about empirical comparisons of neural network optimizers. First, we show that inclusion relationships between optimizers actually matter in practice; in our experiments, more general optimizers never underperform special cases. Despite conventional wisdom (Wilson et al., 2017; Balles and Hennig, 2017), we find that when carefully tuned, ADAM and other adaptive gradient methods never underperform MOMENTUM or SGD. Second, we demonstrate the sensitivity of optimizer comparisons to the hyperparameter tuning protocol. By comparing to previous experimental evaluations, we show how easy it is to change optimizer rankings on a given workload (model and dataset pair) by changing the hyperparameter tuning protocol, with optimizer rankings stabilizing according to inclusion relationships as we spend more and more effort tuning. Our findings raise serious questions about the practical relevance of conclusions drawn from these sorts of empirical comparisons.
+
+The remainder of this paper is structured as follows. In Section 2, we review related work, focusing on papers that make explicit claims about optimizer comparisons in deep learning and application papers that provide evidence about the tuning protocols of practitioners. We develop our definition of first-order optimizers in Section 3 along with a notion of inclusion relationships between optimizers. We present our experimental results in Section 4. Despite thorny methodological issues over how to avoid biases in comparisons due to search spaces that favor one optimizer over another, we believe that our experimental methodology is an acceptable compromise and has substantial practical relevance. Among other results, we show that the inclusion hierarchy of update rules is almost entirely predictive of optimizer comparisons. In particular, NADAM (Dozat, 2016) achieves the best top-1 validation accuracy on ResNet-50 on ImageNet in our experiments. The $7 7 . 1 \%$ we obtain with NADAM, although not as good as the $7 7 . 6 \%$ obtained using learned data augmentation by Cubuk et al. (2018), is better than the best existing published results using any of the more standard pre-processing pipelines $7 6 . 5 \%$ , due to Goyal et al. (2017) using MOMENTUM).
+
+# 2 BACKGROUND AND RELATED WORK
+
+Our work was inspired by the recent studies of neural network optimizers by Wilson et al. (2017) and Schneider et al. (2019). Wilson et al. (2017) constructed a simple classification problem in which adaptive gradient methods (e.g. ADAM) converge to provably worse solutions than standard gradient methods. However, crucially, their analysis ignored the $\epsilon$ parameter in the denominator of some adaptive gradient methods. Wilson et al. (2017) also presented experiments in which ADAM produced worse validation accuracy than SGD across all deep learning workloads considered. However, they only tuned over the learning rate and learning rate decay scheme in their experiments, leaving all other parameters of ADAM at fixed default values. Despite these findings, adaptive gradient methods continue to be popular since the work of Wilson et al. (2017). Schneider et al. (2019) presented a benchmark suite (DEEPOBS) for deep learning optimizers and reported that there was no single best optimizer across the workloads they considered. Yet Schneider et al. (2019) only tuned the learning rate of each optimizer and left all other hyperparameters at some fixed default values.
+
+As we discuss in Section 4.3, the choices of hyperparameter tuning protocols in Wilson et al. (2017) and Schneider et al. (2019) may be the most important factor preventing their results from being relevant to practical choices about which optimizer to use. Hyperparameter tuning is a crucial step of the deep learning pipeline (Bergstra and Bengio, 2012; Snoek et al., 2012; Sutskever et al., 2013; Smith, 2018), so it is critical for papers studying optimizers to match as closely as possible the tuning protocols of an ideal practitioner. Yet, tuning protocols often differ between works studying neural network optimizers and works concerned with training neural networks to solve specific problems.
+
+Recent papers that study or introduce optimization algorithms tend to compare to ADAM and RMSPROP without tuning their respective $\epsilon$ hyperparameters, presumably to simplify their experiments. It is standard to leave $\epsilon$ at the common default value of $1 0 ^ { - 8 }$ for ADAM and $1 0 ^ { - 1 0 ^ { - } }$ for RMSPROP (Tieleman and Hinton, 2012; Kingma and Ba, 2015; Dozat, 2016; Balles and Hennig, 2017; Loshchilov and Hutter, 2017; Zou and Shen, 2018; Ma and Yarats, 2018; Bernstein et al., 2018; Chen et al., 2019; Zou et al., 2019). Others do not even report the value of $\epsilon$ used (Balles and Hennig, 2017; Zhang and Mitliagkas, 2017; Keskar and Socher, 2017; Chen et al., 2018; Zhou et al., 2018; Aitchison, 2018; Reddi et al., 2019; Luo et al., 2019). There are exceptions. Zaheer et al. (2018) and Liu et al. (2019) considered $\epsilon$ values orders of magnitude larger than the standard default. However, the experiments in both papers gave only a limited consideration to $\epsilon$ , testing at most two values while tuning ADAM. De et al. (2018) is the only work we found that considered a broad range of values for $\epsilon$ . Both Zaheer et al. (2018) and De et al. (2018) found that non-default values of $\epsilon$ outperformed the default.
+
+While it is also extremely common in applications to use a default value of $\epsilon$ , some notable papers tuned $\epsilon$ and selected values up to eight orders of magnitude away from the common defaults. Szegedy et al. (2016) used $\epsilon = 1$ for RMSPROP; Liu et al. (2019) reported that their results were sensitive to $\epsilon$ and set $\epsilon = 1 0 ^ { - 6 }$ for ADAM; Tan et al. (2019) and Tan and Le (2019) set $\epsilon = 1 0 ^ { - 3 }$ for RMSPROP, the latter achieving state-of-the-art ImageNet top-1 accuracy. In reinforcement learning, Hessel et al. (2017) set $\epsilon = 1 . 5 \times 1 0 ^ { - 4 }$ .
+
+Despite being introduced solely to prevent division by zero1 , ADAM’s $\epsilon$ can be interpreted in ways that suggest the optimal choice is problem-dependent. If ADAM is interpreted as an empirical, diagonal approximation to natural gradient descent (Kingma and Ba, 2015), $\epsilon$ can be viewed as a multi-purpose damping term whose role is to improve the conditioning of the Fisher, in analogy to the approximate second-order method considered by Becker and Le Cun (1988). We can also view $\epsilon$ as setting a trust region radius (Martens and Grosse, 2015; Adolphs et al., 2019) and controlling an interpolation between momentum and diagonal natural gradient descent, by either diminishing or increasing the effect of $v _ { t }$ on the update direction. Under either interpretation, the best value for $\epsilon$ will be problem-dependent and likely benefit from tuning.
+
+# 3 WHAT IS AN OPTIMIZER?
+
+Optimization algorithms are typically defined by their update rule, which is controlled by hyperparameters that determine its behavior (e.g. the learning rate). Consider a differentiable loss function $\ell :  { \mathbb { R } ^ { d } } \to  { \mathbb { R } }$ whose vector of first partial derivatives is given by $\nabla \ell ( \theta )$ (more generally, $\nabla \ell ( \theta )$ might be a stochastic estimate of the true gradient). In our context, $\ell$ represents the loss function computed over an entire dataset by a neural network and $\theta \in \mathbb { R } ^ { d }$ represents the vector of model parameters. The optimization problem is to find a point that (at least locally) minimizes $\ell$ . First-order iterative methods for this problem (Nesterov, 2018) construct a sequence $\theta _ { t }$ of iterates converging to a local minimum $\theta _ { \star }$ using queries to $\ell$ and $\nabla \ell$ . The sequence $\theta _ { t }$ is constructed by an update rule $\mathcal { M }$ , which determines the next iterate $\theta _ { t + 1 }$ from the history $H _ { t }$ of previous iterates along with their function and gradient values, $H _ { t } = \{ \boldsymbol { \theta _ { s } } , \boldsymbol { \nabla } \ell ( \boldsymbol { \theta _ { s } } ) , \ell ( \boldsymbol { \theta _ { s } } ) \} _ { s = 0 } ^ { t }$ , and a setting of hyperparameters $\phi : \mathbb { N } \to \mathbb { R } ^ { n }$ . Given an initial parameter value $\theta _ { 0 } \in \mathbb { R } ^ { d }$ , the sequence of points visited by an optimizer with update rule $\mathcal { M }$ is given by,
+
+$$
+\theta _ { t + 1 } = \mathcal { M } ( H _ { t } , \phi _ { t } ) .
+$$
+
+The stochastic gradient descent algorithm (SGD; Robbins and Monro, 1951) is one of the simplest such methods used for training neural networks. SGD is initialized with $\theta _ { 0 } \in \mathbb { R } ^ { d }$ , and its hyperparameter is a learning rate schedule $\eta : \mathbb { N } \to ( 0 , \infty )$ . The SGD update rule is given by $\dot { \mathrm { S G D } } \bar { ( } H _ { t } , \eta _ { t } ) = \theta _ { t } - \eta _ { t } \nabla \bar { \ell } ( \theta _ { t } )$ . The MOMENTUM method due to Polyak (1964) generalizes the SGD method by linearly combining the gradient direction with a constant multiple of the previous parameter update. Its hyperparameters are a learning rate schedule $\eta : \mathbb { N } \ \overset { \cdot } {  } \ ( 0 , \infty )$ and a momentum parameter $\gamma \in [ 0 , \infty )$ ,
+
+$$
+\begin{array} { r } { \mathbf { M o M E N T U M } ( H _ { t } , \eta _ { t } , \gamma ) = \theta _ { t } - \eta _ { t } \nabla \ell ( \theta _ { t } ) + \gamma ( \theta _ { t } - \theta _ { t - 1 } ) . } \end{array}
+$$
+
+There has been an explosion of novel first-order methods in deep learning, all of which fall into this standard first-order scheme. In Table 1 we list the first-order update rules considered in this paper.
+
+The difference between optimizers is entirely captured by the choice of update rule $\mathcal { M }$ and hyperparameters $\phi$ . Since the roles of optimizer hyperparameters on neural network loss functions are not well-understood, most practitioners tune a subset of the hyperparameters to maximize performance over a validation set, while leaving some hyperparameters at fixed default values. The choice of which hyperparameters to tune determines an effective family of update rules, and this family is the critical object from a practitioners perspective. Thus, in analogy to (overloaded) function declarations in $\mathrm { C } { + } { + }$ , we define an optimizer by an update rule “signature,” the update rule name together with the free hyperparameter arguments. For example, in this definition $\mathbf { M O M E N T U M } \left( \cdot , \eta _ { t } , \gamma \right)$ is not the same optimizer as MOMENTUM $( \cdot , \eta _ { t } , 0 . 9 )$ , because the latter has two free hyperparameters while the former only has one. ADAM with the default $\epsilon$ is “different” from ADAM with tuned $\epsilon$ .
+
+# 3.1 THE TAXONOMY OF FIRST-ORDER METHODS
+
+The basic observation of this section is that some optimizers can approximately simulate others (i.e., optimizer A might be able to approximately simulate the trajectory of optimizer B for any particular setting of B’s hyperparameters). This is important knowledge because, as a hyperparameter tuning protocol approaches optimality, a more expressive optimizer can never underperform any of its specializations. To capture the concept of one optimizer approximating another, we define the following inclusion relationship between optimizers.
+
+Table 1: Update rules considered in this work. SGD is due to Robbins and Monro (1951), MOMENTUM to Polyak (1964), NESTEROV to Nesterov (1983), RMSPROP to Tieleman and Hinton (2012), and NADAM to Dozat (2016). All operations are taken component-wise for vectors. In particular, for $\boldsymbol { x } \in \mathbb { R } ^ { d }$ , $x ^ { 2 }$ is a component-wise power function.   
+
+<table><tr><td>SGD(Ht, nt)</td><td>ADAM(Ht,αt, β1, β2,∈)</td></tr><tr><td>0t+1=0t-ntVl(0t)</td><td>mo = 0,vo = 0</td></tr><tr><td>MOMENTUM(Ht, ηt, γ)</td><td>mt+1 = βimt +(1- βi)Vl(0t)</td></tr><tr><td>vo =0</td><td>Ut+1 = β2Ut +(1- β2)Vl(0t)²</td></tr><tr><td>Ut+1 = γUt +Vl(0t)</td><td>√1-β+1 bt+1</td></tr><tr><td>0t+1=0t-ntUt+1</td><td>1-β+1 mt+1</td></tr><tr><td>NESTEROV(Ht, nt,γ)</td><td>0t+1=0t-αt -bt+1 VUt+1+∈</td></tr><tr><td>vo=0 Ut+1= γUt + Vl(0t)</td><td>NADAM(Ht,αt, β1, β2,∈)</td></tr><tr><td>0t+1=0t-nt(γUt+1+Vl(0t))</td><td>mo = 0,vo = 0</td></tr><tr><td></td><td>mt+1= βimt+(1- βi)Vl(0t)</td></tr><tr><td>RMSPROP(Ht,It, γ, p,∈) Uo = 1,mo = 0</td><td>Ut+1 = β2Ut+(1-β2)Vl(0t)²</td></tr><tr><td>Ut+1 = pvt +(1- p)Vl(0t)²</td><td>V1-+1</td></tr><tr><td>mt mt+1 =γmt + Ve(0t)</td><td>bt+1 1-βt+1</td></tr><tr><td>√Ut+1+e</td><td>βimt+1+(1-βi)Vl(0t) 0t+1=0t-αt</td></tr><tr><td>0t+1=0t-mt+1</td><td>-bt+1 √Ut+1+E</td></tr></table>
+
+Definition 1 (Inclusion relationship). Let $\mathcal { M } , \mathcal { N }$ be update rules for use in a first-order optimization method. $\mathcal { M }$ is a subset or specialization of $\mathcal { N }$ , if for all $\phi : \mathbb { N } \to \mathbb { R } ^ { n }$ , there exists a sequence $\psi ^ { i } : \mathbb { N }  \mathbb { R } ^ { m }$ , such that for all $t \in [ 0 , \infty )$ and histories $H _ { t }$ ,
+
+$$
+\operatorname* { l i m } _ { i \to \infty } \mathcal { N } ( H _ { t } , \psi _ { t } ^ { i } ) = \mathcal { M } ( H _ { t } , \phi _ { t } )
+$$
+
+This is denoted ${ \mathcal { M } } \subseteq { \mathcal { N } }$ , with equality $\mathcal { M } = \mathcal { N }$ iff ${ \mathcal { M } } \subseteq { \mathcal { N } }$ and $\mathcal { N } \subseteq \mathcal { M }$ .
+
+Evidently SGD $\subseteq$ MOMENTUM, since $\mathrm { S G D } ( H _ { t } , \eta _ { t } ) \ = \ \mathrm { M o M E N T U M } ( H _ { t } , \eta _ { t } , 0 )$ . Many wellknown optimizers fall naturally into this taxonomy. In particular, we consider RMSPROP with momentum (Tieleman and Hinton, 2012), ADAM (Kingma and Ba, 2015) and NADAM (Dozat, 2016) (see Table 1) and show the following inclusions in the appendix.
+
+$$
+\begin{array} { l } { { \mathrm { S G D } } \subseteq { \mathrm { M O M E N T U M } } \subseteq { \mathrm { R M S P R O P } } } \\ { { \mathrm { S G D } } \subseteq { \mathrm { M O M E N T U M } } \subseteq { \mathrm { A D A M } } } \\ { { \mathrm { S G D } } \subseteq { \mathrm { N E S T E R O V } } \subseteq { \mathrm { N A D A M } } } \end{array}
+$$
+
+Note, some of these inclusions make use of the flexibility of hyperparameter schedules (dependence of $\psi ^ { i }$ on $t$ ). In particular, to approximate MOMENTUM with ADAM, one needs to choose a learning rate schedule that accounts for ADAM’s bias correction.
+
+If two optimizers have an inclusion relationship, the more general optimizer can never be worse with respect to any metric of interest, provided the hyperparameters are sufficiently tuned to optimize that metric. Optimally-tuned MOMENTUM cannot underperform optimally-tuned SGD, because setting $\gamma = 0$ in MOMENTUM recovers SGD. However, optimizers with more hyperparameters might be more expensive to tune, so we should have a theoretical or experimental reason for using (or creating) a more general optimizer. For example, MOMENTUM improves local convergence rates over SGD on twice-differentiable functions that are smooth and strongly convex (Polyak, 1964), and
+
+NESTEROV has globally optimal convergence rates within the class of smooth and strongly convex functions (Nesterov, 1983; 2018).
+
+At first glance, the taxonomy of optimizer inclusions appears to resolve many optimizer comparison questions. However, for a deep learning practitioner, there is no guarantee that the inclusion hierarchy is at all meaningful in practice. For example, the hyperparameters that allow ADAM to match or outperform MOMENTUM might not be easily accessible. They might exist only in the limit of very large values, or be so difficult to find that only practitioners with huge computational budgets can hope to discover them. Indeed, empirical studies and conventional wisdom hold that the inclusion hierarchy does not predict optimizer performance for many practical workloads (Wilson et al., 2017; Balles and Hennig, 2017; Schneider et al., 2019). Either these experimental investigations are too limited or the taxonomy of this section is of limited practical interest and provides no guidance about which optimizer to use on a real workload. In the following section we attempt to answer this question experimentally, and show that these inclusion relationships are meaningful in practice.
+
+# 4 EXPERIMENTS
+
+An empirical comparison of optimizers should aim to inform a careful practitioner. Accordingly, we model our protocol on a practitioner that is allowed to vary all optimization hyperparameters for each optimizer (e.g. $\alpha _ { t }$ , $\beta _ { 1 }$ , $\beta _ { 2 }$ , $\epsilon$ for ADAM) in addition to a parameterized learning rate decay schedule, in contrast to studies that fix a subset of the optimization hyperparameters to their default values (e.g. Wilson et al., 2017; Schneider et al., 2019). There is no standard method for selecting the values of these hyperparameters, but most practitioners tune at least a subset of the optimization hyperparameters by running a set of trials to maximize performance over the validation set. In our experiments, we run tens to hundreds of individual trials per workload. Given the variety of workloads we consider, this trial budget covers a wide range of computational budgets.
+
+Selecting the hyperparameter search space for each optimizer is a key methodological choice for any empirical comparison of optimizers. Prior studies have attempted to treat each optimizer fairly by using the same search space for all optimizers (e.g. Wilson et al., 2017; Schneider et al., 2019). However, this requires the assumption that similarly-named hyperparameters should take similar values between optimizers, which is not always true. For example, MOMENTUM and NESTEROV both have similar-looking momentum and learning rate hyperparameters, but NESTEROV tolerates larger values of its momentum hyperparameter (Sutskever et al., 2013), so any fixed search space will likely be more favorable for one of the two. The situation worsens with less closely related optimizers, and designing a search space that is equally appropriate for optimizers with incommensurate hyperparameters is almost impossible. Despite coming with its own set of challenges, it is most informative to compare optimizers assuming the practitioner is allowed to tune hyperparameters for different optimizers independently by way of optimizer-specific search spaces.
+
+In our experiments, we chose the search space for each optimizer by running an initial set of experiments over a relatively large search space. In a typical case, we ran a single set of initial trials per optimizer to select the final search space. However, in some cases we chose the initial search space poorly, so we ran another set of experiments to select the final search space. The effort required to choose each search space cannot simply be quantified by the number of initial trials; the provenance of each search space is difficult to trace exactly. In some cases, our search spaces were informed by published results or prior experience with particular models and optimizers. We note that this is true of all search spaces in the literature: they are hard-won treasures that tend to be refined over many experiments and across many workloads, representing the sum total of our community’s experience. We validated our search spaces by checking that that the optimal hyperparameter values were away from the search space boundaries for all optimizers in all experiments (see Figure 5 in Appendix E). We provide our final search spaces for all experiments in Appendix D. The fact that our final error rates compare favorably to prior published results – including reaching state-of-the-art for our particular configuration of ResNet-50 on ImageNet (see Section 4.2) – supports our claim that our methodology is highly competitive with expert tuning procedures.
+
+Table 2: Summary of workloads used in experiments.   
+
+<table><tr><td>Task</td><td>Evaluation metric</td><td>Model</td><td>Dataset</td><td>Target error</td><td>Batch size</td><td>Budget</td></tr><tr><td rowspan="5">Image classification</td><td rowspan="5">Classification</td><td>Simple CNN</td><td>Fashion MNIST</td><td>6.6%</td><td>256</td><td>10k steps</td></tr><tr><td>ResNet-32</td><td>CIFAR-10</td><td>7%</td><td>256</td><td>50k steps</td></tr><tr><td>CNN</td><td>CIFAR-100</td><td>二</td><td>256</td><td>350 epochs</td></tr><tr><td>VGG-16</td><td>CIFAR-10</td><td>1</td><td>128</td><td>250 epochs</td></tr><tr><td>ResNet-50</td><td>ImageNet</td><td>24%</td><td>1024</td><td>150k steps</td></tr><tr><td rowspan="2">Language modeling</td><td>Classification error</td><td>LSTM</td><td>War and Peace</td><td>1</td><td>50</td><td>200 epochs</td></tr><tr><td>Cross entropy</td><td>Transformer</td><td>LM1B</td><td>3.45</td><td>256</td><td>750k steps</td></tr></table>
+
+# 4.1 OVERVIEW OF WORKLOADS AND EXPERIMENTAL DETAILS
+
+We investigated the relative performance of optimizers across a variety of image classification and language modeling tasks. For image classification, we trained a simple convolutional neural network (Simple CNN) on Fashion MNIST (Xiao et al., 2017); ResNet-32 (He et al., 2016a) on CIFAR-10 (Krizhevsky, 2009); a CNN on CIFAR-100; VGG-16 (Simonyan and Zisserman, 2014) on CIFAR10; and ResNet-50 on ImageNet (Russakovsky et al., 2015). For language modeling, we trained a 2-layer LSTM model (Hochreiter and Schmidhuber, 1997) on Tolstoy’s War and Peace; and Transformer (Vaswani et al., 2017) on LM1B (Chelba et al., 2014). We used a linear learning rate decay schedule parameterized the same way as Shallue et al. (2019) for all workloads. We used a fixed batch size and a fixed budget of training steps for each workload independent of the optimizer. Table 2 summarizes these workloads and Appendix B provides the full details.
+
+Given a hypercube-shaped search space, our tuning protocol sought to model a practitioner with a fixed budget of trials trying to achieve the best outcome using tens of feasible trials (10, 50, or 100 depending on the workload).2 A feasible trial is any trial that achieves finite training loss. We used quasi-random uniform search (Bousquet et al., 2017), and continued the search until we obtained a fixed number of feasible trials. From those trials we considered two statistics. The first, in order to characterize the best outcome, is a metric of interest (e.g. test accuracy) corresponding to the trial achieving the optimum of some other metric (e.g. validation accuracy). The second, in order to characterize the speed of training, is the number of steps required to reach a fixed validation target conditional on at least one trial in the search having reached that target. We chose the target for each workload based on initial experiments and known values from the literature (see Table 2). We estimated means and uncertainties using the bootstrap procedure described in Appendix C.
+
+# 4.2 INCLUSION RELATIONSHIPS MATTER IN PRACTICE
+
+Figure 1 shows the final predictive performance of six optimizers on four different workloads after tuning hyperparameters to minimize validation error. Regardless of whether we compare final validation error or test error, the inclusion relationships hold in all cases – a more general optimizer never underperforms any of its specializations within the error bars. Similar results hold for training error (see Figure 9 in Appendix E). Training speed is also an important consideration, and Figure 2 demonstrates that the inclusion relationships also hold within error bars when we compare the number of steps required to reach a target validation error. Moreover, these results confirming the relevance of optimizer inclusion relationships do not depend on the exact step budgets or error targets we chose (see Figure 10 in Appendix E), although large changes to these values would require new experiments.
+
+![](images/0f8f4c0644525cc30b04f201b64e8223a2a269bf1f7e7672655f6b9f4acd4901.jpg)  
+Figure 1: The relative performance of optimizers is consistent with the inclusion relationships, regardless of whether we compare final validation error (top) or test error (bottom). For all workloads, we tuned the hyperparameters of each optimizer separately, and selected the trial that achieved the lowest final validation error.
+
+![](images/3e7d9f4ec1fea2fd410dcde921669f8cbbefce184cdf1765baaa4970c8a263b6.jpg)  
+Figure 2: The relative training speed of optimizers is consistent with the inclusion relationships. We measured (idealized) training speed as the number of training steps required to reach a target validation error (see Table 2 for the error targets).
+
+Of course, just because a more general optimizer is no worse than any of its specializations doesn’t mean the choice of optimizer makes a large difference on all workloads. For some workloads in Figures 1 and 2, all optimizers perform about the same, while other workloads have a clear ranking or even dramatic differences. For example, the choice of optimizer seems to make little difference for ResNet-32 on CIFAR-10; all optimizers achieve similar predictive performance and training speed. On the other hand, Transformer on LM1B exhibits a clear ranking in terms of predictive performance and training speed. For this workload, ADAM needs roughly half the steps that MOMENTUM requires to reach our target error, and, although not shown in Figure 2, roughly six times fewer steps to get the same result as SGD. These differences are clearly significant enough to matter to a practitioner, and highlight the practical importance of choosing the right optimizer for some workloads.
+
+The most general optimizers we considered were RMSPROP, ADAM, and NADAM, which do not include each other as special cases, and whose relative performance is not predicted by inclusion relationships. Across the workloads we considered, none of these optimizers emerged as the clear winner, although ADAM and NADAM generally seemed to have an edge over RMSPROP. For all of these optimizers, we sometimes had to set the $\epsilon$ parameter orders of magnitude larger than the default value in order to get good results. In particular, we achieved a validation accuracy of $7 7 . 1 \%$ for ResNet-50 on ImageNet using NADAM with $\epsilon = 9 4 7 5$ , a result that exceeds the $7 6 . 5 \%$ achieved by Goyal et al. (2017) using MOMENTUM. Across just these 4 workloads, the range of the optimal values of the $\epsilon$ parameter spanned 10 orders of magnitude. Faced with this reality, a practitioner might reasonably doubt their ability to find a value near the optimum. However, we found that we could reasonably expect to find a suitable value with only tens of trials. When tuning $\epsilon$ for ADAM or NADAM over a large range, we found it more efficient to search over $( \epsilon , \alpha _ { 0 } / \epsilon )$ instead of $( \epsilon , \alpha _ { 0 } )$ ; see Appendix $\mathrm { D }$ for more details.
+
+![](images/c5bc4e80e06d2923098c948b06bde025a18b9124729f376f093ddc1951dc86d6.jpg)  
+Figure 3: Tuning more hyperparameters removes the differences in test error between optimizers observed by Wilson et al. (2017). Tuning a subset of optimizer hyperparameters and the initial learning rate is sufficient to equalize performance between all optimizers (left). More extensive hyperparameter tuning in our setup, including the learning rate schedule, improves results for all optimizers and still does not produce any differences between optimizer performances (right).
+
+# 4.3 RECONCILING DISAGREEMENTS WITH PREVIOUS WORK
+
+In order to confirm that differences in hyperparameter tuning protocols explain the differences between our conclusions and those of Wilson et al. (2017) and Schneider et al. (2019), we reproduced a representative subset of their results and then inverted, or at least collapsed, the ranking over optimizers just by expanding the hyperparameter search space.
+
+The left pane of Figure 3 shows our experiments on VGG on CIFAR-10 using code released by Wilson et al. (2017). When we match their protocol and perform their grid search over the initial learning rate and no other tuning, we reproduce their original result showing worse test error for RMSPROP and ADAM. However, when we tune the momentum parameter and $\epsilon$ with random search, all four optimizers reach nearly identical test error rates.3 With our learning rate schedule search space, merely tuning the learning rate schedule was enough to make all optimizers reach the same test error within error bars. When we additionally tuned the optimization hyperparameters and weight decay in our setup we also get similar results for all optimizers, removing any evidence the inclusion relationships might be violated in practice.
+
+Figure 4 shows our results with different tuning protocols for a CNN on CIFAR-100 and an LSTM language model trained on War and Peace to match the experiments in Schneider et al. (2019). As reported by Schneider et al. (2019), if we only tune the learning rate without tuning the decay schedule or other optimizer hyperparameters, ADAM does worse than MOMENTUM for the CNN and SGD performs slightly better than ADAM and MOMENTUM on the War and Peace dataset, although Schneider et al. (2019) found a larger advantage for SGD. However, once we tune the all the optimizer hyperparameters, ADAM does better than MOMENTUM which does better than SGD, as predicted by the inclusion relationships.
+
+We conclude that the reason both Schneider et al. (2019) and Wilson et al. (2017) observed a ranking that, at first glance, contradicts the inclusion relationships is because they were not tuning enough of the hyperparameters. If we recast their results in our terminology where ADAM with default $\epsilon$ is a different optimizer than ADAM with $\epsilon$ tuned then there is no contradiction with our results and it becomes clear immediately that they do not consider the most interesting form of ADAM for practitioners.
+
+![](images/76084c8ebb639aee222a688bc0d0fde182c3d74a79772507edbabc43c06ce5d5.jpg)  
+Figure 4: Tuning more hyperparameters changes optimizer rankings from Schneider et al. (2019) to rankings that are consistent with the inclusion relationships. The leftmost columns for each workload reproduce the rankings from Schneider et al. (2019), while the remaining columns tune over increasingly general search spaces. All columns use our random search tuning protocol.
+
+# 5 CONCLUSIONS
+
+Inspired by the recent efforts of Wilson et al. (2017) and Schneider et al. (2019), we set out to provide a detailed empirical characterization of the optimizer selection process in deep learning. Our central finding is that inclusion relationships between optimizers are meaningful in practice. When tuning all available hyperparameters under a realistic protocol at scales common in deep learning, we find that more general optimizers never underperform their special cases. In particular, we found that RMSPROP, ADAM, and NADAM never underperformed SGD, NESTEROV, or MOMENTUM under our most exhaustive tuning protocol. We did not find consistent trends when comparing optimizers that could not approximate each other. We also found workloads for which there was not a statistically significant separation in the optimizer ranking.
+
+Our experiments have some important limitations and we should be careful not to overgeneralize from our results. The first major caveat is that we did not measure the effects of varying the batch size. Recent empirical work (Shallue et al., 2019; Zhang et al., 2019) has shown that increasing the batch size can increase the gaps between training times for different optimizers, with the gap from SGD to MOMENTUM (Shallue et al., 2019) and from MOMENTUM to ADAM (Zhang et al., 2019) increasing with the batch size. Nevertheless, we strongly suspect that the inclusion relations would be predictive at any batch size under a tuning protocol similar to the one we used. The second important caveat of our results is that they inevitably depend on the tuning protocol and workloads that we considered. Although we made every attempt to conduct realistic experiments, we should only expect our detailed findings to hold for similar workloads under similar protocols, namely uniform quasi-random tuning for tens to hundreds of trials, over hypercube search spaces, and with our specific learning rate schedule parameterization. Nevertheless, these caveats reinforce our central point: all empirical comparisons of neural network optimizers depend heavily on the hyperparameter tuning protocol, perhaps far more than we are used to with comparisons between model architectures.
+
+If we were to extract “best practices” from our findings, then we suggest the following. If we can afford tens or more runs of our code, we should tune all of the hyperparameters of the popular adaptive gradient methods. Just because two hyperparameters have a similar role in two different update rules doesn’t mean they should take similar values— optimization hyperparameters tend to be coupled and the optimal value for one may depend on how the others are set. Our results also confirm that the optimal value of Adam’s $\epsilon$ is problem-dependent, so the onus is on empirical studies that fix $\epsilon = 1 0 ^ { - 8 }$ to defend that choice. Finally, we should be skeptical of empirical comparisons of optimizers in papers, especially if an optimizer underperforms any of its specializations. When we do inevitably compare optimizers, we should report search spaces and highlight decisions about what hyperparameters were tuned when interpreting results.
+
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+
+# A OPTIMIZER INCLUSIONS
+
+Table 1 summarizes the update rules for the optimizers we consider in this work. We assume update rules as implemented in TensorFlow r1.15. RMSPROP includes momentum. Here we prove the their inclusion relationships, see Definition 1.
+
+# MOMENTUM can exactly implement SGD
+
+MOMENTUM $\left( I _ { t } , \eta _ { t } , 0 \right) = \mathbf { S } \mathbf { G } \mathbf { D } ( I _ { t } , \eta _ { t } )$ , so SGD ⊆ MOMENTUM.
+
+# NESTEROV can exactly implement SGD
+
+NESTEROV $\left( I _ { t } , \eta _ { t } , 0 \right) = \mathbf { S } \mathbf { G } \mathbf { D } ( I _ { t } , \eta _ { t } )$ , so $\mathrm { S G D } \subseteq \mathrm { N E S T E R O V } .$
+
+# RMSPROP with momentum can exactly implement MOMENTUM
+
+Consider RMSPROP $( I _ { t } , \eta _ { t } , \gamma , \rho = 1 , \epsilon = 0 )$ , so that
+
+$$
+\begin{array} { r l } & { m _ { t + 1 } = \gamma m _ { t } + \eta _ { t } \nabla \ell ( \theta _ { t } ) , } \\ & { \theta _ { t + 1 } = \theta _ { t } - m _ { t + 1 } . } \end{array}
+$$
+
+This is equivalent to MOMENTUM, since
+
+$$
+m _ { t + 1 } ^ { ( \mathrm { R M S P R O P } ) } \equiv \eta _ { t } v _ { t + 1 } ^ { ( \mathrm { M o M E N T U M } ) } .
+$$
+
+Thus $\mathrm { R M S P R O P } ( I _ { t } , \eta _ { t } , \gamma , 1 , 0 ) = \mathbf { M O M E N T U M } ( I _ { t } , \eta _ { t } , \gamma )$ , so MOMENTUM $\subseteq$ RMSPROP.
+
+# RMSTEROV can exactly implement NESTEROV
+
+Consider RMSTEROV $( I _ { t } , \eta _ { t } , \gamma , \rho = 1 , \epsilon = 0 )$ , so that
+
+$$
+\begin{array} { r l } & { m _ { t + 1 } = \gamma m _ { t } + \eta _ { t } \nabla \ell ( \theta _ { t } ) , } \\ & { \theta _ { t + 1 } = \theta _ { t } - [ \gamma m _ { t + 1 } + \eta _ { t } \nabla \ell ( \theta _ { t } ) ] \ . } \end{array}
+$$
+
+This is equivalent to MOMENTUM, since
+
+$$
+m _ { t + 1 } ^ { ( \mathrm { R M S P R o P } ) } \equiv \eta _ { t } v _ { t + 1 } ^ { ( \mathrm { N E S T E R O V } ) } .
+$$
+
+Thus RMSTERO $\begin{array} { r } { \mathsf { V } \left( I _ { t } , \eta _ { t } , \gamma , 1 , 0 \right) = \mathbf { M O M E N T U M } \left( I _ { t } , \eta _ { t } , \gamma \right) } \end{array}$ , so MOMENTUM $\subseteq$ RMSTEROV.
+
+# ADAM can approximate MOMENTUM for large $\epsilon$
+
+Consider $\mathrm { A D A M } ( I _ { t } , \alpha _ { t } = \epsilon \eta _ { t } ( 1 - \gamma ^ { t } ) , \beta _ { 1 } = \gamma , \beta _ { 2 } = 0 , \epsilon ) ,$ so that
+
+$$
+\begin{array} { l } { m _ { t + 1 } = \gamma m _ { t } + ( 1 - \gamma ) \nabla \ell ( \theta _ { t } ) , } \\ { \theta _ { t + 1 } = \theta _ { t } - \displaystyle \frac { \eta _ { t } } { ( 1 - \gamma ) } \left[ \frac { m _ { t + 1 } } { | \nabla \ell ( \theta _ { t } ) | / \epsilon + 1 } \right] . } \end{array}
+$$
+
+If $\epsilon$ is large, so that $| \nabla \ell ( \theta _ { t } ) | / \epsilon \ll 1$ , then
+
+$$
+\begin{array} { l } { { m _ { t + 1 } = \gamma m _ { t } + ( 1 - \gamma ) \nabla \ell ( \theta _ { t } ) , } } \\ { { \theta _ { t + 1 } = \theta _ { t } - \eta _ { t } \displaystyle \frac { m _ { t + 1 } } { 1 - \gamma } . } } \end{array}
+$$
+
+This is equivalent to MOMENTUM, since
+
+$$
+m _ { t } ^ { ( \mathrm { A D A M } ) } \equiv \left( 1 - \gamma \right) v _ { t } ^ { ( \mathrm { M o M E N T U M } ) } .
+$$
+
+Thus $\begin{array} { r } { \operatorname* { l i m } _ { \epsilon \to \infty } \mathrm { A D A M } \big ( I _ { t } , \epsilon \eta _ { t } \big ( 1 - \gamma ^ { t } \big ) , \gamma , 0 , \epsilon \big ) = \mathbf { M O M E N T U M } \big ( I _ { t } , \eta _ { t } , \gamma \big ) , } \end{array}$ so MOMENTUM $\subseteq$ ADAM
+
+# NADAM can approximate NESTEROV for large $\epsilon$
+
+Consider NADAM $\mathrm { t } \big ( I _ { t } , \alpha _ { t } = \epsilon \eta _ { t } ( 1 - \gamma ^ { t } ) , \beta _ { 1 } = \gamma , \beta _ { 2 } = 0 , \epsilon \big )$ , so that
+
+$$
+\begin{array} { l } { m _ { t + 1 } = \gamma m _ { t } + ( 1 - \gamma ) \nabla \ell ( \theta _ { t } ) , } \\ { \theta _ { t + 1 } = \theta _ { t } - \displaystyle \frac { \eta _ { t } } { ( 1 - \gamma ) } \left[ \frac { \gamma m _ { t + 1 } + ( 1 - \gamma ) \nabla \ell ( \theta _ { t } ) } { | \nabla \ell ( \theta _ { t } ) | / \epsilon + 1 } \right] . } \end{array}
+$$
+
+If $\epsilon$ is large, so that $| \nabla \ell ( \theta _ { t } ) | / \epsilon \ll 1$ , then
+
+$$
+\begin{array} { r l } & { m _ { t + 1 } = \gamma m _ { t } + ( 1 - \gamma ) \nabla \ell ( \theta _ { t } ) , } \\ & { \theta _ { t + 1 } = \theta _ { t } - \eta _ { t } \left[ \frac { \gamma m _ { t + 1 } } { 1 - \gamma } + \nabla \ell ( \theta _ { t } ) \right] . } \end{array}
+$$
+
+This is equivalent to NESTEROV, since
+
+$$
+m _ { t } ^ { ( \mathrm { N A D A M } ) } \equiv \left( 1 - \gamma \right) v _ { t } ^ { ( \mathrm { N E S T E R O V } ) }
+$$
+
+Thus $\begin{array} { l l l } { \operatorname* { l i m } _ { \epsilon \to \infty } \mathrm { N A D A M } ( I _ { t } , \epsilon \eta _ { t } ( 1 - \gamma ^ { t } ) , \gamma , 0 , \epsilon ) } & { = } & { \mathrm { N E S T E R O V } ( I _ { t } , \eta _ { t } , \gamma ) } \end{array}$ , so NESTEROV ⊆ NADAM.
+
+# B WORKLOAD DETAILS
+
+This section details the datasets and models summarized in Table 2.
+
+# B.1 DATASET DESCRIPTIONS
+
+For Fashion MNIST, CIFAR-10, ImageNet, and LM1B, our setup was identical to Shallue et al. (2019) except for the image pre-processing details described below. For War and Peace, our setup was identical to the “Tolstoi” dataset of Schneider et al. (2019).
+
+CIFAR-10/100: We pre-processed images by subtracting the average value across all pixels and channels and dividing by the standard deviation.4 For experiments with the ResNet-32 and CNN models, we followed the standard data augmentation scheme used in He et al. (2016a): 4 pixels padded on each side with single random crop from padded image or its horizontal reflection. We did not use random cropping for experiments with VGG for consistency with Wilson et al. (2017).
+
+ImageNet: We augmented images at training time by resizing each image, taking a random crop of $2 2 4 \times 2 2 4$ pixels, randomly horizontally reflecting the cropped images, and randomly distorting the image colors. At evaluation time, we performed a single central crop of $2 2 4 \times 2 2 4$ pixels. In both training and evaluation, we then subtracted the global mean RGB value from each pixel using the values computed by Simonyan and Zisserman (2014).5
+
+# B.2 MODEL DESCRIPTIONS
+
+Simple CNN is identical to the base model described in Shallue et al. (2019). It consists of 2 convolutional layers with max pooling followed by 1 fully connected layer. The convolutional layers use $5 \times 5$ filters with stride 1, “same” padding, and ReLU activation function. Max pooling uses a $2 \times 2$ window with stride 2. Convolutional layers have 32 and 64 filters each and the fully connected layer has 1024 units. It does not use batch normalization.
+
+CNN is the “All-CNN-C” model from Springenberg et al. (2014), as used in Schneider et al. (2019). The model consists of 3 convolutional layer blocks with max pooling. The convolutional layers use $5 \times 5$ filters with stride 1, “same” padding, and ReLU activation function. Max pooling uses a $2 \times 2$ window with stride 2. Convolutional layer blocks have 96, 192 and 192 filters each. As in Schneider et al. (2019), we used $L _ { 2 }$ regularization of $5 \times 1 0 ^ { - 4 }$ .
+
+ResNet is described in He et al. (2016a). We used the improved residual block described in He et al. (2016b). We used batch normalization (Ioffe and Szegedy, 2015) with exponential moving average (EMA) decay of 0.997 for ResNet-32, and ghost batch normalization (Hoffer et al., 2017) with ghost batch size of 32 and EMA decay of 0.9 for ResNet-50.
+
+VGG is based on “model C” from Simonyan and Zisserman (2014). It consists of 13 convolutional layers followed by 3 fully connected hidden layers. We followed the modification used by Wilson et al. (2017) with batch normalization layers.
+
+LSTM is a two hidden-layer LSTM model (Hochreiter and Schmidhuber, 1997) identical to the model used in Schneider et al. (2019). It uses 128 embedding dimensions and 128 hidden units.
+
+Transformer is the “base” model described in (Vaswani et al., 2017). We used it as an autoregressive language model by applying the decoder directly to the sequence of word embeddings for each sentence. Unlike the default implementation, we removed dropout regularization and used separate weight matrices for the input embedding layer and the pre-softmax linear transformation, as we observed these choices led to better performing models.
+
+# C ESTIMATING TRIAL OUTCOMES VIA BOOTSTRAP
+
+Our tuning protocol corresponds to running trials with quasi-random hyperparameter values sampled uniformly from the search space until $K$ feasible trials are obtained, with $K$ depending on the workload. We then select the best trial, based on our statistic of interest, over those $K$ trials.
+
+We used the following bootstrap procedure to estimate means and uncertainties of our tuning protocol. We ran $N > K$ trials, with $N$ depending on the workload. Then, for each bootstrap sample, we resampled the dataset of $N$ trials with replacement and computed our statistic on the first $K$ trials of the resampled dataset. We collected 100 such bootstrap samples each time, and from those computed the means, $5 ^ { \mathrm { t h } }$ percentiles, and $9 5 ^ { \mathrm { t h } }$ percentiles of the bootstrap distribution. We used this procedure to generate the means and error bars for each plot.
+
+Simple CNN on Fashion MNIST used $( K , N ) \ = \ ( 1 0 0 , 5 0 0 )$ ; ResNet-32 on CIFAR-100 used $( K , \mathbf { \bar { \cal { N } } } ) = ( 1 0 0 , 5 0 0 )$ ; ResNet-50 on ImageNet used $( K , N ) = ( 5 0 , 2 5 0 )$ ; Transformer on LM1B used $( K , N ) = ( 5 0 , 1 0 0 )$ ; VGG on CIFAR-10 with our code used $( K , N ) = ( 5 0 , 2 5 0 )$ for tuning the learning rate schedule and $( K , N ) = ( 1 0 0 , 5 0 0 )$ for tuning the learning rate schedule, $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon \}$ , and $L _ { 2 }$ regularization; CNN on CIFAR-10 used $( K , N ) = ( 1 0 0 , 5 0 0 )$ ; LSTM on War and Peace used $( K , \bar { N } ) = ( 1 0 , 5 0 )$ for tuning just the learning rate and $( K , N ) \stackrel { } { = } ( 1 0 0 , 5 0 0 )$ for tuning the learning rate schedule and $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon \}$ .
+
+The sole exceptions to this bootstrap procedure are the two left panels of Figure 3, for which we used a similar procedure to Wilson et al. (2017) to ensure comparability. For each optimizer, we selected the trial that minimized validation error in our final search space and ran the same hyperparameter values 5 times, reporting the mean, minimum, and maximum test error over those 5 runs in Figure 3. This is slightly different to Wilson et al. (2017), who chose the trial that minimized training error and reported validation error. When tuning the learning rate and $\{ \gamma , \epsilon \}$ , we used 24 trials per optimizer in the initial search space (which we used to select the final search space), and 16 trials per optimizer in the final search space.
+
+# D HYPERPARAMETER SEARCH SPACES
+
+When tuning hyperparameters over a large range, we found that our search could sometimes be made more efficient if we parametrized the search space in a way that decorrelated the axes of the space. For example, with MOMENTUM and NESTEROV we observed a clear relationship between the initial learning rate $\eta _ { 0 }$ and the momentum parameter $\gamma$ ; smaller values of $\eta _ { 0 }$ require larger values of $\gamma$ for good performance, and vice versa. Indeed, Shallue et al. (2019) suggested that these optimizers are governed by the “effective learning rate” $\eta _ { \mathrm { e f f } } = \eta _ { 0 } / ( 1 - \gamma )$ , and inspired by this, we found that searching over $( \eta _ { 0 } , \eta _ { \mathrm { e f f } } )$ instead of $( \eta _ { 0 } , \gamma )$ usually led to a more efficient hyperparameter search. Similarly, with ADAM and NADAM we observed a relationship between the initial learning rate $\alpha _ { 0 }$ and the $\epsilon$ parameter; larger values of $\alpha _ { 0 }$ require larger values of $\epsilon$ for good performance, and vice versa. This is not surprising given the analysis in Appendix A that showed that, for large $\epsilon$ , $\alpha _ { 0 } / \epsilon$ is analogous to the effective learning rate of ADAM and NADAM. We found that searching over $( \epsilon , \alpha _ { 0 } / \epsilon \bar { ) }$ was usually more efficient than searching over $( \epsilon , \alpha _ { 0 } )$ . We used these techniques in a subset of our experiments.
+
+Below we report the search spaces used for our experiments. We include both the initial search spaces used to refine the search spaces, and the final spaces used to generate the plots. When only one search space was used, we denote the initial space as final. $\eta _ { 0 } , \alpha _ { 0 } , 1 - \gamma , 1 - \beta _ { 1 } , 1 - \beta _ { 2 } , \epsilon$ $\epsilon$ and combinations thereof are always tuned on a log scale. The number of samples from each search space is specified in Appendix C.
+
+# D.1 CNN ON FASHION MNIST
+
+We used linear learning rate decay for all experiments. We tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor within $\{ 1 0 ^ { - 3 } , \dot { 1 } 0 ^ { - 2 } , 1 \dot { 0 } ^ { - 1 } \}$ . We did not use $L _ { 2 }$ regularization or weight decay.
+
+Table 3: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,102]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,101]</td></tr></table>
+
+Table 4: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-γ</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,102]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-5,101]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr></table>
+
+Table 5: NESTEROV   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-γ</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,102]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr></table>
+
+Table 6: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>1-p</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-5,101]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-5,1]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-10,10-5]</td></tr></table>
+
+Table 7: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-3,5 × 10-1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-9,10-5]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-5,10-1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-10,10-5]</td></tr></table>
+
+Table 8: NADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0/e</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-10,1010]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-6,10-2]</td></tr></table>
+
+# D.2 RESNET-32 ON CIFAR-10
+
+We used linear learning rate decay for all experiments. We tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor $f$ within the values shown in the tables below. $\lambda _ { L _ { 2 } }$ denotes the $L _ { 2 }$ regularization coefficient.
+
+Table 9: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,102]</td><td rowspan=1 colspan=1>{10-5,10-4,10-3,10-2}{</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr></table>
+
+Table 10: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,102]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>{10-5,10-4,10-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-3,10-²,10-1}</td></tr></table>
+
+Table 11: NESTEROV   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,102]</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>10-4</td><td rowspan=1 colspan=1>{10-3,10-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>{10-5,10-4,10-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr></table>
+
+Table 12: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-~</td><td rowspan=1 colspan=1>1-p</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-5,101]</td><td rowspan=1 colspan=1>10-4</td><td rowspan=1 colspan=1>{10-3,10-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-5,101]</td><td rowspan=1 colspan=1>{10-5,10-410-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-310-2,10-1}</td></tr></table>
+
+Table 13: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-3,5 × 10-1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-9,10-5]</td><td rowspan=1 colspan=1>10-4</td><td rowspan=1 colspan=1>{10-3,10-210-1</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-3,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-5,101]</td><td rowspan=1 colspan=1>{10-5,10-410-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-310-2,10-1}</td></tr></table>
+
+Table 14: NADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>a0/e</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>1-β2</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入L2</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>{10-5,10-410-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-310-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-²,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[1,104]</td><td rowspan=1 colspan=1>{10-5,10-410-3,10-2}</td><td rowspan=1 colspan=1>{10-4,10-310-²,10-1}</td></tr></table>
+
+# D.3 RESNET-50 ON IMAGENET
+
+We used linear learning rate decay for all experiments. We tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor $f$ within the values shown in the tables below. $\lambda _ { \mathrm { w d } }$ denotes the weight decay coefficient and $\tau$ denotes the label smoothing coefficient.
+
+Table 15: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,101]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>{0,10-²,10-1}</td><td rowspan=1 colspan=1>{0,10-2,10-1}{10-4,10-3,10-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[1,102]</td><td rowspan=1 colspan=1>[10-4,10-3]</td><td rowspan=1 colspan=1>10-1</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr></table>
+
+Table 16: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>{0,10-2,10-1}</td><td rowspan=1 colspan=1>{0,10-2,10-1]{10-4,10-3,10-2,10-1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,10-3]</td><td rowspan=1 colspan=1>10-2</td><td rowspan=1 colspan=1>{10-4,10-3,10-²,10-1}</td></tr></table>
+
+Table 17: NESTEROV   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>{0,10-2,10-1}</td><td rowspan=1 colspan=1>10-3</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-3]</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>{10-4,10-3,10-²,10-1}</td></tr></table>
+
+Table 18: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no/e</td><td rowspan=1 colspan=1>1-~</td><td rowspan=1 colspan=1>1-p</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>{0,10-2,10-1}</td><td rowspan=1 colspan=1>10-3</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-8,10-3]</td><td rowspan=1 colspan=1>[10-4,10-3]</td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>{10-4,10-310-2,10-1}</td></tr></table>
+
+Table 19: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao/e</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[1,102]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[1,104]</td><td rowspan=1 colspan=1>[10-5,10-3]</td><td rowspan=1 colspan=1>{0,10-2,10-1}</td><td rowspan=1 colspan=1>10-3</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[1,102]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-²,102]</td><td rowspan=1 colspan=1>10-4</td><td rowspan=1 colspan=1>10-1</td><td rowspan=1 colspan=1>{10-4,10-310-2,10-1}</td></tr></table>
+
+Table 20: NADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>a0/∈</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入wd</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-1,103]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-2,1010]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>{0,10-2,10-1}</td><td rowspan=1 colspan=1>10-3</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[1,102]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[103,107]</td><td rowspan=1 colspan=1>10-4</td><td rowspan=1 colspan=1>10-1</td><td rowspan=1 colspan=1>10-3</td></tr></table>
+
+# D.4 TRANSFORMER ON LM1B
+
+We used linear learning rate decay for all experiments. We tuned the number of decay steps within $[ 0 . 5 , 1 . 0 ]$ times the number of training steps and the learning rate decay factor within $\{ 1 0 ^ { - 4 } , \dot { 1 } 0 ^ { - 3 } , 1 \dot { 0 } ^ { - 2 } , 1 0 ^ { - 1 } , 1 \}$ .
+
+Table 21: SGD   
+
+<table><tr><td></td><td>mo</td></tr><tr><td>final</td><td>[10-4,10-1]</td></tr></table>
+
+Table 22: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-~</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr></table>
+
+Table 23: NESTEROV   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-4,1]</td></tr></table>
+
+Table 24: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>1-p</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,101]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-12,1010]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-6,10-2]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-7,10-1]</td></tr></table>
+
+Table 25: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β2</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>[10-3,5×10-1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-9,10-5]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,10-2]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-1]</td><td rowspan=1 colspan=1>[10-7,10-2]</td></tr></table>
+
+Table 26: NADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-1]</td><td rowspan=1 colspan=1>[10-9,10-5]</td></tr></table>
+
+# D.5 VGG ON CIFAR-10 USING CODE FROM WILSON ET AL. (2017)
+
+# D.5.1 GRID SEARCH OVER LEARNING RATE
+
+We tuned over the same grid of initial learning rate values for each optimizer as Wilson et al. (2017). As in Wilson et al. (2017), we decayed the initial learning rate by a factor of 0.5 every 25 epochs and used a fixed $L _ { 2 }$ regularization coefficient of 0.0005.
+
+# D.5.2 TUNING LEARNING RATE & $\{ \gamma , \epsilon \}$
+
+We used our quasi-random tuning protocol to tune over the initial learning rate, MOMENTUM’s $\gamma$ , RMSPROP’s $\epsilon$ , and ADAM’s $\epsilon$ . As in Wilson et al. (2017), we decayed the initial learning rate by a factor of 0.5 every 25 epochs and used a fixed $L _ { 2 }$ regularization coefficient of 0.0005.
+
+Table 27: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,101]</td></tr></table>
+
+Table 28: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>1-γ</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-3,1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,101]</td><td rowspan=1 colspan=1>[10-1,1]</td></tr></table>
+
+Table 29: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>ao/</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>[10-2,104]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,102]</td><td rowspan=1 colspan=1>[10-1,101]</td></tr></table>
+
+Table 30: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>a0/e</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>[10-2,104]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[106,1010]</td><td rowspan=1 colspan=1>[10-1,101]</td></tr></table>
+
+# D.6 VGG ON CIFAR-10 USING OUR CODE
+
+We used linear learning rate decay for all experiments. We tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor within $\{ 1 0 ^ { - 4 } , \dot { 1 } 0 ^ { - 3 } , 1 \dot { 0 } ^ { - 2 } , 1 0 ^ { - 1 } \}$ .
+
+# D.6.1 TUNING LEARNING RATE SCHEDULE
+
+We fixed all optimizer hyperparameters excluding the learning rate to match those specified in Wilson et al. (2017). As in Wilson et al. (2017), we used a fixed $L _ { 2 }$ regularization coefficient of 0.0005.
+
+Table 31: Learning rate search ranges.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo (SGD)</td><td rowspan=1 colspan=1>no (MOMENTUM)</td><td rowspan=1 colspan=1>mo (RMSPROP)</td><td rowspan=1 colspan=1>αo (ADAM)</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,101]</td><td rowspan=1 colspan=1>[10-3,101]</td><td rowspan=1 colspan=1>[10-5,10-1]</td><td rowspan=1 colspan=1>[10-5,10-1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,10-2]</td><td rowspan=1 colspan=1>[10-5,10-1]</td></tr></table>
+
+D.6.2 TUNING LEARNING RATE SCHEDULE & $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon , \lambda _ { L _ { 2 } } \}$   
+Table 32: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>no</td><td rowspan=1 colspan=1>入L2</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,101]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-3,10-1]</td></tr></table>
+
+Table 33: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-2</td><td rowspan=1 colspan=1>入L2</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-1,1]</td><td rowspan=1 colspan=1>[10-3,10-1]</td></tr></table>
+
+Table 34: RMSPROP   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao/</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>1-p</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入L2</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-1,1]</td><td rowspan=1 colspan=1>[10-3,10-2]</td><td rowspan=1 colspan=1>[102,106]</td><td rowspan=1 colspan=1>[10-3,10-1]</td></tr></table>
+
+Table 35: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>a0/e</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>入L2</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-2,101]</td><td rowspan=1 colspan=1>[10-1,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[106,1010]</td><td rowspan=1 colspan=1>[10-3,10-1]</td></tr></table>
+
+# D.7 CNN ON CIFAR-100
+
+# D.7.1 TUNING CONSTANT LEARNING RATE
+
+We fixed all optimizer hyperparameters excluding the learning rate to match those specified in Schneider et al. (2019).
+
+Table 36: Learning rate search ranges.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>n (SGD)</td><td rowspan=1 colspan=1>η (MOMENTUM)</td><td rowspan=1 colspan=1>α (ADAM)</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,1]</td><td rowspan=1 colspan=1>[10-3,10-2]</td><td rowspan=1 colspan=1>[10-4,10-3]</td></tr></table>
+
+# D.7.2 TUNING LEARNING RATE SCHEDULE
+
+We used linear learning rate decay, and tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor within $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , \dot { 1 0 } ^ { - 2 } , 1 0 ^ { - 1 } \}$ .
+
+Table 37: Learning rate search ranges.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo (SGD)</td><td rowspan=1 colspan=1>n0 (MOMENTUM)</td><td rowspan=1 colspan=1>αo (ADAM)</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,1]</td><td rowspan=1 colspan=1>[10-3,10-1]</td><td rowspan=1 colspan=1>[10-4,10-3]</td></tr></table>
+
+# D.7.3 TUNING CONSTANT LEARNING RATE & $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon \}$
+
+For SGD, we reused the results from Appendix D.7.1, since there were no additional hyperparameters to tune.
+
+Table 38: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>1-</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-3,1]</td></tr></table>
+
+Table 39: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>α/e</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β2</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,10-2]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-10,10-10]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,10-1]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10²,106]</td></tr></table>
+
+# D.7.4 TUNING LEARNING RATE SCHEDULE & $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon \}$
+
+We used linear learning rate decay, and tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor within $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , \hat { 1 0 } ^ { - 1 } \}$ . For SGD, we reused the results from Appendix D.7.2, since there were no additional hyperparameters to tune.
+
+Table 40: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-2</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-2,1]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-3,10-1]</td><td rowspan=1 colspan=1>[10-3,10-1]</td></tr></table>
+
+Table 41: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>a0/e</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>1-β2</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-1,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10²,106]</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,101]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>[10²,106]</td></tr></table>
+
+# D.8 LSTM ON WAR AND PEACE
+
+D.8.1 TUNING CONSTANT LEARNING RATE
+
+Table 42: SGD   
+
+<table><tr><td></td><td>m</td></tr><tr><td>final</td><td>[10-2,101]</td></tr></table>
+
+Table 43: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=1>1-Y</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>0.99</td></tr></table>
+
+Table 44: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>α/e</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β</td><td rowspan=1 colspan=1>E</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-5,10-2]</td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>0.999</td><td rowspan=1 colspan=1>10-8</td></tr></table>
+
+# D.8.2 TUNING LEARNING RATE SCHEDULE & $\{ \gamma , \beta _ { 1 } , \beta _ { 2 } , \epsilon \}$
+
+We used linear learning rate decay, and tuned the number of decay steps within [0.5, 1.0] times the number of training steps and the learning rate decay factor $f$ within the values shown in the tables below.
+
+Table 45: SGD   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>m</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-3,101]{</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[1,101]</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr></table>
+
+Table 46: MOMENTUM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>mo</td><td rowspan=1 colspan=1>1-γ</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-4,1]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>{10-4,10-3,10-²,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[10-1,101]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>{10-4,10-3,10-²,10-1}</td></tr></table>
+
+Table 47: ADAM   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>ao/e</td><td rowspan=1 colspan=1>1-β1</td><td rowspan=1 colspan=1>1-β2</td><td rowspan=1 colspan=1>E</td><td rowspan=1 colspan=1>f</td></tr><tr><td rowspan=1 colspan=1>initial</td><td rowspan=1 colspan=1>[10-2,104]</td><td rowspan=1 colspan=1>[10-3,1]</td><td rowspan=1 colspan=1>[10-4,10-1]</td><td rowspan=1 colspan=1>[10-10,1010]</td><td rowspan=1 colspan=1>{10-4,10-3,10-2,10-1}</td></tr><tr><td rowspan=1 colspan=1>final</td><td rowspan=1 colspan=1>[1,102]</td><td rowspan=1 colspan=1>[10-2,1]</td><td rowspan=1 colspan=1>0.999</td><td rowspan=1 colspan=1>[1,104]</td><td rowspan=1 colspan=1>10-3</td></tr></table>
+
+# CNN on CIFAR-100 with Momentum
+
+![](images/8472678df32a7cc8dbcd6f0443a9f25c057f2b9b929704aae5e9c9a7a836b557.jpg)  
+Figure 5: Example plot of final validation error projected onto the axes of the hyperparameter space. We consider this search space to be appropriate because the optimal values are away from the search space boundaries.
+
+![](images/ca51833d8d7fe555faa83251c8777f4319737e8033973082d68a0206ae01dd22.jpg)  
+Figure 6: Validation performance of the best trial mostly converges with as few as $2 ^ { 4 }$ hyperparameter tuning trials for Transformer on LM1B. Shaded regions indicate $5 ^ { \mathrm { t h } }$ and $9 5 ^ { \mathrm { t h } }$ percentiles estimated with bootstrap sampling (see Appendix C). The search spaces can be found in Appendix D.4.
+
+![](images/6b7e7b8662b35c0ec6a3ac133f9a2bdb92f222668f3f5ac43b2d6741ebd207b1.jpg)  
+Figure 7: Validation performance of the best trial mostly converges with as few as $2 ^ { 4 }$ hyperparameter tuning trials for ResNet-50 in ImageNet. Shaded regions indicate $5 ^ { \mathrm { t h } }$ and $9 5 ^ { \mathrm { t h } }$ percentile estimated with bootstrap sampling (see Appendix C). The search spaces can be found in D.3.
+
+![](images/07134ef8150d2363de7b6165f9a3c12f46a33393d296b90396a6728f14ac84f6.jpg)  
+Figure 8: Test performance of the best trial mostly converges with as few as $2 ^ { 3 }$ hyperparameter tuning trials for a 2-layer LSTM on War and Peace. Shaded regions indicate $5 ^ { \mathrm { t h } }$ and ${ \dot { 9 } } { \dot { 5 } } ^ { \mathrm { t h } }$ percentile estimated with bootstrap sampling (see Appendix C). The search spaces can be found in D.8.2.
+
+![](images/88f1d88601e2cbb6e6c0f5129dd451affb1e40aafc9db9b6d46991f75c6e6814.jpg)  
+Figure 9: The relative performance of optimizers is consistent with the inclusion relationships when we select for lowest training loss. Note that SGD, ADAM, and NADAM for ResNet-50 on ImageNet used label smoothing in their final search spaces (see Section D.3), which makes their loss values incommensurate with the other optimizers. This is because their final search spaces were optimized to minimize validation error—if we had optimized their search spaces to minimize training error instead, we would not have used label smoothing, and we expect their training loss values would be consistent with the inclusion relationships.
+
+![](images/db9e1f287f005cf6f34cca6efbba32828c8cbc41c052b1d015cb25deaa63a820.jpg)  
+Figure 10: Our results confirming the relevance of optimizer inclusion relationships do not depend on the exact step budgets or error targets we chose.

parse/train/SylVJTNKDr/SylVJTNKDr.md

@@ -0,0 +1,368 @@
+# ENTROPY MINIMIZATION IN EMERGENT LANGUAGES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+There is a growing interest in studying the languages emerging when neural agents are jointly trained to solve tasks requiring communication through a discrete channel. We investigate here the information-theoretic complexity of such languages, focusing on the basic two-agent, one-exchange setup. We find that, under common training procedures, the emergent languages are subject to an entropy minimization pressure that has also been detected in human language, whereby the mutual information between the communicating agent’s inputs and the messages is minimized, within the range afforded by the need for successful communication. This pressure is amplified as we increase communication channel discreteness. Further, we observe that stronger discrete-channel-driven entropy minimization leads to representations with increased robustness to overfitting and adversarial attacks. We conclude by discussing the implications of our findings for the study of natural and artificial communication systems.
+
+# 1 INTRODUCTION
+
+There has recently been much interest in the analysis of the communication systems arising when deep network agents that interact to accomplish a goal are allowed to exchange language-like discrete messages (Lazaridou et al., 2016; Havrylov & Titov, 2017; Choi et al., 2018; Lazaridou et al., 2018). Understanding the emergent protocol is important if we want to eventually develop agents capable of interacting with each other and with us through language (Mikolov et al., 2016; Chevalier-Boisvert et al., 2019). The pursuit might also provide comparative evidence about how core properties of human language itself have evolved (Kirby, 2002; Hurford, 2014; Graesser et al., 2019). While earlier studies reported ways in which deep agent protocols radically depart from human language (Kottur et al., 2017; Bouchacourt & Baroni, 2018; Chaabouni et al., 2019; Lowe et al., 2019), we show here that emergent communication shares an important property of the latter, namely a tendency towards entropy minimization.
+
+Converging evidence indicates that efficiency pressures are at work in language and other biological communication systems (Ferrer i Cancho et al., 2013; Gibson et al., 2019). One particular aspect of communicative efficiency, that has been robustly observed across many semantic domains, is a tendency to minimize lexicon entropy, to the extent allowed by the counteracting need for accuracy (Zaslavsky et al., 2018; 2019). For example, while most languages distinguish grandmothers from grandfathers, very few have separate words for mother- and father-side grandmothers, as the latter distinction would make communication only slightly more accurate at the cost of an increase in lexicon complexity (Kemp & Regier, 2012). We show here, in two separate games designed to precisely measure such property, that the protocol evolved by interacting deep agents is subject to the same complexity minimization pressure.
+
+Entropy minimization in natural language has been connected to the Information Bottleneck principle (Tishby et al., 1999). In turn, complexity reduction due to the Information Bottleneck provides a beneficial regularization effect on the learned representations (Fischer, 2019; Alemi et al., 2016; Achille & Soatto, 2018a;b). It is difficult to experimentally verify the presence of such effect in human languages, but we can look for it in our emergent language simulations. We confirm that, when relaxing channel discreteness, the entropy minimization property no longer holds, and the system becomes less robust against overfitting and adversarial noise.
+
+# 2 GENERAL FRAMEWORK
+
+We establish our results in the context of signaling games (Lewis, 1969), as introduced to the current language emergence literature by Lazaridou et al. (2016) and adopted in several later studies (Havrylov & Titov, 2017; Bouchacourt & Baroni, 2018; Lazaridou et al., 2018). There are two agents, Sender and Receiver, provided with individual inputs at the beginning of each episode. Sender sends a single message to Receiver, and Receiver has to perform an action based on its own input and the received message. Importantly, there is no direct supervision on the message protocol. We consider agents that are deterministic functions of their inputs (after training).
+
+As an example, consider the task of communicating a $n$ -bit number, sampled uniformly at random from $0 . . . 2 \sp n - 1$ . The full number is shown to Sender, and its $k$ $0 \leq k \leq n \}$ least-significant bits are also revealed to Receiver. Receiver has to output the full number, based on the message from Sender and its own input. Would the Sender transmit the entire number through its message? In this case, the protocol would be “complex,” encoding $n$ bits. Alternatively, Sender could only encode the bits that Receiver does not know, and let Receiver fill in the rest by itself. This emergent protocol would be “simple,” encoding less information about the number. We find experimentally that, once the agents are successfully trained to jointly solve the task, the emergent protocol minimizes the entropy of the messages or, equivalently in our setup, the mutual information between Sender’s input and messages. In other words, the agents consistently approximate the simplest successful protocol (in the current example, the one transmitting $\approx n - k$ bits).
+
+After training, we can connect the entropies of Sender and Receiver inputs $i _ { s }$ and $i _ { r }$ , messages $m =  { \mathit { S } } ( i _ { s } )$ , Receiver’s output (the chosen action) $\pmb { o } = R ( \pmb { m } , \pmb { i } _ { r } )$ , and ground-truth outputs $\imath$ by using standard inequalities (Cover & Thomas, 2012):
+
+$$
+H ( i _ { s } ) \geq H ( S ( i _ { s } ) ) = H ( m ) \geq H ( m | i _ { r } ) \geq H ( R ( m , i _ { r } ) | i _ { r } ) = H ( o | i _ { r } ) \approx H ( l | i _ { r } )
+$$
+
+(Note that, since agents are deterministic after training, $H ( \pmb { { m } } ) = I ( \pmb { { i } } _ { s } ; \pmb { { m } } )$ . We can then use these quantities interchangeably.) Our empirical measurements indicate that the entropy of the messages $_ { \mathbf { \nabla } } \mathbf { m }$ in the emergent protocol tends to approach the lower bound: $H ( { \pmb m } )  \bar { H ( { \pmb l } | { \pmb i } _ { r } ) }$ , even if the upper-bound $H ( i _ { s } )$ is far.
+
+In our experiments, we observe that when the amount of information that Receiver needs is reduced, without changing other parameters, the emergent protocol becomes simpler (lower entropy). In other words, the emergent protocol adapts to minimize the information that passes through it.
+
+We will release the code for our experiments upon acceptance.
+
+# 3 METHODOLOGY
+
+# 3.1 GAMES
+
+We study two signaling games. In Guess Number, the agents are trained to recover an integerrepresenting vector with uniform Bernoulli-distributed components. This simple setup gives us full control over the amount of information needed to solve the task. The second game, Image Classification, uses more naturalistic data, as the agents are jointly trained to classify pairs of hand-written MNIST digits (LeCun et al., 1998b).
+
+Guess Number We draw an 8-bit integer $z$ , $0 \leq z \leq 2 5 5$ uniformly at random, by sampling its 8 bits independently from the uniform Bernoulli distribution. All bits are revealed to Sender as a 8-dimensional binary vector $i _ { s }$ . The last $k$ bits are revealed to Receiver $0 \leq k \leq 8 $ ) as its input $i _ { r }$ . Sender outputs a single-symbol message $_ { \mathbf { \nabla } } \mathbf { m }$ to Receiver. In turn, Receiver outputs a vector $^ o$ that recovers all the bits of $z$ and should be equal to $i _ { s }$ .
+
+In this game, Sender has a linear layer that maps the input vector $i _ { s }$ to a hidden representation of size 10, followed by a leaky ReLU activation. Next is a linear layer followed by a softmax over the vocabulary. Receiver linearly maps both its input $i _ { r }$ and the message to 10-dimensional vectors, concatenates them, applies a fully connected layer with output size 20, followed by a leaky ReLU. Finally, another linear layer and a sigmoid nonlinearity are applied. When training with REINFORCE and the Stochastic Computation graph approach (see Section 3.2), we increase the hidden layer sizes threefold, as this leads to more robust convergence.
+
+Image Classification In this game, the agents are jointly trained to classify 28x56 images of two MNIST digits, stacked side-by-side (more details in Appendix). Unlike Guess Number, Receiver has no side input. Instead, we control the informational complexity of Receiver’s task by controlling the size of its output space, i.e., the number of labels we assign to the images. To do so, we group all two-digit sequences 00..99 into $N _ { l } \in \{ 2 , 4 , 1 0 , 2 0 , 2 5 , 5 0 , 1 0 0 \}$ equally-sized classes.
+
+In Sender, input images are embedded a LeNet-1 instance (LeCun et al., 1990) into 400-dimensional vectors. These embedded vectors are passed to a fully connected layer, followed by a softmax selecting a vocabulary symbol. Receiver embeds the received messages into 400-dimensional vectors, passed to a fully connected layer with a softmax activation returning the class probabilities.
+
+We report hyperparameter grids in Appendix. In the following experiments, we fix vocabulary to 1024 symbols (experiments with other vocabulary sizes, multi-symbol messages, and larger architectures are reported in Appendix). No parts of the agents are pre-trained or shared. The optimized loss depends on the gradient estimation method used (see Section 3.2). We denote it $\mathcal { L } ( o , l )$ , and it is a function of Receiver’s output $^ o$ and the ground-truth output $\imath$ . When training in Guess Number with REINFORCE, we use a $_ { 0 / 1 }$ loss: the agents get 0 only if all bits of $z$ were correctly recovered. When training with Gumbel-Softmax relaxation or the Stochastic Computation Graph approach, we use binary cross-entropy (Guess Number) and negative log-likelihood (Image Classification).
+
+# 3.2 TRAINING WITH DISCRETE CHANNEL
+
+Training to communicate with discrete messages is non-trivial, as we cannot back-propagate through the messages. Current language emergence work mostly uses Gumbel-Softmax relaxation (e.g. (Havrylov & Titov, 2017)) or REINFORCE (e.g. (Lazaridou et al., 2016)) to get gradient estimates. We also explore the Stochastic Computation Graph optimization approach. We plug the obtained gradient estimates into the Adam optimizer (Kingma & Ba, 2014).
+
+Gumbel-Softmax relaxation Samples from the Gumbel-Softmax (Maddison et al., 2016; Jang et al., 2016) distribution (a) are reperameterizable, hence allow gradient-based training, and (b) approximate samples from the corresponding Categorical distribution. To get a sample that approximates an $n$ -dimensional Categorical distribution with probabilities $p _ { i }$ , we draw $n$ i.i.d. samples $g _ { i }$ from Gumbel(0,1) and use them to calculate a vector $\textbf {  { y } }$ with components:
+
+$$
+y _ { i } = \frac { e x p \left[ ( g _ { i } + \log p _ { i } ) / \tau \right] } { \sum _ { j } e x p \left[ ( g _ { j } + \log p _ { j } ) / \tau \right] }
+$$
+
+where $\tau$ is the temperature hyperparameter. As $\tau$ tends to 0, the samples $\textbf {  { y } }$ get closer to one-hot samples; as $\tau \to + \infty$ , the components $y _ { i }$ become uniform. During training, we use these relaxed samples as messages from Sender, making the entire Sender/Receiver setup differentiable.
+
+REINFORCE by Williams (1992) is a standard reinforcement learning algorithm. In our setup, it estimates the gradient of the expectation of the loss $\mathcal { L } ( o , l )$ w.r.t. the parameter vector $\pmb \theta$ as follows:
+
+$$
+\mathbb { E } _ { i _ { s } , i _ { r } } \mathbb { E } _ { m \sim S ( i _ { s } ) , o \sim R ( m , i _ { r } ) } \left[ ( \mathcal { L } ( o ; l ) - b ) \nabla _ { \theta } \log P _ { \theta } ( m , o ) \right]
+$$
+
+The expectations are estimated by sampling $_ { m }$ from Sender and, after that, sampling $^ o$ from Receiver. We use the running mean baseline $b$ (Greensmith et al., 2004; Williams, 1992) as a control variate. We adopt the common trick to add an entropy regularization term (Williams & Peng, 1991; Mnih et al., 2016) that favors higher entropy. We impose entropy regularization on the outputs of the agents with coefficients $\lambda _ { s }$ (Sender) and $\lambda _ { r }$ (Receiver).
+
+Stochastic Computation Graph In our setup, the gradient estimate approach of Schulman et al. (2015) reduces to computing the gradient of the following surrogate function:
+
+$$
+\mathbb { E } _ { i _ { s } , i _ { r } } \mathbb { E } _ { m \sim S ( i _ { s } ) } \left[ \mathcal { L } ( o ; l ) + s t o p \_ g r a d i e n t \left( \mathcal { L } ( o ; l ) - b \right) \log P _ { \theta } ( m ) \right]
+$$
+
+Here, we do not sample Receiver actions: Its parameter gradients are obtained with standard backpropagation (the first term in Eq. 4). Sender’s messages are sampled, and its gradient are calculated akin to REINFORCE (the second term in Eq. 4). As in REINFORCE, we apply entropy-favoring regularization on Sender’s output (with coefficient $\lambda _ { s }$ ) and use the mean baseline $b$ .
+
+Role of entropy regularization As we mentioned above, when training with REINFORCE and Stochastic Computation Graph, we include a (standard) entropy regularization term in the loss which explicitly maximizes entropy of Sender’s output. Clearly, this term is at odds with the entropy minimization effect we observe. In our experiments, we found that high values of $\lambda _ { s }$ prevent communication success; on the other hand, small non-zero $\lambda _ { s }$ is crucial for successful training. In Section 4 we investigate the effect of $\lambda _ { s }$ on entropy minimization.
+
+![](images/7bf01b11262fe43d461e678755c7da46c1502316eb31e2937253de79c4f47923.jpg)  
+Figure 1: Guess Number: entropy of the messages $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ . Shaded regions mark standard deviation.
+
+# 3.3 EXPERIMENTAL PROTOCOL
+
+In Guess Number, we use all $2 ^ { 8 }$ possible inputs for training, early stopping and analysis. In Image Classification, we train on random image pairs from the MNIST training data, and use image pairs from the MNIST held-out set for validation. We select the runs that achieved a high level of performance (training accuracy above 0.99 for Guess Number and validation accuracy above 0.98 for Image Classification), thus studying typical agent behavior provided they succeeded at the game.
+
+At test time, we select the Sender’s message symbol greedily, hence the messages are discrete and Sender represents a (deterministic) function $S$ of its input $i _ { s }$ , $m = S ( i )$ . Calculating the entropy $H ( m )$ of the distribution of discrete messages $_ { \mathbf { \nabla } } \mathbf { m }$ is straightforward. In Guess Number, we enumerate all 256 possible values of $z$ as inputs, save the messages from Sender and calculate entropy $H ( m )$ For Image Classification, we sample image pairs from the MNIST hold-out set.
+
+The upper bound on $H ( m )$ is as follow: $H _ { m a x } = 8$ bits (bounded by $H ( i _ { s } ) )$ in Guess Number, and $H _ { m a x } = 1 0$ bits (bounded by vocabulary size) in Image Classification. Its lower bound is equal to $H _ { m i n } = H ( l | i _ { r } ) = 8 - k$ bits for Guess number. In Image Classification, communication can only succeed if $H ( m )$ is not less than $H ( l )$ , i.e., $H _ { m i n } = H ( l ) = \log _ { 2 } N _ { l }$ , with $N _ { l }$ the number of equally-sized classes we split the images into.
+
+# 4 EXPERIMENTS
+
+# 4.1 ENTROPY MINIMIZATION
+
+Guess Number In Figure 1, the horizontal axes span the number of bits of $z$ that Receiver lacks, $8 - k$ . The vertical axis reports the information content of the protocol, measured by messages entropy
+
+![](images/17264c5bd247f11740017af7d8f364cd642984e0b43cb1f2fa16e76d6dc93c85.jpg)  
+Figure 2: Image Classification: entropy of the messages $_ { m }$ in function of log number of target classes, $N _ { l }$ . Shaded regions mark standard deviation.
+
+$H ( m )$ . Each integer on the horizontal axis corresponds to a game configuration, and for each such configuration we aggregate multiple (successful) runs with different hyperparameters and random seeds. $H _ { m i n }$ indicates the minimal amount of bits Sender has to send in a particular configuration for the task to be solvable. The upper bound (not shown) is $H _ { m a x } = 8$ bits.
+
+Consider first the configurations where Receiver’s input is insufficient to answer correctly (at least one binary digit hidden, $k \leq 7 ,$ ). From Figure 1a, we observe that the transmitted information is strictly monotonically increasing with the number of binary digits hidden from Receiver. Thus, even if Sender sees the very same input in all configurations, a more nuanced protocol is only developed when it is necessary. Moreover, the entropy $H ( m )$ (equivalently: the transmitted information) stays close to the lower bound. This entropy minimization property holds for all the considered training approaches across all configurations.
+
+Consider next the configuration where Receiver is getting the whole integer $z$ as its input $k = 8$ , the leftmost configuration in Figure 1, corresponding to 0 on $\mathbf { X }$ axis). Based on the observations above, one would expect that the protocol would approach zero entropy in this case (as no information needs to be transmitted). However, the measurements indicate that the protocol is encoding considerably more information. It turns out that this information is entirely ignored by Receiver. To demonstrate this, we fed all possible distinct inputs to Sender, retrieved the corresponding messages, and shuffled them to destroy any information about the inputs they might carry. The shuffled messages were then passed to Receiver alongside with its own (un-shuffled) inputs. The overall performance was not affected by this manipulation, confirming the hypothesis that Receiver ignores messages. We conclude that in this case there is no apparent entropy minimization pressure on Sender simply because there is no communication. The full experiment is reported in Appendix.
+
+We further consider the effect of various hyperparameters. In Figure 1b, we split the results obtained with Gumbel-Softmax by relaxation temperature. As discussed in Section 3.2, lower temperatures more closely approximate discrete communication, hence providing a convenient control of the level of discreteness imposed during training (recall that at test time we select the symbol greedily). The figure shows that lower temperatures consistently lead to lower $H ( m )$ values. This implies that, as we increase the “level of discreteness” at training, we get stronger entropy minimization pressures.
+
+In Figures 1c & 1d, we report $H ( m )$ when training with Stochastic Graph Optimization and REINFORCE across degrees of entropy regularization. We report curves corresponding to $\lambda _ { s }$ values1 which converged in more than three configurations. With REINFORCE, we see a weak tendency for a higher $\lambda _ { s }$ to trigger higher entropy in the protocol (only violated at $\lambda _ { s } = 0 . 5$ ). However, message entropy stays generally close to the lower bound even in presence of strong exploration, which favors higher entropy in Sender’s output distribution.
+
+Image Classification As the models are more complex, we only had consistent success when training with Gumbel-Softmax. In Figure 2a we aggregate all successful runs. The information encoded by the protocol grows as Receiver’s output requires more information. However, in all configurations, the transmitted information stays well below the 10-bit upper bound and tends to be close to $H _ { m i n }$
+
+![](images/6fa8e7b9ba43e806e95fc16fc1116f453973a8f37add915cf0b2443524f00517.jpg)  
+Figure 3: Learning in presence of random labels. GS (SM) indicates models trained with GumbelSoftmax (Softmax) channel. Linear are models with the channel removed.
+
+A natural interpretation is that Sender prefers to take charge of image classification and directly pass information about the output label, rather than sending along a presumably more information-heavy description of the input. In Figure 2b, we split the runs by temperature. Again, we see that lower temperatures consistently lead to stronger entropy minimization pressures.
+
+Summarizing, when communicating through a discrete channel, there is consistent pressure for the emergent protocol to encode as little information as necessary. This holds across games, training methods and hyperparameters. When training with Gumbel-Softmax, temperature controls the strength of this pressure, confirming the relation between entropy minimization and discreteness.
+
+# 4.2 REPRESENTATION DISCRETENESS AND ROBUSTNESS
+
+The entropy minimization effect we established in Section 4.1 indicates that a discrete representation will only store as much information as necessary to solve the task. This emergent behavior respects the “information bottleneck” principle (Tishby et al., 1999; Achille & Soatto, 2018a). The fact that lower training-time temperatures in Gumbel-Softmax optimization correlate with both higher discreteness and a tighter bottleneck (see Section 4.1) makes us further conjecture that discreteness is causally connected to the emergent bottleneck.
+
+Interestingly, the same information-bottleneck principle has also been claimed to govern entropy minimization in natural language (Zaslavsky et al., 2018; 2019). The information-bottleneck effects in neural agents and natural language might be due to the same cause, namely discreteness of the communication. Further, we hypothesize that the emergent discrete information-bottleneck might have useful properties, since existing (continuous) architectures that explicitly impose a bottleneck pressure to provide a form of beneficial regularization are more robust to overfitting (Fischer, 2019) and adversarial attacks (Alemi et al., 2016; Fischer, 2019). We test here whether the expected regularization properties also emerge in our computational simulations (without any explicit pressure imposed through the cost function), and whether they are correlated with the degree of discreteness of the communication channel. If this connection exists, this also suggests that discreteness might be “beneficial” to human languages for the same reasons.
+
+To assess our hypotheses, we consider the Image Classification game $N _ { l } = 1 0 $ ) in presence of randomly-shuffled training labels (the test set is untouched) (Zhang et al., 2016). This task allows us to explore whether the discrete communication bottleneck is associated to robustness to overfitting, and whether the latter depends on the level of discreteness (controlled by the temperature $\tau$ of the Gumbel-Softmax relaxation).2
+
+We use the same architecture as before. The agents are trained with Gumbel-Softmax relaxation. However, for practicality, we do not switch to fully discrete communication at test time, only removing the noise component, thus effectively reducing Sender’s output to softmax with temperature. We refer to this architecture as GS.
+
+We also consider two baseline architectures without relaxed discrete channel. In Linear, the fully connected output layer of Sender is directly connected to the linear embedding input of Receiver. Softmax (SM) places a softmax activation (with temperature) after Sender’s output layer and passes the result to Receiver. At test time, SM coincides with GS with the same temperature, but there was no discrete-sampling approximation during SM training.
+
+We vary temperature and proportion of training examples with shuffled labels. We use temperatures $\tau = 1 . 0$ and $\tau = 1 0 . 0$ (the agents reach a test accuracy of 0.98 when trained with these temperatures on the original training set). SM with $\tau = 1 . 0$ and $\tau = 1 0 . 0$ behave similarly, hence we only report SM with $\tau = 1 . 0$ .
+
+In Figure 3a we report training accuracy when all labels are shuffled. Linear and SM fit the random labels almost perfectly within the first 150 epochs. With $\tau = 1 0 . 0$ , GS achieves 0.8 accuracy within 200 epochs. When GS with $\tau = 1 . 0$ is considered, the agents only start to improve over random guessing after 150 epochs, and accuracy is well below 0.2 after 200 epochs. As expected, test set performance is at chance level (Figure 3b). In the next experiment, we shuffle labels for a randomly selected half of the training instances. Train and test accuracies are shown in Figures 3c and 3d, respectively. All models initially fit the true-label examples (train accuracy $\approx 0 . 5$ , test accuracy $\approx 0 . 9 7 ,$ ). With more training, the baselines and GS with $\tau = 1 0 . 0$ start (over)fitting the randomly labeled examples, too: train accuracy grows, while test accuracy falls. In contrast, GS with $\tau = 1 . 0$ does not fit random labels, and its test accuracy stays high. Note that SM patterns with Linear and high-temperature GS, showing that the training-time discretization noise in GS is instrumental for robustness to over-fitting.
+
+We interpret the results as follows. To fully exploit their joint capacity for “successful” over-fitting, the agents need to coordinate label memorization. This requires passing large amounts of information through the channel. With a low temperature (more closely approximating a discrete channel), this is hard, due to stronger entropy minimization pressure. To test the hypothesis, we ran an experiment where all labels are shuffled and a layer of size $4 0 0 { \bf x } 4 0 0$ is either added to Sender (just before the channel) or to Receiver (just after the channel). We predict that, with higher $\tau$ (less discrete, less entropy minimization pressure), the training curves will be close, as the extra capacity can be used for memorization equally easy in both cases. With lower $\tau$ (better discrete approximation, more pressure), the accuracy curves will be more distant, as the extra capacity can only be successfully exploited for memorization when placed before the channel. Figures 3e & 3f borne out the prediction.
+
+These experiments showed that increased channel discreteness makes it harder to pass large amounts of information through, and leads to increased robustness against overfitting. This supports our hypotheses that discreteness brings about a bottleneck that in turn has some beneficial properties, which might ultimately provide a motivation for why an emergent communication system should evolve towards discreteness.
+
+# 5 RELATED WORK
+
+We briefly reviewed studies of emergent deep agent communication and entropy minimization in human language in the introduction. We are not aware of earlier work that looks for this property in emergent communication, although Evtimova et al. (2018) used information theory to study protocol development during learning, and, closer to us, Kågebäck et al. (2018) studied the effect of explicitly adding a complexity minimization term to the cost function on an emergent color-naming system.
+
+Discrete representations are explored in many places (e.g., van den Oord et al., 2017; Jang et al., 2016; Rolfe, 2016). However, these works focus on ways to learn discrete representations, rather than analyzing the properties of representations that are independently emerging on the side. Furthermore, our study also extends to the agents communicating with variable-length messages, produced and consumed by GRU (Cho et al., 2014) and Transformer (Vaswani et al., 2017) cells (see Appendix C.3). The sequential setup is specific to language, clearly distinguished from the settings studied in generic sparse-representation work.
+
+Other studies, inspired by the informational bottleneck principle, control the complexity of neural representations by regulating their information content (Strouse & Schwab, 2017; Fischer, 2019; Alemi et al., 2016; Achille & Soatto, 2018a;b). While they externally impose the bottleneck, we observe that it is an intrinsic feature when learning to communicate through a discrete channel.
+
+# 6 DISCUSSION
+
+Entropy minimization is pervasive in human language, where it constitutes a specific facet of the more general pressure towards communication efficiency. We found that the same property consistently characterizes the protocol emerging in simulations where two neural networks learn to solve a task jointly through a discrete communication code.
+
+In a comparative perspective, our results suggest that entropy minimization is a general property of discrete communication systems, independent of specific biological constraints humans are subject to. In particular, our analysis tentatively establishes a link between this property and the inherent difficulty of encoding information in discrete form (cf. the effect of adding a layer before or after the communication bottleneck in the overfitting experiment above).
+
+Exploring entropy minimization in computational simulations provides a flexibility we lack when studying humans. For example, we uncovered here initial evidence that the communication bottleneck is acting as a good regularizer, making the joint agent system more robust to noise. This leads to an intriguing conjecture on the origin of language. Its discrete nature is often traced back to the fact that it allows us to produce an infinite number of expressions by combining a finite set of primitives (e.g., Berwick & Chomsky, 2016). However, it is far from clear that the need to communicate an infinite number of concepts could have provided the initial pressure to develop a discrete code. More probably, once such code independently emerged, it laid the conditions to develop an infinitely expressive language (Bickerton, 2014; Collier et al., 2014). Our work suggests that, because of its inherent regularizing effect, discrete coding is advantageous already when communication is about a limited number of concepts, providing an alternative explanation for its origin.
+
+In the future, we would like to study more continuous domains, such as color maps, where perfect accuracy is not easily attainable, nor desirable. Will the networks find an accuracy/complexity trade-off similar to those attested in human languages? Will other core language properties claimed to be related to this trade-off, such as Zipfian frequency distributions (Ferrer i Cancho & Díaz-Guilera, 2007), concurrently emerge? We would also like to compare the performance of human subjects equipped with novel continuous vs. discrete communication protocols, adopting the methods of experimental semiotics (Galantucci, 2009). We expect discrete protocols to favor generalization and robustness.
+
+Our results have implications for the efforts to evolve agents interacting with each other and with humans through a discrete channel. First, because of entropy minimization, we should not expect the agents to develop a richer protocol than the simplest one that will ensure accurate communication. For example, Bouchacourt & Baroni (2018) found that agents trained to discriminate pairs of natural images depicting instances of about 500 high-level categories, such as cats and dogs, developed a lexicon that does not denote such categories, but low-level properties of the image themselves. This makes sense from an entropy-minimization perspective, as talking about the 500 high-level categories demands $\log _ { 2 } 5 0 0$ bits of information, whereas many low-level strategies (e.g., discriminating average pixel intensity in the images) will only require transmitting a few bits. To have agents developing rich linguistic protocols, we must face them with varied challenges that truly demand them.
+
+Second, the focus on a discrete protocol is typically motivated by the goal to develop machines eventually able to communicate with humans. Indeed, discrete messages are not required in multiagent scenarios where no human in the loop is foreseen (Sukhbaatar et al., 2016). Our results suggest that, long before agents reach the level of complexity necessary to converse with humans, there are independent reasons to encourage discreteness, as it provides a source of robustness in a noisy world. An exciting direction for future applied work will be to test, in more practical settings, the effectiveness of discrete communication as a general form of representation learning.
+
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+
+![](images/2e19b1010d43d3c12875a93c74cec80b63ff24122260fae0268ff3fb167820d5.jpg)  
+Figure 4: Robustness to adversarial examples: higher accuracy given fixed $\epsilon$ implies more robustness.
+
+![](images/8e91081d64406159c83cb9042cdb5a174448d618f5f343dc2a39f5d144c8d6dd.jpg)  
+Figure 5: Guess Number: Receiver’s dependence on messages, measured as performance drop under message intervention.
+
+# A ROBUSTNESS TO ADVERSARIAL ATTACKS
+
+In this Section, we study robustness of the agents equipped with a relaxed discrete channel against adversarial attacks. We use the same architectures as in Section 4.2 of the main paper. Specifically, by GS we indicate the architecture where agents are trained with Gumbel-Softmax relaxation, which at test-time is replaced by (noiseless) softmax with the same temperature. SM is an architecture where the communication channel is replaced by a Softmax layer with temperature. The Linear baseline has no “channel”: the output of Sender is directly plugged as input to the Receiver.
+
+We train agents with different random seeds and implement white-box attacks on the trained models, varying temperature $\tau$ and the allowed perturbation norm, $\epsilon$ . We use the standard Fast Gradient Sign Method (FGSM) of Goodfellow et al. (2014). The original image $i _ { s }$ is perturbed to $i _ { s } ^ { * }$ along the direction that maximizes the loss of Receiver’s output $\pmb { o } = R \big ( S ( i _ { s } ) \big )$ w.r.t. ground-truth class $\imath$ :
+
+$$
+i _ { s } ^ { * } = c l i p \left[ i _ { s } + \epsilon \cdot s i g n \left[ \nabla _ { i _ { s } } \mathcal { L } ( o , l ) \right] , 0 , 1 \right]
+$$
+
+where $\epsilon$ controls the $L _ { \infty }$ norm of the perturbation. Under an attack with a fixed $\epsilon$ , a more robust method would have a smaller accuracy drop. To avoid numerical stability issues akin to those reported by Carlini & Wagner (2016), all computations are done in 64-bit floats.
+
+As earlier in Section 4.2, we observed that SM behaves similarly with different temperatures (we experimented with $\tau \in \{ 0 . 1 , 1 . 0 , 1 0 . 0 \} )$ , we report only results with $\tau = 1 . 0$ . Figure 4a shows that, as the relaxation temperature decreases, the accuracy drop also decreases. The highest robustness is achieved with $\tau = 0 . 1$ . Comparison with the baselines (Figure 4b) confirms that relaxed discrete training with $\tau = 0 . 1$ improves robustness. However, this robustness comes at the cost of harder training: 2 out of 5 random seeds did not reach the desired performance level (0.98) after 200 epochs.
+
+# B HOW MUCH DOES RECEIVER RELY ON MESSAGES IN GUESS NUMBER?
+
+We supplement the experiments of Section 3 of the main text by studying the degree to which Receiver relies on messages in Guess Number. In particular, we show that when Receiver has the full input $( i _ { s } = i _ { r }$ ), it ignores the messages.
+
+We measure the degree to which Receiver relies on the messages from Sender by constructing a setup where we break communication, but still let Receiver rely on its own input. More precisely, we first enumerate all test inputs for Sender $i _ { s }$ and Receiver $i _ { r }$ . We obtain messages that correspond to Sender’s inputs, and shuffle them. Next, we feed the shuffled messages alongside Receiver’s own (un-shuffled) inputs and compute accuracy, as a measure of Receiver’s dependence on the messages. This procedure preserves the marginal distribution of the Receiver input messages, but destroys all the information Sender transmits.
+
+Without messages, Receiver, given $k$ input bits, can only reach an accuracy of $2 ^ { 8 - k }$ . In Figure 5, we report results aggregated by training method. Receiver is extremely close to the accuracy’s higher bound in all configurations. Moreover, when Receiver gets the entire input, the drop in accuracy after shuffling is tiny, proving that Receiver’s reliance on the message is minimal in that setting.
+
+# C INFLUENCE OF ARCHITECTURE CHOICES
+
+# C.1 DOES VOCABULARY SIZE AFFECT THE RESULTS?
+
+We repeat the same experiments as in Section 3 of the main text while varying vocabulary size. Note that, to make Guess Number solvable across each configuration, the vocabulary has to contain at least 256 symbols. Similarly, for Image Classification, vocabulary size must be of at least 100. We tried vocabulary sizes of 256, 1024, 4096 for Guess Number, and 512, 1024, 2048 for Image Classification. The results are reported in Figures 6 (Guess Number) and 7 (Image Classification). We observe that there is little qualitative variation over vocabulary size, hence the conclusions we had in Section 3 of the main paper are robust to variations of this parameter.
+
+# C.2 DOES RECEIVER’S CAPACITY AFFECT THE RESULTS?
+
+One potential confounding variable is the capacity of Receiver. Indeed, if Receiver is very simple, then, for the task to be solved, Sender would have to calculate the answer itself and feed it to Receiver. To investigate this, we repeat the Image Classification experiment from Section 4 of the main paper while controlling the power of Receiver’s architecture: we put two additional fully-connected 400x400 hidden layers between the input embedding and the output layer, while in Section 4, Receiver had a single hidden layer.
+
+In Figure 8 we compare the results obtained with these two variations of Receiver. The reported entropy minimization effect holds: even in presence of additional layers, the entropy of messages $H ( m )$ is far from the upper-bound $H _ { m a x } = 1 0$ bits and closely follows the lower bound, $H _ { m i n } =$ $\log _ { 2 } N _ { l }$ . Thus, again, a more nuanced protocol only appears when it is needed. Finally, we see that results for both architectures are close, although in three out of seven task setups (the number of classes $N _ { l }$ is 2, 10, and 20) a deeper model results in a slightly higher entropy of the protocol, on average. Overall, we conclude that Receiver’s capacity does not play a major role in the entropy minimization effect and the latter also takes place with a more powerful Receiver.
+
+# C.3 WHAT IF COMMUNICATION TAKES PLACE THROUGH SEQUENCES OF SYMBOLS?
+
+We also experiment with Guess Number in a setup where the agents communicate via variable-length messages. The general architecture of the agents is same as in Section 3.1, however the output of Sender is used as the initial hidden state of a GRU cell (Cho et al., 2014). In turn, this GRU is unrolled to generate the message. The message is produced until the GRU outputs a special $< e o s >$ token or until the maximal length is reached. In the latter case, ${ < } e o s { > }$ is appended to the message. The produced message is consumed by a Receiver’s GRU unit and the hidden state corresponding to ${ < } e o s { > }$ is used by Receiver as input to further processing. We use the Stochastic Computation Graph estimator as described in Section 3.2, as it provided fastest convergence.
+
+![](images/880a1381c927999dfa608b78060d21b3fbfc22c630e00fdd9526dd473345df65.jpg)  
+Figure 6: Guess Number: Entropy of the messages $_ { m }$ , depending on vocabulary size, training method, and relaxation temperature $\tau$ (when trained with Gumbel-Softmax) or Sender’s entropy regularization coefficient $\lambda _ { s }$ . Shaded regions mark standard deviation.
+
+We consider the entire variable-length message as the realization of a random variable $_ { m }$ when calculating the entropy of the messages, $H ( m )$ . The results are reported in Figure 9, arranged in function of maximal message length and vocabulary size. As before, we aggregate the successful runs according to the entropy regularization coefficient $\lambda _ { s }$ applied to Sender’s output layer.
+
+From Figure 9 we observe that the results are in line with those obtained in the one-symbol scenario. Entropy minimization still holds: a more nuanced (high-entropy) protocol only develops when more digits are hidden from Receiver, which hence requires more information to perform the task. The approximation to the lower bound is however less tight as the overall number of possible messages grows (higher maximum length and/or vocabulary size). There is also a weak tendency for lower $\lambda _ { s }$ to encourage a tighter bottleneck.
+
+In preliminary experiments, we have similar results when the variable-length communication is performed via Transformer cells (Vaswani et al., 2017) instead of GRUs (not reported here).
+
+# D TWO-DIGIT MNIST DATASET
+
+As discussed in Section 3, to ensure high output informational complexity in the Image Classification task, we use a two-digit variant of the MNIST dataset (LeCun et al., 1998a). We construct it as follows. When iterating over the original MNIST dataset, we take a batch $b$ and (a) select the first $| b | / 2$ and last $| b | / 2$ images, refer to them as $b _ { 1 }$ and $b _ { 2 }$ , respectively; (b) create a new batch where the ith image from $b _ { 1 }$ is placed to the left of the ith image from $b _ { 2 }$ and then vice versa. As a result, we obtain a new stream of images, where each MNIST digit is seen twice, on the left and on the right side. Note that not all possible pairwise combinations of the original images are generated (there are $6 0 0 0 0 ^ { 2 }$ of those in the training set alone) and the exact combinations change across epochs. As labels, we use the depicted two-digit number modulo $N _ { l }$ , where $N _ { l }$ is the required number of classes. All pixels are scaled into [0, 1]. We use the same process to generate training and test sets, based on the training and test images of the original MNIST dataset, respectively.
+
+![](images/960a1cc8dface251f1f8f8ca8cd2aab06beb57b3185a545c72a677a3a41a7048.jpg)  
+Figure 7: Image Classification: entropy of the messages $H ( m )$ across vocabulary sizes. Successful runs are pooled together. Shaded regions mark standard deviation.
+
+![](images/bfe57e96396202bf263b1209a7444cb917cd4d4a12307438ffcc32e50e86db7c.jpg)  
+Figure 8: Image Classification: entropy of the messages $H ( m )$ across Receiver model sizes. Successful runs are pooled together. Shaded regions mark standard deviation.
+
+![](images/40a177662c62a29e55feb03cc83b7813716dd47d1ac6b5a694034a541ca214e4.jpg)  
+Figure 9: Guess Number: Entropy of the emergent protocol when communication is performed with variable-length messages. Shaded regions mark standard deviation.
+
+# E HYPERPARAMETERS
+
+In our experiments, we used the following hyperparameter grids.
+
+Guess Number (Gumbel-Softmax) Vocab. size: [256, 1024, 4096]; temperature, $\tau$ : [0.5, 0.75, 1.0, 1.25, 1.5]; learning rate: [0.001, 0.0001]; max. number of epochs: 250; random seeds: [0, 1, 2, 3]; batch size: 8; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8].
+
+Guess Number (REINFORCE) Vocab. size: [256, 1024, 4096]; Sender entropy regularization coef., $\lambda _ { s }$ : [0.01, 0.05, 0.025, 0.1, 0.5, 1.0]; Receiver entropy regularization coef., $\lambda _ { r }$ : [0.01, 0.1, 0.5, 1.0]; learning rate: [0.0001, 0.001, 0.01]; max. number of epochs: 1000; random seeds: [0, 1, 2, 3]; batch size: 2048; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8].
+
+Guess Number (Stochastic Computation Graph approach): Vocab. size: [256, 1024, 4096]; Sender entropy regularization coef., $\lambda _ { s }$ : [0.01, 0.025, 0.05, 0.075, 0.1, 0.25]; learning rate: [0.0001, 0.001]; max. number of epochs: 1000; random seeds: [0, 1, 2, 3]; batch size: 2048; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8].
+
+Image Classification experiments Vocab. size: [512, 1024, 2048]; temperature, $\tau$ : [0.5, 0.75, 1.0, 1.5, 2.0]; learning rate: [0.01, 0.001, 0.0001], max. number of epochs: 100; random seeds: [0, 1, 2]; batch size: 32; early stopping thr.: 0.98; number of classes: [2, 4, 10, 20, 25, 50, 100].
+
+Fitting random labels experiments Vocab. size: 1024; temperature, $\tau { } .$ : [1.0, 10.0]; learning rate: 1e-4, max. number of epochs: 200; random seeds: [0, 1, 2, 3, 4]; batch size: 32; early stopping thr.: $\infty$ ; prob. of label corruption: [0.0, 0.5, 1.0].
+
+Adversarial attack experiments Vocab. size: 1024; temperature, $\tau$ : [0.1, 1.0, 10.0]; learning rate: 1e-4, max. number of epochs: 200; random seeds: [0, 1, 2, 3, 4]; batch size: 32; early stopping thr.: 0.98.
+
+# F EVOLUTION OF MESSAGE ENTROPY DURING TRAINING
+
+In this Section, we aim to gain additional insight into development of the communication protocol by measuring its entropy during training. We concentrate on Guess Number and use the same experimental runs summarized in Figure 1 of the main text.
+
+For each game configuration (that is, number of bits hidden from Receiver), we randomly select one successful run and plot the evolution of Sender message entropy and accuracy over training epochs.3 We also plot entropy and accuracy curves for a randomly selected failed run, to verify to what extent entropy development depends on task success.
+
+We report results for runs where training was performed with Gumbel-Softmax relaxation and with the Stochastic Graph Computation approach in Figures 10 and 11, respectively. The reported entropy and accuracy values are calculated in evaluation mode, where Sender’s output is selected greedily, without sampling. A higher entropy of such deterministic Sender indicates that the latter can encode more information about inputs in its messages.
+
+From these results, we firstly observe that the initial entropy of Sender’s messages (before training) can be both higher than required for communication success (Figures 10a and 11a) and lower (the rest). When it starts higher than needed, it generally falls closer to the minimum level required for the solution. When the initial value is low, it increases during training. The failed runs can have message entropy above (Figures 10a, 10b & 11a) and below (e.g. Figures 10c, 10d & 11d) successful runs, suggesting that there is no systematic relation between degree of entropy and task success.
+
+The fact that the entropy can be reduced with no decrease in accuracy or even with accuracy growth (e.g. Figure 10a, red line, epochs 5..30) indicates that the tendency to discover new messages (increasing entropy) is counter-balanced by the complexity of mutual coordination with Receiver when entropy is larger. In our interpretation, it is this interplay that serves as a source of the natural bottleneck.
+
+Finally, while in some runs the entropy is effectively increased w.r.t. its initialization level, the resulting protocol’s entropy is at, or slightly above the lower bound of what the task allows. In this sense, we argue that the reported effect can be correctly denoted as a “minimization” result.
+
+![](images/ec673b2b4b5c6f3a8028c0d4d329bebe77be350d900efebd2016ce16aad210b3.jpg)  
+Figure 10: Evolution of $H ( m )$ over training epochs. Gumbel Softmax-based optimization, Guess Number. For each game configuration, specified by the number of bits Receiver lacks, we sample one successful (black line) and one failed (red line) training trajectory. The blue line marks $H _ { m i n }$ , minimal entropy for a successful solution.
+
+![](images/88e877bacfe4f92d5898e1513423df0cef7e71b5ecb42210d345e73c1dfd650a.jpg)  
+Figure 11: Evolution of $H ( m )$ over training epochs. Stochastic Computation Graph-based optimization, Guess Number. For each game configuration, specified by the number of bits Receiver lacks, we sample one successful (black line) and one failed (red line) training trajectory. The blue line marks $H _ { m i n }$ , minimal entropy for a successful solution.

parse/train/SylVJTNKDr/SylVJTNKDr_content_list.json

@@ -0,0 +1,2026 @@
+[
+    {
+        "type": "text",
+        "text": "ENTROPY MINIMIZATION IN EMERGENT LANGUAGES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            818,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            209,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There is a growing interest in studying the languages emerging when neural agents are jointly trained to solve tasks requiring communication through a discrete channel. We investigate here the information-theoretic complexity of such languages, focusing on the basic two-agent, one-exchange setup. We find that, under common training procedures, the emergent languages are subject to an entropy minimization pressure that has also been detected in human language, whereby the mutual information between the communicating agent’s inputs and the messages is minimized, within the range afforded by the need for successful communication. This pressure is amplified as we increase communication channel discreteness. Further, we observe that stronger discrete-channel-driven entropy minimization leads to representations with increased robustness to overfitting and adversarial attacks. We conclude by discussing the implications of our findings for the study of natural and artificial communication systems. ",
+        "bbox": [
+            233,
+            250,
+            766,
+            430
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            481,
+            334,
+            496
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There has recently been much interest in the analysis of the communication systems arising when deep network agents that interact to accomplish a goal are allowed to exchange language-like discrete messages (Lazaridou et al., 2016; Havrylov & Titov, 2017; Choi et al., 2018; Lazaridou et al., 2018). Understanding the emergent protocol is important if we want to eventually develop agents capable of interacting with each other and with us through language (Mikolov et al., 2016; Chevalier-Boisvert et al., 2019). The pursuit might also provide comparative evidence about how core properties of human language itself have evolved (Kirby, 2002; Hurford, 2014; Graesser et al., 2019). While earlier studies reported ways in which deep agent protocols radically depart from human language (Kottur et al., 2017; Bouchacourt & Baroni, 2018; Chaabouni et al., 2019; Lowe et al., 2019), we show here that emergent communication shares an important property of the latter, namely a tendency towards entropy minimization. ",
+        "bbox": [
+            174,
+            520,
+            825,
+            672
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Converging evidence indicates that efficiency pressures are at work in language and other biological communication systems (Ferrer i Cancho et al., 2013; Gibson et al., 2019). One particular aspect of communicative efficiency, that has been robustly observed across many semantic domains, is a tendency to minimize lexicon entropy, to the extent allowed by the counteracting need for accuracy (Zaslavsky et al., 2018; 2019). For example, while most languages distinguish grandmothers from grandfathers, very few have separate words for mother- and father-side grandmothers, as the latter distinction would make communication only slightly more accurate at the cost of an increase in lexicon complexity (Kemp & Regier, 2012). We show here, in two separate games designed to precisely measure such property, that the protocol evolved by interacting deep agents is subject to the same complexity minimization pressure. ",
+        "bbox": [
+            174,
+            680,
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+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Entropy minimization in natural language has been connected to the Information Bottleneck principle (Tishby et al., 1999). In turn, complexity reduction due to the Information Bottleneck provides a beneficial regularization effect on the learned representations (Fischer, 2019; Alemi et al., 2016; Achille & Soatto, 2018a;b). It is difficult to experimentally verify the presence of such effect in human languages, but we can look for it in our emergent language simulations. We confirm that, when relaxing channel discreteness, the entropy minimization property no longer holds, and the system becomes less robust against overfitting and adversarial noise. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 GENERAL FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            403,
+            117
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We establish our results in the context of signaling games (Lewis, 1969), as introduced to the current language emergence literature by Lazaridou et al. (2016) and adopted in several later studies (Havrylov & Titov, 2017; Bouchacourt & Baroni, 2018; Lazaridou et al., 2018). There are two agents, Sender and Receiver, provided with individual inputs at the beginning of each episode. Sender sends a single message to Receiver, and Receiver has to perform an action based on its own input and the received message. Importantly, there is no direct supervision on the message protocol. We consider agents that are deterministic functions of their inputs (after training). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As an example, consider the task of communicating a $n$ -bit number, sampled uniformly at random from $0 . . . 2 \\sp n - 1$ . The full number is shown to Sender, and its $k$ $0 \\leq k \\leq n \\}$ least-significant bits are also revealed to Receiver. Receiver has to output the full number, based on the message from Sender and its own input. Would the Sender transmit the entire number through its message? In this case, the protocol would be “complex,” encoding $n$ bits. Alternatively, Sender could only encode the bits that Receiver does not know, and let Receiver fill in the rest by itself. This emergent protocol would be “simple,” encoding less information about the number. We find experimentally that, once the agents are successfully trained to jointly solve the task, the emergent protocol minimizes the entropy of the messages or, equivalently in our setup, the mutual information between Sender’s input and messages. In other words, the agents consistently approximate the simplest successful protocol (in the current example, the one transmitting $\\approx n - k$ bits). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "After training, we can connect the entropies of Sender and Receiver inputs $i _ { s }$ and $i _ { r }$ , messages $m =  { \\mathit { S } } ( i _ { s } )$ , Receiver’s output (the chosen action) $\\pmb { o } = R ( \\pmb { m } , \\pmb { i } _ { r } )$ , and ground-truth outputs $\\imath$ by using standard inequalities (Cover & Thomas, 2012): ",
+        "bbox": [
+            174,
+            397,
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+            440
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a0907ebacd5cebe03d1160c09076deb5b35abdf4ac99b2d357565cf4551a20a.jpg",
+        "text": "$$\nH ( i _ { s } ) \\geq H ( S ( i _ { s } ) ) = H ( m ) \\geq H ( m | i _ { r } ) \\geq H ( R ( m , i _ { r } ) | i _ { r } ) = H ( o | i _ { r } ) \\approx H ( l | i _ { r } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            445,
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+            463
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(Note that, since agents are deterministic after training, $H ( \\pmb { { m } } ) = I ( \\pmb { { i } } _ { s } ; \\pmb { { m } } )$ . We can then use these quantities interchangeably.) Our empirical measurements indicate that the entropy of the messages $_ { \\mathbf { \\nabla } } \\mathbf { m }$ in the emergent protocol tends to approach the lower bound: $H ( { \\pmb m } )  \\bar { H ( { \\pmb l } | { \\pmb i } _ { r } ) }$ , even if the upper-bound $H ( i _ { s } )$ is far. ",
+        "bbox": [
+            174,
+            468,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we observe that when the amount of information that Receiver needs is reduced, without changing other parameters, the emergent protocol becomes simpler (lower entropy). In other words, the emergent protocol adapts to minimize the information that passes through it. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We will release the code for our experiments upon acceptance. ",
+        "bbox": [
+            173,
+            580,
+            581,
+            595
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            614,
+            341,
+            632
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 GAMES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            266,
+            661
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We study two signaling games. In Guess Number, the agents are trained to recover an integerrepresenting vector with uniform Bernoulli-distributed components. This simple setup gives us full control over the amount of information needed to solve the task. The second game, Image Classification, uses more naturalistic data, as the agents are jointly trained to classify pairs of hand-written MNIST digits (LeCun et al., 1998b). ",
+        "bbox": [
+            174,
+            672,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Guess Number We draw an 8-bit integer $z$ , $0 \\leq z \\leq 2 5 5$ uniformly at random, by sampling its 8 bits independently from the uniform Bernoulli distribution. All bits are revealed to Sender as a 8-dimensional binary vector $i _ { s }$ . The last $k$ bits are revealed to Receiver $0 \\leq k \\leq 8 $ ) as its input $i _ { r }$ . Sender outputs a single-symbol message $_ { \\mathbf { \\nabla } } \\mathbf { m }$ to Receiver. In turn, Receiver outputs a vector $^ o$ that recovers all the bits of $z$ and should be equal to $i _ { s }$ . ",
+        "bbox": [
+            174,
+            750,
+            825,
+            819
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this game, Sender has a linear layer that maps the input vector $i _ { s }$ to a hidden representation of size 10, followed by a leaky ReLU activation. Next is a linear layer followed by a softmax over the vocabulary. Receiver linearly maps both its input $i _ { r }$ and the message to 10-dimensional vectors, concatenates them, applies a fully connected layer with output size 20, followed by a leaky ReLU. Finally, another linear layer and a sigmoid nonlinearity are applied. When training with REINFORCE and the Stochastic Computation graph approach (see Section 3.2), we increase the hidden layer sizes threefold, as this leads to more robust convergence. ",
+        "bbox": [
+            174,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Image Classification In this game, the agents are jointly trained to classify 28x56 images of two MNIST digits, stacked side-by-side (more details in Appendix). Unlike Guess Number, Receiver has no side input. Instead, we control the informational complexity of Receiver’s task by controlling the size of its output space, i.e., the number of labels we assign to the images. To do so, we group all two-digit sequences 00..99 into $N _ { l } \\in \\{ 2 , 4 , 1 0 , 2 0 , 2 5 , 5 0 , 1 0 0 \\}$ equally-sized classes. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In Sender, input images are embedded a LeNet-1 instance (LeCun et al., 1990) into 400-dimensional vectors. These embedded vectors are passed to a fully connected layer, followed by a softmax selecting a vocabulary symbol. Receiver embeds the received messages into 400-dimensional vectors, passed to a fully connected layer with a softmax activation returning the class probabilities. ",
+        "bbox": [
+            174,
+            180,
+            825,
+            236
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We report hyperparameter grids in Appendix. In the following experiments, we fix vocabulary to 1024 symbols (experiments with other vocabulary sizes, multi-symbol messages, and larger architectures are reported in Appendix). No parts of the agents are pre-trained or shared. The optimized loss depends on the gradient estimation method used (see Section 3.2). We denote it $\\mathcal { L } ( o , l )$ , and it is a function of Receiver’s output $^ o$ and the ground-truth output $\\imath$ . When training in Guess Number with REINFORCE, we use a $_ { 0 / 1 }$ loss: the agents get 0 only if all bits of $z$ were correctly recovered. When training with Gumbel-Softmax relaxation or the Stochastic Computation Graph approach, we use binary cross-entropy (Guess Number) and negative log-likelihood (Image Classification). ",
+        "bbox": [
+            174,
+            242,
+            825,
+            354
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 TRAINING WITH DISCRETE CHANNEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            372,
+            470,
+            386
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Training to communicate with discrete messages is non-trivial, as we cannot back-propagate through the messages. Current language emergence work mostly uses Gumbel-Softmax relaxation (e.g. (Havrylov & Titov, 2017)) or REINFORCE (e.g. (Lazaridou et al., 2016)) to get gradient estimates. We also explore the Stochastic Computation Graph optimization approach. We plug the obtained gradient estimates into the Adam optimizer (Kingma & Ba, 2014). ",
+        "bbox": [
+            174,
+            397,
+            825,
+            467
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gumbel-Softmax relaxation Samples from the Gumbel-Softmax (Maddison et al., 2016; Jang et al., 2016) distribution (a) are reperameterizable, hence allow gradient-based training, and (b) approximate samples from the corresponding Categorical distribution. To get a sample that approximates an $n$ -dimensional Categorical distribution with probabilities $p _ { i }$ , we draw $n$ i.i.d. samples $g _ { i }$ from Gumbel(0,1) and use them to calculate a vector $\\textbf {  { y } }$ with components: ",
+        "bbox": [
+            173,
+            473,
+            825,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/306bc4067206ac6d23e91e49d3c008ec1b09de44dae0838c906a6fb28a308493.jpg",
+        "text": "$$\ny _ { i } = \\frac { e x p \\left[ ( g _ { i } + \\log p _ { i } ) / \\tau \\right] } { \\sum _ { j } e x p \\left[ ( g _ { j } + \\log p _ { j } ) / \\tau \\right] }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau$ is the temperature hyperparameter. As $\\tau$ tends to 0, the samples $\\textbf {  { y } }$ get closer to one-hot samples; as $\\tau \\to + \\infty$ , the components $y _ { i }$ become uniform. During training, we use these relaxed samples as messages from Sender, making the entire Sender/Receiver setup differentiable. ",
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+    {
+        "type": "text",
+        "text": "REINFORCE by Williams (1992) is a standard reinforcement learning algorithm. In our setup, it estimates the gradient of the expectation of the loss $\\mathcal { L } ( o , l )$ w.r.t. the parameter vector $\\pmb \\theta$ as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/cab360000e4c7229b2765f51b658063552c9b6aac7e197301548927f11ce7b6f.jpg",
+        "text": "$$\n\\mathbb { E } _ { i _ { s } , i _ { r } } \\mathbb { E } _ { m \\sim S ( i _ { s } ) , o \\sim R ( m , i _ { r } ) } \\left[ ( \\mathcal { L } ( o ; l ) - b ) \\nabla _ { \\theta } \\log P _ { \\theta } ( m , o ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The expectations are estimated by sampling $_ { m }$ from Sender and, after that, sampling $^ o$ from Receiver. We use the running mean baseline $b$ (Greensmith et al., 2004; Williams, 1992) as a control variate. We adopt the common trick to add an entropy regularization term (Williams & Peng, 1991; Mnih et al., 2016) that favors higher entropy. We impose entropy regularization on the outputs of the agents with coefficients $\\lambda _ { s }$ (Sender) and $\\lambda _ { r }$ (Receiver). ",
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+    {
+        "type": "text",
+        "text": "Stochastic Computation Graph In our setup, the gradient estimate approach of Schulman et al. (2015) reduces to computing the gradient of the following surrogate function: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/fd2763b852e1d7e3733c618c258a0059c8b0927c4cce57d6b0a674ccc6735f6b.jpg",
+        "text": "$$\n\\mathbb { E } _ { i _ { s } , i _ { r } } \\mathbb { E } _ { m \\sim S ( i _ { s } ) } \\left[ \\mathcal { L } ( o ; l ) + s t o p \\_ g r a d i e n t \\left( \\mathcal { L } ( o ; l ) - b \\right) \\log P _ { \\theta } ( m ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we do not sample Receiver actions: Its parameter gradients are obtained with standard backpropagation (the first term in Eq. 4). Sender’s messages are sampled, and its gradient are calculated akin to REINFORCE (the second term in Eq. 4). As in REINFORCE, we apply entropy-favoring regularization on Sender’s output (with coefficient $\\lambda _ { s }$ ) and use the mean baseline $b$ . ",
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+    {
+        "type": "text",
+        "text": "Role of entropy regularization As we mentioned above, when training with REINFORCE and Stochastic Computation Graph, we include a (standard) entropy regularization term in the loss which explicitly maximizes entropy of Sender’s output. Clearly, this term is at odds with the entropy minimization effect we observe. In our experiments, we found that high values of $\\lambda _ { s }$ prevent communication success; on the other hand, small non-zero $\\lambda _ { s }$ is crucial for successful training. In Section 4 we investigate the effect of $\\lambda _ { s }$ on entropy minimization. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7bf01b11262fe43d461e678755c7da46c1502316eb31e2937253de79c4f47923.jpg",
+        "image_caption": [
+            "Figure 1: Guess Number: entropy of the messages $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ . Shaded regions mark standard deviation. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.3 EXPERIMENTAL PROTOCOL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Guess Number, we use all $2 ^ { 8 }$ possible inputs for training, early stopping and analysis. In Image Classification, we train on random image pairs from the MNIST training data, and use image pairs from the MNIST held-out set for validation. We select the runs that achieved a high level of performance (training accuracy above 0.99 for Guess Number and validation accuracy above 0.98 for Image Classification), thus studying typical agent behavior provided they succeeded at the game. ",
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+    {
+        "type": "text",
+        "text": "At test time, we select the Sender’s message symbol greedily, hence the messages are discrete and Sender represents a (deterministic) function $S$ of its input $i _ { s }$ , $m = S ( i )$ . Calculating the entropy $H ( m )$ of the distribution of discrete messages $_ { \\mathbf { \\nabla } } \\mathbf { m }$ is straightforward. In Guess Number, we enumerate all 256 possible values of $z$ as inputs, save the messages from Sender and calculate entropy $H ( m )$ For Image Classification, we sample image pairs from the MNIST hold-out set. ",
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+        "type": "text",
+        "text": "The upper bound on $H ( m )$ is as follow: $H _ { m a x } = 8$ bits (bounded by $H ( i _ { s } ) )$ in Guess Number, and $H _ { m a x } = 1 0$ bits (bounded by vocabulary size) in Image Classification. Its lower bound is equal to $H _ { m i n } = H ( l | i _ { r } ) = 8 - k$ bits for Guess number. In Image Classification, communication can only succeed if $H ( m )$ is not less than $H ( l )$ , i.e., $H _ { m i n } = H ( l ) = \\log _ { 2 } N _ { l }$ , with $N _ { l }$ the number of equally-sized classes we split the images into. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 ENTROPY MINIMIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Guess Number In Figure 1, the horizontal axes span the number of bits of $z$ that Receiver lacks, $8 - k$ . The vertical axis reports the information content of the protocol, measured by messages entropy ",
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+    {
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+        "img_path": "images/17264c5bd247f11740017af7d8f364cd642984e0b43cb1f2fa16e76d6dc93c85.jpg",
+        "image_caption": [
+            "Figure 2: Image Classification: entropy of the messages $_ { m }$ in function of log number of target classes, $N _ { l }$ . Shaded regions mark standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "$H ( m )$ . Each integer on the horizontal axis corresponds to a game configuration, and for each such configuration we aggregate multiple (successful) runs with different hyperparameters and random seeds. $H _ { m i n }$ indicates the minimal amount of bits Sender has to send in a particular configuration for the task to be solvable. The upper bound (not shown) is $H _ { m a x } = 8$ bits. ",
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+    {
+        "type": "text",
+        "text": "Consider first the configurations where Receiver’s input is insufficient to answer correctly (at least one binary digit hidden, $k \\leq 7 ,$ ). From Figure 1a, we observe that the transmitted information is strictly monotonically increasing with the number of binary digits hidden from Receiver. Thus, even if Sender sees the very same input in all configurations, a more nuanced protocol is only developed when it is necessary. Moreover, the entropy $H ( m )$ (equivalently: the transmitted information) stays close to the lower bound. This entropy minimization property holds for all the considered training approaches across all configurations. ",
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+    {
+        "type": "text",
+        "text": "Consider next the configuration where Receiver is getting the whole integer $z$ as its input $k = 8$ , the leftmost configuration in Figure 1, corresponding to 0 on $\\mathbf { X }$ axis). Based on the observations above, one would expect that the protocol would approach zero entropy in this case (as no information needs to be transmitted). However, the measurements indicate that the protocol is encoding considerably more information. It turns out that this information is entirely ignored by Receiver. To demonstrate this, we fed all possible distinct inputs to Sender, retrieved the corresponding messages, and shuffled them to destroy any information about the inputs they might carry. The shuffled messages were then passed to Receiver alongside with its own (un-shuffled) inputs. The overall performance was not affected by this manipulation, confirming the hypothesis that Receiver ignores messages. We conclude that in this case there is no apparent entropy minimization pressure on Sender simply because there is no communication. The full experiment is reported in Appendix. ",
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+    {
+        "type": "text",
+        "text": "We further consider the effect of various hyperparameters. In Figure 1b, we split the results obtained with Gumbel-Softmax by relaxation temperature. As discussed in Section 3.2, lower temperatures more closely approximate discrete communication, hence providing a convenient control of the level of discreteness imposed during training (recall that at test time we select the symbol greedily). The figure shows that lower temperatures consistently lead to lower $H ( m )$ values. This implies that, as we increase the “level of discreteness” at training, we get stronger entropy minimization pressures. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figures 1c & 1d, we report $H ( m )$ when training with Stochastic Graph Optimization and REINFORCE across degrees of entropy regularization. We report curves corresponding to $\\lambda _ { s }$ values1 which converged in more than three configurations. With REINFORCE, we see a weak tendency for a higher $\\lambda _ { s }$ to trigger higher entropy in the protocol (only violated at $\\lambda _ { s } = 0 . 5$ ). However, message entropy stays generally close to the lower bound even in presence of strong exploration, which favors higher entropy in Sender’s output distribution. ",
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+        "type": "text",
+        "text": "Image Classification As the models are more complex, we only had consistent success when training with Gumbel-Softmax. In Figure 2a we aggregate all successful runs. The information encoded by the protocol grows as Receiver’s output requires more information. However, in all configurations, the transmitted information stays well below the 10-bit upper bound and tends to be close to $H _ { m i n }$ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6fa8e7b9ba43e806e95fc16fc1116f453973a8f37add915cf0b2443524f00517.jpg",
+        "image_caption": [
+            "Figure 3: Learning in presence of random labels. GS (SM) indicates models trained with GumbelSoftmax (Softmax) channel. Linear are models with the channel removed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A natural interpretation is that Sender prefers to take charge of image classification and directly pass information about the output label, rather than sending along a presumably more information-heavy description of the input. In Figure 2b, we split the runs by temperature. Again, we see that lower temperatures consistently lead to stronger entropy minimization pressures. ",
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+    {
+        "type": "text",
+        "text": "Summarizing, when communicating through a discrete channel, there is consistent pressure for the emergent protocol to encode as little information as necessary. This holds across games, training methods and hyperparameters. When training with Gumbel-Softmax, temperature controls the strength of this pressure, confirming the relation between entropy minimization and discreteness. ",
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+    {
+        "type": "text",
+        "text": "4.2 REPRESENTATION DISCRETENESS AND ROBUSTNESS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The entropy minimization effect we established in Section 4.1 indicates that a discrete representation will only store as much information as necessary to solve the task. This emergent behavior respects the “information bottleneck” principle (Tishby et al., 1999; Achille & Soatto, 2018a). The fact that lower training-time temperatures in Gumbel-Softmax optimization correlate with both higher discreteness and a tighter bottleneck (see Section 4.1) makes us further conjecture that discreteness is causally connected to the emergent bottleneck. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, the same information-bottleneck principle has also been claimed to govern entropy minimization in natural language (Zaslavsky et al., 2018; 2019). The information-bottleneck effects in neural agents and natural language might be due to the same cause, namely discreteness of the communication. Further, we hypothesize that the emergent discrete information-bottleneck might have useful properties, since existing (continuous) architectures that explicitly impose a bottleneck pressure to provide a form of beneficial regularization are more robust to overfitting (Fischer, 2019) and adversarial attacks (Alemi et al., 2016; Fischer, 2019). We test here whether the expected regularization properties also emerge in our computational simulations (without any explicit pressure imposed through the cost function), and whether they are correlated with the degree of discreteness of the communication channel. If this connection exists, this also suggests that discreteness might be “beneficial” to human languages for the same reasons. ",
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+    {
+        "type": "text",
+        "text": "To assess our hypotheses, we consider the Image Classification game $N _ { l } = 1 0 $ ) in presence of randomly-shuffled training labels (the test set is untouched) (Zhang et al., 2016). This task allows us to explore whether the discrete communication bottleneck is associated to robustness to overfitting, and whether the latter depends on the level of discreteness (controlled by the temperature $\\tau$ of the Gumbel-Softmax relaxation).2 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use the same architecture as before. The agents are trained with Gumbel-Softmax relaxation. However, for practicality, we do not switch to fully discrete communication at test time, only removing the noise component, thus effectively reducing Sender’s output to softmax with temperature. We refer to this architecture as GS. ",
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+    {
+        "type": "text",
+        "text": "We also consider two baseline architectures without relaxed discrete channel. In Linear, the fully connected output layer of Sender is directly connected to the linear embedding input of Receiver. Softmax (SM) places a softmax activation (with temperature) after Sender’s output layer and passes the result to Receiver. At test time, SM coincides with GS with the same temperature, but there was no discrete-sampling approximation during SM training. ",
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+    {
+        "type": "text",
+        "text": "We vary temperature and proportion of training examples with shuffled labels. We use temperatures $\\tau = 1 . 0$ and $\\tau = 1 0 . 0$ (the agents reach a test accuracy of 0.98 when trained with these temperatures on the original training set). SM with $\\tau = 1 . 0$ and $\\tau = 1 0 . 0$ behave similarly, hence we only report SM with $\\tau = 1 . 0$ . ",
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+    {
+        "type": "text",
+        "text": "In Figure 3a we report training accuracy when all labels are shuffled. Linear and SM fit the random labels almost perfectly within the first 150 epochs. With $\\tau = 1 0 . 0$ , GS achieves 0.8 accuracy within 200 epochs. When GS with $\\tau = 1 . 0$ is considered, the agents only start to improve over random guessing after 150 epochs, and accuracy is well below 0.2 after 200 epochs. As expected, test set performance is at chance level (Figure 3b). In the next experiment, we shuffle labels for a randomly selected half of the training instances. Train and test accuracies are shown in Figures 3c and 3d, respectively. All models initially fit the true-label examples (train accuracy $\\approx 0 . 5$ , test accuracy $\\approx 0 . 9 7 ,$ ). With more training, the baselines and GS with $\\tau = 1 0 . 0$ start (over)fitting the randomly labeled examples, too: train accuracy grows, while test accuracy falls. In contrast, GS with $\\tau = 1 . 0$ does not fit random labels, and its test accuracy stays high. Note that SM patterns with Linear and high-temperature GS, showing that the training-time discretization noise in GS is instrumental for robustness to over-fitting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We interpret the results as follows. To fully exploit their joint capacity for “successful” over-fitting, the agents need to coordinate label memorization. This requires passing large amounts of information through the channel. With a low temperature (more closely approximating a discrete channel), this is hard, due to stronger entropy minimization pressure. To test the hypothesis, we ran an experiment where all labels are shuffled and a layer of size $4 0 0 { \\bf x } 4 0 0$ is either added to Sender (just before the channel) or to Receiver (just after the channel). We predict that, with higher $\\tau$ (less discrete, less entropy minimization pressure), the training curves will be close, as the extra capacity can be used for memorization equally easy in both cases. With lower $\\tau$ (better discrete approximation, more pressure), the accuracy curves will be more distant, as the extra capacity can only be successfully exploited for memorization when placed before the channel. Figures 3e & 3f borne out the prediction. ",
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+    {
+        "type": "text",
+        "text": "These experiments showed that increased channel discreteness makes it harder to pass large amounts of information through, and leads to increased robustness against overfitting. This supports our hypotheses that discreteness brings about a bottleneck that in turn has some beneficial properties, which might ultimately provide a motivation for why an emergent communication system should evolve towards discreteness. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We briefly reviewed studies of emergent deep agent communication and entropy minimization in human language in the introduction. We are not aware of earlier work that looks for this property in emergent communication, although Evtimova et al. (2018) used information theory to study protocol development during learning, and, closer to us, Kågebäck et al. (2018) studied the effect of explicitly adding a complexity minimization term to the cost function on an emergent color-naming system. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Discrete representations are explored in many places (e.g., van den Oord et al., 2017; Jang et al., 2016; Rolfe, 2016). However, these works focus on ways to learn discrete representations, rather than analyzing the properties of representations that are independently emerging on the side. Furthermore, our study also extends to the agents communicating with variable-length messages, produced and consumed by GRU (Cho et al., 2014) and Transformer (Vaswani et al., 2017) cells (see Appendix C.3). The sequential setup is specific to language, clearly distinguished from the settings studied in generic sparse-representation work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other studies, inspired by the informational bottleneck principle, control the complexity of neural representations by regulating their information content (Strouse & Schwab, 2017; Fischer, 2019; Alemi et al., 2016; Achille & Soatto, 2018a;b). While they externally impose the bottleneck, we observe that it is an intrinsic feature when learning to communicate through a discrete channel. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Entropy minimization is pervasive in human language, where it constitutes a specific facet of the more general pressure towards communication efficiency. We found that the same property consistently characterizes the protocol emerging in simulations where two neural networks learn to solve a task jointly through a discrete communication code. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In a comparative perspective, our results suggest that entropy minimization is a general property of discrete communication systems, independent of specific biological constraints humans are subject to. In particular, our analysis tentatively establishes a link between this property and the inherent difficulty of encoding information in discrete form (cf. the effect of adding a layer before or after the communication bottleneck in the overfitting experiment above). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Exploring entropy minimization in computational simulations provides a flexibility we lack when studying humans. For example, we uncovered here initial evidence that the communication bottleneck is acting as a good regularizer, making the joint agent system more robust to noise. This leads to an intriguing conjecture on the origin of language. Its discrete nature is often traced back to the fact that it allows us to produce an infinite number of expressions by combining a finite set of primitives (e.g., Berwick & Chomsky, 2016). However, it is far from clear that the need to communicate an infinite number of concepts could have provided the initial pressure to develop a discrete code. More probably, once such code independently emerged, it laid the conditions to develop an infinitely expressive language (Bickerton, 2014; Collier et al., 2014). Our work suggests that, because of its inherent regularizing effect, discrete coding is advantageous already when communication is about a limited number of concepts, providing an alternative explanation for its origin. ",
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+    {
+        "type": "text",
+        "text": "In the future, we would like to study more continuous domains, such as color maps, where perfect accuracy is not easily attainable, nor desirable. Will the networks find an accuracy/complexity trade-off similar to those attested in human languages? Will other core language properties claimed to be related to this trade-off, such as Zipfian frequency distributions (Ferrer i Cancho & Díaz-Guilera, 2007), concurrently emerge? We would also like to compare the performance of human subjects equipped with novel continuous vs. discrete communication protocols, adopting the methods of experimental semiotics (Galantucci, 2009). We expect discrete protocols to favor generalization and robustness. ",
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+    },
+    {
+        "type": "text",
+        "text": "Our results have implications for the efforts to evolve agents interacting with each other and with humans through a discrete channel. First, because of entropy minimization, we should not expect the agents to develop a richer protocol than the simplest one that will ensure accurate communication. For example, Bouchacourt & Baroni (2018) found that agents trained to discriminate pairs of natural images depicting instances of about 500 high-level categories, such as cats and dogs, developed a lexicon that does not denote such categories, but low-level properties of the image themselves. This makes sense from an entropy-minimization perspective, as talking about the 500 high-level categories demands $\\log _ { 2 } 5 0 0$ bits of information, whereas many low-level strategies (e.g., discriminating average pixel intensity in the images) will only require transmitting a few bits. To have agents developing rich linguistic protocols, we must face them with varied challenges that truly demand them. ",
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+    {
+        "type": "text",
+        "text": "Second, the focus on a discrete protocol is typically motivated by the goal to develop machines eventually able to communicate with humans. Indeed, discrete messages are not required in multiagent scenarios where no human in the loop is foreseen (Sukhbaatar et al., 2016). Our results suggest that, long before agents reach the level of complexity necessary to converse with humans, there are independent reasons to encourage discreteness, as it provides a source of robustness in a noisy world. An exciting direction for future applied work will be to test, in more practical settings, the effectiveness of discrete communication as a general form of representation learning. ",
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+            "Figure 4: Robustness to adversarial examples: higher accuracy given fixed $\\epsilon$ implies more robustness. "
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+        "type": "text",
+        "text": "A ROBUSTNESS TO ADVERSARIAL ATTACKS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this Section, we study robustness of the agents equipped with a relaxed discrete channel against adversarial attacks. We use the same architectures as in Section 4.2 of the main paper. Specifically, by GS we indicate the architecture where agents are trained with Gumbel-Softmax relaxation, which at test-time is replaced by (noiseless) softmax with the same temperature. SM is an architecture where the communication channel is replaced by a Softmax layer with temperature. The Linear baseline has no “channel”: the output of Sender is directly plugged as input to the Receiver. ",
+        "bbox": [
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "We train agents with different random seeds and implement white-box attacks on the trained models, varying temperature $\\tau$ and the allowed perturbation norm, $\\epsilon$ . We use the standard Fast Gradient Sign Method (FGSM) of Goodfellow et al. (2014). The original image $i _ { s }$ is perturbed to $i _ { s } ^ { * }$ along the direction that maximizes the loss of Receiver’s output $\\pmb { o } = R \\big ( S ( i _ { s } ) \\big )$ w.r.t. ground-truth class $\\imath$ : ",
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+        "page_idx": 11
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+        "type": "equation",
+        "img_path": "images/b87e8b8074018fe2234a3db1269e1cd7da74a41761101d08cfbb54ea709af41b.jpg",
+        "text": "$$\ni _ { s } ^ { * } = c l i p \\left[ i _ { s } + \\epsilon \\cdot s i g n \\left[ \\nabla _ { i _ { s } } \\mathcal { L } ( o , l ) \\right] , 0 , 1 \\right]\n$$",
+        "text_format": "latex",
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ controls the $L _ { \\infty }$ norm of the perturbation. Under an attack with a fixed $\\epsilon$ , a more robust method would have a smaller accuracy drop. To avoid numerical stability issues akin to those reported by Carlini & Wagner (2016), all computations are done in 64-bit floats. ",
+        "bbox": [
+            173,
+            790,
+            825,
+            833
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "As earlier in Section 4.2, we observed that SM behaves similarly with different temperatures (we experimented with $\\tau \\in \\{ 0 . 1 , 1 . 0 , 1 0 . 0 \\} )$ , we report only results with $\\tau = 1 . 0$ . Figure 4a shows that, as the relaxation temperature decreases, the accuracy drop also decreases. The highest robustness is achieved with $\\tau = 0 . 1$ . Comparison with the baselines (Figure 4b) confirms that relaxed discrete training with $\\tau = 0 . 1$ improves robustness. However, this robustness comes at the cost of harder training: 2 out of 5 random seeds did not reach the desired performance level (0.98) after 200 epochs. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B HOW MUCH DOES RECEIVER RELY ON MESSAGES IN GUESS NUMBER? ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            795,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We supplement the experiments of Section 3 of the main text by studying the degree to which Receiver relies on messages in Guess Number. In particular, we show that when Receiver has the full input $( i _ { s } = i _ { r }$ ), it ignores the messages. ",
+        "bbox": [
+            174,
+            133,
+            823,
+            176
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We measure the degree to which Receiver relies on the messages from Sender by constructing a setup where we break communication, but still let Receiver rely on its own input. More precisely, we first enumerate all test inputs for Sender $i _ { s }$ and Receiver $i _ { r }$ . We obtain messages that correspond to Sender’s inputs, and shuffle them. Next, we feed the shuffled messages alongside Receiver’s own (un-shuffled) inputs and compute accuracy, as a measure of Receiver’s dependence on the messages. This procedure preserves the marginal distribution of the Receiver input messages, but destroys all the information Sender transmits. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            280
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Without messages, Receiver, given $k$ input bits, can only reach an accuracy of $2 ^ { 8 - k }$ . In Figure 5, we report results aggregated by training method. Receiver is extremely close to the accuracy’s higher bound in all configurations. Moreover, when Receiver gets the entire input, the drop in accuracy after shuffling is tiny, proving that Receiver’s reliance on the message is minimal in that setting. ",
+        "bbox": [
+            176,
+            287,
+            825,
+            343
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C INFLUENCE OF ARCHITECTURE CHOICES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            366,
+            545,
+            381
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.1 DOES VOCABULARY SIZE AFFECT THE RESULTS? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            397,
+            553,
+            411
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We repeat the same experiments as in Section 3 of the main text while varying vocabulary size. Note that, to make Guess Number solvable across each configuration, the vocabulary has to contain at least 256 symbols. Similarly, for Image Classification, vocabulary size must be of at least 100. We tried vocabulary sizes of 256, 1024, 4096 for Guess Number, and 512, 1024, 2048 for Image Classification. The results are reported in Figures 6 (Guess Number) and 7 (Image Classification). We observe that there is little qualitative variation over vocabulary size, hence the conclusions we had in Section 3 of the main paper are robust to variations of this parameter. ",
+        "bbox": [
+            174,
+            422,
+            825,
+            521
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.2 DOES RECEIVER’S CAPACITY AFFECT THE RESULTS? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            540,
+            583,
+            554
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "One potential confounding variable is the capacity of Receiver. Indeed, if Receiver is very simple, then, for the task to be solved, Sender would have to calculate the answer itself and feed it to Receiver. To investigate this, we repeat the Image Classification experiment from Section 4 of the main paper while controlling the power of Receiver’s architecture: we put two additional fully-connected 400x400 hidden layers between the input embedding and the output layer, while in Section 4, Receiver had a single hidden layer. ",
+        "bbox": [
+            174,
+            565,
+            825,
+            648
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8 we compare the results obtained with these two variations of Receiver. The reported entropy minimization effect holds: even in presence of additional layers, the entropy of messages $H ( m )$ is far from the upper-bound $H _ { m a x } = 1 0$ bits and closely follows the lower bound, $H _ { m i n } =$ $\\log _ { 2 } N _ { l }$ . Thus, again, a more nuanced protocol only appears when it is needed. Finally, we see that results for both architectures are close, although in three out of seven task setups (the number of classes $N _ { l }$ is 2, 10, and 20) a deeper model results in a slightly higher entropy of the protocol, on average. Overall, we conclude that Receiver’s capacity does not play a major role in the entropy minimization effect and the latter also takes place with a more powerful Receiver. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            767
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.3 WHAT IF COMMUNICATION TAKES PLACE THROUGH SEQUENCES OF SYMBOLS? ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            786,
+            763,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We also experiment with Guess Number in a setup where the agents communicate via variable-length messages. The general architecture of the agents is same as in Section 3.1, however the output of Sender is used as the initial hidden state of a GRU cell (Cho et al., 2014). In turn, this GRU is unrolled to generate the message. The message is produced until the GRU outputs a special $< e o s >$ token or until the maximal length is reached. In the latter case, ${ < } e o s { > }$ is appended to the message. The produced message is consumed by a Receiver’s GRU unit and the hidden state corresponding to ${ < } e o s { > }$ is used by Receiver as input to further processing. We use the Stochastic Computation Graph estimator as described in Section 3.2, as it provided fastest convergence. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/880a1381c927999dfa608b78060d21b3fbfc22c630e00fdd9526dd473345df65.jpg",
+        "image_caption": [
+            "Figure 6: Guess Number: Entropy of the messages $_ { m }$ , depending on vocabulary size, training method, and relaxation temperature $\\tau$ (when trained with Gumbel-Softmax) or Sender’s entropy regularization coefficient $\\lambda _ { s }$ . Shaded regions mark standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            109,
+            803,
+            558
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We consider the entire variable-length message as the realization of a random variable $_ { m }$ when calculating the entropy of the messages, $H ( m )$ . The results are reported in Figure 9, arranged in function of maximal message length and vocabulary size. As before, we aggregate the successful runs according to the entropy regularization coefficient $\\lambda _ { s }$ applied to Sender’s output layer. ",
+        "bbox": [
+            174,
+            640,
+            825,
+            696
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "From Figure 9 we observe that the results are in line with those obtained in the one-symbol scenario. Entropy minimization still holds: a more nuanced (high-entropy) protocol only develops when more digits are hidden from Receiver, which hence requires more information to perform the task. The approximation to the lower bound is however less tight as the overall number of possible messages grows (higher maximum length and/or vocabulary size). There is also a weak tendency for lower $\\lambda _ { s }$ to encourage a tighter bottleneck. ",
+        "bbox": [
+            174,
+            703,
+            825,
+            787
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In preliminary experiments, we have similar results when the variable-length communication is performed via Transformer cells (Vaswani et al., 2017) instead of GRUs (not reported here). ",
+        "bbox": [
+            173,
+            794,
+            823,
+            821
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D TWO-DIGIT MNIST DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            847,
+            455,
+            863
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 3, to ensure high output informational complexity in the Image Classification task, we use a two-digit variant of the MNIST dataset (LeCun et al., 1998a). We construct it as follows. When iterating over the original MNIST dataset, we take a batch $b$ and (a) select the first $| b | / 2$ and last $| b | / 2$ images, refer to them as $b _ { 1 }$ and $b _ { 2 }$ , respectively; (b) create a new batch where the ith image from $b _ { 1 }$ is placed to the left of the ith image from $b _ { 2 }$ and then vice versa. As a result, we obtain a new stream of images, where each MNIST digit is seen twice, on the left and on the right side. Note that not all possible pairwise combinations of the original images are generated (there are $6 0 0 0 0 ^ { 2 }$ of those in the training set alone) and the exact combinations change across epochs. As labels, we use the depicted two-digit number modulo $N _ { l }$ , where $N _ { l }$ is the required number of classes. All pixels are scaled into [0, 1]. We use the same process to generate training and test sets, based on the training and test images of the original MNIST dataset, respectively. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/960a1cc8dface251f1f8f8ca8cd2aab06beb57b3185a545c72a677a3a41a7048.jpg",
+        "image_caption": [
+            "Figure 7: Image Classification: entropy of the messages $H ( m )$ across vocabulary sizes. Successful runs are pooled together. Shaded regions mark standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            117,
+            802,
+            241
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/bfe57e96396202bf263b1209a7444cb917cd4d4a12307438ffcc32e50e86db7c.jpg",
+        "image_caption": [
+            "Figure 8: Image Classification: entropy of the messages $H ( m )$ across Receiver model sizes. Successful runs are pooled together. Shaded regions mark standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            349,
+            319,
+            630,
+            474
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/40a177662c62a29e55feb03cc83b7813716dd47d1ac6b5a694034a541ca214e4.jpg",
+        "image_caption": [
+            "Figure 9: Guess Number: Entropy of the emergent protocol when communication is performed with variable-length messages. Shaded regions mark standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            246,
+            546,
+            738,
+            873
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            236,
+            379,
+            251
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we used the following hyperparameter grids. ",
+        "bbox": [
+            174,
+            267,
+            598,
+            281
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Guess Number (Gumbel-Softmax) Vocab. size: [256, 1024, 4096]; temperature, $\\tau$ : [0.5, 0.75, 1.0, 1.25, 1.5]; learning rate: [0.001, 0.0001]; max. number of epochs: 250; random seeds: [0, 1, 2, 3]; batch size: 8; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8]. ",
+        "bbox": [
+            176,
+            287,
+            825,
+            330
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Guess Number (REINFORCE) Vocab. size: [256, 1024, 4096]; Sender entropy regularization coef., $\\lambda _ { s }$ : [0.01, 0.05, 0.025, 0.1, 0.5, 1.0]; Receiver entropy regularization coef., $\\lambda _ { r }$ : [0.01, 0.1, 0.5, 1.0]; learning rate: [0.0001, 0.001, 0.01]; max. number of epochs: 1000; random seeds: [0, 1, 2, 3]; batch size: 2048; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8]. ",
+        "bbox": [
+            174,
+            337,
+            825,
+            393
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Guess Number (Stochastic Computation Graph approach): Vocab. size: [256, 1024, 4096]; Sender entropy regularization coef., $\\lambda _ { s }$ : [0.01, 0.025, 0.05, 0.075, 0.1, 0.25]; learning rate: [0.0001, 0.001]; max. number of epochs: 1000; random seeds: [0, 1, 2, 3]; batch size: 2048; early stopping thr.: 0.99; bits shown to Receiver: [0, 1, 2, 3, 4, 5, 6, 7, 8]. ",
+        "bbox": [
+            173,
+            400,
+            826,
+            457
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Image Classification experiments Vocab. size: [512, 1024, 2048]; temperature, $\\tau$ : [0.5, 0.75, 1.0, 1.5, 2.0]; learning rate: [0.01, 0.001, 0.0001], max. number of epochs: 100; random seeds: [0, 1, 2]; batch size: 32; early stopping thr.: 0.98; number of classes: [2, 4, 10, 20, 25, 50, 100]. ",
+        "bbox": [
+            173,
+            463,
+            825,
+            506
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Fitting random labels experiments Vocab. size: 1024; temperature, $\\tau { } .$ : [1.0, 10.0]; learning rate: 1e-4, max. number of epochs: 200; random seeds: [0, 1, 2, 3, 4]; batch size: 32; early stopping thr.: $\\infty$ ; prob. of label corruption: [0.0, 0.5, 1.0]. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            554
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Adversarial attack experiments Vocab. size: 1024; temperature, $\\tau$ : [0.1, 1.0, 10.0]; learning rate: 1e-4, max. number of epochs: 200; random seeds: [0, 1, 2, 3, 4]; batch size: 32; early stopping thr.: 0.98. ",
+        "bbox": [
+            174,
+            561,
+            826,
+            603
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F EVOLUTION OF MESSAGE ENTROPY DURING TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            626,
+            661,
+            640
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In this Section, we aim to gain additional insight into development of the communication protocol by measuring its entropy during training. We concentrate on Guess Number and use the same experimental runs summarized in Figure 1 of the main text. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            698
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For each game configuration (that is, number of bits hidden from Receiver), we randomly select one successful run and plot the evolution of Sender message entropy and accuracy over training epochs.3 We also plot entropy and accuracy curves for a randomly selected failed run, to verify to what extent entropy development depends on task success. ",
+        "bbox": [
+            174,
+            704,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We report results for runs where training was performed with Gumbel-Softmax relaxation and with the Stochastic Graph Computation approach in Figures 10 and 11, respectively. The reported entropy and accuracy values are calculated in evaluation mode, where Sender’s output is selected greedily, without sampling. A higher entropy of such deterministic Sender indicates that the latter can encode more information about inputs in its messages. ",
+        "bbox": [
+            174,
+            767,
+            825,
+            838
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "From these results, we firstly observe that the initial entropy of Sender’s messages (before training) can be both higher than required for communication success (Figures 10a and 11a) and lower (the rest). When it starts higher than needed, it generally falls closer to the minimum level required for the solution. When the initial value is low, it increases during training. The failed runs can have message entropy above (Figures 10a, 10b & 11a) and below (e.g. Figures 10c, 10d & 11d) successful runs, suggesting that there is no systematic relation between degree of entropy and task success. ",
+        "bbox": [
+            174,
+            844,
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+            887
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The fact that the entropy can be reduced with no decrease in accuracy or even with accuracy growth (e.g. Figure 10a, red line, epochs 5..30) indicates that the tendency to discover new messages (increasing entropy) is counter-balanced by the complexity of mutual coordination with Receiver when entropy is larger. In our interpretation, it is this interplay that serves as a source of the natural bottleneck. ",
+        "bbox": [
+            173,
+            152,
+            825,
+            222
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Finally, while in some runs the entropy is effectively increased w.r.t. its initialization level, the resulting protocol’s entropy is at, or slightly above the lower bound of what the task allows. In this sense, we argue that the reported effect can be correctly denoted as a “minimization” result. ",
+        "bbox": [
+            174,
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+            272
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec673b2b4b5c6f3a8028c0d4d329bebe77be350d900efebd2016ce16aad210b3.jpg",
+        "image_caption": [
+            "Figure 10: Evolution of $H ( m )$ over training epochs. Gumbel Softmax-based optimization, Guess Number. For each game configuration, specified by the number of bits Receiver lacks, we sample one successful (black line) and one failed (red line) training trajectory. The blue line marks $H _ { m i n }$ , minimal entropy for a successful solution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            75,
+            761,
+            852
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/88e877bacfe4f92d5898e1513423df0cef7e71b5ecb42210d345e73c1dfd650a.jpg",
+        "image_caption": [
+            "Figure 11: Evolution of $H ( m )$ over training epochs. Stochastic Computation Graph-based optimization, Guess Number. For each game configuration, specified by the number of bits Receiver lacks, we sample one successful (black line) and one failed (red line) training trajectory. The blue line marks $H _ { m i n }$ , minimal entropy for a successful solution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
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+        ],
+        "page_idx": 18
+    }
+]

parse/train/kziQtP-nGqzDb/kziQtP-nGqzDb.md

@@ -0,0 +1,185 @@
+# Learning Human Pose Estimation Features with Convolutional Networks
+
+Arjun Jain New York University ajain@nyu.edu
+
+Jonathan Tompson New York University tompson@cims.nyu.edu
+
+Mykhaylo Andriluka MPI Saarbruecken andriluk@mpi-inf.mpg.de
+
+Graham W. Taylor University of Guelph gwtaylor@uoguelph.ca
+
+Christoph Bregler New York University chris.bregler@nyu.edu
+
+# Abstract
+
+This paper introduces a new architecture for human pose estimation using a multilayer convolutional network architecture and a modified learning technique that learns low-level features and a higher-level weak spatial model. Unconstrained human pose estimation is one of the hardest problems in computer vision, and our new architecture and learning schema shows improvement over the current stateof-the-art. The main contribution of this paper is showing, for the first time, that a specific variation of deep learning is able to meet the performance, and in many cases outperform, existing traditional architectures on this task. The paper also discusses several lessons learned while researching alternatives, most notably, that it is possible to learn strong low-level feature detectors on regions that might only cover a few pixels in the image. Higher-level spatial models improve somewhat the overall result, but to a much lesser extent than expected. Many researchers previously argued that the kinematic structure and top-down information are crucial for this domain, but with our purely bottom-up, and weak spatial model, we improve on other more complicated architectures that currently produce the best results. This echos what many other researchers, like those in the speech recognition, object recognition, and other domains have experienced [26].
+
+![](images/9c14247104c73dc1d4734f534db3640f451ff31bbb03df6711e6f09004866cf0.jpg)  
+Figure 1: The green cross is our new technique’s wrist locator, the red cross is the state-of-the-art CVPR13 MODEC detector [38] on the FLIC database.
+
+# 1 Introduction
+
+One of the hardest tasks in computer vision is determining the high degree-of-freedom configuration of a human body with all its limbs, complex self-occlusion, self-similar parts, and large variations due to clothing, body-type, lighting, and many other factors. The most challenging scenario for this problem is from a monocular RGB image and with no prior assumptions made using motion models, pose models, background models, or any other common heuristics that current state-of-theart systems utilize. Finding a face in frontal or side view is relatively simple, but determining the exact location of body parts such as hands, elbows, shoulders, hips, knees and feet, each of which sometimes only occupy a few pixels in the image in front of an arbitrary cluttered background, is significantly harder.
+
+The best performing pose estimation methods, including those based on deformable part models, typically are based on body part detectors. Such body part detectors commonly consist of multiple stages of processing. The first stage of processing in a typical pipeline consists of extracting sets of low-level features such as SIFT [25], HoG [11], or other filters that describe orientation statistics in local image patches. Next, these features are pooled over local spatial regions and sometimes across multiple scales to reduce the size of the representation and also develop local shift/scale invariance. Finally, the aggregate features are mapped to a vector, which is then either input to 1) a standard classifier such as a support vector machine (SVM) or 2) the next stage of processing (e.g. assembling the parts into a whole). Much work is devoted to engineering the system to produce a vector representation that is sensitive to class (e.g. head, hands, torso) while remaining invariant to the various nuisance factors (lighting, viewpoint, scale, etc.)
+
+An alternative approach is representation learning: relying on the data instead of feature engineering, to learn a good representation that is invariant to nuisance factors. For a recent review, see [6]. It is common to learn multiple layers of representation, which is referred to as deep learning. Several such techniques have used unsupervised or semi-supervised learning to extract multi-layer domain-specific invariant representations, however, it is purely supervised techniques that have won several recent challenges by large margins, including ImageNet LSVRC 2012 and 2013 [23, 51]. These end-to-end learning systems have capitalized on advances in computing hardware (notably GPUs), larger datasets like ImageNet, and algorithmic advances (specifically gradient-based training methods and regularization).
+
+While these methods are now proven in generic object recognition, their use in pose estimation has been limited. Part of the challenge in making end-to-end learning work for human pose estimation is related to the nonrigid structure of the body, the necessity for precision (deep recognition systems often throw away precise location information through pooling), and the complex, multi-modal nature of pose.
+
+In this paper, we present the first end-to-end learning approach for full-body human pose estimation. While our approach is based on convolutional networks (convnets) [24], we want to stress that the na¨ıve implementation of applying this model “off-the-shelf” will not work. Therefore, the contribution of this work is in both a model that outperforms state of the art deformable part models (DPMs) on a modern, challenging dataset, and also an analysis of what is needed to make convnets work in human pose estimation. In particular, we present a two-stage filtering approach whereby the response maps of convnet part detectors are denoised by a second process informed by the part hierarchy.
+
+# 2 Related Work
+
+Detecting people and their pose has been investigated for decades. Many early techniques rely on sliding-window part detectors based on hand-crafted or learned features or silhouette extraction techniques applied to controlled recording conditions. Examples include [14, 49, 5, 30]. We refer to [35] for a complete survey of this era. More recently, several new approaches have been proposed that are applied to unconstrained domains. In such domains, good performance has been achieved with so-called “bag of features” followed by regression-based, nearest neighbor or SVM-based architectures. Examples include “shape-context” edge-based histograms from the human body [28, 1] or just silhouette features [19]. Shakhnarovich et al. [39] learn a parameter sensitive hash function to perform example-based pose estimation. Many relevant techniques have also been applied to hand tracking such as [48]. A more general survey of the large field of hand tracking can be found in [12].
+
+Many techniques have been proposed that extract, learn, or reason over entire body features. Some use a combination of local detectors and structural reasoning (see [36] for coarse tracking and [10] for person-dependent tracking). In a similar spirit, more general techniques using pictorial structures [2, 3, 17, 37, 33, 34], “poselets” [9], and other part-models [16, 50] have received increased attention. We will focus on these techniques and their latest incarnations in the following sections.
+
+Further examples come from the HumanEva dataset competitions [41], or approaches that use higher-resolution shape models such as SCAPE [4] and further extensions [20, 8]. These differ from our domain in that the images considered are of higher quality and less cluttered. Also many of these techniques work on images from a single camera, but need video sequence input (not single images) to achieve impressive results [42, 52].
+
+As an example of a technique that works for single images against cluttered backgrounds, Shotton et al.’s Kinect based body part detector [40] uses a random forest of decision trees trained on synthetic depth data to create simple body part detectors. In the proposed work, we also adopt simple partbased detectors, however, we focus on a different learning strategy.
+
+There are a number of successful end-to-end representation learning techniques which perform pose estimation on a limited subset of body parts or body poses. One of the earliest examples of this type was Nowlan and Platt’s convolutional neural network hand tracker [30], which tracked a single hand. Osadchy et al. applied a convolutional network to simultaneously detect and estimate the pitch, yaw and roll of a face [31]. Taylor et al. [44] trained a convolutional neural network to learn an embedding in which images of people in similar pose lie nearby. They used a subset of body parts, namely, the head and hand locations to learn the “gist” of a pose, and resorted to nearest-neighbour matching rather than explicitly modeling pose. Perhaps most relevant to our work is Taylor et al.’s work on tracking people in video [45], augmenting a particle filter with a structured prior over human pose and dynamics based on learning representations. While they estimated a posterior over the whole body (60 joint angles), their experiments were limited to the HumanEva dataset [41], which was collected in a controlled laboratory setting. The datasets we consider in our experiments are truly poses “in the wild”, though we do not consider dynamics.
+
+A factor limiting earlier methods from tacking full pose-estimation with end-to-end learning methods, in particular deep networks, was the limited amount of labeled data. Such techniques, with millions or more parameters, require more data than structured techniques that have more a priori knowledge, such as DPMs. We attack this issue on two fronts. First, directly, by using larger labeled training sets which have become available in the past year or two, such as FLIC [38]. Second, indirectly, by better exploiting the data we have. The annotations provided by typical pose estimation datasets contain much richer information compared to the class labels in object recognition datasets In particular, we show that the relationships among parts contained in these annotations can be used to build better detectors.
+
+# 3 Model
+
+To perform pose estimation with a convolutional network architecture [24] (convnet), the most obvious approach would be to map the image input directly to a vector coding the articulated pose: i.e. the type of labels found in pose datasets. The convnet output would represent the unbounded 2-D or 3-D positions of joints, or alternatively a hierarchy of joint angles. However, we found that this worked very poorly. One issue is that pooling, while useful for improving translation invariance during object recognition, destroys precise spatial information which is necessary to accurately predict pose. Convnets that produce segmentation maps, for example, avoid pooling completely [47, 13]. Another issue is that the direct mapping from input space to kinematic body pose coefficients is highly non-linear and not one-to-one. However, even if we took this route, there is a deeper issue with attempting to map directly to a representation of full body pose. Valid poses represent a much lower-dimensional manifold in the high-dimensional space in which they are captured. It seems troublesome to make a discriminative network map to a space in which the majority of configurations do not represent valid poses. In other words, it makes sense to restrict the net’s output to a much smaller class of valid configurations.
+
+Rather than perform multiple-output regression using a single convnet to learn pose coefficients directly, we found that training multiple convnets to perform independent binary body-part classification, with one network per feature, resulted in improved performance on our dataset. These convnets are applied as sliding windows to overlapping regions of the input, and map a window of pixels to a single binary output: the presence or absence of that body part. The result of applying the convnet is a response-map indicating the confidence of the body part at that location. This lets us use much smaller convnets, and retain the advantages of pooling, at the expense of having to maintain a separate set of parameters for each body part. Of course, a series of independent part detectors cannot enforce consistency in pose in the same way as a structured output model, which produces valid full-body configurations. In the following sections, we first describe in detail the convolutional network architecture and then a method of enforcing pose consistency using parent-child relationships.
+
+![](images/0c57a43c96117a1f22c399c632755454928706adc26fa15333cd335d6768f45a.jpg)  
+Figure 2: The convolutional network architecture used in our experiments.
+
+# 3.1 Convolutional Network Architecture
+
+The lowest level of our two-stage feature detection pipeline is based on a standard convnet architecture, an overview of which is shown in Figure 2. Convnets, like their fully-connected, deep neural network counterparts, perform end-to-end feature learning and are trained with the back-propagation algorithm. However, they differ in a number of respects, most notably local connectivity, weight sharing, and local pooling. The first two properties significantly reduce the number of free parameters, and reduce the need to learn repeated feature detectors at different locations of the input. The third property makes the learned representation invariant to small translations of the input.
+
+The convnet pipeline shown in Figure 2 starts with a $6 4 \times 6 4$ pixel RGB input patch which has been local contrast normalized (LCN) [22] to emphasize geometric discontinuities and improve generalization performance [32]. The LCN layer is comprised of a $9 \times 9$ pixel local subtractive normalization, followed by a $9 \times 9$ local divisive normalization. The input is then processed by three convolution and subsampling layers, which use rectified linear units (ReLUs) [18] and max-pooling.
+
+As expected, we found that internal pooling layers help to a) reduce computational complexity1 and b) improve classification tolerance to small input image translations. Unfortunately, pooling also results in a loss of spatial precision. Since the target application for this convnet was offline (rather than real-time) body-pose detection, and since we found that with sufficient training exemplars, invariance to input translations can be learned, we choose to use only 2 stages of $2 \times 2$ pooling (where the total image downsampling rate is $4 \times 4$ ).
+
+Following the three stages of convolution and subsampling, the top-level pooled map is flattened to a vector and processed by three fully connected layers, analogous to those used in deep neural networks. Each of these output stages is composed of a linear matrix-vector multiplication with learned bias, followed by a point-wise non-linearity (ReLU). The output layer has a single logistic unit, representing the probability of the body part being present in that patch.
+
+To train the convnet, we performed standard batch stochastic gradient descent. From the training set images, we set aside a validation set to tune the network hyper-parameters, such as number and size of features, learning rate, momentum coefficient, etc. We used Nesterov momentum [43] as well as RMSPROP [46] to accelerate learning and we used L2 regularization and dropout [21] on the input to each of the fully-connected linear stages to reduce over-fitting the restricted-size training set.
+
+![](images/b897c3b4b74b20fbf081a67ec1f554fb1ec63a45784f14ca28d6610bc592eff5.jpg)  
+Figure 3: Spatial Model Connectivity with Spatial Priors
+
+# 3.2 Enforcing Global Pose Consistency with a Spatial Model
+
+When applied to the validation set, the raw output of the network presented in Section 3.1 produces many false-positives. We believe this is due to two factors: 1) the small image context as input to the convnet $6 4 \times 6 4$ pixels or approximately $5 \%$ of the input image area) does not give the model enough contextual information to perform anatomically consistent joint position inference and 2) the training set size is limited. We therefore use a higher-level spatial model with simple body-pose priors to remove strong outliers from the convnet output. We do not expect this model to improve the performance of poses that are close to the ground truth labels (within 10 pixels for instance), but rather it functions as a post processing step to de-emphasize anatomically impossible poses due to strong outliers.
+
+The inter-node connectivity of our simple spatial model is displayed in Figure 3. It consists of a linear chain of kinematic 2D nodes for a single side of the human body. Throughout our experiments we used the left shoulder, elbow and wrist; however we could have used the right side joints without loss of generality (since detection of the right body parts simply requires a horizontal mirror of the input image). For each node in the chain, our convnet detector generates response-map unary distributions $p _ { \mathrm { f a c } } \left( x \right)$ , $p _ { \mathrm { s h o } } \left( x \right)$ , $p _ { \mathrm { e l b } } \left( x \right)$ , $p _ { \mathrm { w r i } } \left( x \right)$ over the dense pixel positions $x$ , for the face, shoulder, elbow and wrist joints respectively. For the remainder of this section, all distributions are assumed to be a function over the pixel position, and so the $x$ notation will be dropped. The output of our spatial model will produce filtered response maps: $\hat { p } _ { \mathrm { f a c } }$ , $\hat { p } _ { \mathrm { s h o } }$ , $\hat { p } _ { \mathrm { e l b } }$ , and $\hat { p } _ { \mathrm { w r i } }$ .
+
+The body part priors for a pair of joints $( a , b )$ , $p _ { a \mid b = \vec { 0 } }$ , are calculated by creating a histogram of joint $a$ locations over the training set, given that the adjacent joint $b$ is located at the image center $\boldsymbol { x } = \vec { 0 }$ ). The histograms are then smoothed (using a gaussian filter) and normalized. The learned priors for $p _ { \mathrm { s h o } | \mathrm { f a c } = \vec { 0 } }$ , $p _ { \mathrm { e l b | s h o = } } \vec { 0 }$ , and $p _ { \mathrm { w r i l e l b { = } } \vec { 0 } }$ are shown in Figure 4. Note that due to symmetry, the prior for $p _ { \mathrm { e l b | w r i = } } \vec { 0 }$ is a $1 8 0 ^ { \circ }$ rotation of $p _ { \mathrm { w r i l e l b { = } } \vec { 0 } }$ (as is the case of other adjacent pairs). Rather than assume a simple Gaussian distribution for modeling pairwise interactions of adjacent nodes, as is standard in many parts-based detector implementations, we have found that the these non-parametric spatial priors lead to improved detection performance.
+
+![](images/b8f2a8dcb8c6bf47b5ff5f98798ab05770c71c336c282ef2b649e04771ee00c3.jpg)  
+Figure 4: Part priors for left body parts
+
+Given the full set of prior conditional distributions and the convnet unary distributions, we can now construct the filtered distribution for each part by using an approach that is analogous to the sumproduct belief propagation algorithm. For body part $i$ , with a set of neighbouring nodes $U$ , the final distribution is defined as:
+
+![](images/0adef6ef0b207b617c98fd4d726ef18c3945dee5dddf6ba7e3bc85554f07b216.jpg)  
+Figure 5: Global prior for the face: $h _ { \mathrm { f a c } }$
+
+$$
+\hat { p } _ { i } \propto { p _ { i } } ^ { \lambda } \prod _ { u \in U } \left( p _ { i | u = \vec { 0 } } * p _ { u } \right)
+$$
+
+where $\lambda$ is a mixing parameter and controls the confidence of each joint’s unary distribution towards its final filtered distribution (we used $\lambda = 1$ for our experiments). The final joint distribution is therefore a product of the unary distribution for that joint, as well as the beliefs from neighbouring nodes (as with standard sum-product belief propagation). In log space, the above product for the shoulder joint becomes:
+
+$$
+\begin{array} { r } { \log \left( \hat { p } _ { \mathrm { s h o } } \right) \propto \lambda \log \left( p _ { \mathrm { s h o } } \right) + \log \left( p _ { \mathrm { s h o } | \mathrm { f a c } = \vec { 0 } } * p _ { \mathrm { f a c } } \right) + \log \left( p _ { \mathrm { s h o } | \mathrm { e l b } = \vec { 0 } } * p _ { \mathrm { e l b } } \right) } \end{array}
+$$
+
+We also perform an equivalent computation for the elbow and wrist joints. The face joint is treated as a special case. Empirically, we found that incorporating image evidence from the shoulder joint to the filtered face distribution resulted in poor performance. This is likely due to the fact that the convnet does a very good job of localizing the face position, and so incorporating noisy evidence from the shoulder detector actually increases uncertainty. Instead, we use a global position prior for the face, $h _ { \mathrm { f a c } }$ , which is obtained by learning a location histogram over the face positions in the training set images, as shown in Figure 5. In log space, the output distribution for the face is then given by:
+
+$$
+\log \left( \hat { p } _ { \mathrm { f a c } } \right) \propto \lambda ~ \log \left( p _ { \mathrm { f a c } } \right) + \log \left( h _ { \mathrm { f a c } } \right)
+$$
+
+Lastly, since the learned neural network convolution features and the spatial priors are not explicitly invariant to scale, we must run the convnet and spatial model on images at multiple scales at test time, and then use the most likely joint location across those scales as the final joint location. For datasets containing examples with multiple persons (known a priori), we use non-maximal suppression [29] to find multiple local maxima across the filtered response-maps from each scale, and we then take the top $n$ most likely joint candidates from each person in the scene.
+
+# 4 Results
+
+We evaluated our architecture on the FLIC [38] dataset, which is comprised of 5003 still RGB images taken from an assortment of Hollywood movies. Each frame in the dataset contains at least one person in a frontal pose (facing the camera), and each frame was processed by Amazon Mechanical Turk to obtain ground truth labels for the joint positions of the upper body of a single person. The FLIC dataset is very challenging for state-of-the-art pose estimation methodologies because the poses are unconstrained, body parts are often occluded, and clothing and background are not consistent.
+
+We use 3987 training images from the dataset, which we also mirror horizontally to obtain a total of $3 9 8 7 \times 2 = 7 9 7 4$ examples. Since the training images are not at the same scale, we also manually annotate the bounding box for the head in these training set images, and bring them to canonical scale. Further, we crop them to $3 2 0 \times 2 4 0$ such that the center of the shoulder annotations lies at (160 px, $8 0 \ \mathrm { p x }$ ). We do not perform this image normalization at test time. Following the methodology of Felzenszwalb et al. [15], at test time we run our model on images with only one person (351 images of the 1016 test examples). As stated in Section 3, the model is run on 6 different input image scales and we then use the joint location with highest confidence across those scales as the final location.
+
+For training the convnet we use Theano [7], which provides a Python-based framework for efficient GPU processing and symbolic differentiation of complex compound functions. To reduce GPU memory usage while training, we cache only 100 mini-batches on the GPU; this allows us to use larger convnet models and keep all training data on a single GPU. As part of this framework, our system has two main threads of execution: 1) a training function which runs on the GPU evaluating the batched-SGD updates, and 2) a data dispatch function which preprocesses the data on the CPU and transfers it on the GPU when thread 1) is finished processing the $1 0 0 \ \mathrm { m i n i }$ batches. Training each convnet on an NVIDIA TITAN GPU takes $1 . 9 \mathrm { m s }$ per patch (fprop $+ \mathrm { b p r o p } ) = 4 1 \mathrm { m i n }$ total. We test on a cpu cluster with 5000 nodes. Testing takes: 0.49sec per image $( 0 . 9 4 \mathrm { x \ s c a l e } ) = 2 . 8 \mathrm { m i n }$ total. NMS and spatial model take negligible time.
+
+For testing, because of the shared nature of weights for all windows in each image, we convolve the learned filters with the full image instead of individual windows. This dramatically reduces the time to perform forward propagation on the full test set.
+
+# 4.1 Evaluation
+
+To evaluate our model on the FLIC dataset we use a measure of accuracy suggested by Sapp et al. [38]: for a given joint precision radius we report the percentage of joints in the test set correct within the radius threshold (where distance is defined as 2D Euclidean distance in pixels). In Figure 4.1 we evaluate this performance measure on the the wrist, elbow and shoulder joints. We also compare our detector to the DPM [15] and MODEC [38] architectures. Note that we use the same subset of 351 images when testing all detectors.
+
+![](images/1343c8190857b623a1191eb1705e348829d55f24f85c6caf1aab64957b8739c7.jpg)  
+Figure 6: Comparison of Detector Performance on the Test set
+
+Figure 4.1 shows that our architecture out-performs or is equal to the MODEC and DPM detectors for all three body parts. For the wrist and elbow joints our simple spatial model improves joint localization for approximately $5 \%$ of the test set cases (at a 5 pixel threshold), which enables us to outperform all other detectors. However, for the shoulder joint our spatial model actual decreases the joint location accuracy for large thresholds. This is likely due to the poor performance of the convnet on the elbow.
+
+As expected, the spatial model cannot improve the joint accuracy of points that are already close to the correct value, however it is never-the-less successful in removing outliers for the wrist and elbow joints. Figure 4.1 is an example where a strong false positive results in an incorrect part location before the spatial model is applied, which is subsequently removed after applying our spatial model.
+
+# 5 Conclusion
+
+We have shown successfully how to improve the state-of-the-art on one of the most complex computer vision tasks: unconstrained human pose estimation. Convnets are impressive low-level feature detectors, which when combined with a global position prior is able to outperform much more complex and popular models. We explored many different higher level structural models with the aim to further improve the results, but the most generic higher level spatial model achieved the best results. As mentioned in the introduction, this is counter-intuitive to common belief for human kinematic structures, but it mirrors results in other domains. For instance in speech recognition, researchers observed, if the learned transition probabilities (higher level structure) are reset to equal probabilities, the recognition performance, now mainly driven by the emission probabilities does not reduce significantly [27]. Other domains are discussed in more detail by [26].
+
+![](images/d1edd0ce185d7d5c2edbace1e927b7ac102d505bbad2d0a84e0493ca61841f62.jpg)  
+Figure 7: Impact of Our Spatial Model: Red cross is MODEC, Blue cross is before our Spatial Model, Green cross is after our Spatial Model
+
+![](images/fa85f684e5afbef1fc69264f2c8501eb54e04d30d6b0d2daf508b2d9a1fcf7ff.jpg)  
+Figure 8: Failure cases: The green cross is our new technique’s wrist locator, the red cross is the state-of-the-art CVPR13 MODEC detector [38] on the FLIC database.
+
+We expect to obtain further improvement by enlarging the training set with a new pose-based warping technique that we are currently investigating. Furthermore, we are also currently experimenting with multi-resolution input representations, that take a larger spatial context into account.
+
+# 6 Acknowledgements
+
+This research was funded in part by the Office of Naval Research ONR Award N000141210327 and by a Google award.
+
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+[44] G. Taylor, R. Fergus, I. Spiro, G. Williams, and C. Bregler. Pose-sensitive embedding by nonlinear NCA regression. In Advances in Neural Information Processing Systems 23 (NIPS), pages 2280–2288, 2010. 3   
+[45] G. Taylor, L. Sigal, D. Fleet, and G. Hinton. Dynamical binary latent variable models for 3d human pose tracking. In Proc. of the 23rd IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), 2010. 3   
+[46] T. Tieleman and G. Hinton. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 2012. 4   
+[47] S. C. Turaga, J. F. Murray, V. Jain, F. Roth, M. Helmstaedter, K. Briggman, W. Denk, and H. S. Seung. Convolutional networks can learn to generate affinity graphs for image segmentation. Neural Computation, 22:511–538, 2010. 3   
+[48] R. Y. Wang and J. Popovic. Real-time hand-tracking with a color glove. In ´ ACM Transactions on Graphics (TOG), volume 28, page 63. ACM, 2009. 2   
+[49] C. Wren, A. Azarbayejani, T. Darrell, and A. Pentland. Pfinder: Real-time tracking of the human body. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(7):780–785, 1997. 2   
+[50] Y. Yang and D. Ramanan. Articulated pose estimation with flexible mixtures-of-parts. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pages 1385–1392. IEEE, 2011. 2   
+[51] M. D. Zeiler and R. Fergus. Visualizing and understanding convolutional neural networks. arXiv preprint arXiv:1311.2901, 2013. 2   
+[52] S. Zuffi, J. Romero, C. Schmid, and M. J. Black. Estimating human pose with flowing puppets. 3
+
+![](images/6dd19421889f64764d36d3e84dcebe6cc876021aa96dad892c44b2b09c4b050c.jpg)  
+Figure 9: Success cases: The green cross is our new technique’s wrist locator, the red cross is the state-of-the-art CVPR13 MODEC detector [38] on the FLIC database.

parse/train/kziQtP-nGqzDb/kziQtP-nGqzDb_content_list.json

@@ -0,0 +1,859 @@
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+    {
+        "type": "text",
+        "text": "Learning Human Pose Estimation Features with Convolutional Networks ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Arjun Jain New York University ajain@nyu.edu ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Jonathan Tompson New York University tompson@cims.nyu.edu ",
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+        ],
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+    {
+        "type": "text",
+        "text": "Mykhaylo Andriluka MPI Saarbruecken andriluk@mpi-inf.mpg.de ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Graham W. Taylor University of Guelph gwtaylor@uoguelph.ca ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Christoph Bregler New York University chris.bregler@nyu.edu ",
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+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper introduces a new architecture for human pose estimation using a multilayer convolutional network architecture and a modified learning technique that learns low-level features and a higher-level weak spatial model. Unconstrained human pose estimation is one of the hardest problems in computer vision, and our new architecture and learning schema shows improvement over the current stateof-the-art. The main contribution of this paper is showing, for the first time, that a specific variation of deep learning is able to meet the performance, and in many cases outperform, existing traditional architectures on this task. The paper also discusses several lessons learned while researching alternatives, most notably, that it is possible to learn strong low-level feature detectors on regions that might only cover a few pixels in the image. Higher-level spatial models improve somewhat the overall result, but to a much lesser extent than expected. Many researchers previously argued that the kinematic structure and top-down information are crucial for this domain, but with our purely bottom-up, and weak spatial model, we improve on other more complicated architectures that currently produce the best results. This echos what many other researchers, like those in the speech recognition, object recognition, and other domains have experienced [26]. ",
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+        "img_path": "images/9c14247104c73dc1d4734f534db3640f451ff31bbb03df6711e6f09004866cf0.jpg",
+        "image_caption": [
+            "Figure 1: The green cross is our new technique’s wrist locator, the red cross is the state-of-the-art CVPR13 MODEC detector [38] on the FLIC database. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "One of the hardest tasks in computer vision is determining the high degree-of-freedom configuration of a human body with all its limbs, complex self-occlusion, self-similar parts, and large variations due to clothing, body-type, lighting, and many other factors. The most challenging scenario for this problem is from a monocular RGB image and with no prior assumptions made using motion models, pose models, background models, or any other common heuristics that current state-of-theart systems utilize. Finding a face in frontal or side view is relatively simple, but determining the exact location of body parts such as hands, elbows, shoulders, hips, knees and feet, each of which sometimes only occupy a few pixels in the image in front of an arbitrary cluttered background, is significantly harder. ",
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+    {
+        "type": "text",
+        "text": "The best performing pose estimation methods, including those based on deformable part models, typically are based on body part detectors. Such body part detectors commonly consist of multiple stages of processing. The first stage of processing in a typical pipeline consists of extracting sets of low-level features such as SIFT [25], HoG [11], or other filters that describe orientation statistics in local image patches. Next, these features are pooled over local spatial regions and sometimes across multiple scales to reduce the size of the representation and also develop local shift/scale invariance. Finally, the aggregate features are mapped to a vector, which is then either input to 1) a standard classifier such as a support vector machine (SVM) or 2) the next stage of processing (e.g. assembling the parts into a whole). Much work is devoted to engineering the system to produce a vector representation that is sensitive to class (e.g. head, hands, torso) while remaining invariant to the various nuisance factors (lighting, viewpoint, scale, etc.) ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An alternative approach is representation learning: relying on the data instead of feature engineering, to learn a good representation that is invariant to nuisance factors. For a recent review, see [6]. It is common to learn multiple layers of representation, which is referred to as deep learning. Several such techniques have used unsupervised or semi-supervised learning to extract multi-layer domain-specific invariant representations, however, it is purely supervised techniques that have won several recent challenges by large margins, including ImageNet LSVRC 2012 and 2013 [23, 51]. These end-to-end learning systems have capitalized on advances in computing hardware (notably GPUs), larger datasets like ImageNet, and algorithmic advances (specifically gradient-based training methods and regularization). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While these methods are now proven in generic object recognition, their use in pose estimation has been limited. Part of the challenge in making end-to-end learning work for human pose estimation is related to the nonrigid structure of the body, the necessity for precision (deep recognition systems often throw away precise location information through pooling), and the complex, multi-modal nature of pose. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present the first end-to-end learning approach for full-body human pose estimation. While our approach is based on convolutional networks (convnets) [24], we want to stress that the na¨ıve implementation of applying this model “off-the-shelf” will not work. Therefore, the contribution of this work is in both a model that outperforms state of the art deformable part models (DPMs) on a modern, challenging dataset, and also an analysis of what is needed to make convnets work in human pose estimation. In particular, we present a two-stage filtering approach whereby the response maps of convnet part detectors are denoised by a second process informed by the part hierarchy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Detecting people and their pose has been investigated for decades. Many early techniques rely on sliding-window part detectors based on hand-crafted or learned features or silhouette extraction techniques applied to controlled recording conditions. Examples include [14, 49, 5, 30]. We refer to [35] for a complete survey of this era. More recently, several new approaches have been proposed that are applied to unconstrained domains. In such domains, good performance has been achieved with so-called “bag of features” followed by regression-based, nearest neighbor or SVM-based architectures. Examples include “shape-context” edge-based histograms from the human body [28, 1] or just silhouette features [19]. Shakhnarovich et al. [39] learn a parameter sensitive hash function to perform example-based pose estimation. Many relevant techniques have also been applied to hand tracking such as [48]. A more general survey of the large field of hand tracking can be found in [12]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Many techniques have been proposed that extract, learn, or reason over entire body features. Some use a combination of local detectors and structural reasoning (see [36] for coarse tracking and [10] for person-dependent tracking). In a similar spirit, more general techniques using pictorial structures [2, 3, 17, 37, 33, 34], “poselets” [9], and other part-models [16, 50] have received increased attention. We will focus on these techniques and their latest incarnations in the following sections. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Further examples come from the HumanEva dataset competitions [41], or approaches that use higher-resolution shape models such as SCAPE [4] and further extensions [20, 8]. These differ from our domain in that the images considered are of higher quality and less cluttered. Also many of these techniques work on images from a single camera, but need video sequence input (not single images) to achieve impressive results [42, 52]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As an example of a technique that works for single images against cluttered backgrounds, Shotton et al.’s Kinect based body part detector [40] uses a random forest of decision trees trained on synthetic depth data to create simple body part detectors. In the proposed work, we also adopt simple partbased detectors, however, we focus on a different learning strategy. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There are a number of successful end-to-end representation learning techniques which perform pose estimation on a limited subset of body parts or body poses. One of the earliest examples of this type was Nowlan and Platt’s convolutional neural network hand tracker [30], which tracked a single hand. Osadchy et al. applied a convolutional network to simultaneously detect and estimate the pitch, yaw and roll of a face [31]. Taylor et al. [44] trained a convolutional neural network to learn an embedding in which images of people in similar pose lie nearby. They used a subset of body parts, namely, the head and hand locations to learn the “gist” of a pose, and resorted to nearest-neighbour matching rather than explicitly modeling pose. Perhaps most relevant to our work is Taylor et al.’s work on tracking people in video [45], augmenting a particle filter with a structured prior over human pose and dynamics based on learning representations. While they estimated a posterior over the whole body (60 joint angles), their experiments were limited to the HumanEva dataset [41], which was collected in a controlled laboratory setting. The datasets we consider in our experiments are truly poses “in the wild”, though we do not consider dynamics. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A factor limiting earlier methods from tacking full pose-estimation with end-to-end learning methods, in particular deep networks, was the limited amount of labeled data. Such techniques, with millions or more parameters, require more data than structured techniques that have more a priori knowledge, such as DPMs. We attack this issue on two fronts. First, directly, by using larger labeled training sets which have become available in the past year or two, such as FLIC [38]. Second, indirectly, by better exploiting the data we have. The annotations provided by typical pose estimation datasets contain much richer information compared to the class labels in object recognition datasets In particular, we show that the relationships among parts contained in these annotations can be used to build better detectors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 Model ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To perform pose estimation with a convolutional network architecture [24] (convnet), the most obvious approach would be to map the image input directly to a vector coding the articulated pose: i.e. the type of labels found in pose datasets. The convnet output would represent the unbounded 2-D or 3-D positions of joints, or alternatively a hierarchy of joint angles. However, we found that this worked very poorly. One issue is that pooling, while useful for improving translation invariance during object recognition, destroys precise spatial information which is necessary to accurately predict pose. Convnets that produce segmentation maps, for example, avoid pooling completely [47, 13]. Another issue is that the direct mapping from input space to kinematic body pose coefficients is highly non-linear and not one-to-one. However, even if we took this route, there is a deeper issue with attempting to map directly to a representation of full body pose. Valid poses represent a much lower-dimensional manifold in the high-dimensional space in which they are captured. It seems troublesome to make a discriminative network map to a space in which the majority of configurations do not represent valid poses. In other words, it makes sense to restrict the net’s output to a much smaller class of valid configurations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Rather than perform multiple-output regression using a single convnet to learn pose coefficients directly, we found that training multiple convnets to perform independent binary body-part classification, with one network per feature, resulted in improved performance on our dataset. These convnets are applied as sliding windows to overlapping regions of the input, and map a window of pixels to a single binary output: the presence or absence of that body part. The result of applying the convnet is a response-map indicating the confidence of the body part at that location. This lets us use much smaller convnets, and retain the advantages of pooling, at the expense of having to maintain a separate set of parameters for each body part. Of course, a series of independent part detectors cannot enforce consistency in pose in the same way as a structured output model, which produces valid full-body configurations. In the following sections, we first describe in detail the convolutional network architecture and then a method of enforcing pose consistency using parent-child relationships. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "image",
+        "img_path": "images/0c57a43c96117a1f22c399c632755454928706adc26fa15333cd335d6768f45a.jpg",
+        "image_caption": [
+            "Figure 2: The convolutional network architecture used in our experiments. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3.1 Convolutional Network Architecture ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The lowest level of our two-stage feature detection pipeline is based on a standard convnet architecture, an overview of which is shown in Figure 2. Convnets, like their fully-connected, deep neural network counterparts, perform end-to-end feature learning and are trained with the back-propagation algorithm. However, they differ in a number of respects, most notably local connectivity, weight sharing, and local pooling. The first two properties significantly reduce the number of free parameters, and reduce the need to learn repeated feature detectors at different locations of the input. The third property makes the learned representation invariant to small translations of the input. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The convnet pipeline shown in Figure 2 starts with a $6 4 \\times 6 4$ pixel RGB input patch which has been local contrast normalized (LCN) [22] to emphasize geometric discontinuities and improve generalization performance [32]. The LCN layer is comprised of a $9 \\times 9$ pixel local subtractive normalization, followed by a $9 \\times 9$ local divisive normalization. The input is then processed by three convolution and subsampling layers, which use rectified linear units (ReLUs) [18] and max-pooling. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "As expected, we found that internal pooling layers help to a) reduce computational complexity1 and b) improve classification tolerance to small input image translations. Unfortunately, pooling also results in a loss of spatial precision. Since the target application for this convnet was offline (rather than real-time) body-pose detection, and since we found that with sufficient training exemplars, invariance to input translations can be learned, we choose to use only 2 stages of $2 \\times 2$ pooling (where the total image downsampling rate is $4 \\times 4$ ). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Following the three stages of convolution and subsampling, the top-level pooled map is flattened to a vector and processed by three fully connected layers, analogous to those used in deep neural networks. Each of these output stages is composed of a linear matrix-vector multiplication with learned bias, followed by a point-wise non-linearity (ReLU). The output layer has a single logistic unit, representing the probability of the body part being present in that patch. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To train the convnet, we performed standard batch stochastic gradient descent. From the training set images, we set aside a validation set to tune the network hyper-parameters, such as number and size of features, learning rate, momentum coefficient, etc. We used Nesterov momentum [43] as well as RMSPROP [46] to accelerate learning and we used L2 regularization and dropout [21] on the input to each of the fully-connected linear stages to reduce over-fitting the restricted-size training set. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/b897c3b4b74b20fbf081a67ec1f554fb1ec63a45784f14ca28d6610bc592eff5.jpg",
+        "image_caption": [
+            "Figure 3: Spatial Model Connectivity with Spatial Priors "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Enforcing Global Pose Consistency with a Spatial Model ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When applied to the validation set, the raw output of the network presented in Section 3.1 produces many false-positives. We believe this is due to two factors: 1) the small image context as input to the convnet $6 4 \\times 6 4$ pixels or approximately $5 \\%$ of the input image area) does not give the model enough contextual information to perform anatomically consistent joint position inference and 2) the training set size is limited. We therefore use a higher-level spatial model with simple body-pose priors to remove strong outliers from the convnet output. We do not expect this model to improve the performance of poses that are close to the ground truth labels (within 10 pixels for instance), but rather it functions as a post processing step to de-emphasize anatomically impossible poses due to strong outliers. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The inter-node connectivity of our simple spatial model is displayed in Figure 3. It consists of a linear chain of kinematic 2D nodes for a single side of the human body. Throughout our experiments we used the left shoulder, elbow and wrist; however we could have used the right side joints without loss of generality (since detection of the right body parts simply requires a horizontal mirror of the input image). For each node in the chain, our convnet detector generates response-map unary distributions $p _ { \\mathrm { f a c } } \\left( x \\right)$ , $p _ { \\mathrm { s h o } } \\left( x \\right)$ , $p _ { \\mathrm { e l b } } \\left( x \\right)$ , $p _ { \\mathrm { w r i } } \\left( x \\right)$ over the dense pixel positions $x$ , for the face, shoulder, elbow and wrist joints respectively. For the remainder of this section, all distributions are assumed to be a function over the pixel position, and so the $x$ notation will be dropped. The output of our spatial model will produce filtered response maps: $\\hat { p } _ { \\mathrm { f a c } }$ , $\\hat { p } _ { \\mathrm { s h o } }$ , $\\hat { p } _ { \\mathrm { e l b } }$ , and $\\hat { p } _ { \\mathrm { w r i } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The body part priors for a pair of joints $( a , b )$ , $p _ { a \\mid b = \\vec { 0 } }$ , are calculated by creating a histogram of joint $a$ locations over the training set, given that the adjacent joint $b$ is located at the image center $\\boldsymbol { x } = \\vec { 0 }$ ). The histograms are then smoothed (using a gaussian filter) and normalized. The learned priors for $p _ { \\mathrm { s h o } | \\mathrm { f a c } = \\vec { 0 } }$ , $p _ { \\mathrm { e l b | s h o = } } \\vec { 0 }$ , and $p _ { \\mathrm { w r i l e l b { = } } \\vec { 0 } }$ are shown in Figure 4. Note that due to symmetry, the prior for $p _ { \\mathrm { e l b | w r i = } } \\vec { 0 }$ is a $1 8 0 ^ { \\circ }$ rotation of $p _ { \\mathrm { w r i l e l b { = } } \\vec { 0 } }$ (as is the case of other adjacent pairs). Rather than assume a simple Gaussian distribution for modeling pairwise interactions of adjacent nodes, as is standard in many parts-based detector implementations, we have found that the these non-parametric spatial priors lead to improved detection performance. ",
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+        "img_path": "images/b8f2a8dcb8c6bf47b5ff5f98798ab05770c71c336c282ef2b649e04771ee00c3.jpg",
+        "image_caption": [
+            "Figure 4: Part priors for left body parts "
+        ],
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+        "text": "Given the full set of prior conditional distributions and the convnet unary distributions, we can now construct the filtered distribution for each part by using an approach that is analogous to the sumproduct belief propagation algorithm. For body part $i$ , with a set of neighbouring nodes $U$ , the final distribution is defined as: ",
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+            "Figure 5: Global prior for the face: $h _ { \\mathrm { f a c } }$ "
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+        "text": "$$\n\\hat { p } _ { i } \\propto { p _ { i } } ^ { \\lambda } \\prod _ { u \\in U } \\left( p _ { i | u = \\vec { 0 } } * p _ { u } \\right)\n$$",
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+        "text": "where $\\lambda$ is a mixing parameter and controls the confidence of each joint’s unary distribution towards its final filtered distribution (we used $\\lambda = 1$ for our experiments). The final joint distribution is therefore a product of the unary distribution for that joint, as well as the beliefs from neighbouring nodes (as with standard sum-product belief propagation). In log space, the above product for the shoulder joint becomes: ",
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+        "text": "$$\n\\begin{array} { r } { \\log \\left( \\hat { p } _ { \\mathrm { s h o } } \\right) \\propto \\lambda \\log \\left( p _ { \\mathrm { s h o } } \\right) + \\log \\left( p _ { \\mathrm { s h o } | \\mathrm { f a c } = \\vec { 0 } } * p _ { \\mathrm { f a c } } \\right) + \\log \\left( p _ { \\mathrm { s h o } | \\mathrm { e l b } = \\vec { 0 } } * p _ { \\mathrm { e l b } } \\right) } \\end{array}\n$$",
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+        "type": "text",
+        "text": "We also perform an equivalent computation for the elbow and wrist joints. The face joint is treated as a special case. Empirically, we found that incorporating image evidence from the shoulder joint to the filtered face distribution resulted in poor performance. This is likely due to the fact that the convnet does a very good job of localizing the face position, and so incorporating noisy evidence from the shoulder detector actually increases uncertainty. Instead, we use a global position prior for the face, $h _ { \\mathrm { f a c } }$ , which is obtained by learning a location histogram over the face positions in the training set images, as shown in Figure 5. In log space, the output distribution for the face is then given by: ",
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+        "type": "equation",
+        "img_path": "images/93993bbf2b6d2a78bb818a8e6e2b550712a4539f33579f9f275e466925dcc684.jpg",
+        "text": "$$\n\\log \\left( \\hat { p } _ { \\mathrm { f a c } } \\right) \\propto \\lambda ~ \\log \\left( p _ { \\mathrm { f a c } } \\right) + \\log \\left( h _ { \\mathrm { f a c } } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Lastly, since the learned neural network convolution features and the spatial priors are not explicitly invariant to scale, we must run the convnet and spatial model on images at multiple scales at test time, and then use the most likely joint location across those scales as the final joint location. For datasets containing examples with multiple persons (known a priori), we use non-maximal suppression [29] to find multiple local maxima across the filtered response-maps from each scale, and we then take the top $n$ most likely joint candidates from each person in the scene. ",
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+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "We evaluated our architecture on the FLIC [38] dataset, which is comprised of 5003 still RGB images taken from an assortment of Hollywood movies. Each frame in the dataset contains at least one person in a frontal pose (facing the camera), and each frame was processed by Amazon Mechanical Turk to obtain ground truth labels for the joint positions of the upper body of a single person. The FLIC dataset is very challenging for state-of-the-art pose estimation methodologies because the poses are unconstrained, body parts are often occluded, and clothing and background are not consistent. ",
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+        "type": "text",
+        "text": "We use 3987 training images from the dataset, which we also mirror horizontally to obtain a total of $3 9 8 7 \\times 2 = 7 9 7 4$ examples. Since the training images are not at the same scale, we also manually annotate the bounding box for the head in these training set images, and bring them to canonical scale. Further, we crop them to $3 2 0 \\times 2 4 0$ such that the center of the shoulder annotations lies at (160 px, $8 0 \\ \\mathrm { p x }$ ). We do not perform this image normalization at test time. Following the methodology of Felzenszwalb et al. [15], at test time we run our model on images with only one person (351 images of the 1016 test examples). As stated in Section 3, the model is run on 6 different input image scales and we then use the joint location with highest confidence across those scales as the final location. ",
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+    {
+        "type": "text",
+        "text": "For training the convnet we use Theano [7], which provides a Python-based framework for efficient GPU processing and symbolic differentiation of complex compound functions. To reduce GPU memory usage while training, we cache only 100 mini-batches on the GPU; this allows us to use larger convnet models and keep all training data on a single GPU. As part of this framework, our system has two main threads of execution: 1) a training function which runs on the GPU evaluating the batched-SGD updates, and 2) a data dispatch function which preprocesses the data on the CPU and transfers it on the GPU when thread 1) is finished processing the $1 0 0 \\ \\mathrm { m i n i }$ batches. Training each convnet on an NVIDIA TITAN GPU takes $1 . 9 \\mathrm { m s }$ per patch (fprop $+ \\mathrm { b p r o p } ) = 4 1 \\mathrm { m i n }$ total. We test on a cpu cluster with 5000 nodes. Testing takes: 0.49sec per image $( 0 . 9 4 \\mathrm { x \\ s c a l e } ) = 2 . 8 \\mathrm { m i n }$ total. NMS and spatial model take negligible time. ",
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+    {
+        "type": "text",
+        "text": "For testing, because of the shared nature of weights for all windows in each image, we convolve the learned filters with the full image instead of individual windows. This dramatically reduces the time to perform forward propagation on the full test set. ",
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+    {
+        "type": "text",
+        "text": "4.1 Evaluation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate our model on the FLIC dataset we use a measure of accuracy suggested by Sapp et al. [38]: for a given joint precision radius we report the percentage of joints in the test set correct within the radius threshold (where distance is defined as 2D Euclidean distance in pixels). In Figure 4.1 we evaluate this performance measure on the the wrist, elbow and shoulder joints. We also compare our detector to the DPM [15] and MODEC [38] architectures. Note that we use the same subset of 351 images when testing all detectors. ",
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+        "image_caption": [
+            "Figure 6: Comparison of Detector Performance on the Test set "
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+    {
+        "type": "text",
+        "text": "Figure 4.1 shows that our architecture out-performs or is equal to the MODEC and DPM detectors for all three body parts. For the wrist and elbow joints our simple spatial model improves joint localization for approximately $5 \\%$ of the test set cases (at a 5 pixel threshold), which enables us to outperform all other detectors. However, for the shoulder joint our spatial model actual decreases the joint location accuracy for large thresholds. This is likely due to the poor performance of the convnet on the elbow. ",
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+    {
+        "type": "text",
+        "text": "As expected, the spatial model cannot improve the joint accuracy of points that are already close to the correct value, however it is never-the-less successful in removing outliers for the wrist and elbow joints. Figure 4.1 is an example where a strong false positive results in an incorrect part location before the spatial model is applied, which is subsequently removed after applying our spatial model. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We have shown successfully how to improve the state-of-the-art on one of the most complex computer vision tasks: unconstrained human pose estimation. Convnets are impressive low-level feature detectors, which when combined with a global position prior is able to outperform much more complex and popular models. We explored many different higher level structural models with the aim to further improve the results, but the most generic higher level spatial model achieved the best results. As mentioned in the introduction, this is counter-intuitive to common belief for human kinematic structures, but it mirrors results in other domains. For instance in speech recognition, researchers observed, if the learned transition probabilities (higher level structure) are reset to equal probabilities, the recognition performance, now mainly driven by the emission probabilities does not reduce significantly [27]. Other domains are discussed in more detail by [26]. ",
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+    {
+        "type": "image",
+        "img_path": "images/d1edd0ce185d7d5c2edbace1e927b7ac102d505bbad2d0a84e0493ca61841f62.jpg",
+        "image_caption": [
+            "Figure 7: Impact of Our Spatial Model: Red cross is MODEC, Blue cross is before our Spatial Model, Green cross is after our Spatial Model "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
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+        "img_path": "images/fa85f684e5afbef1fc69264f2c8501eb54e04d30d6b0d2daf508b2d9a1fcf7ff.jpg",
+        "image_caption": [
+            "Figure 8: Failure cases: The green cross is our new technique’s wrist locator, the red cross is the state-of-the-art CVPR13 MODEC detector [38] on the FLIC database. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We expect to obtain further improvement by enlarging the training set with a new pose-based warping technique that we are currently investigating. Furthermore, we are also currently experimenting with multi-resolution input representations, that take a larger spatial context into account. ",
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+    {
+        "type": "text",
+        "text": "6 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This research was funded in part by the Office of Naval Research ONR Award N000141210327 and by a Google award. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "type": "text",
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parse/train/rJgsskrFwH/rJgsskrFwH.md

@@ -0,0 +1,353 @@
+# SCALING AUTOREGRESSIVE VIDEO MODELS
+
+Dirk Weissenborn∗ Google Research diwe@google.com
+
+Oscar Tackstr¨ om¨ ∗† Sana Labs oscar@sanalabs.com
+
+Jakob Uszkoreit Google Research usz@google.com
+
+# ABSTRACT
+
+Due to the statistical complexity of video, the high degree of inherent stochasticity, and the sheer amount of data, generating natural video remains a challenging task. State-of-the-art video generation models often attempt to address these issues by combining sometimes complex, usually video-specific neural network architectures, latent variable models, adversarial training and a range of other methods. Despite their often high complexity, these approaches still fall short of generating high quality video continuations outside of narrow domains and often struggle with fidelity. In contrast, we show that conceptually simple autoregressive video generation models based on a three-dimensional self-attention mechanism achieve competitive results across multiple metrics on popular benchmark datasets, for which they produce continuations of high fidelity and realism. We also present results from training our models on Kinetics, a large scale action recognition dataset comprised of YouTube videos exhibiting phenomena such as camera movement, complex object interactions and diverse human movement. While modeling these phenomena consistently remains elusive, we hope that our results, which include occasional realistic continuations encourage further research on comparatively complex, large scale datasets such as Kinetics.
+
+# 1 INTRODUCTION
+
+Generative modeling of video holds promise for applications such as content creation, forecasting, transfer learning and model-based reinforcement learning (Srivastava et al., 2015; Carl Vondrick, 2016; Oh et al., 2015; Kaiser et al., 2019). While recently there has been a lot of progress on generative models for text, audio and images, video generation remains challenging. To some extent this is simply due to the large amount of data that needs to be produced. Autoregressive models suffer from this particularly in their generation speed. On the other hand, they have a number of desirable attributes, such as their conceptual simplicity and tractable likelihood, which enables straightforward evaluation of their ability to model the entire data distribution.
+
+Moreover, recent results on image generation by Menick & Kalchbrenner (2019) show that pixellevel autoregressive models are capable of generating images with high fidelity. These findings motivate the question of how far one can push such autoregressive models in the more general task of video generation when scaling recent advances in neural architectures to modern hardware accelerators.
+
+In this work, we introduce a generalization of the Transformer architecture of Vaswani et al. (2017) using three-dimensional, block-local self-attention. In contrast to the block-local attention mechanism of Parmar et al. (2018), our formulation can be implemented efficiently on Tensor Processing Units, or TPUs (Jouppi et al., 2017). To further reduce the memory footprint, we combine this with a three-dimensional generalization of methods from Menick & Kalchbrenner (2019), who generate images as sequences of smaller, sub-scaled image slices.
+
+Together, these techniques allow us to efficiently model videos as 3D volumes instead of sequences of still image frames, with direct interactions between representations of pixels across the spatial and temporal dimensions.
+
+We obtain strong results on popular benchmarks (Section 4.2, Appendix A) and produce high fidelity video continuations on the BAIR robot pushing dataset (Ebert et al., 2017) exhibiting plausible object interactions. Furthermore, our model achieves an almost $50 \%$ reduction in perplexity compared to prior work on autoregressive models on another robot pushing dataset.
+
+Finally, we apply our models to down-sampled videos from the Kinetics-600 dataset (Carreira et al., 2018) (Section 4.3). While modeling the full range of Kinetics-600 videos still poses a major challenge, we see encouraging video continuations for a more limited subset, namely cooking videos. These feature camera movement, complex object interactions and still cover diverse subjects.
+
+We hope that these initial results will encourage future video generation work to evaluate models on more challenging datasets such as Kinetics.
+
+# 2 RELATED WORK
+
+Our setup is closely related to that of Kalchbrenner et al. (2016), who extend work on pixel-level autoregressive image generation (van den Oord et al., 2016b;a) to videos. However, whereas they model the temporal and spatial dimensions separately with dilated convolutions and convolutional LSTMs, respectively, our model is conceptually simpler in that we do not make any distinction between temporal and spatial dimensions and instead rely almost entirely on multi-head self-attention (Vaswani et al., 2017) within the 3D video volume. For comparability, we provide results on Moving MNIST and another robot pushing dataset (Finn et al., 2016a) on which our model achieves an almost $50 \%$ reduction in perplexity (see Appendix A).
+
+One major drawback of autoregressive models is their notoriously slow generation speed. However, we believe that further research into (partially) parallelizing sampling (Stern et al., 2018) and future hardware accelerators will help alleviate this issue and eventually make autoregressive modeling a viable solution even for extremely high-dimensional data such as videos.
+
+To reduce the generally quadratic space complexity of the self-attention mechanism, we use blocklocal self-attention, generalizing the image generation approaches of Parmar et al. (2018) and Chen et al. (2018) to 3D volumes. In concurrent work, Child et al. (2019) instead use sparse attention after linearizing images to a sequence of pixels.
+
+To further reduce memory requirements, we generalize sub-scaling (Menick & Kalchbrenner, 2019) to video. An alternative approach is optionally hierarchical multi-scale generation, which has recently been explored for both image generation (Reed et al., 2017; De Fauw et al., 2019) as well as video generation (Mathieu et al., 2016).
+
+Earlier work on video generation mostly focused on deterministic approaches (Srivastava et al., 2015; Carl Vondrick, 2016; Xingjian et al., 2015; Liu et al., 2017; Jia et al., 2016), which fail to capture the high degree of stochasticity inherent in video. In response, a popular research direction has been that of generative latent-variable video models. In contrast to pixel-level autoregressive models, these posit an underlying latent process in tandem with the observed pixel values. Work in this category includes variants of variational autoencoders (Babaeizadeh et al., 2018; Denton & Fergus, 2018). To address the issues inherent in these models, most notably the tendency to generate blurry outputs possibly due to restricted modeling power, inadequate prior distributions, or optimization of a lower bound in place of the true likelihood, various directions have been explored, including the use of adversarial objectives (Mathieu et al., 2016; Vondrick et al., 2016; Lee et al., 2018), hierarchical latent-variables (Castrejon et al., 2019), or flow-based models (Kumar et al., ´ 2019). All of these approaches admit significantly faster generation. However, in the adversarial case, they tend to only focus on a subset of the modes in the empirical distribution while flowbased models struggle with limited modeling power even when using a large number of layers and parameters.
+
+A large fraction of earlier work on video generation has encoded specific intuitions about videos, such as explicit modeling of motion (Finn et al., 2016b; Denton & Fergus, 2018) or generation of optical flow (Patr ˘ aucean et al., 2016). The conceptual simplicity of our model, however, is more in ˘ line with recent approaches to video classification that process videos by means of 3D convolutions (Carreira & Zisserman, 2017; Xie et al., 2018) or, similar to this work, spatiotemporal self-attention (Girdhar et al., 2018).
+
+![](images/fbd6814863fa9732b13133acc7273c882206a5ec97702b30f61c6560843ebffc.jpg)  
+Figure 1: Top: Illustration of the subscale video transformer architecture and process flow. We incrementally generate $s = 4 \cdot 2 \cdot 2 = 1 6$ video slices. The video slices and their respective generation order are derived from subscaling. In each iteration, we first process the partially padded video (illustrated for slice index $( 1 , 0 , 1 )$ , black means padding and gray means already generated or visible) by an encoder, the output of which is used as conditioning for decoding the current video slice. After generating a slice we replace the respective padding in the video with the generated output and repeat the process for the next slice. Bottom: Subscaling in 3D (best viewed in color). The 3D volume is evenly divided by a given subscale factor, here $\textbf { \textit { s } } = \ ( 4 , 2 , 2 )$ , and the respective slices are extracted. The whole volume is generated by incrementally predicting the individual, much smaller slices, starting at slice ${ \pmb x } _ { ( 0 , 0 , 0 ) }$ (yellow), followed by ${ \pmb x } _ { ( 0 , 0 , 1 ) }$ (green), $\pmb { x } _ { ( 0 , 1 , 0 ) }$ (red), etc., in rasterscan order.
+
+# 3 VIDEO TRANSFORMER
+
+We generalize the one-dimensional Transformer (Vaswani et al., 2017) to explicitly model videos represented as three-dimensional spatiotemporal volumes, without resorting to sequential linearization of the positions in the volume (Child et al., 2019). This allows for maintaining spatial neighborhoods around positions, which is important as the large number of individual positions to be predicted in a video requires limiting the receptive field of the self-attention mechanism to a neighborhood around every position to avoid the quadratic blow-up in memory consumption of naive fully-connected attention.
+
+We model the distribution $p ( { \pmb x } )$ over videos $\pmb { x } \in \mathbb { R } ^ { T \times H \times W \times N _ { c } }$ — with time, height, width and channel dimensions, respectively — by means of a pixel-channel level autoregressive factorization.1 That is, the joint distribution over pixels is factorized into a product of channel intensities for all $N _ { c }$ channels, for each of the $N _ { p } = T \cdot H \cdot W$ pixels, with respect to an ordering $\pi$ over pixels:
+
+$$
+p ( \pmb { x } ) = \prod _ { i = 0 } ^ { N _ { p } - 1 } \prod _ { k = 0 } ^ { N _ { c } - 1 } p ( \pmb { x } _ { \pi ( i ) } ^ { k } | \pmb { x } _ { \pi ( < i ) } , \pmb { x } _ { \pi ( i ) } ^ { < k } ) .
+$$
+
+The ordering $\pi$ is given by a combination of a subscale- and raster-scan ordering, as detailed in 3.2.
+
+# 3.1 BLOCK-LOCAL SELF-ATTENTION
+
+The attention mechanism of the original Transformer lets each element in a set of $N _ { p }$ elements connect to every other element, via the fully-connected weighted adjacency (attention) matrix $A \in$ $\mathbb { R } ^ { N _ { p } \times N _ { p } }$ , with $A _ { i j }$ representing attention weights from element $i$ to element $j$ . Because $A$ grows quadratically with the number of elements it becomes prohibitively large for objects such as videos, which typically consist of hundreds of thousands of pixels or more. Therefore, similar in spirit to Parmar et al. (2018), we propose to use local self-attention by dividing a video into much smaller non-overlapping sub-volumes, or $3 D$ blocks. We then apply self-attention separately within each block. This approach is conceptually simple and amenable to highly efficient implementation on
+
+TPUs, which enables us to scale our models substantially while maintaining a comparatively high training speed with only a modest sacrifice in expressive power.
+
+The Video Transformer consists of multiple stacked self-attention layers. Each layer divides the overall video volume of shape $( T , H , W )$ into smaller blocks of shape $( t , h , w )$ of length $n _ { p } = t \cdot h$ · $w$ , and performs attention within each block independently. Given a (flattened) block representation $z \in \mathbb { R } ^ { n _ { p } \times d }$ of hidden size $d$ as input, this amounts to:
+
+$$
+\begin{array} { r l r } & { [ \boldsymbol { q } , \boldsymbol { k } , \boldsymbol { v } ] = \mathrm { l a y e r n o r m } ( \boldsymbol { z } ) W _ { q k v } \quad } & { \boldsymbol { q } , \boldsymbol { k } , \boldsymbol { v } \in \mathbb { R } ^ { n _ { p } \times d _ { a } } , W _ { q k v } \in \mathbb { R } ^ { d \times 3 d _ { a } } , } \\ & { \qquad \boldsymbol { A } = \mathrm { s o f t m a x } \left( \boldsymbol { q } \boldsymbol { k } ^ { \top } / \sqrt { d _ { a } } + \boldsymbol { B } \right) \quad } & { \boldsymbol { A } , \boldsymbol { B } \in \mathbb { R } ^ { n _ { p } \times n _ { p } } , } \\ & { \mathrm { a t t e n t i o n } ( \boldsymbol { z } ) = \boldsymbol { A } \boldsymbol { v } . } \end{array}
+$$
+
+The input is first projected to query, key and value representations (Eq. 2). The attention matrix $A$ is then formed as the scaled dot-product between all query-key pairs adding a relative position bias $B$ (Parikh et al., 2016) (Eq. 3). The bias $B _ { i j }$ is defined as the sum of per-dimension relative distance biases between element $i$ and $j$ , along each of the time- and spatial dimensions. Finally, the values are aggregated with respect to the attention weights (Eq. 4).
+
+Following Vaswani et al. (2017), we concatenate the output of $n _ { a }$ parallel attention heads in each layer and project the result by a linear transformation (Eq. 5) before applying a residual connection. Finally, the output of the multi-head self-attention layer is passed through another dense layer with ReLU activation, followed by a final linear transformation and a residual connection (Eq. 6):
+
+$$
+\begin{array} { r l r } & { \tilde { z } = [ \mathrm { a t t e n t i o n } _ { 1 } ( z ) ; \cdots ; \mathrm { a t t e n t i o n } _ { n _ { a } } ( z ) ] W _ { p } + z \qquad } & { \quad } & { W _ { p } \in \mathbb { R } ^ { ( n _ { a } \cdot d _ { a } ) \times d } , } \\ & { z ^ { \prime } = \mathrm { r e l u } ( \mathrm { l a y e r n o r m } ( \tilde { z } ) T _ { 1 } ) T _ { 2 } + \tilde { z } } & { \quad } & { T _ { 1 } , T _ { 2 } \in \mathbb { R } ^ { d \times d } , } \end{array}
+$$
+
+where overloading notation, attention $( z )$ denotes the blockwise application of self-attention to $_ { z }$ . Similar to Baevski & Auli (2019), we found that applying layer normalization before each block, rather than after each block as proposed by Vaswani et al. (2017), improves training.
+
+Connectivity. Operating on 3D sub-volumes (blocks) of videos means that there is no direct information exchange between blocks. However, this can be addressed by varying the block sizes between each layer. To achieve this, we define blocks that stretch over the entire extent of at least a single dimension in each layer. Following this procedure, we can effectively connect all pixel positions in the encoder, but due to masking some dependencies are missed in the decoder. However, in our experiments these did not produce any visible, systematic artifacts. We discuss missing dependencies and potential remedies in Appendix C.
+
+Efficiency. Running block-local self-attention is very efficient in practice as the cost of splitting videos into blocks is negligible. The approach of Parmar et al. (2018) uses overlapping 2D image blocks in each layer. We found this prohibitive as the required data copying is comparatively expensive. To avoid the need for overlaps to connect pixels across blocks, we simply vary block sizes between layers, which is highly efficient and, as our results show, works well in practice.
+
+# 3.2 SPATIOTEMPORAL SUBSCALING
+
+Menick & Kalchbrenner (2019) recently proposed generating images as a sequence of subscaled image slices. We similarly define a subscale factor $\boldsymbol { s } = \left( s _ { t } , s _ { h } , s _ { w } \right)$ which divides a video into $s = \overline { { ( \boldsymbol { s } _ { t } \cdot \boldsymbol { s } _ { h } \cdot \boldsymbol { s } _ { w } ) } }$ sub-sampled videos (slices), each of resolution $( T / s _ { t } , H / s _ { h } , W / s _ { w } )$ , as depicted in the bottom part of Figure 1. The slices are generated in order according to their respective offsets, such that we first generate slice $\pmb { x } _ { ( 0 , 0 , 0 ) }$ , then $\pmb { x } _ { ( 0 , 0 , 1 ) }$ , up until slice $\pmb { x } _ { ( s _ { t } - 1 , s _ { h } - 1 , s _ { w } - 1 ) }$ . Generating all slices one at a time in this way drastically reduces the number of pixels in memory to $N _ { p } / s$ , which enables scaling our architectures by a factor of $s$ . Each slice is internally generated according to the raster-scan order. In the following we explain how slices are generated and how they are conditioned on already decoded slices. An overview is illustrated in the upper part of Figure 1.
+
+Slice Encoder. The current slice $\pmb { x } _ { ( a , b , c ) }$ is generated conditioned on the encoded pixels from preceding slices as follows. First, we create a partially masked video, where only the pixels of preceding slices $\pmb { x } _ { < ( a , b , c ) }$ are visible. The partially masked video is then embedded by concatenating the onehot encoding of the discretized pixel intensities of each channel. Subsequently, a 3D convolution with kernel size $\pmb { k } = ( k _ { 1 } , k _ { 2 } , k _ { 3 } )$ and stride $\pmb { s }$ (the sub-scaling factor) results in an encoded video of resolution $( T / s _ { t } , H / s _ { h } , W / s _ { w } )$ . We apply convolution padding depending on the current slice index $( a , b , c )$ . In particular, we pad with $( \lfloor k _ { 1 } / 2 \rfloor - a , \lfloor k _ { 2 } / 2 \rfloor - b , \lfloor k _ { 3 } / 2 \rfloor - c )$ , which “centers” the convolution kernel on the pixels of the current slice. Finally, we add positional embeddings for each axis, as well as embeddings for the current slice index $( a , b , c )$ , to the output of this strided convolution. The result is an initial encoder representation $\begin{array} { r } { z _ { ( a , b , c ) } ^ { 0 } \in \mathbb { R } ^ { T / s _ { t } \times H / s _ { h } \times W / s _ { w } \times d _ { e } } } \end{array}$ , where $d _ { e }$ is the embedding size. We can optionally condition on auxiliary information, such as per-frame action values of a robot arm, by concatenating this information to the initial encoder representation.
+
+This representation is further transformed by a linear projection to hidden size $d$ , before being fed as input to a stack of terized by a different $L$ block-local self-attention layers as described in ock size and number of attention heads. The res $\ S 3 . 1$ . Each lg output $\dot { z } _ { ( a , b , c ) } ^ { L }$ arame-is used as conditional input to the subscale slice decoder, which generates the pixels of the current slice $( a , b , c )$ .
+
+Slice Decoder. The pixel values of the current slice $\pmb { x } _ { ( a , b , c ) }$ are predicted conditioned on the encoder representation $z _ { ( a , b , c ) } ^ { L }$ . The decoder is almost identical to the encoder in structure, except for the use of masking in the decoder as defined by the generation order. First, we embed $\pmb { x } _ { ( a , b , c ) }$ by summing $N _ { c }$ channel embeddings of size $d _ { e }$ at every pixel, before applying a $3 \mathrm { x } 3 \mathrm { x } 3 $ masked convolution (van den Oord et al., 2016a) on the embedded pixels, effectively representing each pixel by its already generated, immediate neighbors. Similar to the encoder, we add positional embeddings for the space- and time dimensions to the ouresults in an initial decoder representation $\mathbf { \dot { y } } _ { ( a , b , c ) } ^ { 0 } \in \mathbb { R } ^ { T / s _ { t } \times H / s _ { h } \times W / s _ { w } \times d }$ As in the encoder, this.
+
+To condition on the encoder state, a linear projection of $z _ { ( a , b , c ) } ^ { L }$ is added to $\pmb { y } _ { ( a , b , c ) } ^ { 0 }$ and the resulting representation is fed through a stack of block-local self-attention layers, with masking, to produce a state $\pmb { y } _ { ( a , b , c ) } ^ { L }$ on which the final channel predictions are conditioned.
+
+# 3.3 CHANNEL PREDICTION & LOSS FUNCTION.
+
+The per-pixel channel intensities $\pmb { x } _ { ( a , b , c ) } ^ { k }$ (we omit the slice index $( a , b , c )$ in the following) for each channel $k \ < \ N _ { c }$ are predicted by MLPs with a single hidden layer (Eq. 8), conditioned on the flattened final decoder state $\pmb { y } ^ { L } \in \mathbb { R } ^ { n _ { p } \times d }$ — which is itself conditioned on $z _ { ( a , b , c ) } ^ { L }$ and hence on prior slices $\pmb { x } _ { < ( a , b , c ) }$ — as well as the preceding channels $( { \pmb x } ^ { j } ) _ { j = 1 \dots k - 1 }$ for each pixel, encoded as one-hot vectors. Finally, the per video slice loss is defined as the negative log-likelihood as in Eq. 9:
+
+$$
+\begin{array} { r l r } & { \boldsymbol { u } ^ { k } = \left[ \mathrm { l a y e r n o r m } \left( \boldsymbol { y } ^ { L } \right) ; \mathrm { o n e h o t } \left( \boldsymbol { x } ^ { 1 } \right) ; \cdots ; \mathrm { o n e h o t } \left( \boldsymbol { x } ^ { k - 1 } \right) \right] \boldsymbol { U } _ { k } , } & \\ & { p \left( x _ { i } ^ { k } | \boldsymbol { x } _ { i } ^ { < k } , \boldsymbol { x } _ { < i } \right) = \mathrm { s o f t m a x } \left( \mathrm { r e l u } ( \boldsymbol { u } _ { i } ^ { k } ) \boldsymbol { P } \right) , } & { \boldsymbol { P } \in \mathbb { R } ^ { d \times N _ { v } } , \quad \boldsymbol { U } _ { k } \in \mathbb { R } ^ { ( d + ( k - 1 ) \cdot N _ { v } ) \times d } , } \\ & { \boldsymbol { \mathcal { L } } ( \boldsymbol { x } ) = - \displaystyle \sum _ { i = 0 } ^ { n _ { p } - 1 } \sum _ { k = 0 } ^ { N _ { c } - 1 } \ln p ( x _ { i } ^ { k } | \boldsymbol { x } _ { i } ^ { < k } , \boldsymbol { x } _ { < i } ) . } & \end{array}
+$$
+
+We found that splitting the color channel values of the videos into coarse and fine bits helps slightly in terms of performance. Specifically, we split the $3 \times 8$ -bit RGB channels into $6 \times 4$ -bit channels ( $N _ { c } = 6$ , $N _ { v } = 1 6$ ), such that the coarse bits of all three channels are predicted before the fine bits. Furthermore, splitting channels this way at the input level considerably lowers memory footprint when encoding videos as onehot vectors on TPUs.
+
+# 4 EXPERIMENTS
+
+Below, we provide details on the model variants considered, our training setup and the evaluation metrics used. We focus our evaluation on the BAIR Robot Pushing and Kinetics datasets. Additional results on Moving MNIST and another robot pushing dataset are provided in Appendix A for reference. Sample videos strips of each model and dataset can be found in Appendix F and sample videos at https://bit.ly/2Zb017f.
+
+# 4.1 MODELS & SETUP
+
+Unless specified otherwise, we model video slices of 4 frames with a spatial resolution of $3 2 \mathbf { x } 3 2$ . Both the encoder and decoder consist of 8 layers and have a nearly identical structure, except for the use of masking in the decoder, as described in Section 3.2. We apply block-local self-attention with the following block sizes $( t , h , w )$ . Layers 1-4: (4, 8, 4); (4, 4, 8); (1, 32, 4); and (1, 4, 32). Intuitively, layers 1 and 2 are responsible for gathering temporal information whereas layers 3 and 4 gather spatial information of the entire frame. Layer 3 has access to the entire height and layer 4 to the entire width of a frame. The remaining 4 layers have the same block sizes, but in reverse order. However, as discussed in Appendix B, this particular choice of block size ordering is not crucial. There are $n _ { a } = 8$ attention heads, each with hidden size $d _ { a } = 1 2 8$ . Our base models are trained with embedding size $d _ { e } = 1 2 8$ and hidden size of $d = 5 1 2$ (46M parameters). Based on ablations in Appendix B, we observed that increasing the hidden dimension is preferable to using deeper networks. Hence, we increase the hidden size to $d = 2 0 4 8$ and use $n _ { a } = 1 6$ instead of 8 heads for the last 4 encoder/decoder layers in our large models (373M parameters).
+
+Models. To assess the effect of subscaling, we explore the following variants. These differ mainly in the subscaling factor $\pmb { s }$ as well as the context kernel size $\boldsymbol { k }$ , defaulting to $k = s$ :
+
+Spatiotemporal Subscaling. The subscale video transformer with full spatiotemporal subscaling applies subscaling in every dimension. For instance, a $1 6 \mathrm { x } 6 4 \mathrm { x } 6 4$ video is subscaled by factors $s = ( 4 , 2 , 2 )$ to 16 slices of $4 \mathbf { x } 3 2 \mathbf { x } 3 2$ .
+
+Spatial Subscaling. This model uses no temporal subscaling and only subscales individual frames to a resolution of $3 2 \mathbf { x } 3 2$ . For instance, a $4 \mathrm { x } 6 4 \mathrm { x } 6 4$ video is subscaled by factors $s = ( 1 , 2 , 2 )$ to 4 slices of $4 \mathbf { x } 3 2 \mathbf { x } 3 2$ .
+
+Single Frame. This model uses no subscaling. Instead, we here model an entire single frame at a time, conditioned only on the previous three frames to limit memory consumption. The model uses no actual subscaling. Instead, one can imagine a $1 6 \mathrm { x } 6 4 \mathrm { x } 6 4$ video to be subscaled by factors $\pmb { s } = ( 1 6 , 1 , 1 )$ to 16 slices of $1 \mathrm { x } 6 4 \mathrm { x } 6 4$ frames. The context kernel size is $\pmb { k } = ( 6 , 1 , 1 )$ which means that we merely condition on a context of 3 past frames, as the current and future frames are always masked when the temporal subscaling factor equals the full video length. Self-attention blocks are adapted as follows: Layers 1-4: (1, 8, 16); (1, 16, 8); (1, 2, 64); (1, 64, 2). For the remaining 4 layers we use the same blocks, again in reverse order.
+
+Training. All models are trained with RMSProp (Tieleman & Hinton, 2012) with a fixed learning rate of $2 \cdot 1 0 ^ { - 5 }$ , decay of 0.95 and momentum of 0.9. We use a batch size of 64 video slices, if not stated otherwise, and shuffle the slices to avoid having all slices in a batch correspond to the same video. The smaller models are trained for 300K steps and the larger ones for 1M steps. No explicit regularization is applied as we could not observe any form of over-fitting. Videos longer than the training resolution are cropped randomly in time to the defined training length. If not stated otherwise, models are conditioned on the first frame during training, which is achieved by masking the loss corresponding to this frame. In preliminary experiments, this gave a minor improvement over computing the training loss across all frames.
+
+Intrinsic Evaluation. Most results are reported as bits per dimension (bits/dim), the average negative $\log _ { 2 }$ -probability assigned by the model per (RGB) channel, averaged across all pixels in the video. This corresponds directly to the loss optimized by the model. In all experiments, we condition (prime) on a specified number of initial frames. The log-probabilities corresponding to these frames are excluded from this average.
+
+Extrinsic Evaluation. Prior work mainly reported results on the peak signal-to-noise ratio (PSNR) and mean-structural similarity (SSIM) metrics (Wang et al., 2004b). However, these metrics were developed for images and have serious flaws when applied to videos (Wang et al., $2 0 0 4 \mathrm { a }$ ; Wang & Li, 2007; Zhang et al., 2018; Lee et al., 2018). Conceptually, PSNR has a strong preference for blurry videos as it is based on pixel-level mean squared error. Similarly, SSIM does not correlate well with perceptual quality either. For instance, variational autoencoders show very strong performance on this metric despite producing blurry videos (Lee et al., 2018). Hence, we focus on the Frechet Video ´ Distance (FVD), which was recently proposed by Unterthiner et al. (2018) as a qualitative metric sensitive to visual quality, temporal coherence and diversity of samples. This is the spatiotemporal counterpart to the Frechet Inception Distance (Heusel et al., 2017), replacing the ImageNet-trained ´
+
+Table 1: Quantitative results on BAIR Robot Pushing (left) and Kinetics (right).   
+
+<table><tr><td>Models</td><td>Bits/dim</td><td>FVD</td><td>FVD (Avg)</td></tr><tr><td>Single Frame</td><td>1.49</td><td>104±4</td><td>99±2</td></tr><tr><td>Spatial Sub.</td><td>1.57</td><td>111±4</td><td>108±1</td></tr><tr><td>Spatiotemp. Sub.</td><td>1.53</td><td>106±3</td><td>106±2</td></tr><tr><td>Spatiotemp. Sub. (L)</td><td>1.35</td><td>94±2</td><td>96±2</td></tr><tr><td>SV2P[1]+</td><td>1</td><td>263</td><td>1</td></tr><tr><td>SAVP [2]t</td><td>1</td><td>116t</td><td></td></tr><tr><td>VideoFlow [3]</td><td>1.87</td><td>1</td><td></td></tr></table>
+
+(a) BAIR Robot Pushing. Bits/dim averaged across 15 subsequent frames when priming with 1 initial frame, FVD and unrolled average FVD scores. Best results in bold. $^ { \dagger }$ Results from Unterthiner et al. (2018). ‡ Results are not strictly comparable (see text for details). [1] Babaeizadeh et al. (2018), [2] Lee et al. (2018), [3] Kumar et al. (2019).
+
+<table><tr><td>Models</td><td>Bits/dim</td><td>FVD</td><td>FVD (Avg)</td></tr><tr><td>Single Frame</td><td>1.40</td><td>243±6</td><td>413±11</td></tr><tr><td>Spatial Sub.</td><td>1.47</td><td>263±6</td><td>450±15</td></tr><tr><td>Spatiotemp. Sub.</td><td>1.49</td><td>195±7</td><td>375±11</td></tr><tr><td>Single frame (L)</td><td>1.14</td><td>207±8</td><td>353±13</td></tr><tr><td>Spatiotemp. Sub. (L)</td><td>1.19</td><td>170±5</td><td>316±12</td></tr></table>
+
+(b) Kinetics. Bits/dim averaged across 15 subsequent frames when priming with 1 initial frame, FVD and unrolled average FVD scores when priming with 5 frames. Best results in bold.
+
+Inception network of the latter with an I3D Network trained on Kinetics. Despite sharing the known drawbacks of FID (Binkowski et al., 2018), FVD has shown to correlate much stronger with human ´ raters compared to both PSNR and SSIM (Unterthiner et al., 2018). We report the FVD of the first 16 frames, as well as the “unrolled” average FVD across all contiguous subsequences of 16 frames. In each case, we report the mean and standard deviation of 20 trials.
+
+Sampling time. Sampling from autoregressive models is notoriously slow. However, because our decoders are not very deep (8 layers) we are able to sample a batch of four 30x64x64 videos in acceptable time (approx. 8 minutes) with our large models on a Nvidia Tesla V100. Though this might still be impractical we argue that further advances in parallel sampling strategies (Stern et al., 2018) and future hardware will alleviate this disadvantage significantly.
+
+# 4.2 BAIR ROBOT PUSHING
+
+BAIR Robot Pushing (Ebert et al., 2017) shows a robotic arm pushing and grasping objects in a box. It consists of roughly 40K training- and 256 test videos. We prime on the first frame for training and evaluation.
+
+Empirical Results. All variants of the Video Transformer achieve strong results compared to prior work in terms of both intrinsic and extrinsic metrics. From Table 1a, we see that the small models already reduce the perplexity in terms of bits/dim by almost $20 \%$ compared to the recently proposed VideoFlow model (Kumar et al., 2019) with our large model (L) reducing perplexity even further to a $2 5 \%$ improvement. Similar to Menick & Kalchbrenner (2019), we find that subscaling can have a slightly negative effect on bits/dim. In terms of perceptual quality, every incarnation of our model obtains a lower (better) FVD score compared to all models evaluated by Unterthiner et al. (2018), which notably includes adversarial networks with no guarantees of covering the full empirical distribution. These results are not strictly comparable, since prior work has used longer priming sequences of two (Babaeizadeh et al., 2018; Lee et al., 2018) or three (Kumar et al., 2019) frames, whereas our models (to our disadvantage) see a single prime frame. Note that we sample with temperature 0.9 for the extrinsic metrics as we observed improved qualitative results at this temperature on the validation set. This corresponds to a mild form of mode dropping and is common practice to improve sampling quality. For fair comparison we also tweaked the “temperature” of SAVP by scaling the variance of its normal distribution when sampling. This, however, did not result in any improvements for FVD.
+
+Further results on an earlier version of robot pushing (Finn et al., 2016a) and Moving MNIST (Srivastava et al., 2015) can be found in Appendix A for brevity. In summary, like Kalchbrenner et al. (2016), we match the lower bound on Moving MNIST while obtaining an almost $50 \%$ reduction in bits/dim on robotic pushing which demonstrates the superiority of our models against prior work on autoregressive video modeling.
+
+Qualitative Observations. All variants of our model reach similar quantitative results on these benchmarks and we observe no immediate differences in fidelity. However, there are some notable differences. First, whereas the spatiotemporal subscaling model is able to capture temporal dependencies across up to 16 frames (given subscaling in time by a factor four), the remaining models can only capture dependencies across four frames. This can, for example, result in deformation of occluded objects (e.g., Figure 4 of the Appendix). However, due to the simplicity of the benchmark datasets, this is not appropriately reflected in the metrics including better unrolled FVD curves for the single frame base model in Figure 2a. Second, we observe that lowering the sampling temperature from 1.0 to 0.9 consistently improves results. Notably, spatiotemporal subscaling seems more robust to sampling errors as its performance decays less when sampling with temperature 1.0 ( $1 2 2 { \pm } 4$ Avg. FVD) compared to the spatial subscaling $( 1 3 4 \pm 4 )$ and single frame models $( 1 5 3 \pm 7 )$ . We attribute this finding to the difference in generation order when spatiotemporal subscaling is employed as it predicts pixels over the entire extend of the 3D video volume early and thereby effectively anchors future predictions around these pixels. Finally, considering that our results on BAIR Robot Pushing in terms of FVD are on par with those between two ground-truth subsamples (Figure 4 of Unterthiner et al. (2018)), we may be approaching the limit of this benchmark. On the other hand, it could be that FVD suffers out-of-domain and is not sufficiently sensitive to longrange temporal dynamics, since it is trained to perform human action recognition, which is known to predominantly rely on local features (Carreira & Zisserman, 2017; Xie et al., 2018).
+
+![](images/dfa6ccf1b97621553902141f4626b3162f2ec6245a35eafcceb80526bf719de8.jpg)  
+Figure 2: Unrolled FVD metrics on BAIR Robot Pushing (left) and Kinetics (right).
+
+# 4.3 KINETICS
+
+Moving from a constrained to a real world setting, we next apply our models to the Kinetics dataset (Kay et al., 2017), a large scale action-recognition dataset consisting of YouTube videos. Specifically, we use Kinetics-600, which contains roughly 400K training videos ranging over 600 action classes (Carreira et al., 2018). We center-crop and down-sample each frame to 64x64 with a width-3 Lanczos filter and anti-aliasing.
+
+We introduce a slight change to our setup by using a separate decoder for the first slice $\pmb { x } _ { ( 0 , 0 , 0 ) }$ . This decoder can be twice as deep (16 instead of 8 layers) as the original subscale decoder, because it does not rely on any encoder. For all other slices we train a regular subscale model (8 layers in both encoder and decoder) as before. Using a separate first-slice decoder means that there is no wasted encoder computation on the first slice and that there are additional parameters. Furthermore, for our large models we scale the batch size to 256 by training in parallel on 128 TPU v3 instances for 1M steps.
+
+Empirical Results. Results for our base models are shown in the upper part of Table 1b. In line with results on BAIR pushing, we find that the single frame model obtains better performance in terms of bits/dim. In contrast, we observe that the spatiotemporal subscaling model generates better and more robust video continuations which is reflected by its superior FVD scores. Our large models (L) show much stronger performance across the board (see lower half of Table 1b and Figures 2b), lowering the perplexity to 1.14 bits/dim for the single frame model. While the spatiotemporal subscaling model obtains slightly worse perplexity of 1.19 bits/dim, it improves FVD to 170. Despite its good performance on bits/dim, even with a temperature of 0.9, samples from the large single frame model are prone to instability and in many cases we observe color “explosions” (Figure 12 in the Appendix shows an example) which is reflected in its significantly higher FVD score. Although much less pronounced we observed such instability already when sampling with temperature 1.0 on BAIR pushing which clearly indicates the benefits of temporal subscaling for video generation.
+
+![](images/085bae087f32ce532cec998a0c41eea2a6dce29429281a174e5f535139017b79.jpg)  
+Figure 3: Selected Kinetics continuations from a set of 128 videos and 16 samples which showcase a variety of natural, video-specific phenomena our model learns to generate. We used our large spatiotemporal subscaling model and prime generation with 5 frames (0-4) to include the first two frames in subscale order (0, 4). Samples are generated with temperature of 0.9. The examples depict frames 0, 5, 10 and 15.
+
+Qualitative Observations. Figure 3 shows samples from a cooking subset of Kinetics that we describe in Appendix E. These are selected to showcase different aspects of real-world videos learned by the large spatiotemporal subscaling model. Figures 3a and 3c demonstrate the model’s ability to handle camera movement. We find that camera movement seems to be learned early in training, possibly since it is a major source of uncertainty. This requires transforming pixels correctly while hallucinating new pixels at the edges. Similarly, object movement resulting, for instance, in a change of perspective is predicted quite well (Figure 3b). Highly stochastic motion such as fire (Figure 3d) or steam is modeled surprisingly well. Videos in Kinetics sometimes contain scene changes and our model, too, occasionally generates videos with jumps to completely new scenes (Figure 3e). Motion of human fingers and faces seems challenging to model. Nevertheless, in a number of samples the model is able to generate somewhat believable continuations as can be seen in Figures 3g, 3h or 3i.
+
+These selected examples show only a small subset of the interesting phenomena handled by the model and illustrate the sheer complexity involved in modeling this dataset. In Appendix F, we provide multiple samples, primed with the same initial frames to illustrate the diversity of the generated samples.
+
+Limitations. While we obtained the occasional encouraging sampl, we would like to point out that the diversity of Kinetics still poses a major challenge. Failure modes range from freezing movement or object distortions to continuations that ”wash out” entirely after a few frames. We firmly beleive that yet larger datasets and/or models will be required to capture the complexity of even short clips from YouTube videos. With this work we merely provide an initial baseline, hoping to highlight both the potential and the enormous room for improvement.
+
+# 5 CONCLUSION
+
+We presented an autoregressive model of videos based almost entirely on a variant of block-local self-attention that can easily be implemented efficiently on TPUs. Combined with spatiotemporal subscaling, our models can be scaled up substantially while retaining the ability to capture longer range spatiotemporal dependencies.
+
+Empirically, we obtain state-of-the-art results across a range of video generation benchmarks, while the scalability of our approach enables us to make an initial attempt at modeling videos of unusually high complexity and diversity as found in the Kinetics dataset. Our models occasionally generate encouraging continuations, especially on a subset of cooking videos, yet we find modeling the full range of such videos clearly remains a major challenge.
+
+# ACKNOWLEDGEMENTS
+
+This work benefited from numerous conversations with Nal Kalchbrenner, as well as discussions with Jacob Menick, Mohammad Taghi Saffar and Niki Parmar. We would also like to thank Chelsea Finn and Tom Kwiatkowski for thoughtful comments on an earlier draft.
+
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+![](images/f85a708c118abdfbf1adf3d921f784deeaf29204e187183c06887a4dc761f6c6.jpg)  
+Figure 4: Samples (showing every 5th frame horizontally) illustrating occlusion effects on BAIR Robot Pushing. Models without temporal subscaling (rows 3-4) fail on occlusions, whereas the model with temporal subscaling (row 2) correctly maintains objects from the ground truth video (row 1). Notice the green ball deformation on rows 2 and 3 and the hallucinated green ball on the right edge of row 3, which are caused by missing temporal dependencies across the duration of occlusion.
+
+Table 2: Moving MNIST. Nats per frame averaged across 10 subsequent frames when priming with 10 initial frames. Best results in bold. † The lower bound reported in (Kalchbrenner et al., 2016) is slightly higher than ours.   
+
+<table><tr><td>Models</td><td>Nats/Frame (↓)</td></tr><tr><td>Single Frame</td><td>86.2</td></tr><tr><td>Spatial Subscaling</td><td>91.8</td></tr><tr><td>Spatiotemporal Subscaling</td><td>90.0</td></tr><tr><td>VPN (Kalchbrenner et al., 2016)</td><td>87.6</td></tr><tr><td>Lower bound</td><td>85.1 (86.3)‡</td></tr></table>
+
+# A FURTHER BENCHMARKS
+
+# A.1 MOVING MNIST
+
+Moving MNIST (Srivastava et al., 2015) consists of $1 0 0 \mathrm { K }$ training- and 10K validation/test videos of two handwritten digits from the MNIST benchmark that move deterministically across the frame, crossing each other and bouncing off the borders. The partial occlusion of crossing digits makes this dataset challenging. To be comparable with Kalchbrenner et al. (2016), we use the first ten frames as priming and predict the subsequent ten frames.
+
+To allow direct comparison with Kalchbrenner et al. (2016), we change our loss to a “deterministic” loss (and derived nats-per-frame metric) which is defined as: $\begin{array} { r } { H ( \bar { z } , y ) = - \sum _ { i } z _ { i } \ln y _ { i } + ( 1 - } \end{array}$ $z _ { i } ) \ln ( 1 - y _ { i } )$ , where $z _ { i }$ are the gray-scale targets between 0.0 and 1.0, and $y _ { i }$ are the predicted scalar intensities.
+
+From Table 2, we find that like Kalchbrenner et al. (2016) our single frame prediction model (i.e., no subscaling) virtually solves the task in the sense that it almost matches the lower bound of the loss. However, this is not true for our subscaling models. Employing spatial subscaling on this task gives aliasing artifacts that make it harder to predict future frames. Although this finding is limited to Moving MNIST, it suggests that spatial subscaling can potentially hurt generation.
+
+Table 3: Ablation of hyper-parameter settings in terms of bits per dimension for models on 256 BAIR Robot Pushing validation videos. All models were primed on 1 frame and trained for 300K steps with a batch size of 64.   
+
+<table><tr><td colspan="2">Layers</td><td>Heads</td><td>Hidden size 1.65</td></tr><tr><td>4</td><td colspan="3">1.63 4 1.59 256</td></tr><tr><td>8</td><td colspan="3">1.55 8 1.55 512</td></tr><tr><td>16</td><td colspan="3">1.55 1.4816 1.51 1024 1.47</td></tr><tr><td>24</td><td colspan="3">1.4524 1.47 2048 1.40</td></tr></table>
+
+# A.2 ROBOTIC PUSHING.
+
+Robotic Pushing (Finn et al., 2016a) was used in prior work on autoregressive video generation (Kalchbrenner et al., 2016). The videos show a robotic arm pushing and grasping objects in a box and there are roughly 50K training videos and 1500 test videos with seen and novel objects, respectively. Following prior work, we use the initial two frames for priming and condition on the robot arm action for each frame as described in Section 3.2. We use the same setup as (Kalchbrenner et al., 2016) with videos of twenty frames down-sampled to 64x64 with a Lanczos filter and antialiasing.
+
+We report results to compare with prior work on autoregressive video generation by Kalchbrenner et al. (2016), who achieve 0.92 bits/dim (0.64 nats/dim) with 2 frames of priming on each of the test splits (one with objects seen during training and one with novel objects). We trained a large (2048 dimensional) spatiotemporal subscaling model which achieves 0.51 bits/dim on the subset with seen objects and 0.47 bits/dim on the subset with new objects, which corresponds to an almost $50 \%$ reduction in perplexity.
+
+# B HYPER-PARAMETER SWEEPS
+
+Table 3 shows the impact of different architectural settings. We see that the hidden size has the biggest impact followed by the number of layers and heads. This is an interesting as well as important finding because increasing the hidden size (wider networks) requires more parallel compute which modern Deep Learning hardware excels at. Computation time grows sub-linear, memory linear and parameters partially quadratically. In contrast all of these aspects grow linearly with deep networks. For scaling up architectures depth is therefore not the preferred option as we suffer much more in terms of computation time while having less parameters.
+
+In another experiment, we shuffle the arrangement of block sizes between layers and found that it did not really matter, that is, all results were within 0.01 bits/dim. However, our setup had the best overall performance.
+
+Finally, we tried sampling temperature 0.9 and 1.0 only on the BAIR Robot Pushing validation set and found that temperature 0.9 consistently gave more robust predictions and better results on all extrinsic metrics.
+
+# C CONNECTIVITY IN BLOCK-LOCAL SELF-ATTENTION
+
+Blind Spots. Varying block sizes between layers in block-local self-attention can efficiently connect every pixel with every other pixel when no masking is employed. If masking is employed to respect the generation order (as in our slice decoder) block-local self attention produces “blind spots” which leads to independence assumptions. To exemplify these special cases, consider position $( 1 , 0 , 0 )$ , the top-left pixel of the second frame, and its direct predecessor in generation order $( 0 , h - 1 , w - 1 )$ , the bottom-right pixel of the first frame. The only way to establish a connection between these two positions is through a direct connection, because masking prevents any indirect connection. Thus, there has to be one layer in which both of these pixels are in the same block. This block must at least stretch over the entire extent of both width and height (i.e., the full frame) as well as at least 2 time steps. Running full self-attention in such blocks can easily become prohibitive for large $h$ and $w$ .
+
+Remedies. There seems to be no simple solution that solves the problem of blind spots completely. However, we can make sure that local dependencies up to a certain distance are all covered by increasing the kernel size of the initial, masked convolution in the decoder. It is also possible to combine block-local self-attention with its dual form, dilated self-attention in $n$ dimensions which connects all pixels at the same relative position within their respective block with each other. Finally, we find that it is important to avoid blocks of small sizes in any dimension (e.g., 1). That means, even if we stretch a block to the full extent of one dimension it is important to define sizes at least larger than 1 on all other dimensions to limit the number of unconnected pixels.
+
+On the other hand, the independence assumptions due to masking do not seem to produce any systematic, visible artifacts in our samples. We believe this to be an interesting finding by itself as it shows that there is potential for parallelizing autoregressive video generation by systematically exploring further independence assumptions.
+
+# D ADDITIONAL FINDINGS
+
+Below, we summarize some additional findings that may be of interest to some readers:
+
+• We found that using blocks stretching across a single time-/row-/column- dimension, is substantially worse than using blocks that stretch at least to some extent in all directions. This is likely due to the fact that future masking in the decoder imposes strong independence assumptions in this case, as discussed in Appendix C. We found that RMSProp with momentum converges significantly faster than ADAM, which we tried with different learning rates and settings for $\beta _ { 1 }$ and $\beta _ { 2 }$ .   
+• We tried using continuous, rather than discretized one-hot, input channel representations, but this had an overall negative impact on both performance and sample quality.   
+• We experimented with a gating mechanism in Eq. 3, such that the attention matrix $A$ is masked elementwise with $( 1 - I )$ to allow for not attending to any element, similar to sentinel attention (Lu et al., 2017). However, this had no effect on generation quality.
+
+# E KINETICS COOKING
+
+We found that for many video-prefixes in Kinetics it is very hard for our model to predict continuations. For instance, main objects in the videos are too small or movement is too fast which results in very blurry frames or there is little to no movement at all. Figure 13 shows some examples. Therefore, we created a subset of cooking videos that we found to exhibit these problems to a lesser degree.
+
+In particular we filtered videos whose label matched the following regular expression:
+
+. $\star$ (baking|barbequing|breading|cooking|cutting|pancake|vegetables| meat|cake|sandwich|pizza|sushi|tea|peeling|fruit|eggs|salad).\*
+
+Note that we still train on the full Kinetics training set and only use the cooking set to showcase samples in some cases.
+
+# F SAMPLES
+
+Figures 5-8 show samples from our spatiotemporal subscaling and large spatiotemporal subscaling models on BAIR Robot Pushing. Figures 5 and 6 illustrate the fidelity and realism of the generated samples, whereas Figures 7 and 8 illustrate the diversity of samples.
+
+Figures 9-11 show samples from our spatiotemporal subscaling model on cooking videos for Kinetics-600, while Figure 12 depicts samples from the single frame model. In each case, we prime on 5 frames and sample the next 11 frames. Each figure shows 16 different samples from the same model. As can be seen, the model is able to generate diverse continuations while retaining fidelity. For the single frame model we observe strange color artifacts (exploding colors) which we attribute to the standard, raster-scan generation order of this model.
+
+![](images/1da4058c9504b3302e241acbb9fbea6a96582f105e85fc5a50ad10b08fa72b4a.jpg)  
+Figure 5: Samples of 30 future frames (showing every 4th frame) for 12 test videos with the spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing.
+
+![](images/61fed3cceb926177810e03032d2edf0e83a5ee6ebe8688b199292174070ad8fc.jpg)  
+Figure 6: Samples of 30 future frames (showing every 4th frame) for 12 test videos with the large spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing.
+
+![](images/d30b988f5520416ac050282af99f11ea32ad7242a3646a76733791afa802c50a.jpg)  
+Figure 7: 11 samples of 30 future frames (showing every 4th frame) for 1 test video (top row) with the spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing.
+
+![](images/2c3ad68584652166a78279a77193cea938e43256e30fb61d5ffdbc6238ddac65.jpg)  
+Figure 8: 11 samples of 30 future frames (showing every 4th frame) for 1 test video (top row) with the large spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing.
+
+![](images/60eda85ed2c491d55a73e6d6cac2a0c2e777308f7cd59c29862db394849132b9.jpg)  
+Figure 9: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics.
+
+![](images/bd5e27408a3de0729b5402d95d24e82771f5e33f8e4038d972f1b6483b958e8f.jpg)  
+Figure 10: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics.
+
+![](images/c249bb000473202c4ddc516ad52809952700dd5c2c992681504816711af4c5f2.jpg)  
+Figure 11: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics.
+
+![](images/4a130c065bd77737c167189965063fff68ac5665578b82002f122e4c571d2845.jpg)  
+Figure 12: Samples of 11 future frames from the single frame model with 5 prime frames on 64x64 Kinetics exhibiting strange color artifacts.
+
+![](images/378bd4d72631a6cf0342e15dca462a151a7e82f1836ab227d3bf2e7c1622be04.jpg)  
+Figure 13: Ground-truth (top) and 2 samples of 30 future frames (showing every 4th frame) demonstrating that random Kinetics videos do not always lend themselves as good prefixes for generating continuations.

parse/train/rJgsskrFwH/rJgsskrFwH_content_list.json

@@ -0,0 +1,1823 @@
+[
+    {
+        "type": "text",
+        "text": "SCALING AUTOREGRESSIVE VIDEO MODELS ",
+        "text_level": 1,
+        "bbox": [
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+            99,
+            714,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dirk Weissenborn∗ Google Research diwe@google.com ",
+        "bbox": [
+            184,
+            145,
+            334,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Oscar Tackstr¨ om¨ ∗† Sana Labs oscar@sanalabs.com ",
+        "bbox": [
+            392,
+            143,
+            571,
+            186
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jakob Uszkoreit Google Research usz@google.com ",
+        "bbox": [
+            630,
+            145,
+            771,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            224,
+            544,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Due to the statistical complexity of video, the high degree of inherent stochasticity, and the sheer amount of data, generating natural video remains a challenging task. State-of-the-art video generation models often attempt to address these issues by combining sometimes complex, usually video-specific neural network architectures, latent variable models, adversarial training and a range of other methods. Despite their often high complexity, these approaches still fall short of generating high quality video continuations outside of narrow domains and often struggle with fidelity. In contrast, we show that conceptually simple autoregressive video generation models based on a three-dimensional self-attention mechanism achieve competitive results across multiple metrics on popular benchmark datasets, for which they produce continuations of high fidelity and realism. We also present results from training our models on Kinetics, a large scale action recognition dataset comprised of YouTube videos exhibiting phenomena such as camera movement, complex object interactions and diverse human movement. While modeling these phenomena consistently remains elusive, we hope that our results, which include occasional realistic continuations encourage further research on comparatively complex, large scale datasets such as Kinetics. ",
+        "bbox": [
+            233,
+            257,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            334,
+            539
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative modeling of video holds promise for applications such as content creation, forecasting, transfer learning and model-based reinforcement learning (Srivastava et al., 2015; Carl Vondrick, 2016; Oh et al., 2015; Kaiser et al., 2019). While recently there has been a lot of progress on generative models for text, audio and images, video generation remains challenging. To some extent this is simply due to the large amount of data that needs to be produced. Autoregressive models suffer from this particularly in their generation speed. On the other hand, they have a number of desirable attributes, such as their conceptual simplicity and tractable likelihood, which enables straightforward evaluation of their ability to model the entire data distribution. ",
+        "bbox": [
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+            555,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Moreover, recent results on image generation by Menick & Kalchbrenner (2019) show that pixellevel autoregressive models are capable of generating images with high fidelity. These findings motivate the question of how far one can push such autoregressive models in the more general task of video generation when scaling recent advances in neural architectures to modern hardware accelerators. ",
+        "bbox": [
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+            674,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce a generalization of the Transformer architecture of Vaswani et al. (2017) using three-dimensional, block-local self-attention. In contrast to the block-local attention mechanism of Parmar et al. (2018), our formulation can be implemented efficiently on Tensor Processing Units, or TPUs (Jouppi et al., 2017). To further reduce the memory footprint, we combine this with a three-dimensional generalization of methods from Menick & Kalchbrenner (2019), who generate images as sequences of smaller, sub-scaled image slices. ",
+        "bbox": [
+            174,
+            751,
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+            834
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Together, these techniques allow us to efficiently model videos as 3D volumes instead of sequences of still image frames, with direct interactions between representations of pixels across the spatial and temporal dimensions. ",
+        "bbox": [
+            176,
+            840,
+            823,
+            883
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We obtain strong results on popular benchmarks (Section 4.2, Appendix A) and produce high fidelity video continuations on the BAIR robot pushing dataset (Ebert et al., 2017) exhibiting plausible object interactions. Furthermore, our model achieves an almost $50 \\%$ reduction in perplexity compared to prior work on autoregressive models on another robot pushing dataset. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Finally, we apply our models to down-sampled videos from the Kinetics-600 dataset (Carreira et al., 2018) (Section 4.3). While modeling the full range of Kinetics-600 videos still poses a major challenge, we see encouraging video continuations for a more limited subset, namely cooking videos. These feature camera movement, complex object interactions and still cover diverse subjects. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We hope that these initial results will encourage future video generation work to evaluate models on more challenging datasets such as Kinetics. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            282,
+            344,
+            299
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our setup is closely related to that of Kalchbrenner et al. (2016), who extend work on pixel-level autoregressive image generation (van den Oord et al., 2016b;a) to videos. However, whereas they model the temporal and spatial dimensions separately with dilated convolutions and convolutional LSTMs, respectively, our model is conceptually simpler in that we do not make any distinction between temporal and spatial dimensions and instead rely almost entirely on multi-head self-attention (Vaswani et al., 2017) within the 3D video volume. For comparability, we provide results on Moving MNIST and another robot pushing dataset (Finn et al., 2016a) on which our model achieves an almost $50 \\%$ reduction in perplexity (see Appendix A). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One major drawback of autoregressive models is their notoriously slow generation speed. However, we believe that further research into (partially) parallelizing sampling (Stern et al., 2018) and future hardware accelerators will help alleviate this issue and eventually make autoregressive modeling a viable solution even for extremely high-dimensional data such as videos. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To reduce the generally quadratic space complexity of the self-attention mechanism, we use blocklocal self-attention, generalizing the image generation approaches of Parmar et al. (2018) and Chen et al. (2018) to 3D volumes. In concurrent work, Child et al. (2019) instead use sparse attention after linearizing images to a sequence of pixels. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To further reduce memory requirements, we generalize sub-scaling (Menick & Kalchbrenner, 2019) to video. An alternative approach is optionally hierarchical multi-scale generation, which has recently been explored for both image generation (Reed et al., 2017; De Fauw et al., 2019) as well as video generation (Mathieu et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Earlier work on video generation mostly focused on deterministic approaches (Srivastava et al., 2015; Carl Vondrick, 2016; Xingjian et al., 2015; Liu et al., 2017; Jia et al., 2016), which fail to capture the high degree of stochasticity inherent in video. In response, a popular research direction has been that of generative latent-variable video models. In contrast to pixel-level autoregressive models, these posit an underlying latent process in tandem with the observed pixel values. Work in this category includes variants of variational autoencoders (Babaeizadeh et al., 2018; Denton & Fergus, 2018). To address the issues inherent in these models, most notably the tendency to generate blurry outputs possibly due to restricted modeling power, inadequate prior distributions, or optimization of a lower bound in place of the true likelihood, various directions have been explored, including the use of adversarial objectives (Mathieu et al., 2016; Vondrick et al., 2016; Lee et al., 2018), hierarchical latent-variables (Castrejon et al., 2019), or flow-based models (Kumar et al., ´ 2019). All of these approaches admit significantly faster generation. However, in the adversarial case, they tend to only focus on a subset of the modes in the empirical distribution while flowbased models struggle with limited modeling power even when using a large number of layers and parameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A large fraction of earlier work on video generation has encoded specific intuitions about videos, such as explicit modeling of motion (Finn et al., 2016b; Denton & Fergus, 2018) or generation of optical flow (Patr ˘ aucean et al., 2016). The conceptual simplicity of our model, however, is more in ˘ line with recent approaches to video classification that process videos by means of 3D convolutions (Carreira & Zisserman, 2017; Xie et al., 2018) or, similar to this work, spatiotemporal self-attention (Girdhar et al., 2018). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/fbd6814863fa9732b13133acc7273c882206a5ec97702b30f61c6560843ebffc.jpg",
+        "image_caption": [
+            "Figure 1: Top: Illustration of the subscale video transformer architecture and process flow. We incrementally generate $s = 4 \\cdot 2 \\cdot 2 = 1 6$ video slices. The video slices and their respective generation order are derived from subscaling. In each iteration, we first process the partially padded video (illustrated for slice index $( 1 , 0 , 1 )$ , black means padding and gray means already generated or visible) by an encoder, the output of which is used as conditioning for decoding the current video slice. After generating a slice we replace the respective padding in the video with the generated output and repeat the process for the next slice. Bottom: Subscaling in 3D (best viewed in color). The 3D volume is evenly divided by a given subscale factor, here $\\textbf { \\textit { s } } = \\ ( 4 , 2 , 2 )$ , and the respective slices are extracted. The whole volume is generated by incrementally predicting the individual, much smaller slices, starting at slice ${ \\pmb x } _ { ( 0 , 0 , 0 ) }$ (yellow), followed by ${ \\pmb x } _ { ( 0 , 0 , 1 ) }$ (green), $\\pmb { x } _ { ( 0 , 1 , 0 ) }$ (red), etc., in rasterscan order. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 VIDEO TRANSFORMER ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We generalize the one-dimensional Transformer (Vaswani et al., 2017) to explicitly model videos represented as three-dimensional spatiotemporal volumes, without resorting to sequential linearization of the positions in the volume (Child et al., 2019). This allows for maintaining spatial neighborhoods around positions, which is important as the large number of individual positions to be predicted in a video requires limiting the receptive field of the self-attention mechanism to a neighborhood around every position to avoid the quadratic blow-up in memory consumption of naive fully-connected attention. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We model the distribution $p ( { \\pmb x } )$ over videos $\\pmb { x } \\in \\mathbb { R } ^ { T \\times H \\times W \\times N _ { c } }$ — with time, height, width and channel dimensions, respectively — by means of a pixel-channel level autoregressive factorization.1 That is, the joint distribution over pixels is factorized into a product of channel intensities for all $N _ { c }$ channels, for each of the $N _ { p } = T \\cdot H \\cdot W$ pixels, with respect to an ordering $\\pi$ over pixels: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa7c1edf488c479a89c2b21deb0c5bff0e2eba16501d14401e46d4be0063eebe.jpg",
+        "text": "$$\np ( \\pmb { x } ) = \\prod _ { i = 0 } ^ { N _ { p } - 1 } \\prod _ { k = 0 } ^ { N _ { c } - 1 } p ( \\pmb { x } _ { \\pi ( i ) } ^ { k } | \\pmb { x } _ { \\pi ( < i ) } , \\pmb { x } _ { \\pi ( i ) } ^ { < k } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The ordering $\\pi$ is given by a combination of a subscale- and raster-scan ordering, as detailed in 3.2. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 BLOCK-LOCAL SELF-ATTENTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The attention mechanism of the original Transformer lets each element in a set of $N _ { p }$ elements connect to every other element, via the fully-connected weighted adjacency (attention) matrix $A \\in$ $\\mathbb { R } ^ { N _ { p } \\times N _ { p } }$ , with $A _ { i j }$ representing attention weights from element $i$ to element $j$ . Because $A$ grows quadratically with the number of elements it becomes prohibitively large for objects such as videos, which typically consist of hundreds of thousands of pixels or more. Therefore, similar in spirit to Parmar et al. (2018), we propose to use local self-attention by dividing a video into much smaller non-overlapping sub-volumes, or $3 D$ blocks. We then apply self-attention separately within each block. This approach is conceptually simple and amenable to highly efficient implementation on ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "TPUs, which enables us to scale our models substantially while maintaining a comparatively high training speed with only a modest sacrifice in expressive power. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Video Transformer consists of multiple stacked self-attention layers. Each layer divides the overall video volume of shape $( T , H , W )$ into smaller blocks of shape $( t , h , w )$ of length $n _ { p } = t \\cdot h$ · $w$ , and performs attention within each block independently. Given a (flattened) block representation $z \\in \\mathbb { R } ^ { n _ { p } \\times d }$ of hidden size $d$ as input, this amounts to: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e4d9a84f0de966e8ce762ebeb244d53463883d307cb8905346f089887d9062d.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { [ \\boldsymbol { q } , \\boldsymbol { k } , \\boldsymbol { v } ] = \\mathrm { l a y e r n o r m } ( \\boldsymbol { z } ) W _ { q k v } \\quad } & { \\boldsymbol { q } , \\boldsymbol { k } , \\boldsymbol { v } \\in \\mathbb { R } ^ { n _ { p } \\times d _ { a } } , W _ { q k v } \\in \\mathbb { R } ^ { d \\times 3 d _ { a } } , } \\\\ & { \\qquad \\boldsymbol { A } = \\mathrm { s o f t m a x } \\left( \\boldsymbol { q } \\boldsymbol { k } ^ { \\top } / \\sqrt { d _ { a } } + \\boldsymbol { B } \\right) \\quad } & { \\boldsymbol { A } , \\boldsymbol { B } \\in \\mathbb { R } ^ { n _ { p } \\times n _ { p } } , } \\\\ & { \\mathrm { a t t e n t i o n } ( \\boldsymbol { z } ) = \\boldsymbol { A } \\boldsymbol { v } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            200,
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+            267
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The input is first projected to query, key and value representations (Eq. 2). The attention matrix $A$ is then formed as the scaled dot-product between all query-key pairs adding a relative position bias $B$ (Parikh et al., 2016) (Eq. 3). The bias $B _ { i j }$ is defined as the sum of per-dimension relative distance biases between element $i$ and $j$ , along each of the time- and spatial dimensions. Finally, the values are aggregated with respect to the attention weights (Eq. 4). ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Following Vaswani et al. (2017), we concatenate the output of $n _ { a }$ parallel attention heads in each layer and project the result by a linear transformation (Eq. 5) before applying a residual connection. Finally, the output of the multi-head self-attention layer is passed through another dense layer with ReLU activation, followed by a final linear transformation and a residual connection (Eq. 6): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a75855ee33193d2034da69648d4422abde6ff9d6dd7fb3799ff4f76c43ffaeeb.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { \\tilde { z } = [ \\mathrm { a t t e n t i o n } _ { 1 } ( z ) ; \\cdots ; \\mathrm { a t t e n t i o n } _ { n _ { a } } ( z ) ] W _ { p } + z \\qquad } & { \\quad } & { W _ { p } \\in \\mathbb { R } ^ { ( n _ { a } \\cdot d _ { a } ) \\times d } , } \\\\ & { z ^ { \\prime } = \\mathrm { r e l u } ( \\mathrm { l a y e r n o r m } ( \\tilde { z } ) T _ { 1 } ) T _ { 2 } + \\tilde { z } } & { \\quad } & { T _ { 1 } , T _ { 2 } \\in \\mathbb { R } ^ { d \\times d } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            409,
+            759,
+            450
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where overloading notation, attention $( z )$ denotes the blockwise application of self-attention to $_ { z }$ . Similar to Baevski & Auli (2019), we found that applying layer normalization before each block, rather than after each block as proposed by Vaswani et al. (2017), improves training. ",
+        "bbox": [
+            176,
+            454,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Connectivity. Operating on 3D sub-volumes (blocks) of videos means that there is no direct information exchange between blocks. However, this can be addressed by varying the block sizes between each layer. To achieve this, we define blocks that stretch over the entire extent of at least a single dimension in each layer. Following this procedure, we can effectively connect all pixel positions in the encoder, but due to masking some dependencies are missed in the decoder. However, in our experiments these did not produce any visible, systematic artifacts. We discuss missing dependencies and potential remedies in Appendix C. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Efficiency. Running block-local self-attention is very efficient in practice as the cost of splitting videos into blocks is negligible. The approach of Parmar et al. (2018) uses overlapping 2D image blocks in each layer. We found this prohibitive as the required data copying is comparatively expensive. To avoid the need for overlaps to connect pixels across blocks, we simply vary block sizes between layers, which is highly efficient and, as our results show, works well in practice. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 SPATIOTEMPORAL SUBSCALING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Menick & Kalchbrenner (2019) recently proposed generating images as a sequence of subscaled image slices. We similarly define a subscale factor $\\boldsymbol { s } = \\left( s _ { t } , s _ { h } , s _ { w } \\right)$ which divides a video into $s = \\overline { { ( \\boldsymbol { s } _ { t } \\cdot \\boldsymbol { s } _ { h } \\cdot \\boldsymbol { s } _ { w } ) } }$ sub-sampled videos (slices), each of resolution $( T / s _ { t } , H / s _ { h } , W / s _ { w } )$ , as depicted in the bottom part of Figure 1. The slices are generated in order according to their respective offsets, such that we first generate slice $\\pmb { x } _ { ( 0 , 0 , 0 ) }$ , then $\\pmb { x } _ { ( 0 , 0 , 1 ) }$ , up until slice $\\pmb { x } _ { ( s _ { t } - 1 , s _ { h } - 1 , s _ { w } - 1 ) }$ . Generating all slices one at a time in this way drastically reduces the number of pixels in memory to $N _ { p } / s$ , which enables scaling our architectures by a factor of $s$ . Each slice is internally generated according to the raster-scan order. In the following we explain how slices are generated and how they are conditioned on already decoded slices. An overview is illustrated in the upper part of Figure 1. ",
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+    {
+        "type": "text",
+        "text": "Slice Encoder. The current slice $\\pmb { x } _ { ( a , b , c ) }$ is generated conditioned on the encoded pixels from preceding slices as follows. First, we create a partially masked video, where only the pixels of preceding slices $\\pmb { x } _ { < ( a , b , c ) }$ are visible. The partially masked video is then embedded by concatenating the onehot encoding of the discretized pixel intensities of each channel. Subsequently, a 3D convolution with kernel size $\\pmb { k } = ( k _ { 1 } , k _ { 2 } , k _ { 3 } )$ and stride $\\pmb { s }$ (the sub-scaling factor) results in an encoded video of resolution $( T / s _ { t } , H / s _ { h } , W / s _ { w } )$ . We apply convolution padding depending on the current slice index $( a , b , c )$ . In particular, we pad with $( \\lfloor k _ { 1 } / 2 \\rfloor - a , \\lfloor k _ { 2 } / 2 \\rfloor - b , \\lfloor k _ { 3 } / 2 \\rfloor - c )$ , which “centers” the convolution kernel on the pixels of the current slice. Finally, we add positional embeddings for each axis, as well as embeddings for the current slice index $( a , b , c )$ , to the output of this strided convolution. The result is an initial encoder representation $\\begin{array} { r } { z _ { ( a , b , c ) } ^ { 0 } \\in \\mathbb { R } ^ { T / s _ { t } \\times H / s _ { h } \\times W / s _ { w } \\times d _ { e } } } \\end{array}$ , where $d _ { e }$ is the embedding size. We can optionally condition on auxiliary information, such as per-frame action values of a robot arm, by concatenating this information to the initial encoder representation. ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "This representation is further transformed by a linear projection to hidden size $d$ , before being fed as input to a stack of terized by a different $L$ block-local self-attention layers as described in ock size and number of attention heads. The res $\\ S 3 . 1$ . Each lg output $\\dot { z } _ { ( a , b , c ) } ^ { L }$ arame-is used as conditional input to the subscale slice decoder, which generates the pixels of the current slice $( a , b , c )$ . ",
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+    {
+        "type": "text",
+        "text": "Slice Decoder. The pixel values of the current slice $\\pmb { x } _ { ( a , b , c ) }$ are predicted conditioned on the encoder representation $z _ { ( a , b , c ) } ^ { L }$ . The decoder is almost identical to the encoder in structure, except for the use of masking in the decoder as defined by the generation order. First, we embed $\\pmb { x } _ { ( a , b , c ) }$ by summing $N _ { c }$ channel embeddings of size $d _ { e }$ at every pixel, before applying a $3 \\mathrm { x } 3 \\mathrm { x } 3 $ masked convolution (van den Oord et al., 2016a) on the embedded pixels, effectively representing each pixel by its already generated, immediate neighbors. Similar to the encoder, we add positional embeddings for the space- and time dimensions to the ouresults in an initial decoder representation $\\mathbf { \\dot { y } } _ { ( a , b , c ) } ^ { 0 } \\in \\mathbb { R } ^ { T / s _ { t } \\times H / s _ { h } \\times W / s _ { w } \\times d }$ As in the encoder, this. ",
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+        "type": "text",
+        "text": "To condition on the encoder state, a linear projection of $z _ { ( a , b , c ) } ^ { L }$ is added to $\\pmb { y } _ { ( a , b , c ) } ^ { 0 }$ and the resulting representation is fed through a stack of block-local self-attention layers, with masking, to produce a state $\\pmb { y } _ { ( a , b , c ) } ^ { L }$ on which the final channel predictions are conditioned. ",
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+    {
+        "type": "text",
+        "text": "3.3 CHANNEL PREDICTION & LOSS FUNCTION. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The per-pixel channel intensities $\\pmb { x } _ { ( a , b , c ) } ^ { k }$ (we omit the slice index $( a , b , c )$ in the following) for each channel $k \\ < \\ N _ { c }$ are predicted by MLPs with a single hidden layer (Eq. 8), conditioned on the flattened final decoder state $\\pmb { y } ^ { L } \\in \\mathbb { R } ^ { n _ { p } \\times d }$ — which is itself conditioned on $z _ { ( a , b , c ) } ^ { L }$ and hence on prior slices $\\pmb { x } _ { < ( a , b , c ) }$ — as well as the preceding channels $( { \\pmb x } ^ { j } ) _ { j = 1 \\dots k - 1 }$ for each pixel, encoded as one-hot vectors. Finally, the per video slice loss is defined as the negative log-likelihood as in Eq. 9: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f0dddf7079e4b267f08f1f66ce8c0590caf97f3d5551079aac775507d819cc7a.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { \\boldsymbol { u } ^ { k } = \\left[ \\mathrm { l a y e r n o r m } \\left( \\boldsymbol { y } ^ { L } \\right) ; \\mathrm { o n e h o t } \\left( \\boldsymbol { x } ^ { 1 } \\right) ; \\cdots ; \\mathrm { o n e h o t } \\left( \\boldsymbol { x } ^ { k - 1 } \\right) \\right] \\boldsymbol { U } _ { k } , } & \\\\ & { p \\left( x _ { i } ^ { k } | \\boldsymbol { x } _ { i } ^ { < k } , \\boldsymbol { x } _ { < i } \\right) = \\mathrm { s o f t m a x } \\left( \\mathrm { r e l u } ( \\boldsymbol { u } _ { i } ^ { k } ) \\boldsymbol { P } \\right) , } & { \\boldsymbol { P } \\in \\mathbb { R } ^ { d \\times N _ { v } } , \\quad \\boldsymbol { U } _ { k } \\in \\mathbb { R } ^ { ( d + ( k - 1 ) \\cdot N _ { v } ) \\times d } , } \\\\ & { \\boldsymbol { \\mathcal { L } } ( \\boldsymbol { x } ) = - \\displaystyle \\sum _ { i = 0 } ^ { n _ { p } - 1 } \\sum _ { k = 0 } ^ { N _ { c } - 1 } \\ln p ( x _ { i } ^ { k } | \\boldsymbol { x } _ { i } ^ { < k } , \\boldsymbol { x } _ { < i } ) . } & \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We found that splitting the color channel values of the videos into coarse and fine bits helps slightly in terms of performance. Specifically, we split the $3 \\times 8$ -bit RGB channels into $6 \\times 4$ -bit channels ( $N _ { c } = 6$ , $N _ { v } = 1 6$ ), such that the coarse bits of all three channels are predicted before the fine bits. Furthermore, splitting channels this way at the input level considerably lowers memory footprint when encoding videos as onehot vectors on TPUs. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Below, we provide details on the model variants considered, our training setup and the evaluation metrics used. We focus our evaluation on the BAIR Robot Pushing and Kinetics datasets. Additional results on Moving MNIST and another robot pushing dataset are provided in Appendix A for reference. Sample videos strips of each model and dataset can be found in Appendix F and sample videos at https://bit.ly/2Zb017f. ",
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+    {
+        "type": "text",
+        "text": "4.1 MODELS & SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Unless specified otherwise, we model video slices of 4 frames with a spatial resolution of $3 2 \\mathbf { x } 3 2$ . Both the encoder and decoder consist of 8 layers and have a nearly identical structure, except for the use of masking in the decoder, as described in Section 3.2. We apply block-local self-attention with the following block sizes $( t , h , w )$ . Layers 1-4: (4, 8, 4); (4, 4, 8); (1, 32, 4); and (1, 4, 32). Intuitively, layers 1 and 2 are responsible for gathering temporal information whereas layers 3 and 4 gather spatial information of the entire frame. Layer 3 has access to the entire height and layer 4 to the entire width of a frame. The remaining 4 layers have the same block sizes, but in reverse order. However, as discussed in Appendix B, this particular choice of block size ordering is not crucial. There are $n _ { a } = 8$ attention heads, each with hidden size $d _ { a } = 1 2 8$ . Our base models are trained with embedding size $d _ { e } = 1 2 8$ and hidden size of $d = 5 1 2$ (46M parameters). Based on ablations in Appendix B, we observed that increasing the hidden dimension is preferable to using deeper networks. Hence, we increase the hidden size to $d = 2 0 4 8$ and use $n _ { a } = 1 6$ instead of 8 heads for the last 4 encoder/decoder layers in our large models (373M parameters). ",
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+        "type": "text",
+        "text": "Models. To assess the effect of subscaling, we explore the following variants. These differ mainly in the subscaling factor $\\pmb { s }$ as well as the context kernel size $\\boldsymbol { k }$ , defaulting to $k = s$ : ",
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+        "type": "text",
+        "text": "Spatiotemporal Subscaling. The subscale video transformer with full spatiotemporal subscaling applies subscaling in every dimension. For instance, a $1 6 \\mathrm { x } 6 4 \\mathrm { x } 6 4$ video is subscaled by factors $s = ( 4 , 2 , 2 )$ to 16 slices of $4 \\mathbf { x } 3 2 \\mathbf { x } 3 2$ . ",
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+        "type": "text",
+        "text": "Spatial Subscaling. This model uses no temporal subscaling and only subscales individual frames to a resolution of $3 2 \\mathbf { x } 3 2$ . For instance, a $4 \\mathrm { x } 6 4 \\mathrm { x } 6 4$ video is subscaled by factors $s = ( 1 , 2 , 2 )$ to 4 slices of $4 \\mathbf { x } 3 2 \\mathbf { x } 3 2$ . ",
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+    {
+        "type": "text",
+        "text": "Single Frame. This model uses no subscaling. Instead, we here model an entire single frame at a time, conditioned only on the previous three frames to limit memory consumption. The model uses no actual subscaling. Instead, one can imagine a $1 6 \\mathrm { x } 6 4 \\mathrm { x } 6 4$ video to be subscaled by factors $\\pmb { s } = ( 1 6 , 1 , 1 )$ to 16 slices of $1 \\mathrm { x } 6 4 \\mathrm { x } 6 4$ frames. The context kernel size is $\\pmb { k } = ( 6 , 1 , 1 )$ which means that we merely condition on a context of 3 past frames, as the current and future frames are always masked when the temporal subscaling factor equals the full video length. Self-attention blocks are adapted as follows: Layers 1-4: (1, 8, 16); (1, 16, 8); (1, 2, 64); (1, 64, 2). For the remaining 4 layers we use the same blocks, again in reverse order. ",
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+        "type": "text",
+        "text": "Training. All models are trained with RMSProp (Tieleman & Hinton, 2012) with a fixed learning rate of $2 \\cdot 1 0 ^ { - 5 }$ , decay of 0.95 and momentum of 0.9. We use a batch size of 64 video slices, if not stated otherwise, and shuffle the slices to avoid having all slices in a batch correspond to the same video. The smaller models are trained for 300K steps and the larger ones for 1M steps. No explicit regularization is applied as we could not observe any form of over-fitting. Videos longer than the training resolution are cropped randomly in time to the defined training length. If not stated otherwise, models are conditioned on the first frame during training, which is achieved by masking the loss corresponding to this frame. In preliminary experiments, this gave a minor improvement over computing the training loss across all frames. ",
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+    {
+        "type": "text",
+        "text": "Intrinsic Evaluation. Most results are reported as bits per dimension (bits/dim), the average negative $\\log _ { 2 }$ -probability assigned by the model per (RGB) channel, averaged across all pixels in the video. This corresponds directly to the loss optimized by the model. In all experiments, we condition (prime) on a specified number of initial frames. The log-probabilities corresponding to these frames are excluded from this average. ",
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+    {
+        "type": "text",
+        "text": "Extrinsic Evaluation. Prior work mainly reported results on the peak signal-to-noise ratio (PSNR) and mean-structural similarity (SSIM) metrics (Wang et al., 2004b). However, these metrics were developed for images and have serious flaws when applied to videos (Wang et al., $2 0 0 4 \\mathrm { a }$ ; Wang & Li, 2007; Zhang et al., 2018; Lee et al., 2018). Conceptually, PSNR has a strong preference for blurry videos as it is based on pixel-level mean squared error. Similarly, SSIM does not correlate well with perceptual quality either. For instance, variational autoencoders show very strong performance on this metric despite producing blurry videos (Lee et al., 2018). Hence, we focus on the Frechet Video ´ Distance (FVD), which was recently proposed by Unterthiner et al. (2018) as a qualitative metric sensitive to visual quality, temporal coherence and diversity of samples. This is the spatiotemporal counterpart to the Frechet Inception Distance (Heusel et al., 2017), replacing the ImageNet-trained ´ ",
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+    {
+        "type": "table",
+        "img_path": "images/745add292a8c724b3174cc28b1b7b8e961f03659a9ea69bf4a85570638b276f2.jpg",
+        "table_caption": [
+            "Table 1: Quantitative results on BAIR Robot Pushing (left) and Kinetics (right). "
+        ],
+        "table_footnote": [
+            "(a) BAIR Robot Pushing. Bits/dim averaged across 15 subsequent frames when priming with 1 initial frame, FVD and unrolled average FVD scores. Best results in bold. $^ { \\dagger }$ Results from Unterthiner et al. (2018). ‡ Results are not strictly comparable (see text for details). [1] Babaeizadeh et al. (2018), [2] Lee et al. (2018), [3] Kumar et al. (2019). "
+        ],
+        "table_body": "<table><tr><td>Models</td><td>Bits/dim</td><td>FVD</td><td>FVD (Avg)</td></tr><tr><td>Single Frame</td><td>1.49</td><td>104±4</td><td>99±2</td></tr><tr><td>Spatial Sub.</td><td>1.57</td><td>111±4</td><td>108±1</td></tr><tr><td>Spatiotemp. Sub.</td><td>1.53</td><td>106±3</td><td>106±2</td></tr><tr><td>Spatiotemp. Sub. (L)</td><td>1.35</td><td>94±2</td><td>96±2</td></tr><tr><td>SV2P[1]+</td><td>1</td><td>263</td><td>1</td></tr><tr><td>SAVP [2]t</td><td>1</td><td>116t</td><td></td></tr><tr><td>VideoFlow [3]</td><td>1.87</td><td>1</td><td></td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/31287c21cccf7bcc50ec19f38dc9df70e1d5d6fc768cbac1642ebe4be7c107c8.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Kinetics. Bits/dim averaged across 15 subsequent frames when priming with 1 initial frame, FVD and unrolled average FVD scores when priming with 5 frames. Best results in bold. "
+        ],
+        "table_body": "<table><tr><td>Models</td><td>Bits/dim</td><td>FVD</td><td>FVD (Avg)</td></tr><tr><td>Single Frame</td><td>1.40</td><td>243±6</td><td>413±11</td></tr><tr><td>Spatial Sub.</td><td>1.47</td><td>263±6</td><td>450±15</td></tr><tr><td>Spatiotemp. Sub.</td><td>1.49</td><td>195±7</td><td>375±11</td></tr><tr><td>Single frame (L)</td><td>1.14</td><td>207±8</td><td>353±13</td></tr><tr><td>Spatiotemp. Sub. (L)</td><td>1.19</td><td>170±5</td><td>316±12</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Inception network of the latter with an I3D Network trained on Kinetics. Despite sharing the known drawbacks of FID (Binkowski et al., 2018), FVD has shown to correlate much stronger with human ´ raters compared to both PSNR and SSIM (Unterthiner et al., 2018). We report the FVD of the first 16 frames, as well as the “unrolled” average FVD across all contiguous subsequences of 16 frames. In each case, we report the mean and standard deviation of 20 trials. ",
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+    {
+        "type": "text",
+        "text": "Sampling time. Sampling from autoregressive models is notoriously slow. However, because our decoders are not very deep (8 layers) we are able to sample a batch of four 30x64x64 videos in acceptable time (approx. 8 minutes) with our large models on a Nvidia Tesla V100. Though this might still be impractical we argue that further advances in parallel sampling strategies (Stern et al., 2018) and future hardware will alleviate this disadvantage significantly. ",
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+    {
+        "type": "text",
+        "text": "4.2 BAIR ROBOT PUSHING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "BAIR Robot Pushing (Ebert et al., 2017) shows a robotic arm pushing and grasping objects in a box. It consists of roughly 40K training- and 256 test videos. We prime on the first frame for training and evaluation. ",
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+    {
+        "type": "text",
+        "text": "Empirical Results. All variants of the Video Transformer achieve strong results compared to prior work in terms of both intrinsic and extrinsic metrics. From Table 1a, we see that the small models already reduce the perplexity in terms of bits/dim by almost $20 \\%$ compared to the recently proposed VideoFlow model (Kumar et al., 2019) with our large model (L) reducing perplexity even further to a $2 5 \\%$ improvement. Similar to Menick & Kalchbrenner (2019), we find that subscaling can have a slightly negative effect on bits/dim. In terms of perceptual quality, every incarnation of our model obtains a lower (better) FVD score compared to all models evaluated by Unterthiner et al. (2018), which notably includes adversarial networks with no guarantees of covering the full empirical distribution. These results are not strictly comparable, since prior work has used longer priming sequences of two (Babaeizadeh et al., 2018; Lee et al., 2018) or three (Kumar et al., 2019) frames, whereas our models (to our disadvantage) see a single prime frame. Note that we sample with temperature 0.9 for the extrinsic metrics as we observed improved qualitative results at this temperature on the validation set. This corresponds to a mild form of mode dropping and is common practice to improve sampling quality. For fair comparison we also tweaked the “temperature” of SAVP by scaling the variance of its normal distribution when sampling. This, however, did not result in any improvements for FVD. ",
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+    {
+        "type": "text",
+        "text": "Further results on an earlier version of robot pushing (Finn et al., 2016a) and Moving MNIST (Srivastava et al., 2015) can be found in Appendix A for brevity. In summary, like Kalchbrenner et al. (2016), we match the lower bound on Moving MNIST while obtaining an almost $50 \\%$ reduction in bits/dim on robotic pushing which demonstrates the superiority of our models against prior work on autoregressive video modeling. ",
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+    {
+        "type": "text",
+        "text": "Qualitative Observations. All variants of our model reach similar quantitative results on these benchmarks and we observe no immediate differences in fidelity. However, there are some notable differences. First, whereas the spatiotemporal subscaling model is able to capture temporal dependencies across up to 16 frames (given subscaling in time by a factor four), the remaining models can only capture dependencies across four frames. This can, for example, result in deformation of occluded objects (e.g., Figure 4 of the Appendix). However, due to the simplicity of the benchmark datasets, this is not appropriately reflected in the metrics including better unrolled FVD curves for the single frame base model in Figure 2a. Second, we observe that lowering the sampling temperature from 1.0 to 0.9 consistently improves results. Notably, spatiotemporal subscaling seems more robust to sampling errors as its performance decays less when sampling with temperature 1.0 ( $1 2 2 { \\pm } 4$ Avg. FVD) compared to the spatial subscaling $( 1 3 4 \\pm 4 )$ and single frame models $( 1 5 3 \\pm 7 )$ . We attribute this finding to the difference in generation order when spatiotemporal subscaling is employed as it predicts pixels over the entire extend of the 3D video volume early and thereby effectively anchors future predictions around these pixels. Finally, considering that our results on BAIR Robot Pushing in terms of FVD are on par with those between two ground-truth subsamples (Figure 4 of Unterthiner et al. (2018)), we may be approaching the limit of this benchmark. On the other hand, it could be that FVD suffers out-of-domain and is not sufficiently sensitive to longrange temporal dynamics, since it is trained to perform human action recognition, which is known to predominantly rely on local features (Carreira & Zisserman, 2017; Xie et al., 2018). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/dfa6ccf1b97621553902141f4626b3162f2ec6245a35eafcceb80526bf719de8.jpg",
+        "image_caption": [
+            "Figure 2: Unrolled FVD metrics on BAIR Robot Pushing (left) and Kinetics (right). "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.3 KINETICS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Moving from a constrained to a real world setting, we next apply our models to the Kinetics dataset (Kay et al., 2017), a large scale action-recognition dataset consisting of YouTube videos. Specifically, we use Kinetics-600, which contains roughly 400K training videos ranging over 600 action classes (Carreira et al., 2018). We center-crop and down-sample each frame to 64x64 with a width-3 Lanczos filter and anti-aliasing. ",
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+    {
+        "type": "text",
+        "text": "We introduce a slight change to our setup by using a separate decoder for the first slice $\\pmb { x } _ { ( 0 , 0 , 0 ) }$ . This decoder can be twice as deep (16 instead of 8 layers) as the original subscale decoder, because it does not rely on any encoder. For all other slices we train a regular subscale model (8 layers in both encoder and decoder) as before. Using a separate first-slice decoder means that there is no wasted encoder computation on the first slice and that there are additional parameters. Furthermore, for our large models we scale the batch size to 256 by training in parallel on 128 TPU v3 instances for 1M steps. ",
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+    {
+        "type": "text",
+        "text": "Empirical Results. Results for our base models are shown in the upper part of Table 1b. In line with results on BAIR pushing, we find that the single frame model obtains better performance in terms of bits/dim. In contrast, we observe that the spatiotemporal subscaling model generates better and more robust video continuations which is reflected by its superior FVD scores. Our large models (L) show much stronger performance across the board (see lower half of Table 1b and Figures 2b), lowering the perplexity to 1.14 bits/dim for the single frame model. While the spatiotemporal subscaling model obtains slightly worse perplexity of 1.19 bits/dim, it improves FVD to 170. Despite its good performance on bits/dim, even with a temperature of 0.9, samples from the large single frame model are prone to instability and in many cases we observe color “explosions” (Figure 12 in the Appendix shows an example) which is reflected in its significantly higher FVD score. Although much less pronounced we observed such instability already when sampling with temperature 1.0 on BAIR pushing which clearly indicates the benefits of temporal subscaling for video generation. ",
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+        "type": "image",
+        "img_path": "images/085bae087f32ce532cec998a0c41eea2a6dce29429281a174e5f535139017b79.jpg",
+        "image_caption": [
+            "Figure 3: Selected Kinetics continuations from a set of 128 videos and 16 samples which showcase a variety of natural, video-specific phenomena our model learns to generate. We used our large spatiotemporal subscaling model and prime generation with 5 frames (0-4) to include the first two frames in subscale order (0, 4). Samples are generated with temperature of 0.9. The examples depict frames 0, 5, 10 and 15. "
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Qualitative Observations. Figure 3 shows samples from a cooking subset of Kinetics that we describe in Appendix E. These are selected to showcase different aspects of real-world videos learned by the large spatiotemporal subscaling model. Figures 3a and 3c demonstrate the model’s ability to handle camera movement. We find that camera movement seems to be learned early in training, possibly since it is a major source of uncertainty. This requires transforming pixels correctly while hallucinating new pixels at the edges. Similarly, object movement resulting, for instance, in a change of perspective is predicted quite well (Figure 3b). Highly stochastic motion such as fire (Figure 3d) or steam is modeled surprisingly well. Videos in Kinetics sometimes contain scene changes and our model, too, occasionally generates videos with jumps to completely new scenes (Figure 3e). Motion of human fingers and faces seems challenging to model. Nevertheless, in a number of samples the model is able to generate somewhat believable continuations as can be seen in Figures 3g, 3h or 3i. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "These selected examples show only a small subset of the interesting phenomena handled by the model and illustrate the sheer complexity involved in modeling this dataset. In Appendix F, we provide multiple samples, primed with the same initial frames to illustrate the diversity of the generated samples. ",
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+    {
+        "type": "text",
+        "text": "Limitations. While we obtained the occasional encouraging sampl, we would like to point out that the diversity of Kinetics still poses a major challenge. Failure modes range from freezing movement or object distortions to continuations that ”wash out” entirely after a few frames. We firmly beleive that yet larger datasets and/or models will be required to capture the complexity of even short clips from YouTube videos. With this work we merely provide an initial baseline, hoping to highlight both the potential and the enormous room for improvement. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We presented an autoregressive model of videos based almost entirely on a variant of block-local self-attention that can easily be implemented efficiently on TPUs. Combined with spatiotemporal subscaling, our models can be scaled up substantially while retaining the ability to capture longer range spatiotemporal dependencies. ",
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+    },
+    {
+        "type": "text",
+        "text": "Empirically, we obtain state-of-the-art results across a range of video generation benchmarks, while the scalability of our approach enables us to make an initial attempt at modeling videos of unusually high complexity and diversity as found in the Kinetics dataset. Our models occasionally generate encouraging continuations, especially on a subset of cooking videos, yet we find modeling the full range of such videos clearly remains a major challenge. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This work benefited from numerous conversations with Nal Kalchbrenner, as well as discussions with Jacob Menick, Mohammad Taghi Saffar and Niki Parmar. We would also like to thank Chelsea Finn and Tom Kwiatkowski for thoughtful comments on an earlier draft. ",
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+        "img_path": "images/f85a708c118abdfbf1adf3d921f784deeaf29204e187183c06887a4dc761f6c6.jpg",
+        "image_caption": [
+            "Figure 4: Samples (showing every 5th frame horizontally) illustrating occlusion effects on BAIR Robot Pushing. Models without temporal subscaling (rows 3-4) fail on occlusions, whereas the model with temporal subscaling (row 2) correctly maintains objects from the ground truth video (row 1). Notice the green ball deformation on rows 2 and 3 and the hallucinated green ball on the right edge of row 3, which are caused by missing temporal dependencies across the duration of occlusion. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/5bab4680d76bf4fc559dea41d0793b6c72186458b9fd97e3d2bf9bba890d1c49.jpg",
+        "table_caption": [
+            "Table 2: Moving MNIST. Nats per frame averaged across 10 subsequent frames when priming with 10 initial frames. Best results in bold. † The lower bound reported in (Kalchbrenner et al., 2016) is slightly higher than ours. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>Nats/Frame (↓)</td></tr><tr><td>Single Frame</td><td>86.2</td></tr><tr><td>Spatial Subscaling</td><td>91.8</td></tr><tr><td>Spatiotemporal Subscaling</td><td>90.0</td></tr><tr><td>VPN (Kalchbrenner et al., 2016)</td><td>87.6</td></tr><tr><td>Lower bound</td><td>85.1 (86.3)‡</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A FURTHER BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            694,
+            413,
+            710
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 MOVING MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            727,
+            338,
+            742
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Moving MNIST (Srivastava et al., 2015) consists of $1 0 0 \\mathrm { K }$ training- and 10K validation/test videos of two handwritten digits from the MNIST benchmark that move deterministically across the frame, crossing each other and bouncing off the borders. The partial occlusion of crossing digits makes this dataset challenging. To be comparable with Kalchbrenner et al. (2016), we use the first ten frames as priming and predict the subsequent ten frames. ",
+        "bbox": [
+            174,
+            755,
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+            825
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To allow direct comparison with Kalchbrenner et al. (2016), we change our loss to a “deterministic” loss (and derived nats-per-frame metric) which is defined as: $\\begin{array} { r } { H ( \\bar { z } , y ) = - \\sum _ { i } z _ { i } \\ln y _ { i } + ( 1 - } \\end{array}$ $z _ { i } ) \\ln ( 1 - y _ { i } )$ , where $z _ { i }$ are the gray-scale targets between 0.0 and 1.0, and $y _ { i }$ are the predicted scalar intensities. ",
+        "bbox": [
+            174,
+            832,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "From Table 2, we find that like Kalchbrenner et al. (2016) our single frame prediction model (i.e., no subscaling) virtually solves the task in the sense that it almost matches the lower bound of the loss. However, this is not true for our subscaling models. Employing spatial subscaling on this task gives aliasing artifacts that make it harder to predict future frames. Although this finding is limited to Moving MNIST, it suggests that spatial subscaling can potentially hurt generation. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/7185609f1593fac483102e2a9a27e792a671ca781cb3b386a43b0f4252de8dae.jpg",
+        "table_caption": [
+            "Table 3: Ablation of hyper-parameter settings in terms of bits per dimension for models on 256 BAIR Robot Pushing validation videos. All models were primed on 1 frame and trained for 300K steps with a batch size of 64. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Layers</td><td>Heads</td><td>Hidden size 1.65</td></tr><tr><td>4</td><td colspan=\"3\">1.63 4 1.59 256</td></tr><tr><td>8</td><td colspan=\"3\">1.55 8 1.55 512</td></tr><tr><td>16</td><td colspan=\"3\">1.55 1.4816 1.51 1024 1.47</td></tr><tr><td>24</td><td colspan=\"3\">1.4524 1.47 2048 1.40</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A.2 ROBOTIC PUSHING. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            325,
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+            339
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Robotic Pushing (Finn et al., 2016a) was used in prior work on autoregressive video generation (Kalchbrenner et al., 2016). The videos show a robotic arm pushing and grasping objects in a box and there are roughly 50K training videos and 1500 test videos with seen and novel objects, respectively. Following prior work, we use the initial two frames for priming and condition on the robot arm action for each frame as described in Section 3.2. We use the same setup as (Kalchbrenner et al., 2016) with videos of twenty frames down-sampled to 64x64 with a Lanczos filter and antialiasing. ",
+        "bbox": [
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+            352,
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We report results to compare with prior work on autoregressive video generation by Kalchbrenner et al. (2016), who achieve 0.92 bits/dim (0.64 nats/dim) with 2 frames of priming on each of the test splits (one with objects seen during training and one with novel objects). We trained a large (2048 dimensional) spatiotemporal subscaling model which achieves 0.51 bits/dim on the subset with seen objects and 0.47 bits/dim on the subset with new objects, which corresponds to an almost $50 \\%$ reduction in perplexity. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B HYPER-PARAMETER SWEEPS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            560,
+            450,
+            577
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 3 shows the impact of different architectural settings. We see that the hidden size has the biggest impact followed by the number of layers and heads. This is an interesting as well as important finding because increasing the hidden size (wider networks) requires more parallel compute which modern Deep Learning hardware excels at. Computation time grows sub-linear, memory linear and parameters partially quadratically. In contrast all of these aspects grow linearly with deep networks. For scaling up architectures depth is therefore not the preferred option as we suffer much more in terms of computation time while having less parameters. ",
+        "bbox": [
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+            592,
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In another experiment, we shuffle the arrangement of block sizes between layers and found that it did not really matter, that is, all results were within 0.01 bits/dim. However, our setup had the best overall performance. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Finally, we tried sampling temperature 0.9 and 1.0 only on the BAIR Robot Pushing validation set and found that temperature 0.9 consistently gave more robust predictions and better results on all extrinsic metrics. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C CONNECTIVITY IN BLOCK-LOCAL SELF-ATTENTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            824
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Blind Spots. Varying block sizes between layers in block-local self-attention can efficiently connect every pixel with every other pixel when no masking is employed. If masking is employed to respect the generation order (as in our slice decoder) block-local self attention produces “blind spots” which leads to independence assumptions. To exemplify these special cases, consider position $( 1 , 0 , 0 )$ , the top-left pixel of the second frame, and its direct predecessor in generation order $( 0 , h - 1 , w - 1 )$ , the bottom-right pixel of the first frame. The only way to establish a connection between these two positions is through a direct connection, because masking prevents any indirect connection. Thus, there has to be one layer in which both of these pixels are in the same block. This block must at least stretch over the entire extent of both width and height (i.e., the full frame) as well as at least 2 time steps. Running full self-attention in such blocks can easily become prohibitive for large $h$ and $w$ . ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Remedies. There seems to be no simple solution that solves the problem of blind spots completely. However, we can make sure that local dependencies up to a certain distance are all covered by increasing the kernel size of the initial, masked convolution in the decoder. It is also possible to combine block-local self-attention with its dual form, dilated self-attention in $n$ dimensions which connects all pixels at the same relative position within their respective block with each other. Finally, we find that it is important to avoid blocks of small sizes in any dimension (e.g., 1). That means, even if we stretch a block to the full extent of one dimension it is important to define sizes at least larger than 1 on all other dimensions to limit the number of unconnected pixels. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, the independence assumptions due to masking do not seem to produce any systematic, visible artifacts in our samples. We believe this to be an interesting finding by itself as it shows that there is potential for parallelizing autoregressive video generation by systematically exploring further independence assumptions. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D ADDITIONAL FINDINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
+            406,
+            398
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Below, we summarize some additional findings that may be of interest to some readers: ",
+        "bbox": [
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+            414,
+            743,
+            429
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "• We found that using blocks stretching across a single time-/row-/column- dimension, is substantially worse than using blocks that stretch at least to some extent in all directions. This is likely due to the fact that future masking in the decoder imposes strong independence assumptions in this case, as discussed in Appendix C. We found that RMSProp with momentum converges significantly faster than ADAM, which we tried with different learning rates and settings for $\\beta _ { 1 }$ and $\\beta _ { 2 }$ .   \n• We tried using continuous, rather than discretized one-hot, input channel representations, but this had an overall negative impact on both performance and sample quality.   \n• We experimented with a gating mechanism in Eq. 3, such that the attention matrix $A$ is masked elementwise with $( 1 - I )$ to allow for not attending to any element, similar to sentinel attention (Lu et al., 2017). However, this had no effect on generation quality. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E KINETICS COOKING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            626,
+            377,
+            643
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We found that for many video-prefixes in Kinetics it is very hard for our model to predict continuations. For instance, main objects in the videos are too small or movement is too fast which results in very blurry frames or there is little to no movement at all. Figure 13 shows some examples. Therefore, we created a subset of cooking videos that we found to exhibit these problems to a lesser degree. ",
+        "bbox": [
+            174,
+            657,
+            825,
+            728
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In particular we filtered videos whose label matched the following regular expression: ",
+        "bbox": [
+            174,
+            734,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": ". $\\star$ (baking|barbequing|breading|cooking|cutting|pancake|vegetables| meat|cake|sandwich|pizza|sushi|tea|peeling|fruit|eggs|salad).\\* ",
+        "bbox": [
+            176,
+            762,
+            813,
+            790
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Note that we still train on the full Kinetics training set and only use the cooking set to showcase samples in some cases. ",
+        "bbox": [
+            171,
+            801,
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+            830
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F SAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            851,
+            285,
+            866
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figures 5-8 show samples from our spatiotemporal subscaling and large spatiotemporal subscaling models on BAIR Robot Pushing. Figures 5 and 6 illustrate the fidelity and realism of the generated samples, whereas Figures 7 and 8 illustrate the diversity of samples. ",
+        "bbox": [
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+            882,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figures 9-11 show samples from our spatiotemporal subscaling model on cooking videos for Kinetics-600, while Figure 12 depicts samples from the single frame model. In each case, we prime on 5 frames and sample the next 11 frames. Each figure shows 16 different samples from the same model. As can be seen, the model is able to generate diverse continuations while retaining fidelity. For the single frame model we observe strange color artifacts (exploding colors) which we attribute to the standard, raster-scan generation order of this model. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/1da4058c9504b3302e241acbb9fbea6a96582f105e85fc5a50ad10b08fa72b4a.jpg",
+        "image_caption": [
+            "Figure 5: Samples of 30 future frames (showing every 4th frame) for 12 test videos with the spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/61fed3cceb926177810e03032d2edf0e83a5ee6ebe8688b199292174070ad8fc.jpg",
+        "image_caption": [
+            "Figure 6: Samples of 30 future frames (showing every 4th frame) for 12 test videos with the large spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 17
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+    {
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+        "img_path": "images/d30b988f5520416ac050282af99f11ea32ad7242a3646a76733791afa802c50a.jpg",
+        "image_caption": [
+            "Figure 7: 11 samples of 30 future frames (showing every 4th frame) for 1 test video (top row) with the spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 18
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+    {
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+        "img_path": "images/2c3ad68584652166a78279a77193cea938e43256e30fb61d5ffdbc6238ddac65.jpg",
+        "image_caption": [
+            "Figure 8: 11 samples of 30 future frames (showing every 4th frame) for 1 test video (top row) with the large spatiotemporal subscaling model, using 1 prime frame and temperature 0.9 on BAIR Robot Pushing. "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 9: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 10: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/c249bb000473202c4ddc516ad52809952700dd5c2c992681504816711af4c5f2.jpg",
+        "image_caption": [
+            "Figure 11: Samples of 11 future frames from the spatiotemporal subscaling model with 5 prime frames on 64x64 Kinetics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/4a130c065bd77737c167189965063fff68ac5665578b82002f122e4c571d2845.jpg",
+        "image_caption": [
+            "Figure 12: Samples of 11 future frames from the single frame model with 5 prime frames on 64x64 Kinetics exhibiting strange color artifacts. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "image_caption": [
+            "Figure 13: Ground-truth (top) and 2 samples of 30 future frames (showing every 4th frame) demonstrating that random Kinetics videos do not always lend themselves as good prefixes for generating continuations. "
+        ],
+        "image_footnote": [],
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+]

parse/train/rkgHY0NYwr/rkgHY0NYwr.md

@@ -0,0 +1,260 @@
+# DISCOVERING MOTOR PROGRAMS BY RECOMPOSING DEMONSTRATIONS
+
+Tanmay Shankar Facebook AI Research tanmayshankar@fb.com
+
+Shubham Tulsiani Facebook AI Research shubhtuls@fb.com
+
+Lerrel Pinto Robotics Institute, CMU lerrelp@cs.cmu.edu
+
+Abhinav Gupta Facebook AI Research gabhinav@fb.com
+
+# ABSTRACT
+
+In this paper, we present an approach to learn recomposable motor primitives across large-scale and diverse manipulation demonstrations. Current approaches to decomposing demonstrations into primitives often assume manually defined primitives and bypass the difficulty of discovering these primitives. On the other hand, approaches in primitive discovery put restrictive assumptions on the complexity of a primitive, which limit applicability to narrow tasks. Our approach attempts to circumvent these challenges by jointly learning both the underlying motor primitives and recomposing these primitives to form the original demonstration. Through constraints on both the parsimony of primitive decomposition and the simplicity of a given primitive, we are able to learn a diverse set of motor primitives, as well as a coherent latent representation for these primitives. We demonstrate, both qualitatively and quantitatively, that our learned primitives capture semantically meaningful aspects of a demonstration. This allows us to compose these primitives in a hierarchical reinforcement learning setup to efficiently solve robotic manipulation tasks like reaching and pushing.
+
+# 1 INTRODUCTION
+
+We have seen impressive progress over the recent years in learning based approaches to perform a plethora of manipulation tasks (Levine et al., 2016; Andrychowicz et al., 2018; Levine et al., 2018; Pinto & Gupta, 2016; Agrawal et al., 2016). However, these systems are typically task-centric savants – able to only execute a single task that they were trained for. This is because these systems, whether leveraging demonstrations or environmental rewards, attempt to learn each task tabula rasa, where low to high level motor behaviours, are all acquired from scratch in context of the specified task. In contrast, we humans are adept at a variety of basic manipulation skills e.g. picking, pushing, grasping etc., and can effortlessly perform these diverse tasks via a unified manipulation system.
+
+![](images/b5feeea84c9869e3d3f573006e584d2216ccc197dabf598ae72c0fa007cf45c7.jpg)  
+Figure 1: Sample motor programs that emerge by discovering the space of motor programs from a diverse set of robot demonstration data in an unsupervised manner. These motor programs facilitate understanding the commonalities across various demonstrations, and accelerate learning for downstream tasks.
+
+How can we step-away from the paradigm of learning task-centric savants, and move towards building similar unified manipulation systems? We can begin by not treating these tasks independently, but via instead exploiting the commonalities across them. One such commonality relates to the primitive actions executed to accomplish the tasks – while the high-level semantics of tasks may differ significantly, the low and mid-level motor programs across them are often shared e.g. to either pick or push an object, one must move the hand towards it. This concept of motor programs can be traced back to the work of Lashley, who noted that human motor movements consist of ‘orderly sequences’ that are not simply sequences of stimulus-response patterns. The term ‘motor programs’ is however better attributed to Keele (1968) as being representative of ‘muscle commands that execute a movement sequence uninfluenced by peripheral feedback’, though later works shifted the focus from muscle commands to the movement itself, while allowing for some feedback (Adams, 1971). More directly relevant to our motivation is Schmidt’s notion of ‘generalized’ motor programs (Schmidt, 1975) that can allow abstracting a class of movement patterns instead of a singular one. In this work, we present an approach to discover the shared space of (generalized) motor programs underlying a variety of tasks, and show that elements from this space can be composed to accomplish diverse tasks. Not only does this allow understanding the commonalities and shared structure across diverse skills, the discovered space of motor programs can provide a high-level abstraction using which new skills can be acquired quickly by simply learning the set of desired motor programs to compose.
+
+We are not the first to advocate the use of such mid-level primitives for efficient learning or generalization, and there have been several reincarnations of this idea over the decades, from ‘operators’ in the classical STRIPS algorithm (Fikes & Nilsson, 1971), to ‘options’ (Sutton et al., 1999) or ‘primitives’ (Schaal et al., 2005) in modern usage. These previous approaches however assume a set of manually defined/programmed primitives and therefore bypass the difficulty of discovering them. While some attempts have been made to simultaneously learn the desired skill and the underlying primitives, learning both from scratch is difficult, and are therefore restricted to narrow tasks. Towards overcoming this difficulty, we observe that instead of learning the primitives from scratch in the context of a specific task, we can instead discover them using demonstrations of a diverse set of tasks. Concretely, by leveraging demonstrations for different skills e.g. pouring, grasping, opening etc., we discover the motor programs (or movement primitives) that occur across these.
+
+We present an approach to discover movement primitives from a set of unstructured robot demonstration i.e. demonstrations without additional parsing or segmentation labels available. This is a challenging task as each demonstration is composed of a varying number of unknown primitives, and therefore the process of learning entails both, learning the space of primitives as well as understanding the available demonstrations in context of these. Our approach is based on the insight that an abstraction of a demonstrations into a sequence of motor programs or primitives, each of which correspond to an implied movement sequence, and must yield back the demonstration when the inferred primitives are ‘recomposed’. We build on this and formulate an unsupervised approach to jointly learn the space of movement primitives, as well as a parsing of the available demonstrations into a high-level sequence of these primitives.
+
+We demonstrate that our method allows us to learn a primitive space that captures the shared motions required across diverse skills, and that these motor programs can be adapted and composed to further perform specific tasks. Furthermore, we show that these motor programs are semantically meaningful, and can be recombined to solved robotic tasks using reinforcement learning. Specifically, solving reaching and pushing tasks with reinforcement learning over the space of primitives achieves 2 orders of magnitude faster training than reinforcement learning in the low-level control space.
+
+# 2 RELATED WORK
+
+Our work is broadly related to several different lines of work which either learn task policies from demonstrations, or leverage known primitives for various applications, or learn primitives in context of known segments. While we discuss these relations in more detail below, we note that previous primitive based approaches either require: a) a known and fixed primitive space, or b) annotated segments each corresponding to a primitive. In contrast, we learn the space of primitives without requiring this segmentation annotation, and would like to emphasize that ours is the first work to do so for a diverse set of demonstrations spanning multiple tasks.
+
+Learning from Demonstration: The field of learning from demonstrations (LfD) (Nicolescu & Mataric, 2003) has sought to learn to perform tasks from a set of demonstrated behaviors. A number of techniques exist to do so, including cloning the demonstrated behavior (Esmaili et al., 1995), fitting a parametric model to the demonstrations (Kober & Peters, 2009; Peters et al., 2013), or first segmenting the demonstrations and fitting a model to each of the resultant segments (Niekum et al., 2012; Krishnan et al., 2018; Murali et al., 2016; Meier et al., 2011). We point the reader to Argall et al. (2009) for a comprehensive overview of the field. Rather than directly learn to perform tasks, we use demonstrations to learn a diverse set of composable and reusable primitives, or motor-programs that may be used to perform a variety of downstream tasks.
+
+Learning and Sequencing Motion Primitives: Several works (Kober & Peters, 2009; Peters et al., 2013) learn motion primitives from demonstrations using predetermined representations of skills such as Dynamic Movement Primitives (DMPs) (Schaal et al., 2005). A few other works have approached the problem from an optimization perspective (Dragan et al., 2015). Given these primitives, a question that arises is how to then sequence learned skills to perform downstream tasks. Several works have attempted to answer this question - (Neumann et al., 2014) builds a layered approach to adapt, select, and sequence DMPs. Konidaris et al. (2012); Konidaris & Barto (2009) segments demonstrations into sequences of skills, and also merge these skills into skill-trees. However, the predetermined representations of primitives adopted in these works can prove restrictive. In particular, it prevents learning arbitrarily expressive motions and adapting these motions to a generic downstream task. We seek to move away from these fixed representations of primitives, and instead learn representations of primitives along with the primitives themselves.
+
+Latent Variable Models: The family of latent variable models (LVMs) provides us with fitting machinery to do so. Indeed, the success of LVMs in learning representations in deep learning has inspired several recent works (Co-Reyes et al., 2018; Kipf et al., 2019; Haarnoja et al., 2018; Lynch et al., 2019) to learn latent representations of trajectories. SeCTAR (Co-Reyes et al., 2018) builds a latent variable conditioned policy and model that are constrained to be consistent with one another, and uses the learned policies and model for hierarchical reinforcement learning. The CompILE framework (Kipf et al., 2019) seeks to learn variable length trajectory segments from demonstrations instead, and uses latent variables to represent these trajectory segments, but is evaluated in relatively low-dimensional domains. We adopt a similar perspective to these works and learn continuous latent variable representations (or abstractions) of trajectory segments.
+
+Hierarchical RL and the Options Framework: The related field of hierarchical reinforcement learning (HRL) learns a layering of policies that each abstract away details of control of the policies of a lower level. The Options framework (Sutton et al., 1999) also learns similar temporal abstractions over sequences of atomic actions. While promising, its application has traditionally been restricted to simple domains due to the difficulty of jointly learning internal option policies along with a policy over those options. Recent works have managed to do so with only a reward function as feedback. The Option-Critic framework (Bacon et al., 2017) employs a policy-gradient formulation of options to do so, while the Option-Gradient (Smith et al., 2018) learns options in an off-policy manner. In contrast with most prior work in the options framework, (Daniel et al., 2016; Fox et al., 2017; Krishnan et al., 2017) learn options from a set of demonstrations, rather than in the RL setting. In similar spirit to these works, we too seek to learn abstractions from a given set of demonstrations, however unlike DDO (Fox et al., 2017), DDCO (Krishnan et al., 2017), and CompILE (Kipf et al., 2019), we can learn primitives beyond a discrete set of options in a relatively high dimensional domain.
+
+Hierarchical representations of demonstrations: The idea of hierarchical task representations has permeated into LfD as well. In contrast to reasoning about demonstrations in a flat manner, one may also infer the hierarchical structure of tasks performed in demonstrations. A few recent works have striven to do so, by representing these tasks as programs (Xu et al., 2018; Sun et al., 2018), or as task graphs (Huang et al., 2019). Both Xu et al. (2018) and Huang et al. (2019) address generalizing to new instances of manipulation tasks in the low-shot regime by abstracting away low-level controls.
+
+The idea of policy sketches, i.e. a sketch of the sub-tasks to be accomplished in a particular task, has become popular (Andreas et al., 2017; Shiarlis et al., 2018). Andreas et al. (2017) learn modular policies in the RL setting provided with such policy sketches. Shiarlis et al. (2018) provides a modular LfD framework based on this idea of policy sketches. While all of these works address learning policies at various levels from demonstrations, unlike our approach, they each assume access to heavy supervision over demonstrations to do so.
+
+![](images/9c3d5e5cdfc94f10327df91a3b7379269d256c40ccd7551f868ee4a85134bc2b.jpg)  
+Figure 2: An overview of our approach. Our abstraction network takes in an observed demonstration $\tau _ { o b s }$ and predict a sequence of latent variables $\{ z \}$ . These $\{ z \}$ are each decoded into their corresponding motor programs via the motor program network. We finally recompose these motor programs into the recomposed trajectory.
+
+# 3 APPROACH
+
+We seek to discover the space of motor programs directly from unstructured demonstrations in an unsupervised manner, and show that these can help understanding similarities across tasks as well as quickly quickly adapting to and solving new tasks. Building on ideas of Keele (1968) and Schmidt (1975), we define a motor program $M$ as a movement pattern that may be executed in and of itself, without access to sensory feedback. Concretely, a ‘movement sequence’ or a motor program $M$ is a sequence of robot joint configurations. Our goal is to learn the space of such movement patterns that are present across a diverse set of tasks. We do so via learning a ‘motor program network’ that maps elements $z \in R ^ { n }$ to corresponding movement sequences i.e. $\mathcal { M } : \boldsymbol { z } \longrightarrow M$ .
+
+Given a set of $N$ unlabelled demonstrations $\{ \tau _ { i } \} _ { i = 1 , 2 , \dots , N }$ that consist of sequences of robot states $\{ s _ { 1 } , s _ { 2 } , . . . , s _ { T } \} \in S$ , our aim is to learn the shared space of motor programs via the motor program network $\mathcal { M }$ . However, as each demonstration $\tau _ { i }$ is unannotated, we do not apriori know what subsequences of the demonstration correspond to a distinct motor program. Therefore, to learn motor programs from these demonstrations, we need to simultaneously learn to understand the demonstrated trajectories in terms of composition of motor programs. Thus in addition to learning the network $\mathcal { M }$ , we also learn a mapping $\mathcal { A }$ from each demonstrated trajectory $\tau _ { i }$ to the underlying sequence of motor programs $\left\{ M _ { 1 } , M _ { 2 } , . . . , M _ { K } \right\}$ (and associated latent variables $\{ z _ { 1 } , z _ { 2 } , . . . , z _ { K } \} )$ executed during the span of the trajectory. We call this mapping $\mathcal { A } : \tau _ { i } \longrightarrow \{ M _ { 1 } , M _ { 2 } , . . . , M _ { K } \}$ the abstraction network, as it abstracts away the details of the trajectory into the set of motor programs (i.e., abstractions). Note that both the abstraction and motor program networks are learned using only a set of demonstrations from across diverse tasks.
+
+# 3.1 PRIMITIVE DISCOVERY VIA RECOMPOSITION
+
+Our central insight is that we can jointly learn the space of motor programs and the abstraction of the demonstration by enforcing that the implied ‘recomposition’ is faithful to the original demonstration. Concretely, an abstraction of a demonstration into a sequence of motor programs, each of which corresponds to an implied motion sequence, must yield back the original demonstration when the inferred motor programs are decoded and ‘recomposed’. We operationalize this insight to jointly train the motor program and abstraction networks from demonstrations.
+
+Learning Overview and Objective: Our approach is outlined in Fig 2, where given an input demonstration trajectory $\tau _ { \mathrm { o b s } }$ , we use the abstraction network to predict a sequence of (a variable number of) latent codes $\left\{ z _ { k } \right\}$ . These are each decoded into corresponding sub-trajectories via the learned motor program network $\mathcal { M }$ .
+
+$$
+\{ z _ { k } \} = A ( \tau _ { o b s } ) ; ~ \bar { \tau } _ { k } = \mathcal { M } ( z _ { k } )
+$$
+
+Given the decoding of the predicted motor programs, we can recompose these sub-trajectories to obtain a recomposed demonstration trajectory $\tau _ { r e c }$ , and penalize the discrepancy between the observed and the recomposed demonstrations. Denoting by $\oplus$ the concatenation operator, our loss for a given demonstration $\tau _ { o b s }$ is therefore characterized as:
+
+$$
+\tau _ { r e c } = \bar { \tau } _ { 1 } \oplus \bar { \tau } _ { 2 } \cdot \cdot \cdot \oplus \bar { \tau } _ { K } ; L ( \tau _ { o b s } ; { \mathcal { M } } , A ) = \Delta ( \tau _ { o b s } , \tau _ { r e c } )
+$$
+
+As the trajectories $\tau _ { o b s }$ , $\tau _ { r e c }$ are possibly of different lengths, we use a pairwise matching cost between the two trajectories, where the optimal alignment is computed via dynamic time warping (Berndt &
+
+Clifford, 1994). This provides us with a more robust cost measure that handles different prediction lengths, is invariant to minor velocity perturbations, and enables the model to discard regions of inactivity in the demonstrations. Given two trajectories $\tau _ { a } \equiv ( s _ { 1 } \cdots s _ { M } )$ , $\tau _ { b } \equiv ( s _ { 1 } \cdots s _ { N } )$ , and a distance metric $\delta$ over the state space, the discrepancy measure between trajectories can be defined as the matching cost for the optimal matching path $P$ among all possible valid matching paths $\mathcal { P }$ (i.e. paths satisfying monotonicity, continuity, and boundary conditions (Berndt & Clifford, 1994)):
+
+$$
+\Delta ( \tau _ { a } , \tau _ { b } ) = \operatorname* { m i n } _ { P \in \mathcal { P } } \sum _ { ( m , n ) \in P } \delta ( \tau _ { a } [ m ] , \tau _ { b } [ n ] )
+$$
+
+As the recomposed trajectory comprises of distinct primitives, each of which implies a sequence of states, we sometimes observe discontinuities i.e. large state changes between the boundaries of these primitives. To prevent this, we additionally incorporate a smoothness loss $L _ { s m } ( \tau _ { r e c } )$ that penalizes the state change across consecutive time-steps if they are larger than a certain margin. Our overall objective, comprising of the reconstruction objective and the smoothness prior, can allow us to jointly learn the space of motor programs and the abstraction of trajectories in an unsupervised manner.
+
+Network Architecture and Implementation Details. We parameterize our motor program network $\mathcal { M }$ and our abstraction network $\mathcal { A }$ as neural networks. In particular, the motor program network is a 4 layer LSTM (Graves et al., 2013) that takes a single 64 dimensional latent variable $z$ as input, and predicts a sequence of 16 dimensional states. For our abstraction network, we adopt the Transformer (Vaswani et al., 2017) architecture to take in a varying length 16 dimensional continuous joint angle trajectory $\tau$ as input, and predict a variable number of latent variables $\{ z \}$ , that correspond to the sequence of motor programs $\{ M \}$ executed during trajectory $\tau$ . We find the transformer architecture to be superior to LSTMs for processing long trajectories, due to its capacity to attend to parts of the trajectory as required.
+
+Our abstraction network $\mathcal { A }$ predicts a varying number of primitives by additionally predicting a ‘continuation probability’ $p _ { k }$ after each motor program variable $z _ { k }$ . We then predict an additional primitive only if the sampled discrete variable from $p _ { k }$ is 1, and therefore also need to learn the prediction of these probabilities. While the loss function above can directly yield gradients to the predicted motor program encoding $z _ { k }$ via $\mathcal { M }$ , we use gradients using REINFORCE Williams (1992) (with $\Delta ( \tau _ { o b s } , \tau _ { r e c } ) + L _ { s m } ( \tau _ { r e c } )$ as negative reward) to learn prediction of $p _ { k }$ .
+
+# 3.2 ENFORCING SIMPLICITY AND PARSIMONY
+
+While the objective described so far can in principle allow us to jointly learn the space of motor programs and understand the demonstration trajectories as a composition of these, there are additional properties we would wish to enforce to the bias the learning towards more desirable solutions. As an example, our framework presented so far can allow a solution where each demonstration is a motor program by itself i.e. the abstraction network can learn to map each demonstration to a unique $z$ and the primitive decoder can then decode this back. However, this is not a suitable solution as the learned motor programs are not ‘simple’. On the other extreme, a very simple notion of a motor program is one that models each transition independently. However, this is again undesirable as this does not ‘abstract’ away the details of control or represent the demonstration as a smaller number of motor programs. Therefore, in addition to enforcing that the learned motor programs recompose the demonstrations, we also need to enforce simplicity and parsimony of these motor programs.
+
+We incorporate these additional biases by adding priors in the objective or model space. To encourage the abstraction model to learn parsimonious abstractions of the input demonstrations, we penalize the number of motor primitives used to recompose the trajectory, by adding a small constant to the negative reward used to train the continuation probability $p _ { k }$ if the corresponding sample yielded an additional primitive. To enforce simplicity of motor primitives, we observe that the trajectories yielded by a classical planner (in our case, RRT-Connect) e.g. to go from an initial to final state are ‘simple’ and we therefore initialize the motor primitive network using the decoder of a pretrained autoencoder on random planner trajectories for (start, goal) state pairs. Note that this notion of ‘plannability’ as ‘simplicity’ is merely one plausible alternative, and alternate ones can be explored e.g. ‘linearity’ (Kroemer et al., 2015) or ‘predictability of motion’ (Wolpert & Kawato, 1998).
+
+![](images/1b4e964f97c73037c0014220c6b8bf5c59c637b3bf9b1f78c15f681200dad406.jpg)  
+Figure 3: Visualization of the embedding of the latent representation of motor programs learned by our model (depicted on the left are the start configurations of each motor program, displayed at the corresponding position in the embedded space), and a set of sample primitives unrolled in time (depicted on the right). Each row corresponds to one primitive, annotated with semantic labels of what this motor primitive resembles.
+
+# 4 EXPERIMENTS
+
+We would like to ascertain how well our approach is capable of achieving our objective of successfully discovering and learning a good representation of the space of motor programs. Further, we seek to verify whether despite being learned in an unsupervised manner without semantic grounding, the learned primitive space is semantically meaningful. We would also like to evaluate how well they can be used to solve downstream RL tasks. We first provide a description of the data we wish to learn primitives from, followed by describing our quantitative and qualitative experiments towards verifying these three axes.
+
+Dataset: We use the MIME dataset (Sharma et al., 2018) to train and evaluate our model. The dataset consists of over 8000 kinesthetic demonstrations of 20 tasks (such as pouring, pushing, bottle opening, stacking objects, etc.) collected on a real-world Baxter Robot. While the dataset has head and hand-mounted RGBD data, we use the Baxter joint angle trajectories to train our model. We consider a 16 dimensional space as our input and prediction space, consisting of 7 joints for each of the 2 arms, along with a scalar value for each gripper (we ignore torso and head joints). We emphasize this is a higher dimensional domain than most other related works consider. The gripper values are re-scaled to a $0 - 1$ range, while the joint angles are unnormalized. We temporally down-sample all joint angle data by a constant factor of 20 from the original data frequency of $1 0 0 \mathrm { H z }$ .
+
+We randomly sample a train set of 5900 demonstrations from all 20 tasks, with a validation set of 1600 trajectories, and a held-out test set of 850 trajectories. To help evaluate the learned motor programs, we manually annotate a set of 60 test trajectories (3 trajectories from each task) with temporal segmentation annotations, as well as semantic labels of 10 primitives (such as reaching, twisting, pushing, etc.) that occur in these 60 trajectories. Note that these labels are purely for evaluation, our model does not have access to these annotations during training.
+
+# 4.1 VISUALIZING THE SPACE OF PRIMITIVES
+
+We would first like to evaluate the quality of the learned abstractions in and of themselves, i.e. Is our approach able to discover the space of motor programs, and learn a good representation of this space? We qualitatively answer this question by visualizing the latent representation space of motor primitives learned by our model. We first randomly sample a set of 500 trajectories unseen during training, then pass these trajectories through our model, and retrieve the predicted latent variables $\{ z \}$ for each of these trajectories and their corresponding movement sequences $\{ { \overline { { \tau } } } \}$ . We then embed the latent variables in a 2-dimensional space using T-SNE (van der Maaten & Hinton, 2008), and visualize the corresponding movement sequences at their corresponding position in this 2-D embedded space, as in Fig. 3. We provide a GIF version of Fig. 3 (and other visualizations) at https://sites.google.com/view/discovering-motor-programs/home.
+
+![](images/4160295329ebb5b2736f44a43130310e2f7d0abdf62aba383d1f7de0954fd3bd.jpg)  
+Figure 4: Depiction of execution of the learned primitives on real world Baxter robot. Each row is a single primitive, while columns show progress of the primitive over time. Row 1 depicts a left handed reaching primitive, while row 2 shows a right handed returning primitive. More visualizations provided in supplementary material, and videos are provided in the webpage.
+
+We observe that clusters of movement sequences emerge in this embedded space based on the relative motions executed during the course of these trajectory segments (similar latent variables correspond to similar movement sequences and vice versa). While these clusters are not explicitly semantically labelled by our approach, the motions in these clusters correlate highly with traditional notions of skills in robot manipulation, i.e. reaching motions (top and top-left clusters), returning motions (bottom cluster), bi-manual motions (visible to the bottom right of the left most cluster and the bottom of the right most cluster), etc. We visualize a few such primitives (reaching, twisting, grasping, bi-manual pulling, etc.) among these to the right of Fig. 3. This shows our model learns smooth mappings $\mathcal { M }$ and $\mathcal { A }$ , and is capable of discovering such primitives in an unsupervised manner without explicit temporal segmentation or semantic labels, which we believe is an encouraging result.
+
+Interestingly, the model learns abstractions that pick up on the trend of the motion, rather than distinguishing between whether the left or right hand is used for the motion. This is particularly notable in the case of reaching and returning motions, where both left and right-handed reaching and returning motions appear alongside each other in their respective clusters in the embedded space.
+
+# 4.1.1 EXECUTING PRIMITIVES ON A REAL ROBOT
+
+We would ideally like the learned primitives from our model to be useful on a real Baxter robot platform, and be suitably smooth, feasible, and correspond to the motions executed in simulation (i.e. be largely unaffected by the noise of execution on a real robot). To verify whether our model is indeed able to learn such primitives, we execute a small set of learned primitives on a real Baxter robot, by feeding in the trajectory predicted by the model into a simple position controller. We visualize the results of this execution in Fig. 4, (see project webpage for videos). Despite not explicitly optimizing for feasibility or transfer to a real robot, the use of real-world Baxter data to train our model heavily biases the model towards primitives that are inherently feasible, relatively smooth and can be executed on a real robot without any subsequent modifications.
+
+# 4.2 SEMANTIC SEGMENTATION TRANSFER USING LEARNED ABSTRACTIONS
+
+As the ‘recomposed’ trajectory can be aligned to the original demonstration via sequence alignment, our predicted abstractions induce a partitioning of the demonstrated trajectory (corresponding to the aligned boundaries of the predicted primitives). We test whether the predicted abstraction and induced partitions are consistent across different demonstrations from the same task. To this end, we select 3 instances of the “Drop Object” task (Sharma et al., 2018) from the annotated test set, and retrieve the induced segmentations of the demonstration. We then visualize these segmentations and motor programs predicted for each of these 3 demonstrations as depicted in Fig. 5, along with the ground truth semantic labels of primitives for each of the demonstrations.
+
+The alignment between a recomposed trajectory and the original demonstration also allows us to transfer semantic annotations from a demonstration to the predicted primitives, by simply copying labels from the demonstration to their aligned timepoints in the primitives. Therefore using our small set of annotated demonstrations, we can construct a small library of semantically annotated primitives. Given a novel, unseen demonstration, we can compute its predicted primitives and assign each a semantic label by copying the label of the nearest primitive from the library. This allows us to transfer semantic segmentations from our small annotated test set to unseen demonstrations. Our model’s transfer of semantic segmentation achieves label accuracy on the set of 30 held out trajectories (across all 20 tasks) of $58 \%$ , while a supervised LSTM baseline that predicts semantic labels from trajectories (trained on 30 annotated test trajectories) achieves $54 \%$ accuracy.
+
+![](images/d4e449ab823b48a4e05b1f6ad9d15c26b964461e271c2af26c7a74dc4e6bb47d.jpg)  
+Figure 5: Visualization of consistent segmentations. Each row represents a different instance of a “Drop Objects” task from the MIME Dataset, while each column represents a time-step in the demonstration. White frames represent predicted segmentation points, while colored boxes represent ground truth semantic annotations. Red boxes are reaching primitives, blue boxes are grasping, orange boxes are placing, and green boxes are returning primitives. We see our model predicts 4 motor programs - reaching, grasping, placing the object a small distance away, and returning. This is consistent with the true semantic annotations of these demonstrations, and the overall sequence of primitives expected of the “Drop Box” task.
+
+![](images/775e83c070560161058c437dc3dfea7186217837d18d04253770463cc9b0c65f.jpg)  
+Figure 6: RL training curves with and without motor programs. Solid lines denote mean success rate, while shaded region denotes $\pm 1$ standard deviation across 10 random seeds.
+
+The consistency of our abstractions coupled with the ability to transfer semantic segmentations across demonstrations shows our model is capable of understanding commonalities across demonstrations, and is able to reason about various demonstrations in terms of motor programs shared across them, despite being trained in an unsupervised manner.
+
+# 4.3 COMPOSING PRIMITIVES FOR HIERARCHICAL RL
+
+One of our primary motivations for learning motor programs is that they can be composed together to solve downstream robotic tasks. To evaluate whether the motor programs learned by our model are indeed useful for such downstream tasks, we adopt a hierarchical reinforcement learning setup (as described in detail in the supplementary). For a given task, we train a policy to predict the sequence of motor programs to execute. Given the predicted latent representations, each motor program is decoded into its corresponding motion sequence using the previously learned motor program network. We retrieve a sequence of desired joint velocities from this motion sequence and use a joint velocity controller to execute these “low-level” actions on the robot. The motor program network and the joint velocity controller together serve as an “abstraction” of the low-level control that is executed on the robot. Hence, the policy must learn to predict motor programs that correspond to motion sequences useful for solving the task at hand.
+
+As demonstrated in Fig. 6, training a policy using motor programs is several orders of magnitude more efficient than training with direct low-level actions. For the sparse reaching task, the motor program policy learns within 50 motor program queries, 2 orders of magnitude speedup in low-level control time-steps. For the sparse pushing task, the motor program policy learns within 1000 motor program queries, or a 2X speedup with respect to low-level control time-steps. We note that executing a motor program corresponds to 50 low level control steps (see appendix for details). However, as these motor programs are executed without environment feedback, the improvement in efficiency in terms of environment interactions is all the more significant.
+
+# 5 CONCLUSION
+
+We have presented an unsupervised approach to discover motor programs from a set of unstructured robot demonstrations. Through the insight that learned motor programs should recompose into the original demonstration while being simplistic, we discover a coherent and diverse latent space of primitives on the MIME (Sharma et al., 2018) dataset. We also observed that the learned primitives were semantically meaningful, and useful for efficiently learning downstream tasks in simulation. We hope that the contributions from our work enable learning and executing primitives in a plethora of real-world robotic tasks. It would also be interesting to leverage the learned motor programs in context of continual learning, to investigate how the discovered space can be adapted and expanded in context of novel robotic tasks.
+
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+# A APPENDIX
+
+# A.1 EXECUTING PRIMITIVES ON A REAL ROBOT
+
+We provide additional visualizations of primitives being executed on the real robot below. As mentioned in the main paper, dynamic GIFs of these visualizations may be found at https://sites.google.com/view/discovering-motor-programs/home.
+
+![](images/cab74fb8e8d904bb0f8dc85196e36e5ae7698f599953b3da2c77612c8b2206f9.jpg)  
+Figure 7: Depiction of execution of additional learned primitives on real world Baxter robot. As in Fig. 4, each row is a single primitive, while columns show progress of the primitive over time. Row 1 depicts a left handed returning primitive, row 2 depicts a right handed pushing primitive, row 3 depicts a left handed pushing primitive (in a different configuration to the left handed one), and finally row 4 depicts a right handed twisting primitive.
+
+Our results webpage https://sites.google.com/view/discovering-motor-programs/home also contains visualizations of predicted combinations of primitives being executed on the real robot (as well as in simulation), along with corresponding demonstrations for comparison. We note that both the individual primitives, as well as the combinations of primitives are executed quite smoothly and naturally, and lead to motions that correspond highly to the original demonstrations. This indicates such combinations of primitives can indeed be used towards downstream tasks on the real robot.
+
+A.2 DETAILS OF HIERARCHICAL REINFORCEMENT LEARNING SETUP
+
+For our hierarchical reinforcement learning experiments, we perform policy learning on two sparse reward (Andrychowicz et al., 2017) tasks on a simulated Baxter Robot: (a) Reaching, and (b) Pushing. For the reaching task, the robot’s end-effector needs to reach a specific location in space, while for the pushing task, the robot needs to push a block on the table to a specific location.
+
+For Baxter-Reaching task, the goal is to get the right-hand’s end-effector to a pre-defined goal (x,y,z) state. The reward is a sparse reward with epsilon $\scriptstyle 1 = 0 . 0 5 \mathrm { m }$ ; So if the end-effector reaches within $5 \mathrm { c m }$ of the goal, it gets a reward of $+ 1$ , otherwise it gets a reward of 0. For Baxter-Push, the goal is to get a block (cube) to the desired goal with epsilon $= 0 . 0 5 \mathrm { m }$ . To do this, the robot needs to hit/push the block to the goal. There is no other dense reward to encourage the robot to hit the block.
+
+We train both our motor program policy and the baseline control policy using Proximal Policy Optimization (Schulman et al., 2017). Note that the motor program policy outputs the latent representation $z$ . Each $z$ expands into a 10 length trajectory according the motor program network. To reach each of these trajectory states, a PD velocity controller is used for 5 time-steps. The baseline control policy directly outputs the velocity control action.
+
+We note that our PPO baseline implements the same action for 10 timesteps, in a manner similar to frame-skipping, as is common in RL. We found that varying the number of timesteps frame-skipping was applied for did not have a significant effect on the performance of PPO.

parse/train/rkgHY0NYwr/rkgHY0NYwr_content_list.json

@@ -0,0 +1,1449 @@
+[
+    {
+        "type": "text",
+        "text": "DISCOVERING MOTOR PROGRAMS BY RECOMPOSING DEMONSTRATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tanmay Shankar Facebook AI Research tanmayshankar@fb.com ",
+        "bbox": [
+            184,
+            170,
+            383,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shubham Tulsiani Facebook AI Research shubhtuls@fb.com ",
+        "bbox": [
+            516,
+            170,
+            676,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lerrel Pinto Robotics Institute, CMU lerrelp@cs.cmu.edu ",
+        "bbox": [
+            184,
+            233,
+            362,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abhinav Gupta Facebook AI Research gabhinav@fb.com ",
+        "bbox": [
+            521,
+            233,
+            671,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present an approach to learn recomposable motor primitives across large-scale and diverse manipulation demonstrations. Current approaches to decomposing demonstrations into primitives often assume manually defined primitives and bypass the difficulty of discovering these primitives. On the other hand, approaches in primitive discovery put restrictive assumptions on the complexity of a primitive, which limit applicability to narrow tasks. Our approach attempts to circumvent these challenges by jointly learning both the underlying motor primitives and recomposing these primitives to form the original demonstration. Through constraints on both the parsimony of primitive decomposition and the simplicity of a given primitive, we are able to learn a diverse set of motor primitives, as well as a coherent latent representation for these primitives. We demonstrate, both qualitatively and quantitatively, that our learned primitives capture semantically meaningful aspects of a demonstration. This allows us to compose these primitives in a hierarchical reinforcement learning setup to efficiently solve robotic manipulation tasks like reaching and pushing. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We have seen impressive progress over the recent years in learning based approaches to perform a plethora of manipulation tasks (Levine et al., 2016; Andrychowicz et al., 2018; Levine et al., 2018; Pinto & Gupta, 2016; Agrawal et al., 2016). However, these systems are typically task-centric savants – able to only execute a single task that they were trained for. This is because these systems, whether leveraging demonstrations or environmental rewards, attempt to learn each task tabula rasa, where low to high level motor behaviours, are all acquired from scratch in context of the specified task. In contrast, we humans are adept at a variety of basic manipulation skills e.g. picking, pushing, grasping etc., and can effortlessly perform these diverse tasks via a unified manipulation system. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b5feeea84c9869e3d3f573006e584d2216ccc197dabf598ae72c0fa007cf45c7.jpg",
+        "image_caption": [
+            "Figure 1: Sample motor programs that emerge by discovering the space of motor programs from a diverse set of robot demonstration data in an unsupervised manner. These motor programs facilitate understanding the commonalities across various demonstrations, and accelerate learning for downstream tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "How can we step-away from the paradigm of learning task-centric savants, and move towards building similar unified manipulation systems? We can begin by not treating these tasks independently, but via instead exploiting the commonalities across them. One such commonality relates to the primitive actions executed to accomplish the tasks – while the high-level semantics of tasks may differ significantly, the low and mid-level motor programs across them are often shared e.g. to either pick or push an object, one must move the hand towards it. This concept of motor programs can be traced back to the work of Lashley, who noted that human motor movements consist of ‘orderly sequences’ that are not simply sequences of stimulus-response patterns. The term ‘motor programs’ is however better attributed to Keele (1968) as being representative of ‘muscle commands that execute a movement sequence uninfluenced by peripheral feedback’, though later works shifted the focus from muscle commands to the movement itself, while allowing for some feedback (Adams, 1971). More directly relevant to our motivation is Schmidt’s notion of ‘generalized’ motor programs (Schmidt, 1975) that can allow abstracting a class of movement patterns instead of a singular one. In this work, we present an approach to discover the shared space of (generalized) motor programs underlying a variety of tasks, and show that elements from this space can be composed to accomplish diverse tasks. Not only does this allow understanding the commonalities and shared structure across diverse skills, the discovered space of motor programs can provide a high-level abstraction using which new skills can be acquired quickly by simply learning the set of desired motor programs to compose. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We are not the first to advocate the use of such mid-level primitives for efficient learning or generalization, and there have been several reincarnations of this idea over the decades, from ‘operators’ in the classical STRIPS algorithm (Fikes & Nilsson, 1971), to ‘options’ (Sutton et al., 1999) or ‘primitives’ (Schaal et al., 2005) in modern usage. These previous approaches however assume a set of manually defined/programmed primitives and therefore bypass the difficulty of discovering them. While some attempts have been made to simultaneously learn the desired skill and the underlying primitives, learning both from scratch is difficult, and are therefore restricted to narrow tasks. Towards overcoming this difficulty, we observe that instead of learning the primitives from scratch in the context of a specific task, we can instead discover them using demonstrations of a diverse set of tasks. Concretely, by leveraging demonstrations for different skills e.g. pouring, grasping, opening etc., we discover the motor programs (or movement primitives) that occur across these. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We present an approach to discover movement primitives from a set of unstructured robot demonstration i.e. demonstrations without additional parsing or segmentation labels available. This is a challenging task as each demonstration is composed of a varying number of unknown primitives, and therefore the process of learning entails both, learning the space of primitives as well as understanding the available demonstrations in context of these. Our approach is based on the insight that an abstraction of a demonstrations into a sequence of motor programs or primitives, each of which correspond to an implied movement sequence, and must yield back the demonstration when the inferred primitives are ‘recomposed’. We build on this and formulate an unsupervised approach to jointly learn the space of movement primitives, as well as a parsing of the available demonstrations into a high-level sequence of these primitives. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We demonstrate that our method allows us to learn a primitive space that captures the shared motions required across diverse skills, and that these motor programs can be adapted and composed to further perform specific tasks. Furthermore, we show that these motor programs are semantically meaningful, and can be recombined to solved robotic tasks using reinforcement learning. Specifically, solving reaching and pushing tasks with reinforcement learning over the space of primitives achieves 2 orders of magnitude faster training than reinforcement learning in the low-level control space. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work is broadly related to several different lines of work which either learn task policies from demonstrations, or leverage known primitives for various applications, or learn primitives in context of known segments. While we discuss these relations in more detail below, we note that previous primitive based approaches either require: a) a known and fixed primitive space, or b) annotated segments each corresponding to a primitive. In contrast, we learn the space of primitives without requiring this segmentation annotation, and would like to emphasize that ours is the first work to do so for a diverse set of demonstrations spanning multiple tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Learning from Demonstration: The field of learning from demonstrations (LfD) (Nicolescu & Mataric, 2003) has sought to learn to perform tasks from a set of demonstrated behaviors. A number of techniques exist to do so, including cloning the demonstrated behavior (Esmaili et al., 1995), fitting a parametric model to the demonstrations (Kober & Peters, 2009; Peters et al., 2013), or first segmenting the demonstrations and fitting a model to each of the resultant segments (Niekum et al., 2012; Krishnan et al., 2018; Murali et al., 2016; Meier et al., 2011). We point the reader to Argall et al. (2009) for a comprehensive overview of the field. Rather than directly learn to perform tasks, we use demonstrations to learn a diverse set of composable and reusable primitives, or motor-programs that may be used to perform a variety of downstream tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning and Sequencing Motion Primitives: Several works (Kober & Peters, 2009; Peters et al., 2013) learn motion primitives from demonstrations using predetermined representations of skills such as Dynamic Movement Primitives (DMPs) (Schaal et al., 2005). A few other works have approached the problem from an optimization perspective (Dragan et al., 2015). Given these primitives, a question that arises is how to then sequence learned skills to perform downstream tasks. Several works have attempted to answer this question - (Neumann et al., 2014) builds a layered approach to adapt, select, and sequence DMPs. Konidaris et al. (2012); Konidaris & Barto (2009) segments demonstrations into sequences of skills, and also merge these skills into skill-trees. However, the predetermined representations of primitives adopted in these works can prove restrictive. In particular, it prevents learning arbitrarily expressive motions and adapting these motions to a generic downstream task. We seek to move away from these fixed representations of primitives, and instead learn representations of primitives along with the primitives themselves. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Latent Variable Models: The family of latent variable models (LVMs) provides us with fitting machinery to do so. Indeed, the success of LVMs in learning representations in deep learning has inspired several recent works (Co-Reyes et al., 2018; Kipf et al., 2019; Haarnoja et al., 2018; Lynch et al., 2019) to learn latent representations of trajectories. SeCTAR (Co-Reyes et al., 2018) builds a latent variable conditioned policy and model that are constrained to be consistent with one another, and uses the learned policies and model for hierarchical reinforcement learning. The CompILE framework (Kipf et al., 2019) seeks to learn variable length trajectory segments from demonstrations instead, and uses latent variables to represent these trajectory segments, but is evaluated in relatively low-dimensional domains. We adopt a similar perspective to these works and learn continuous latent variable representations (or abstractions) of trajectory segments. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hierarchical RL and the Options Framework: The related field of hierarchical reinforcement learning (HRL) learns a layering of policies that each abstract away details of control of the policies of a lower level. The Options framework (Sutton et al., 1999) also learns similar temporal abstractions over sequences of atomic actions. While promising, its application has traditionally been restricted to simple domains due to the difficulty of jointly learning internal option policies along with a policy over those options. Recent works have managed to do so with only a reward function as feedback. The Option-Critic framework (Bacon et al., 2017) employs a policy-gradient formulation of options to do so, while the Option-Gradient (Smith et al., 2018) learns options in an off-policy manner. In contrast with most prior work in the options framework, (Daniel et al., 2016; Fox et al., 2017; Krishnan et al., 2017) learn options from a set of demonstrations, rather than in the RL setting. In similar spirit to these works, we too seek to learn abstractions from a given set of demonstrations, however unlike DDO (Fox et al., 2017), DDCO (Krishnan et al., 2017), and CompILE (Kipf et al., 2019), we can learn primitives beyond a discrete set of options in a relatively high dimensional domain. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hierarchical representations of demonstrations: The idea of hierarchical task representations has permeated into LfD as well. In contrast to reasoning about demonstrations in a flat manner, one may also infer the hierarchical structure of tasks performed in demonstrations. A few recent works have striven to do so, by representing these tasks as programs (Xu et al., 2018; Sun et al., 2018), or as task graphs (Huang et al., 2019). Both Xu et al. (2018) and Huang et al. (2019) address generalizing to new instances of manipulation tasks in the low-shot regime by abstracting away low-level controls. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The idea of policy sketches, i.e. a sketch of the sub-tasks to be accomplished in a particular task, has become popular (Andreas et al., 2017; Shiarlis et al., 2018). Andreas et al. (2017) learn modular policies in the RL setting provided with such policy sketches. Shiarlis et al. (2018) provides a modular LfD framework based on this idea of policy sketches. While all of these works address learning policies at various levels from demonstrations, unlike our approach, they each assume access to heavy supervision over demonstrations to do so. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c3d5e5cdfc94f10327df91a3b7379269d256c40ccd7551f868ee4a85134bc2b.jpg",
+        "image_caption": [
+            "Figure 2: An overview of our approach. Our abstraction network takes in an observed demonstration $\\tau _ { o b s }$ and predict a sequence of latent variables $\\{ z \\}$ . These $\\{ z \\}$ are each decoded into their corresponding motor programs via the motor program network. We finally recompose these motor programs into the recomposed trajectory. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We seek to discover the space of motor programs directly from unstructured demonstrations in an unsupervised manner, and show that these can help understanding similarities across tasks as well as quickly quickly adapting to and solving new tasks. Building on ideas of Keele (1968) and Schmidt (1975), we define a motor program $M$ as a movement pattern that may be executed in and of itself, without access to sensory feedback. Concretely, a ‘movement sequence’ or a motor program $M$ is a sequence of robot joint configurations. Our goal is to learn the space of such movement patterns that are present across a diverse set of tasks. We do so via learning a ‘motor program network’ that maps elements $z \\in R ^ { n }$ to corresponding movement sequences i.e. $\\mathcal { M } : \\boldsymbol { z } \\longrightarrow M$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given a set of $N$ unlabelled demonstrations $\\{ \\tau _ { i } \\} _ { i = 1 , 2 , \\dots , N }$ that consist of sequences of robot states $\\{ s _ { 1 } , s _ { 2 } , . . . , s _ { T } \\} \\in S$ , our aim is to learn the shared space of motor programs via the motor program network $\\mathcal { M }$ . However, as each demonstration $\\tau _ { i }$ is unannotated, we do not apriori know what subsequences of the demonstration correspond to a distinct motor program. Therefore, to learn motor programs from these demonstrations, we need to simultaneously learn to understand the demonstrated trajectories in terms of composition of motor programs. Thus in addition to learning the network $\\mathcal { M }$ , we also learn a mapping $\\mathcal { A }$ from each demonstrated trajectory $\\tau _ { i }$ to the underlying sequence of motor programs $\\left\\{ M _ { 1 } , M _ { 2 } , . . . , M _ { K } \\right\\}$ (and associated latent variables $\\{ z _ { 1 } , z _ { 2 } , . . . , z _ { K } \\} )$ executed during the span of the trajectory. We call this mapping $\\mathcal { A } : \\tau _ { i } \\longrightarrow \\{ M _ { 1 } , M _ { 2 } , . . . , M _ { K } \\}$ the abstraction network, as it abstracts away the details of the trajectory into the set of motor programs (i.e., abstractions). Note that both the abstraction and motor program networks are learned using only a set of demonstrations from across diverse tasks. ",
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+    {
+        "type": "text",
+        "text": "3.1 PRIMITIVE DISCOVERY VIA RECOMPOSITION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our central insight is that we can jointly learn the space of motor programs and the abstraction of the demonstration by enforcing that the implied ‘recomposition’ is faithful to the original demonstration. Concretely, an abstraction of a demonstration into a sequence of motor programs, each of which corresponds to an implied motion sequence, must yield back the original demonstration when the inferred motor programs are decoded and ‘recomposed’. We operationalize this insight to jointly train the motor program and abstraction networks from demonstrations. ",
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+    {
+        "type": "text",
+        "text": "Learning Overview and Objective: Our approach is outlined in Fig 2, where given an input demonstration trajectory $\\tau _ { \\mathrm { o b s } }$ , we use the abstraction network to predict a sequence of (a variable number of) latent codes $\\left\\{ z _ { k } \\right\\}$ . These are each decoded into corresponding sub-trajectories via the learned motor program network $\\mathcal { M }$ . ",
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+    {
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+        "img_path": "images/2bf7269283e20c7dbbe47cac30c1cdfe0ce22f43c73f411c838acf55b5895b71.jpg",
+        "text": "$$\n\\{ z _ { k } \\} = A ( \\tau _ { o b s } ) ; ~ \\bar { \\tau } _ { k } = \\mathcal { M } ( z _ { k } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Given the decoding of the predicted motor programs, we can recompose these sub-trajectories to obtain a recomposed demonstration trajectory $\\tau _ { r e c }$ , and penalize the discrepancy between the observed and the recomposed demonstrations. Denoting by $\\oplus$ the concatenation operator, our loss for a given demonstration $\\tau _ { o b s }$ is therefore characterized as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/9b8c172457088fb9968de1d7f5c510504b223546fcd523ae9e6ced416d1b2e52.jpg",
+        "text": "$$\n\\tau _ { r e c } = \\bar { \\tau } _ { 1 } \\oplus \\bar { \\tau } _ { 2 } \\cdot \\cdot \\cdot \\oplus \\bar { \\tau } _ { K } ; L ( \\tau _ { o b s } ; { \\mathcal { M } } , A ) = \\Delta ( \\tau _ { o b s } , \\tau _ { r e c } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "As the trajectories $\\tau _ { o b s }$ , $\\tau _ { r e c }$ are possibly of different lengths, we use a pairwise matching cost between the two trajectories, where the optimal alignment is computed via dynamic time warping (Berndt & ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Clifford, 1994). This provides us with a more robust cost measure that handles different prediction lengths, is invariant to minor velocity perturbations, and enables the model to discard regions of inactivity in the demonstrations. Given two trajectories $\\tau _ { a } \\equiv ( s _ { 1 } \\cdots s _ { M } )$ , $\\tau _ { b } \\equiv ( s _ { 1 } \\cdots s _ { N } )$ , and a distance metric $\\delta$ over the state space, the discrepancy measure between trajectories can be defined as the matching cost for the optimal matching path $P$ among all possible valid matching paths $\\mathcal { P }$ (i.e. paths satisfying monotonicity, continuity, and boundary conditions (Berndt & Clifford, 1994)): ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/9fcaac0f94c629223ae0df8c4ef30bfd0e914e3d1bcf1ab15dbffe8971b28e8d.jpg",
+        "text": "$$\n\\Delta ( \\tau _ { a } , \\tau _ { b } ) = \\operatorname* { m i n } _ { P \\in \\mathcal { P } } \\sum _ { ( m , n ) \\in P } \\delta ( \\tau _ { a } [ m ] , \\tau _ { b } [ n ] )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "As the recomposed trajectory comprises of distinct primitives, each of which implies a sequence of states, we sometimes observe discontinuities i.e. large state changes between the boundaries of these primitives. To prevent this, we additionally incorporate a smoothness loss $L _ { s m } ( \\tau _ { r e c } )$ that penalizes the state change across consecutive time-steps if they are larger than a certain margin. Our overall objective, comprising of the reconstruction objective and the smoothness prior, can allow us to jointly learn the space of motor programs and the abstraction of trajectories in an unsupervised manner. ",
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+    {
+        "type": "text",
+        "text": "Network Architecture and Implementation Details. We parameterize our motor program network $\\mathcal { M }$ and our abstraction network $\\mathcal { A }$ as neural networks. In particular, the motor program network is a 4 layer LSTM (Graves et al., 2013) that takes a single 64 dimensional latent variable $z$ as input, and predicts a sequence of 16 dimensional states. For our abstraction network, we adopt the Transformer (Vaswani et al., 2017) architecture to take in a varying length 16 dimensional continuous joint angle trajectory $\\tau$ as input, and predict a variable number of latent variables $\\{ z \\}$ , that correspond to the sequence of motor programs $\\{ M \\}$ executed during trajectory $\\tau$ . We find the transformer architecture to be superior to LSTMs for processing long trajectories, due to its capacity to attend to parts of the trajectory as required. ",
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+    {
+        "type": "text",
+        "text": "Our abstraction network $\\mathcal { A }$ predicts a varying number of primitives by additionally predicting a ‘continuation probability’ $p _ { k }$ after each motor program variable $z _ { k }$ . We then predict an additional primitive only if the sampled discrete variable from $p _ { k }$ is 1, and therefore also need to learn the prediction of these probabilities. While the loss function above can directly yield gradients to the predicted motor program encoding $z _ { k }$ via $\\mathcal { M }$ , we use gradients using REINFORCE Williams (1992) (with $\\Delta ( \\tau _ { o b s } , \\tau _ { r e c } ) + L _ { s m } ( \\tau _ { r e c } )$ as negative reward) to learn prediction of $p _ { k }$ . ",
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+    {
+        "type": "text",
+        "text": "3.2 ENFORCING SIMPLICITY AND PARSIMONY ",
+        "text_level": 1,
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+        "text": "While the objective described so far can in principle allow us to jointly learn the space of motor programs and understand the demonstration trajectories as a composition of these, there are additional properties we would wish to enforce to the bias the learning towards more desirable solutions. As an example, our framework presented so far can allow a solution where each demonstration is a motor program by itself i.e. the abstraction network can learn to map each demonstration to a unique $z$ and the primitive decoder can then decode this back. However, this is not a suitable solution as the learned motor programs are not ‘simple’. On the other extreme, a very simple notion of a motor program is one that models each transition independently. However, this is again undesirable as this does not ‘abstract’ away the details of control or represent the demonstration as a smaller number of motor programs. Therefore, in addition to enforcing that the learned motor programs recompose the demonstrations, we also need to enforce simplicity and parsimony of these motor programs. ",
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+        "type": "text",
+        "text": "We incorporate these additional biases by adding priors in the objective or model space. To encourage the abstraction model to learn parsimonious abstractions of the input demonstrations, we penalize the number of motor primitives used to recompose the trajectory, by adding a small constant to the negative reward used to train the continuation probability $p _ { k }$ if the corresponding sample yielded an additional primitive. To enforce simplicity of motor primitives, we observe that the trajectories yielded by a classical planner (in our case, RRT-Connect) e.g. to go from an initial to final state are ‘simple’ and we therefore initialize the motor primitive network using the decoder of a pretrained autoencoder on random planner trajectories for (start, goal) state pairs. Note that this notion of ‘plannability’ as ‘simplicity’ is merely one plausible alternative, and alternate ones can be explored e.g. ‘linearity’ (Kroemer et al., 2015) or ‘predictability of motion’ (Wolpert & Kawato, 1998). ",
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+        "img_path": "images/1b4e964f97c73037c0014220c6b8bf5c59c637b3bf9b1f78c15f681200dad406.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of the embedding of the latent representation of motor programs learned by our model (depicted on the left are the start configurations of each motor program, displayed at the corresponding position in the embedded space), and a set of sample primitives unrolled in time (depicted on the right). Each row corresponds to one primitive, annotated with semantic labels of what this motor primitive resembles. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We would like to ascertain how well our approach is capable of achieving our objective of successfully discovering and learning a good representation of the space of motor programs. Further, we seek to verify whether despite being learned in an unsupervised manner without semantic grounding, the learned primitive space is semantically meaningful. We would also like to evaluate how well they can be used to solve downstream RL tasks. We first provide a description of the data we wish to learn primitives from, followed by describing our quantitative and qualitative experiments towards verifying these three axes. ",
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+        "type": "text",
+        "text": "Dataset: We use the MIME dataset (Sharma et al., 2018) to train and evaluate our model. The dataset consists of over 8000 kinesthetic demonstrations of 20 tasks (such as pouring, pushing, bottle opening, stacking objects, etc.) collected on a real-world Baxter Robot. While the dataset has head and hand-mounted RGBD data, we use the Baxter joint angle trajectories to train our model. We consider a 16 dimensional space as our input and prediction space, consisting of 7 joints for each of the 2 arms, along with a scalar value for each gripper (we ignore torso and head joints). We emphasize this is a higher dimensional domain than most other related works consider. The gripper values are re-scaled to a $0 - 1$ range, while the joint angles are unnormalized. We temporally down-sample all joint angle data by a constant factor of 20 from the original data frequency of $1 0 0 \\mathrm { H z }$ . ",
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+    {
+        "type": "text",
+        "text": "We randomly sample a train set of 5900 demonstrations from all 20 tasks, with a validation set of 1600 trajectories, and a held-out test set of 850 trajectories. To help evaluate the learned motor programs, we manually annotate a set of 60 test trajectories (3 trajectories from each task) with temporal segmentation annotations, as well as semantic labels of 10 primitives (such as reaching, twisting, pushing, etc.) that occur in these 60 trajectories. Note that these labels are purely for evaluation, our model does not have access to these annotations during training. ",
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+    {
+        "type": "text",
+        "text": "4.1 VISUALIZING THE SPACE OF PRIMITIVES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We would first like to evaluate the quality of the learned abstractions in and of themselves, i.e. Is our approach able to discover the space of motor programs, and learn a good representation of this space? We qualitatively answer this question by visualizing the latent representation space of motor primitives learned by our model. We first randomly sample a set of 500 trajectories unseen during training, then pass these trajectories through our model, and retrieve the predicted latent variables $\\{ z \\}$ for each of these trajectories and their corresponding movement sequences $\\{ { \\overline { { \\tau } } } \\}$ . We then embed the latent variables in a 2-dimensional space using T-SNE (van der Maaten & Hinton, 2008), and visualize the corresponding movement sequences at their corresponding position in this 2-D embedded space, as in Fig. 3. We provide a GIF version of Fig. 3 (and other visualizations) at https://sites.google.com/view/discovering-motor-programs/home. ",
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+        "img_path": "images/4160295329ebb5b2736f44a43130310e2f7d0abdf62aba383d1f7de0954fd3bd.jpg",
+        "image_caption": [
+            "Figure 4: Depiction of execution of the learned primitives on real world Baxter robot. Each row is a single primitive, while columns show progress of the primitive over time. Row 1 depicts a left handed reaching primitive, while row 2 shows a right handed returning primitive. More visualizations provided in supplementary material, and videos are provided in the webpage. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "We observe that clusters of movement sequences emerge in this embedded space based on the relative motions executed during the course of these trajectory segments (similar latent variables correspond to similar movement sequences and vice versa). While these clusters are not explicitly semantically labelled by our approach, the motions in these clusters correlate highly with traditional notions of skills in robot manipulation, i.e. reaching motions (top and top-left clusters), returning motions (bottom cluster), bi-manual motions (visible to the bottom right of the left most cluster and the bottom of the right most cluster), etc. We visualize a few such primitives (reaching, twisting, grasping, bi-manual pulling, etc.) among these to the right of Fig. 3. This shows our model learns smooth mappings $\\mathcal { M }$ and $\\mathcal { A }$ , and is capable of discovering such primitives in an unsupervised manner without explicit temporal segmentation or semantic labels, which we believe is an encouraging result. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, the model learns abstractions that pick up on the trend of the motion, rather than distinguishing between whether the left or right hand is used for the motion. This is particularly notable in the case of reaching and returning motions, where both left and right-handed reaching and returning motions appear alongside each other in their respective clusters in the embedded space. ",
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+    {
+        "type": "text",
+        "text": "4.1.1 EXECUTING PRIMITIVES ON A REAL ROBOT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would ideally like the learned primitives from our model to be useful on a real Baxter robot platform, and be suitably smooth, feasible, and correspond to the motions executed in simulation (i.e. be largely unaffected by the noise of execution on a real robot). To verify whether our model is indeed able to learn such primitives, we execute a small set of learned primitives on a real Baxter robot, by feeding in the trajectory predicted by the model into a simple position controller. We visualize the results of this execution in Fig. 4, (see project webpage for videos). Despite not explicitly optimizing for feasibility or transfer to a real robot, the use of real-world Baxter data to train our model heavily biases the model towards primitives that are inherently feasible, relatively smooth and can be executed on a real robot without any subsequent modifications. ",
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+    {
+        "type": "text",
+        "text": "4.2 SEMANTIC SEGMENTATION TRANSFER USING LEARNED ABSTRACTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As the ‘recomposed’ trajectory can be aligned to the original demonstration via sequence alignment, our predicted abstractions induce a partitioning of the demonstrated trajectory (corresponding to the aligned boundaries of the predicted primitives). We test whether the predicted abstraction and induced partitions are consistent across different demonstrations from the same task. To this end, we select 3 instances of the “Drop Object” task (Sharma et al., 2018) from the annotated test set, and retrieve the induced segmentations of the demonstration. We then visualize these segmentations and motor programs predicted for each of these 3 demonstrations as depicted in Fig. 5, along with the ground truth semantic labels of primitives for each of the demonstrations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The alignment between a recomposed trajectory and the original demonstration also allows us to transfer semantic annotations from a demonstration to the predicted primitives, by simply copying labels from the demonstration to their aligned timepoints in the primitives. Therefore using our small set of annotated demonstrations, we can construct a small library of semantically annotated primitives. Given a novel, unseen demonstration, we can compute its predicted primitives and assign each a semantic label by copying the label of the nearest primitive from the library. This allows us to transfer semantic segmentations from our small annotated test set to unseen demonstrations. Our model’s transfer of semantic segmentation achieves label accuracy on the set of 30 held out trajectories (across all 20 tasks) of $58 \\%$ , while a supervised LSTM baseline that predicts semantic labels from trajectories (trained on 30 annotated test trajectories) achieves $54 \\%$ accuracy. ",
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+    {
+        "type": "image",
+        "img_path": "images/d4e449ab823b48a4e05b1f6ad9d15c26b964461e271c2af26c7a74dc4e6bb47d.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of consistent segmentations. Each row represents a different instance of a “Drop Objects” task from the MIME Dataset, while each column represents a time-step in the demonstration. White frames represent predicted segmentation points, while colored boxes represent ground truth semantic annotations. Red boxes are reaching primitives, blue boxes are grasping, orange boxes are placing, and green boxes are returning primitives. We see our model predicts 4 motor programs - reaching, grasping, placing the object a small distance away, and returning. This is consistent with the true semantic annotations of these demonstrations, and the overall sequence of primitives expected of the “Drop Box” task. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/775e83c070560161058c437dc3dfea7186217837d18d04253770463cc9b0c65f.jpg",
+        "image_caption": [
+            "Figure 6: RL training curves with and without motor programs. Solid lines denote mean success rate, while shaded region denotes $\\pm 1$ standard deviation across 10 random seeds. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The consistency of our abstractions coupled with the ability to transfer semantic segmentations across demonstrations shows our model is capable of understanding commonalities across demonstrations, and is able to reason about various demonstrations in terms of motor programs shared across them, despite being trained in an unsupervised manner. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 COMPOSING PRIMITIVES FOR HIERARCHICAL RL ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One of our primary motivations for learning motor programs is that they can be composed together to solve downstream robotic tasks. To evaluate whether the motor programs learned by our model are indeed useful for such downstream tasks, we adopt a hierarchical reinforcement learning setup (as described in detail in the supplementary). For a given task, we train a policy to predict the sequence of motor programs to execute. Given the predicted latent representations, each motor program is decoded into its corresponding motion sequence using the previously learned motor program network. We retrieve a sequence of desired joint velocities from this motion sequence and use a joint velocity controller to execute these “low-level” actions on the robot. The motor program network and the joint velocity controller together serve as an “abstraction” of the low-level control that is executed on the robot. Hence, the policy must learn to predict motor programs that correspond to motion sequences useful for solving the task at hand. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As demonstrated in Fig. 6, training a policy using motor programs is several orders of magnitude more efficient than training with direct low-level actions. For the sparse reaching task, the motor program policy learns within 50 motor program queries, 2 orders of magnitude speedup in low-level control time-steps. For the sparse pushing task, the motor program policy learns within 1000 motor program queries, or a 2X speedup with respect to low-level control time-steps. We note that executing a motor program corresponds to 50 low level control steps (see appendix for details). However, as these motor programs are executed without environment feedback, the improvement in efficiency in terms of environment interactions is all the more significant. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented an unsupervised approach to discover motor programs from a set of unstructured robot demonstrations. Through the insight that learned motor programs should recompose into the original demonstration while being simplistic, we discover a coherent and diverse latent space of primitives on the MIME (Sharma et al., 2018) dataset. We also observed that the learned primitives were semantically meaningful, and useful for efficiently learning downstream tasks in simulation. We hope that the contributions from our work enable learning and executing primitives in a plethora of real-world robotic tasks. It would also be interesting to leverage the learned motor programs in context of continual learning, to investigate how the discovered space can be adapted and expanded in context of novel robotic tasks. ",
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+    {
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+        "text": "A APPENDIX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A.1 EXECUTING PRIMITIVES ON A REAL ROBOT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We provide additional visualizations of primitives being executed on the real robot below. As mentioned in the main paper, dynamic GIFs of these visualizations may be found at https://sites.google.com/view/discovering-motor-programs/home. ",
+        "bbox": [
+            174,
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/cab74fb8e8d904bb0f8dc85196e36e5ae7698f599953b3da2c77612c8b2206f9.jpg",
+        "image_caption": [
+            "Figure 7: Depiction of execution of additional learned primitives on real world Baxter robot. As in Fig. 4, each row is a single primitive, while columns show progress of the primitive over time. Row 1 depicts a left handed returning primitive, row 2 depicts a right handed pushing primitive, row 3 depicts a left handed pushing primitive (in a different configuration to the left handed one), and finally row 4 depicts a right handed twisting primitive. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            215,
+            821,
+            402
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Our results webpage https://sites.google.com/view/discovering-motor-programs/home also contains visualizations of predicted combinations of primitives being executed on the real robot (as well as in simulation), along with corresponding demonstrations for comparison. We note that both the individual primitives, as well as the combinations of primitives are executed quite smoothly and naturally, and lead to motions that correspond highly to the original demonstrations. This indicates such combinations of primitives can indeed be used towards downstream tasks on the real robot. ",
+        "bbox": [
+            174,
+            482,
+            825,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 DETAILS OF HIERARCHICAL REINFORCEMENT LEARNING SETUP ",
+        "bbox": [
+            176,
+            582,
+            669,
+            597
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For our hierarchical reinforcement learning experiments, we perform policy learning on two sparse reward (Andrychowicz et al., 2017) tasks on a simulated Baxter Robot: (a) Reaching, and (b) Pushing. For the reaching task, the robot’s end-effector needs to reach a specific location in space, while for the pushing task, the robot needs to push a block on the table to a specific location. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            664
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For Baxter-Reaching task, the goal is to get the right-hand’s end-effector to a pre-defined goal (x,y,z) state. The reward is a sparse reward with epsilon $\\scriptstyle 1 = 0 . 0 5 \\mathrm { m }$ ; So if the end-effector reaches within $5 \\mathrm { c m }$ of the goal, it gets a reward of $+ 1$ , otherwise it gets a reward of 0. For Baxter-Push, the goal is to get a block (cube) to the desired goal with epsilon $= 0 . 0 5 \\mathrm { m }$ . To do this, the robot needs to hit/push the block to the goal. There is no other dense reward to encourage the robot to hit the block. ",
+        "bbox": [
+            174,
+            671,
+            825,
+            741
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We train both our motor program policy and the baseline control policy using Proximal Policy Optimization (Schulman et al., 2017). Note that the motor program policy outputs the latent representation $z$ . Each $z$ expands into a 10 length trajectory according the motor program network. To reach each of these trajectory states, a PD velocity controller is used for 5 time-steps. The baseline control policy directly outputs the velocity control action. ",
+        "bbox": [
+            174,
+            748,
+            825,
+            818
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We note that our PPO baseline implements the same action for 10 timesteps, in a manner similar to frame-skipping, as is common in RL. We found that varying the number of timesteps frame-skipping was applied for did not have a significant effect on the performance of PPO. ",
+        "bbox": [
+            174,
+            825,
+            825,
+            867
+        ],
+        "page_idx": 11
+    }
+]